School of Mathematical Sciences

Queensland University of Technology

Addressing Issues in Sparseness, Ecological Bias and

Formulation of the Adjacency Matrix in Bayesian

Spatio-temporal Analysis of Disease Counts

Arul Earnest

B.Soc.Sc (Hons) in Statistics, National University of Singapore MSc in Medical Statistics, London School of Hygiene and Tropical Medicine,

University of London

A thesis submitted for the degree of Doctor of Philosophy in the Faculty of Science and

Technology, Queensland University of Technology according to QUT requirements

Principal Supervisor: Professor Kerrie Mengersen

Associate Supervisors: Associate Professor Geoff Morgan

Professor Tony Pettitt

2010

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KEYWORDS

Spatial, autoregressive, disease mapping, CAR model, birth defects, ecological bias,

neighbourhood weight matrix, forecasting, priors, Bayesian, MCMC, joint modeling.

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ABSTRACT

The main objective of this PhD was to further develop Bayesian spatio-temporal models

(specifically the Conditional Autoregressive (CAR) class of models), for the analysis of

sparse disease outcomes such as birth defects. The motivation for the thesis arose from

problems encountered when analyzing a large birth defect registry in New South Wales.

The specific components and related research objectives of the thesis were developed

from gaps in the literature on current formulations of the CAR model, and health service

planning requirements. Data from a large probabilistically-linked database from 1990 to

2004, consisting of fields from two separate registries: the Birth Defect Registry (BDR)

and Midwives Data Collection (MDC) were used in the analyses in this thesis.

The main objective was split into smaller goals. The first goal was to determine how the

specification of the neighbourhood weight matrix will affect the smoothing properties of

the CAR model, and this is the focus of chapter 6. Secondly, I hoped to evaluate the

usefulness of incorporating a zero-inflated Poisson (ZIP) component as well as a shared-

component model in terms of modeling a sparse outcome, and this is carried out in

chapter 7. The third goal was to identify optimal sampling and sample size schemes

designed to select individual level data for a hybrid ecological spatial model, and this is

done in chapter 8. Finally, I wanted to put together the earlier improvements to the CAR

model, and along with demographic projections, provide forecasts for birth defects at the

SLA level. Chapter 9 describes how this is done.

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For the first objective, I examined a series of neighbourhood weight matrices, and

showed how smoothing the relative risk estimates according to similarity by an

important covariate (i.e. maternal age) helped improve the model’s ability to recover the

underlying risk, as compared to the traditional adjacency (specifically the Queen)

method of applying weights.

Next, to address the sparseness and excess zeros commonly encountered in the analysis

of rare outcomes such as birth defects, I compared a few models, including an extension

of the usual Poisson model to encompass excess zeros in the data. This was achieved via

a mixture model, which also encompassed the shared component model to improve on

the estimation of sparse counts through borrowing strength across a shared component

(e.g. latent risk factor/s) with the referent outcome (caesarean section was used in this

example). Using the Deviance Information Criteria (DIC), I showed how the proposed

model performed better than the usual models, but only when both outcomes shared a

strong spatial correlation.

The next objective involved identifying the optimal sampling and sample size strategy

for incorporating individual-level data with areal covariates in a hybrid study design. I

performed extensive simulation studies, evaluating thirteen different sampling schemes

along with variations in sample size. This was done in the context of an ecological

regression model that incorporated spatial correlation in the outcomes, as well as

accommodating both individual and areal measures of covariates. Using the Average

Mean Squared Error (AMSE), I showed how a simple random sample of 20% of the

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SLAs, followed by selecting all cases in the SLAs chosen, along with an equal number

of controls, provided the lowest AMSE.

The final objective involved combining the improved spatio-temporal CAR model with

population (i.e. women) forecasts, to provide 30-year annual estimates of birth defects at

the Statistical Local Area (SLA) level in New South Wales, Australia. The projections

were illustrated using sixteen different SLAs, representing the various areal measures of

socio-economic status and remoteness. A sensitivity analysis of the assumptions used in

the projection was also undertaken.

By the end of the thesis, I will show how challenges in the spatial analysis of rare

diseases such as birth defects can be addressed, by specifically formulating the

neighbourhood weight matrix to smooth according to a key covariate (i.e. maternal age),

incorporating a ZIP component to model excess zeros in outcomes and borrowing

strength from a referent outcome (i.e. caesarean counts). An efficient strategy to sample

individual-level data and sample size considerations for rare disease will also be

presented. Finally, projections in birth defect categories at the SLA level will be made.

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TABLE OF CONTENTS

1 INTRODUCTION 1 1.1 Primary research aims and motivation 1 1.2 Content and scope of thesis 4 1.3 Structure of thesis 9 1.4 List of publications and conferences arising from thesis 10 2 DATA 12 2.1 Summary 12 2.2 Sources of data 12 2.2.1 Birth defects 12 2.2.2 Births and maternal characteristics 13 2.2.3 Areal-level indices of socio-economic status 14 2.3 Definition and classification of birth defects 16 2.4 Spatial and temporal trends of birth defects in New South Wales,

Australia 18 3 LITERATURE REVIEW 21 3.1 Summary 21 3.2 Spatial analysis of birth defects 22 3.3 Risk factors for birth defects 25 3.3.1 Maternal age at delivery 25 3.3.2 Maternal smoking during pregnancy 26 3.3.3 Socio-economic indicators 28 3.3.4 Maternal diabetes mellitus 30 3.3.5 Common risk factors for caesarean section rates/ spatial variation 31 4 CONDITIONAL AUTOREGRESSIVE (CAR) MODEL 33 4.1 Summary 33 4.2 Spatial epidemiology 34 4.3 Disease mapping 35 4.4 Geographical correlation studies 40 4.5 Formulation of the CAR model 41 4.6 Comparison of single disease CAR models 43 4.7 Studies that have applied CAR models 55 4.8 Studies that have compared single disease CAR models 64 4.9 Comparison of multiple disease CAR models 67 5 CAR MODELLING ISSUES 74 5.1 Summary 74 5.2 Bayesian Theory 74 5.3 Markov chain Monte Carlo (MCMC) 76 5.4 MCMC Convergence 77

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5.5 Specifying the hyperprior distribution 78 5.6 Conjugate priors and improper priors 81 5.7 Sensitivity analysis on priors 82 5.8 Model selection techniques for spatial models 85 5.9 Modifiable Areal Unit Problem (MAUP) 87 5.10 Boundary analysis 89 6 NEIGHBOURHOOD WEIGHT MATRIX SPECIFICATION 93 6.1 Background 97 6.2 Aims 1026.3 Methods 1026.4 Results 1116.5 Discussion 1146.6 Conclusion 118 7 MODELLING SPARSE DISEASE COUNTS 1277.1 Introduction 1307.2 Methods 1337.3 Results 1387.4 Discussion 1417.5 Conclusion 145 8 STRATEGIES FOR COMBINING AREAL WITH INDIVIDUAL

DATA 1608.1 Introduction 1648.1.1 Ecological bias 1668.1.2 Addressing ecological bias 1678.1.3 Sampling techniques and sample size 1688.2 Methods 1718.2.1 Data 1718.2.2 Statistical model 1728.2.3 Model comparison 1758.2.4 Simulation 1768.2.5 Example 1788.3 Results 1798.4 Discussion 1818.5 Conclusion 185 9 FORECASTING BIRTH DEFECTS AT THE SMALL AREA

LEVEL 1979.1 Introduction 2019.2 Aim 2049.3 Methods 2049.4 Results 2109.5 Discussion 2129.6 Conclusion 214

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10 CONCLUSION 23210.1 Summary of results 23210.1 Implications of research 23310.2 Limitations 23610.3 Directions for future research 239 REFERENCES 243

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STATEMENT OF ORIGINAL AUTHORSHIP

"The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best of

my knowledge and belief, the thesis contains no material previously published or written

by another person except where due reference is made”.

________________

Arul Earnest

26th February 2010

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ACKNOWLEDGEMENTS

I would like to thank my principal supervisor, Professor Kerrie Mengersen, from

Queensland University of Technology (QUT), for her unlimited guidance and

supervision throughout the course of my PhD candidature. I am indebted to her for

introducing the field of Bayesian statistics to me. My appreciation also goes out to

Professor Tony Pettitt for facilitating the smooth flow of my PhD studies. I would also

like to express my gratitude to my associate supervisor, Associate Professor Geoff

Morgan, from the Northern Rivers University Department of Rural Health (University of

Sydney) for constantly providing input on my PhD, in particular the epidemiological,

study design and clinical implication aspects of the thesis. I have certainly enjoyed the

numerous thought-provoking discussions we had in his office in Lismore. I am equally

indebted to Professor John Beard, director of Ageing and Lifecourse at the World Health

Organisation, who was my previous supervisor. I would like to credit him with

providing me with the opportunity to start on this PhD studies, and also for his generous

advice and guidance on the manuscripts resulting from this thesis. My sincere gratitude

goes to Dr Lee Taylor and Dr David Muscatello from the New South Wales Department

of Health for providing me with useful advice on the data upon which this thesis is built

on, and also valuable opinion on the practical applications resulting from this thesis. I

would like to show my appreciation to the internal review panel from QUT and the

external examiners, whose comments and suggestions have strengthened the quality of

this thesis. Most importantly, I would like to thank my family members, especially my

wife Josephine for sacrificing those few important years and helping take care of our

two lovely young girls single-handedly. I am eternally indebted to her.

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CHAPTER 1. INTRODUCTION

1.1 . Primary research aims and motivation

This thesis aims to answer questions related to the small area analysis of sparse disease

counts in a geographical region. The first question relates to the formulation of the

Conditional Autoregressive (CAR) model, a commonly used statistical model in the

analysis of geographically aggregated data. Specifically, I wanted to evaluate whether

the formulation of the neighbourhood weight matrix has any impact on the smoothing

properties of the CAR model. In addition, I wished to examine whether there were any

differences between the adjacency and distance-based methods of assigning neighbours

in terms of recovering the underlying relative risk estimates.

The second hypothesis relates to the modeling or estimation of a sparse outcome, such as

birth defects. The questions I wished to answer were: “Can we better estimate the

outcome with a sparse count by jointly modelling it with another related outcome that

may share some latent risk factors?” and “Can we improve on the estimates by

incorporating a component (zero-inflated Poisson component through a mixture model)

to model the excess zeros in the data?”

The third broad question relates to a CAR regression model, and includes both

individual-level and areal measures of covariates. The question is, for sparse outcomes

like birth defects, what is the optimal sampling scheme to select individual-level data for

analysis in the hybrid model? Also, does the sample size have any impact on the

regression coefficient estimates from the hybrid model? To answer these questions, I

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performed an extensive simulation analysis to evaluate 13 different scenarios, including

various sampling schemes and variations in sample size.

The fourth aim of this thesis was to provide a method for forecasting sparse outcomes at

a small-area level, which took into account spatial correlation in the data, optimal

neighbourhood weight matrix formulation, consideration of excess zeros in the data, as

well as population (women) forecasts for the next 30 years at the Statistical Local Area

(SLA) level in New South Wales, Australia. Sensitivity analysis based on different

population scenarios was also assessed.

The motivation for this thesis came about from the challenges faced when analyzing

birth defects from a large registry in New South Wales (NSW), Australia, as part of an

Australian Research Council (ARC) linkage grant. The first challenge faced was

sparseness of the disease outcome, especially when individual birth defects were

mapped in geographical locations, or even when defects were analysed in broader

groupings, according to the International Classification of Disease- British Pediatric

Association (ICD9-BPA) coding system. The problem was compounded when there

were a large number of areas with zero counts of particular defects. Secondly, one had to

contend with possible spatial correlations in the disease rates across geographical

regions. On a broader level, I found that the spatial analysis of birth defects was under-

researched in the medical literature, and I hypothesize that this could be attributable to

the problems mentioned above, as well as inaccessibility to suitable statistical models

and software.

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The impetus to fine-tune the CAR model was primarily driven by gaps in literature,

identified after an extensive literature review on single-disease and multiple-disease

CAR models was performed. Both spatial only and spatio-temporal models were

evaluated and compared. The review revealed that most of the models were applied to

outcomes that were not rare, and applied to data across broad time intervals, thus

ensuring that there were enough cases in each time point. Disease mapping studies

involving birth defects were few, and none of them actually accounted for spatial

correlation in the data. Almost all the spatial studies used the simpler formulation of the

Queen adjacency method of assigning neighbours, which I suspect was done out of

convenience. The various formulations of the CAR models also failed to incorporate

sparseness in the data, implicitly or explicitly.

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1.2. Content and scope of thesis

This section details what is covered in this thesis and areas which are not within the

scope of this manuscript. This section also provides the links between the various

chapters.

In chapter 1, the motivation for undertaking the study is stated, along with the main aims

of this thesis. The content, scope and structure of the thesis are also presented in this

chapter. The source of data used in the analysis is described in chapter 2. Here, I also

provide the definition and classification of birth defects. A description of the current

state of birth defects in New South Wales, in terms of spatial and temporal trends is

given in this chapter.

A comprehensive literature review is provided in chapter 3. Summarized components of

the literature review are included in subsequent chapters, which are structured as

manuscripts to be submitted for publication. Firstly, I summarise spatial analytical

studies in relation to birth defects, to identify gaps in literature. Secondly, I provide a

review on selected risk factors, with a view to inform the analytical models for a

subsequent analysis that combines both areal with individual risk factors (i.e. chapter 8).

The chapter also reviews the literature on risk factors common to both birth defects and

caesarean section rates and spatial analysis of caesarean section rates for inclusion in

chapter 7, which looks at modelling the two outcomes jointly.

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Chapter 4 introduces the CAR model. I provide readers with an understanding of the

context upon which the CAR model is applied and describe the two main fields of

application: namely disease mapping and geographical correlation studies. The

mathematical properties of the CAR model are also described, along with a brief section

on the adjacency matrix, which introduces a subsequent chapter which examines in

detail the impact of various neighbourhood weight matrices on the smoothing properties

of the CAR model (i.e. chapter 6).

In the same chapter 4, I also discuss the strengths and limitations of the various types of

CAR models commonly used. In addition, I examine the properties of spatial and spatio-

temporal models, including specific comparisons about the nature of data (sparseness of

outcome) used in the studies reviewed, along with the priors and model selection

techniques. The results from these comparisons inform the modelling strategy adopted in

subsequent chapters. Comparisons were made within the multivariate (i.e. models

examining more than one disease outcome simultaneously) classes of models, and the

results used in chapter 7.

The CAR model predominantly uses the Bayesian framework of analysis. To help

readers familiarize with the context of Bayesian modeling, an introduction to Bayesian

theory in general, and the Markov Chain Monte Carlo (MCMC) algorithm is provided in

chapter 5. Here, I also discuss issues pertaining to the choice of prior distributions, and

conduct a sensitivity analysis including commonly suggested values for the prior

formulation. Other issues such as boundary analysis, the Modifiable Areal Unit Problem

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(MAUP) and Bayesian model convergence diagnostics are discussed briefly, as these do

not relate to the main objective of the thesis.

In chapter 6, I examine in detail, the effect of various choices of neighbourhood weight

matrices (ranging from adjacency to distance-based functions, as well as weights based

on key covariates) on the smoothing properties of the CAR model. Addressing the issue

of sparse disease count is the focus of chapter 7, where I investigate the performance of

a CAR model with a zero-inflated Poisson extension, in terms of its ability to recover the

underlying risk surface of specific birth defects, such as Spina Bifida and Trisomy 21. I

also demonstrate how the model can be strengthened by incorporating a shared

component, via jointly modeling birth defects with a referent outcome (caesarean

counts).

Chapter 8 discusses in detail the major drawback of ecological analysis (i.e. potential

ecological bias) and reviews the literature for suggested strategies to incorporate

individual-level data with areal level data, in order to minimize this potential bias.

Through extensive simulation studies, I investigate the performance of various sampling

strategies, along with modifications in sample sizes, and examine how they fare for

sparse outcomes such as birth defects. The findings from the simulation studies are

illustrated using cardiovascular and nervous system birth defect categories. Chapter 9

synthesises what has been learnt from the earlier analysis of the CAR model, and I apply

the modified and enhanced CAR model to provide forecasts of birth defect categories at

the Statistical Local Area (SLA) level. Using population forecasts for areas in NSW at

the SLA, the CAR model is used to make predictions on the number of cases expected to

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be seen in the next 30 years from 2001 for sixteen randomly selected SLAs. The

strengths and limitations of this thesis, as well as areas for future research, are the focus

of the discussion in chapter 10.

In this thesis, I have excluded discussions on other seemingly related models such as

multi-level models and statistical models to analyse point process data, as my main

focus is the CAR model. The aims and objectives as well as the nature of data utilised by

the other models are generally different from studies which use the CAR model, as I will

briefly describe here. Multi-level models, or random effects model as they are

commonly known, are often used to study variables which can vary at more than one

level. The levels can be nested hierarchicaly, and the models can be formulated within

both the frequentist and Bayesian frameworks. Gelman provides details on the theory

behind these models, as well as various formulations and applications of multi-level

models(1). In the context of our spatial analysis, the CAR convolution prior (to be

discussed later in the thesis) is a more specific formulation of a multi-level model, where

the variance of the relative risk estimates is partitioned into both spatially structured and

spatially unstructured random effects.

As for point-process models, one basic goal is to determine whether cases occur at

random or whether there is any form of geographical clustering or pattern in the data. To

this end, various cluster detection models have been developed and applied in practice,

including K-function(2), nearest-neighbour function(3) and hotspot analysis(4). Studies

which analyse point-process data have also included stochastic epidemic models, as well

as spatial prediction techniques. As an example, point-process level spatio-temporal

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models have been used to fit stochastic epidemic models to study measles epidemics in

one study(5). Gelfand and colleagues have also used spatiotemporally varying

coefficient models to study and make predictions of climate data, such as precipitation

and temperature, which are measured at fixed locations(6). The fundamental difference

between these models and CAR models is that for the latter, data is available at an

aggregate level, as opposed to fixed locations or at continuous geographical scales.

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1.3. Structure of thesis

The thesis is structured in the following way. It consists of a series of chapters that are

either published or submitted for publication and unpublished. Chapter 6 “Addressing

the Neighbourhood Weight Matrix” has been published in the International Journal of

Health Geographics. Chapter 7 “Modelling Sparse Disease Counts” has been accepted

for publication in the Health and Place Journal. Chapter 8 “Strategies for Combining

Areal with Individual Data” and chapter 9 “Forecasting Birth Defects at the Small Area

Level, NSW” have been submitted to the Statistics in Medicine journal and the BMC

Health Services Research journal respectively. These chapters have been included in the

same format as they were submitted for publication. This explains the variations in the

way the chapters are presented, the different sub-headings used in the various chapters,

and the distinct format of the bibliographies required by the various journal. The rest of

the chapters consist of unpublished works. I have included the bibliographies separately

at the end of each chapter for the published works, and one overall bibliography for the

rest of the unpublished chapters at the end of the thesis.

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1.4. List of publications and conferences arising from thesis

Arul Earnest, Geoff Morgan, Kerrie Mengersen, Louise Ryan, Richard Summerhayes,

John Beard. Evaluating the effect of neighbourhood weight matrices on smoothing

properties of Conditional Autoregressive (CAR) models. International Journal of Health

Geographics, November 2007, Volume 29;6: pp 54-65.

Arul Earnest, John Beard, Geoff Morgan, Douglas Lincoln, Richard Summerhayes,

Deborah Donoghue, Therese Dunn, David Muscatello, Kerrie Mengersen. Small Area

Estimation of Sparse Disease Counts using Shared Component Models- Application to

Birth Defect Registry Data in New South Wales, Australia. Health and Place Journal

(Accepted for publication 23 February 2010).

Arul Earnest, John Beard, Geoff Morgan, Deborah Donoghue, Therese Dunn, David

Muscatello, Danielle Taylor, Kerrie Mengersen. Sampling and sample size strategies for

including individual with areal-level covariates in the spatial analysis of a sparse disease

outcome . Submitted to Statistics in Medicine Journal, Oct 2009.

Arul Earnest, Kerrie Mengersen, Geoff Morgan, John Beard. Forecasting Birth Defects

at the Small-Area Level in New South Wales, Incorporating Spatial Correlation and

Changes in Demography. Submitted to BMC Health Services Research Journal, Oct

2009.

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Arul Earnest. Evaluating the effect of neighbourhood weight matrices on smoothing

properties of Conditional Autoregressive models. Contributed talk for Spring Bayes 27-

29 September 2006, Queensland University of Technology.

Arul Earnest, John Beard, Geoff Morgan, Douglas Lincoln, Richard Summerhayes,

Deborah Donoghue, Therese Dunn, David Muscatello, Kerrie Mengersen. Modelling

Sparse Disease Counts Using the Shared Component Model. Poster presentation at the

International Society for Bayesian Analysis, 9th World Meeting, Hamilton Island,

Australia, July 20-25 2008.

Arul Earnest, John Beard, Geoff Morgan, Douglas Lincoln, Richard Summerhayes,

Deborah Donoghue, Therese Dunn, David Muscatello, Kerrie Mengersen. Modelling

Sparse Disease Counts Using the Shared Component Model. Poster presentation at the

National Healthcare Group Annual Scientific Congress. 7-8 November 2008, Singapore.

The poster won the first prize in the best poster competition for the Quality/ Health

Services Research section.

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CHAPTER 2. DATA

2.1. Summary

The aim of this chapter is to provide readers with an understanding of the sources of data

used in subsequent analyses in this thesis. Selected birth defects are also described,

along with the classification or grouping of birth defects. A background description of

current spatial and temporal trends of birth defects in New South Wales is provided as a

precursor to subsequent work in this area. It is clear from existing official health

department reports that birth defects do indeed exhibit clear spatial relationships as well

as a time gradient.

2.2. Sources of data

2.2.1. Birth defects

De-identified birth defect records were obtained from the NSW Birth Defects Register

(BDR). The register has been operational since 1990, and in the early years, reporting of

defects was done on a voluntary basis. Since 1998, doctors, hospitals and laboratories

have been required by law to report all birth defects. These defects included those

observed during pregnancy, at birth or up to one year of life. Each birth defect is

recorded as a separate record, so the total number of congenital abnormalities reported is

considerably greater than the number of children born with a birth defect. The study

period was from 1990 to 2004.

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2.2.2. Births and maternal characteristics

Information on births in NSW from 1990 to 2004 was obtained from the NSW

Midwives Data Collection (MDC), which is a population-based register just like the

BDR. Covering all births in NSW (including public, private and home-births), the MDC

is dependent on the attending midwife or doctor to complete and submit a notification

form whenever a birth occurs(7). The registry includes all livebirths and stillbirths of at

least 20 weeks gestation or at least 400 grams birth weight. I also obtained maternal

demographic information (e.g. residential address at time of birth, maternal age at

delivery, maternal smoking during pregnancy, maternal diabetes, delivery in private

versus public hospital), pregnancy, labour, delivery and perinatal outcomes from the

MDC.

Each of the birth records in NSW within the study period was geocoded (i.e. given a

longitude and latitude) based on the mother’s residential address at the time of birth.

This geocoding was done by Mr Richard Summerhayes from the Northern Rivers

University Department of Rural Health using geocoding software developed by the

NSW Health and Australian National University. Further details on the software called

FEBRYL, can be found in this reference(8). Each record was then assigned to the 2001

Census Collectors Districts (CCDs) within which they fell in. There are 11,706 CCDs in

NSW. This assignment was again performed by Mr Summerhayes using the ARC GIS

software. Subsequently, I aggregated the data at an appropriate higher level of grouping,

SLA, or Statistical Local Area, which has 198 areas in NSW, as this level of aggregation

was found to be most useful for policy-makers. The individual records from BDR were

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also probabilistically linked to the MDC, and this was carried out by the Department of

Health, NSW. The combined data was used in a subsequent analysis in the thesis,

involving the association between birth defects and individual maternal characteristics

along with areal covariates, such as socio-economic status of the area that the mother

was living in.

In 1998 a 2% sample of Midwives Data Collection records (N=1703) was validated

against other hospital records(9). The excellent quality of this database is reflected in

high correlations, including a 99.1% agreement on gestational diabetes (kappa 0.87),

94.9% agreement for smoking in pregnancy (kappa 0.85), 96.5% agreement for

birthweight (kappa not calculated) and 84.8% agreement for gestational age (kappa

0.81). This study, and access to both BDR and MDC databases, was approved by the

New South Wales Population & Health Services Research Ethics Committee.

2.2.3. Areal-level indices of socio-economic status

I used data from the Australian Bureau of Statistics (ABS) to describe the level of social

and economic well-being in areal levels of NSW. This data was freely available on the

ABS website, and a technical paper can be found here (10). The following 4 indices

were available to us:

1. Index of Advantage/ Disadvantage. Higher values reflect areas with a greater

advantage. Variables such as income, education, occupation, wealth and living

conditions were used to compute this index.

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2. Index of Relative Social Disadvantage. Higher values reflect lack of

disadvantage, which has a subtle difference from the index above. The variables

that were used to compute this index included income, educational attainment,

unemployment, and dwellings without motor vehicles.

3. Index of Economic Resources. Variables such as income, expenditure and assets

of families, such as family income, rent paid, mortgage repayments, and dwelling

size went into computing this index.

4. Index of Education and Occupation. This index took into account the proportion

of people with a higher qualification or those employed in a skilled occupation.

The data were available at the various Australian Standard Geographical Classification

(ASGC) levels, starting from the most basic Census Collection District (CCD) to the

Statistical Local Area (SLA) level. There are problems associated with the simple

averaging up of the indices from CCD to SLA level, and I used an index that was

calculated at the SLA level and population-weighted. This was performed by the ABS.

Data on the four indices were standardised by the ABS to have a mean of 1000 units and

a standard deviation of 100. For the purposes of my analysis, I used the more general

Index of Relative Social Disadvantage (IRSD). Compared to the other indices, this index

only included variables that are measures of or indicators of disadvantage (rather than

advantage). This index was derived from variables that reflect rather than measure

disadvantage. A decision-tree process, along with principal component analysis was

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used to derive the final index score for each CD. Further details on how these indices

were derived from variables obtained from the 2001 census is described in an

information paper available from the ABS(10). It should be noted that the indices

measure the socio-economic well-being of a region, and not the individual, and this

subtle difference is exemplified in a subsequent analysis presented later in the paper.

2.3. Definition and classification of birth defects

A birth defect can be thought of as a physiological or structural abnormality that is

present at birth and is significant enough to be considered a problem. According to the

US Centers for Disease Control and Prevention, most birth defects are thought to be

caused by a complex mix of factors including genetics, environment, and behaviors(11).

Much of the analysis for this thesis draws on data from the NSW Birth Defects Register.

The Register uses the following definition for a birth defect: ‘Any structural defect or

chromosomal abnormality detected during pregnancy, at birth, or in the first year of life,

excluding birth injuries and minor anomalies such as skin tags, talipes, birthmarks, or

clicky hips(7).

Birth defects take many forms and currently there is no one universal method of coding

and classifying them. Even within Australia, there are considerable differences between

the various states in terms of how the birth defects are classified(12). Tasmania, for

instance, used a combination of the ICD-9-BPA and the ICD-10-AM (Australian

Modification) to classify congenital anomalies recorded between 1998 and 2001. NSW,

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the state from which the data for this study was drawn, relies on the BPA classification

system(13) that is basically organised by body system(7). Table 1 shows a list of birth

defects recorded by the Registry using this approach, together with a short description.

In the United States, the Centers for Disease Control and Prevention uses a classification

system that is modified from the original BPA system(14). The key advantage of this

system is that it allows researchers to describe more specific details about the birth

defects and related conditions. In particular, it describes the laterality of the defect (i.e.

whether the defect was on the right or left part of the body) and provides greater

specificity for a defect. One disadvantage of this approach is that the analysis of such

data becomes more challenging due to the sparseness of the defects as one becomes

more specific.

Table 1. Description of selected birth defects from the NSW birth defect registry Defect Description Anencephaly Absence of the cranial vault, with the brain tissue

completely missing or markedly reduced. Spina bifida Defective closure of the bony encasement of the spinal

cord, through which the spinal cord may protrude. Encephalocele Protrusion of brain through a congenital opening in the

skull Hydrocephalus Dilatation of the cerebral ventricles accompanied by an

accumulation of cerebral fluid within the skull. Buphthalmos Enlargement and distension of the fibrous coats of the

eye. Hypospadias The opening of the urethra lies on the underside of the

penis or on the perineum. Epispadias Absence of the upper wall of the urethra. The opening of

the urethra lies on the dorsum of the penis in males, and anterior to or onto the clitoris in females.

Chordee Downward bowing of the penis. Talipes equinovarus A deformity of the foot in which the heel is elevated and

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turned outward. Polydactyly Presence of additional fingers or toes on hands or feet. Syndactyly Attachment of adjacent fingers or toes on hands or feet. Craniosynostosis Premature closure of the sutures of the skull. Exomphalos Herniation of the abdominal contents into the umbilical

cord. Gastroschisis A defect in the abdominal wall not involving the

umbilicus and through which the abdominal contents herniate.

Cystic hygroma A sac, cyst or bursa distended with fluid. Centre for Epidemiology and Research. NSW Department of Health. New South Wales Mothers and Babies 2005. N S W Public Health Bull 2006; 18(S-1); pp 135.

2.4. Spatial and temporal trends of birth defects in New South Wales,

Australia

Across all states in Australia, there has been considerable variation in the reported rates

of birth defects over the past 20 years. For instance, the rate of all malformations ranged

from 159.4 per 10,000 births (1981-1995) to about 175.2 per 10,000 births (1997). There

was also gross spatial variation in the reported rates between states in the period 1991-

1997, with highest rates found in Victoria (229.2 per 10,000 births), followed by ACT

(222.3 per 10,000 births) and Queensland (194.3 per 10,000 births)(15).

However, since the criteria and source of notification varies by state, these trends may

not reflect underlying changes in disease incidence, and researchers need to be cautious

about attributing differences between time periods or states to true disease trends

Changing clinical practices such as termination of pregnancy for defects picked up by

ultrasound (e.g. neural tube defects) or amniocentesis (e.g. Down syndrome) across

states may also help explain the variation. The problem is compounded for many of the

defects we are interested in, because of imprecision in the reported rates and

19

comparisons, due to small numbers. At this stage, we try not to be unduly concerned

about the reasons for spatial variation, except to note that defects exhibit both spatial and

temporal variation, even at the broader scales of analysis.

In NSW, state-wide surveillance of birth defects is monitored through the Birth Defects

Register (BDR), which is administered through the NSW Department of Health. The

overall rate of birth defects appears to have been stable between 1999 and 2004.

However, when the defects were examined by individual diagnostic categories, there

was considerable year to year variation. Ventricular septal defect, for instance, saw rates

ranging from 2.1 per 1,000 births in 2002, to 0.9 per 1,000 births in 2003 and 2.1 per

1,000 births in 2004(7, 16).

Within NSW, there was spatial variation in the reported birth defect rates for the 8

different administrative health areas between 1999 and 2005. For example, the NSW

Mothers and Babies Report 2005 found elevated rates of birth defects in the Hunter and

New England area(7). However, it should be noted that there are some issues to note

when making this sort of comparisons across regions. The first involves mothers

residing near state borders and the possibly of them going interstate for treatment, where

they may nominate an interstate place of residence for the duration of treatment. It is

possible that they can be under-countered in the register. Secondly, the breakdown of

defects by region, and by individual defects results in a small number for many defects,

which makes the calculation of rates imprecise. Finally, it is unclear how much of these

differences can be attributed to environmental (and/or other) causes, and how much of it

is a result of differences in reporting patterns. Once again, we try not to explain the

20

sources of spatial variation at this stage of the project, but rather make the point that

spatial variation in reported rates of birth defects at a smaller aggregate level is inherent

in the data. Health data in NSW can be grouped at various geographical scales, including

the Census Collection District (CCD) and the Statistical Local Area (SLA). The CCDs

are the smallest spatial unit, and there are 11,706 units in NSW. These CCDs can be

aggregated up to broader groupings, including the SLA and Local Government Area

(LGA). In urban areas, the average number of dwellings per CCD is about 220, whereas

this number drops considerably for CCDs in rural areas. There is also considerable

variation in the geographical size of these CCDs (i.e. interquartile range of 1 km2 to 62

km2).

21

CHAPTER 3. LITERATURE REVIEW

3.1. Summary

The aim of this chapter is to provide a comprehensive literature review in a few selected

important topics. Firstly, current research on the spatial analysis of birth defects is

summarized. Next, I examine the prior evidence on the relationship between selected

risk factors such as maternal age, maternal smoking, maternal diabetes mellitus status

and socio-economic indicators (both areal and individual measures) and birth defects.

The aim is to identify which particular defects are associated with the risk factors. This

chapter also identifies risk factors that are common to birth defects and caesarean rates,

as well as describing spatial variation in caesarean rates. The results are used in

subsequent chapters examining the risk factors for birth defects, as well as the joint

modeling of two related outcomes.

The search strategy used to identify studies for discussion in this chapter is described

here. I searched for all relevant research articles in MEDLINE, which contains

bibliographic citations and author abstracts from more than 4,000 biomedical journals

published in the United States and 70 other countries. The PubMed on-line search

engine tool was used for this purpose. In addition, I also went through the bibliographic

lists of relevant journal articles to identify additional pieces of research to include in my

literature review. For the section on spatial analysis of birth defects, I used the following

search terms: “spatial” and “birth defects” as well as “geographic” and “birth defects”

and “environment” and “birth defects”. I looked through the abstracts and excluded

studies which did not involve spatial analysis of birth defects. I also excluded studies

22

that were performed in the laboratories, as well as those not published in English. For

the section on risk factors for birth defects, I used the following search terms. For

maternal age, as an example, I used “maternal age” and “birth defect”, “age” and “birth

defect”, as well as “risk factor” and “birth defect” more generally. For caesarean rates,

the following key-words were used: “caesarean” and “risk factor”, “caesarean” and

spatial”, as well as “caesarean” and “geographic”. I would like to add that this was not a

systematic review exercise, and hence I did not provide a summary of the results from

the literature. Rather, the studies identified in the literature were used to justify the use

of the selected risk factors, in studying their association with birth defects in my thesis.

3.2. Spatial analysis of birth defects

In many countries, information on birth defects is obtained and analysed from national

or regional birth defect registries. These registries often have data on the mother’s

residential address. This location data enables researchers to undertake various forms of

spatial analyses on the epidemiology of birth defects. Application of spatial analyses

ranges from simple mapping of defects, to identifying clusters and exploring the

influence of environmental factors, such as air pollution, contaminated sites, disinfection

byproducts from water chlorination on the occurrence of birth defects. Ecological

regression models are used to study the exposure variables (e.g. related to social and

environmental), which are assigned to each birth record, based on the maternal

residential address at the time of birth.

23

Few studies have examined the spatial distribution of birth defects and their association

with possible spatially varying risk factors. High altitude has been implicated in at least

three studies in South America. In the first, looking at 53 hospitals across Latin

America(17), adjusted relative risks were found to be significantly higher among those

living in the highland, specifically for cleft lip (RR=1.57, 95%CI: 1.27-1.94), microtia

(RR=3.21, 95%CI: 2.35-4.79), preauricular tag (RR=2.09, 95%CI: 1.86-2.36), branchial

arch anomaly complex (RR=1.79, 95%CI: 1.23-2.61), constriction band complex

(RR=1.92, 95%CI: 1.11-3.31) and anal atresia (craniofacial defects) (RR=1.61, 95% CI:

1.01-2.57). On the other hand, risks were lower for spina bifida (RR=0.57, 95% CI:

0.37-0.78), anencephaly (RR=0.33, 95% CI: 0.20-0.54), hydrocephaly (RR=0.41, 95%

CI: 0.22-0.77) and pes equinovarus (neural tube defects) (RR=0.70, 95%CI: 0.51-0.96).

The second study also linked altitude with the risk of microtia, with a relative risk of

2.66 (p<0.01) comparing those living more than 1000m above sea-level versus those

living less than 500m(18), whilst the third study(19) found that cleft lip/ palate birth

prevalence rates were higher for those living at high altitude above sea-level (effect size

not provided).

In another study, researchers examined indicators of exposure to industrial activities(20)

and found significant associations between textile industry and anencephaly (RR=1.59,

95%CI: 1.23-2.08), along with manufacture of engines and turbines and microcephaly

(RR=4.26, 95%CI: 1.74-11.10).

Two studies looked at the effect of changing residences on birth defects. The first study

examined records of the first and second infants born to mothers. It found that the

24

relative risk of having the second infant with the same defect was higher among those

who lived in the same municipality (RR=11.6, 95%CI: 9.3-14.0) during both

pregnancies, as compared to those who moved to another municipality (RR=5.1, 95%CI:

3.4-6.7)(21). In contrast, the second study did not find any significant change in the

frequency of facial-cleft defects among mothers who changed municipality of residence

(RR=0.9, 95%CI: 0.6-1.5)(22).

In studies that examined the spatial variation (in particular geographical difference) in

risks of specific birth defects, neural tube defects seemed to be the most common defect

that was found to be spatially correlated(23-28), followed by clefts(19, 29),

anophthalmia and microphthalmia(30), where prevalence was found to be higher in rural

versus urban areas as well as diaphragmatic hernia and gastroschisis(27). Birth defects

were also found to vary by the level of aggregation: e.g. across register areas and

hospital catchments, but not below this level(31). A study from England found variation

in reported rates by local register and hospital catchment area (p<0.001), but not by area

deprivation scores (p>0.1, effect size not provided)(32). Proximity of maternal residence

to landfill sites was found to be associated with certain birth defects such as neural tube

defects (RR=1.05, 95%CI: 1.01-1.10), hypospadias and epispadias (RR=1.07, 95%CI:

1.04-1.10) and abdominal wall defects (RR=1.08, 95%CI: 1.01-1.15) (33). A few other

studies examined spatial distribution of birth defects in general (34-36).

Two studies did not find any geographical variation in birth defect rates: the first study

looked at Down syndrome across 9 South American countries(37), while the other on

25

nonsyndromic cleft lip/ palate across public health region of residence across Texas,

USA(38).

3.3. Risk factors for birth defects

There have been numerous studies that have looked at the association between various

risk factors and the occurrence of congenital abnormalities at birth. Study designs have

included case-control (including the use of sick controls), cohort studies, and more

commonly the use of birth defect registries to explore associations with risk factors.

Variables of interest have included socio-demographic covariates, genetic, nutritional,

infectious, and other environmental factors. It is not the aim of this thesis to undertake

an evaluation of the various risk factors, as this has been carried out by other authors.

Here, I discuss specific risk factors that were commonly found in the literature and

available to us for analysis, and these include maternal age, maternal smoking, maternal

diabetes status and socio-economic status. I also examine studies that have analysed the

defects spatially/ geographically, as this project’s interest is in studying risk factors from

the spatial perspective.

3.3.1. Maternal age at delivery

Maternal age is the most commonly studied risk factor in most studies involving birth

defects. This could be due to the ready availability of this demographic variable in most

birth defect registries. The effect of maternal age on the occurrence of birth defects is

not uniform across all defect categories. In addition, both very young maternal age and

old maternal age have been found to be associated with a different range of birth defects.

26

Younger mothers have been shown to have a higher risk of giving birth to babies with

gastroschisis(39-43), chromosomal abnormalities(43, 44), cystic hygroma, autosomal

recessive disorders, monogenic disorders, ventricular septal disorders(43), anencephaly,

all ear defects, female genital defects, polydactyly, omphalocele(42). Older mothers, on

the other hand, are known to have a higher risk of having babies with various types of

atresia(30, 42), anophthalmia, microphthalmia(45), heart defects, right outflow tract

defects, males genital defects including hypospadias, craniosynostosis(42), trisomy 18,

trisomy 21, dysplasia of hip, chromosomal abnormalities(43, 46), pancreas, down

syndrome(47) and cleft lip/ palate(48).

Some defects like chromosomal abnormalities are related to risk factors, associated with

both older and younger mothers, as we can see from above. Neural tube defects also

display such a U-shaped relationship with maternal age (26, 28, 49, 50). On the other

end of the spectrum, other studies have shown that there is no significant relationship

between maternal age and defects such as cleft palate and lip(29, 51, 52) along with

ventricular septal defect(53), severe birth defects(54) and anencephalus and spina

bifida(55).

3.3.2 Maternal smoking during pregnancy

Maternal smoking, during pregnancy has been implicated in a number of studies, to be

associated with births with congenital abnormalities. Defects have included Talipes

equinovarus(56), terminal transverse defect(57), truncus abnormalities, atrial septal

27

defects, persistent ductus arteriosus(58), isolated craniosynostosis(59), kidney

malformations(60), oral clefts(48, 61-66), limb reduction birth defects(67), deformities

of the foot(68), gastroschisis(40, 41, 69, 70), defects of the cardiovascular system(71),

hydrocephaly, polydactyly/ syndactyly/ adactyly(69), clubfoot(69, 72, 73) and defects in

general(47).

Most of the studies have involved the case-control study design and included data from

birth defect registries. However, there were two meta-analyses (74, 75) that looked at the

combined (across studies) effect of maternal smoking on the occurrence of oral cleft

birth defects. Both studies found a relationship between smoking and risk of oral clefts.

Some studies (48, 62) managed to show a dose-response relationship between smoking

and birth defects, a sign of the causality criteria met for an observational study (48, 59,

62, 65).

The use of self-reported smoking as a risk factor variable in epidemiological studies has

been criticized because of the possibility of recall bias. This is especially true in birth

defect studies, which use normal controls (i.e. mothers with babies without any birth

defect who may not be as motivated to accurately recall past behaviours). In the

literature that we reviewed, we found some studies that used affected or sick controls

(i.e. mothers with babies having a birth defect, other than the one of interest)(57, 65, 68).

In these studies, the significant effect of maternal smoking was still evident. There were

some other studies though, which did not find maternal smoking to be a risk factor for

birth defects, specifically neural tube defects(24, 76, 77), central nervous system defects,

28

oral clefts, musculoskeletal malformations(78), conotruncal heart defects, limb

deficiencies(77, 79), Down’s syndrome(80), and nonsyndromic oral cleft(55).

3.3.3. Socio-economic indicators

There is growing evidence that socio-economic disadvantage is associated with higher

risk of a range of adverse health outcomes. Epidemiological research in this field is often

hampered by difficulties in eliciting data on socio-economic status from study

participants. This is particularly because such information may be sensitive to study

participants, or not be collected by routine data collection systems. Consequently,

researchers often resort to using a wide variety of surrogate indicators of socio-economic

status, including occupation, income, race, education, health insurance type, etc.

Gonzalez provides a systematic review of studies that looked at the relationship between

socio-economic status (as measured by education or occupation) and ischemic heart

disease (IHD), and they report a clear relationship between the two variables and risk of

IHD (81). A strong inverse relationship between socioeconomic status (SES) and risk of

cardiovascular disease and mortality has also been highlighted in another study (82).

Sometimes, socioeconomic information may not be collected from the individual

themselves but may be assessed from data on individuals living in the same residential

area. Thus, for example, study participants may be assigned a socioeconomic status

based on measures of SES of the neighbourhood in which they reside. Using ecological

data on socioeconomic status at a fine spatial level has been argued to be a good

29

substitute for individual level information and any bias introduced by this approach is

likely to lead to conservative estimates of association (83).

These areal measures are often derived from data available from the census or other ad-

hoc independent surveys. Composite indices measuring some form of disadvantage are

then formed using statistical techniques such as factor analysis or principal component

analysis. The areal measures are postulated to measure contextual socio-economic

effects of a person’s residential area. I reviewed literature on birth defects to examine the

various types of socio-economic status measured and their subsequent relationship to

specific birth defects. As I will show in a subsequent chapter, the effect of socio-

economic status on the occurrence of birth defects seems to operate at both the

individual and areal-level, and complex statistical models are needed to incorporate this

hierarchical structure in the data (i.e. data measured at disparate scales).

Among the various individual-level socio-economic indicators, occupation(20, 84-90)

seems to be the most commonly studied covariate, followed by education(26, 28, 48, 84,

85, 91, 92), race(23, 38, 50, 93-95), income(49, 70, 84) and insurance status(26, 47).

Areal-level measures of socio-economic status have been evaluated in a number of

studies of occurrences of birth defects(23, 30, 85, 96-101).

I next narrowed my review to the two most commonly studied classes of birth defects,

namely neural tube defects and oral clefts. For neural tube defects, occupation(20, 84),

race(23, 50, 93, 94), income(49, 84), education(26, 28, 84, 91, 92), insurance status(26)

and socio-economic status in general(25, 99) have been shown to be related to the risk of

30

having babies with the defect. For facial clefts, education (48, 85) and occupation(85,

87, 88) were the two main covariates implicated. Areal-level measures were shown to be

related to neural tube defects in a number of studies (23, 97, 99) along with facial clefts

(85, 100, 101). Two particular studies evaluated both individually measured socio-

economic status, along with areal measures. The results were mixed. The first study

found a significant effect of lower individual socio-economic status and residence in a

SES-lower neighbourhood on the occurrence of neural tube defects (OR=1.7, 95%CI:

1.1-2.5). This was when we looked at maternal employment and neighbourhood

unemployment as an indicator of SES (99). The other study (101) found an increased

risk of spina bifida (OR=2.3, 95%CI: 1.0-5.5) and cleft palate (OR=2.3, 95%CI: 1.4-3.8)

with a household SES index, but not with an individual SES measure such as maternal

unemployment, with an OR=1.2, 95%CI: 0.7-2.2 for spina bifida and OR=1.2, 95%CI:

0.9-1.7 for cleft palate respectively.

3.3.4. Maternal diabetes mellitus

There are two general types of diabetes(102). Diabetes type 1 is when the body produces

too little insulin that the body can’t make use of blood sugar for energy. Type 2 diabetes

happens when the body makes too little insulin or is unable to use the insulin to produce

energy from blood sugar. Mothers can enter pregnancy already suffering from controlled

or uncontrolled type 1 or type 2 diabetes. Mothers not previously diagnosed with

diabetes can also develop diabetes during pregnancy. This is often termed ‘gestational

diabetes’ and usually resolves after pregnancy, but could sometimes be a precursor of

subsequent type 2 diabetes.

31

Both chronic and gestational diabetes have been found to be associated with a range of

birth defects. Defects of the cardiovascular system seem to be the most common(43, 47,

103-108), followed by cleft lip/ palate(48, 109), spina bifida and anencephaly(110),

defects of the central nervous system(104, 108), skeletal system(108), and defects in

general(90, 111-113). Among these studies, gestational diabetes, in particular, has been

specifically linked(108-113). On the other hand, the role of diabetes in birth defects has

been specifically ruled out in some other studies(93, 114).

3.3.5 Common risk factors for caesarean section rates/ spatial variation

I next reviewed literature and examined the risk factors that were common to both birth

defects and caesarean section rates. The motivation for doing so comes from a

hypothesis I wished to test (discussed in detail in chapter 7). Specifically, I wanted to

examine whether improvements in the estimation of birth defects can be achieved by

simultaneously modelling a related birth outcome (i.e. caesarean counts). I found

maternal age to be the most commonly identified factor associated with caesarean

section in a number of studies (115-123). Socio-economic indicators were another group

of variables that were related to both birth defects and caesarean counts(117, 124)

followed by maternal diabetes(125).

Spatial variation in the distribution of caesarean section rates have been established in a

number of studies (122, 124, 126, 127). One study found a significant geographical

variation in caesarean delivery: a fourfold variation between low and high use areas

(126). That same study found that the rates were greatly influenced by nonmedical

32

factors such as provider density, capacity of local health care system and malpractice

pressure. This may have implications for its utility for the shared component models,

which we will discuss in a later chapter. Another American study found spatial variation

in the reported caesarean birth rates even after adjustments were made for differences in

maternal age and birth order distributions across states(127).

33

CHAPTER 4. CONDITIONAL AUTOREGRESSIVE (CAR) MODEL

4.1. Summary

In this chapter, I provide an evaluation of the Conditional Autoregressive (CAR) model,

which is the primary model of interest in this thesis. I give a brief description of the

model formulation, including specification of the spatial dependency via the

neighbourhood adjacency and weight matrix. Prior to that, I discuss the major

applications of the CAR model, including disease mapping and geographical regression

studies, which collectively fall under the purview of spatial epidemiology. The aim is to

provide the reader with an appreciation of the applications of these statistical models.

Comparison between the various formulations of the CAR model is performed in the

following ways. Firstly, I compare models that are designed to analyse just single

disease outcomes. Specifically, comparisons are made as to whether they allow for

spatial, temporal or even spatio-temporal interaction effects. The priors used for the

variance components of the random effects are also reviewed. Attention is then focused

on studies that have applied the CAR models in the health area. I make comparisons in

terms of their prior specification and the nature of the outcome analyzed, as well as the

level of spatial and temporal units for which data were available. Finally, I compare

models that were designed to analyse more than one disease outcome simultaneously.

This critical review of the literature forms the basis for the amendments that we propose

in subsequent chapters for the CAR model. In particular, they point to the need to fine-

tune existing formulations of the CAR model to incorporate sparseness in the outcome.

34

4.2. Spatial epidemiology

Spatial epidemiology is the description and analysis of geographically indexed health

data with respect to demographic, environmental, behavioral, socioeconomic, genetic,

and infectious risk factors(128). It encompasses areas such as disease mapping and

ecological regression analysis. Clustering and disease surveillance also fall under the

class of spatial epidemiology, but I do not discuss these topics, as they fall outside the

purview of this thesis. Also excluded from my discussion is the topic of multi-level

models, which do not incorporate spatial correlation easily in the model.

Spatial epidemiology has been used generally to study non-communicable diseases such

as cancer, heart disease and birth defects. In comparison, for infectious diseases such as

influenza or tuberculosis, modeling of disease counts is more complex, because of the

non-independence of disease counts in each area. These models also need to incorporate

transmission dynamics, and this makes the formulation of such statistical models

different and slightly complex.

The CAR model that I discuss in this project, considers data that are available at the

areal level (i.e. aggregated at some geographical level such as Statistical Local Area or

Postal Code), as compared to data at the point-process level (i.e. exact address location

of diseases, referenced from their longitude and latitude) and geostatistical information

(for instance, environmental exposures like air pollution) which are often collected at

particular sampled locations. The aims are generally different in the analysis of these

data. In geostatistical spatial models for instance, the main objective includes making

35

predictions or interpolations over the entire geographical area under study. Describing

the various statistical models available for these other kinds of spatial data, is beyond the

scope of this PhD.

4.3. Disease Mapping

One common method of displaying disease occurrence at an areal level is to aggregate

the counts of disease at some geographical areal level, and present them as choropleth

maps. Some population (denominator) data are often needed to make meaningful

comparisons of disease counts across regions. For instance, an area may simply have a

higher disease count mainly because there are more people at risk living in that area.

Expected counts of disease are usually calculated, and these expected counts are often

standardized for main demographics such as sex and age-group, which are collectively

termed as confounding variables.

Standardization is performed to sieve out the effects of these main confounders. Two

types of standardization are commonly available: external (or direct) and internal

(indirect) standardization. In the latter, one would apply the sex-age rates of the overall

region, to the specific counts of population at risk of individual strata studied, to

calculate the expected counts in each strata. The sum of these expected counts would

provide us with the age-sex standardized expected counts for each area. The ratio of the

observed and expected count of disease in each area, gives us the standardized mortality

or morbidity ratios (SMRs). The SMRs are also referred to as the maximum likelihood

estimators of the area-specific relative risks is. An area with a relative risk of 1 would

36

have neither an increased nor a decreased risk of disease compared to the other areas.

Areas with relative risk greater than one and lower than one have an increased and

decreased risk respectively, as compared to the overall group average.

Example of internal standardisation:

Oi= observed count of heart disease cases in Statistical Local Area (SLA) i in NSW

Ei= expected count of heart disease cases in Statistical Local Area (SLA) i in NSW

The expected counts can be calculated from the population count Ni in each SLA.

e.g. )(

n

ii

n

ii

ii

N

ONE

where n denotes the number of SLAs in NSW (e.g. 198) and the SMR is easily

calculated as: i

ii E

O , where θi is the risk of being in the disease group rather than the

background group. Morbidity outcomes could include counts of diseases such as asthma,

prostate cancer or birth defects (data upon which this thesis will focus attention on). The

SMRs are traditionally portrayed as choropleth maps, where different shades of colour

or grayscale are used to classify the risks in some broad categories. The cut-offs for

these categories can be decided by using the quartiles, epidemiologically meaningful

breaks or other methods. The use of different cut-offs or colour shadings can provide

rather varying visual results, as we demonstrate using data on the Index of Relative

Social Disadvantage (IRSD) in NSW, Australia (Figures 1-3).

37

Figure 1. Map of Index of Relative Social Disadvantage (IRSD) in NSW,

categorized by quartiles in grayscale

Figure 2. Map of Index of Relative Social Disadvantage (IRSD) in NSW,

categorized by quartiles in shades of colour

38

Figure 3. Map of Index of Relative Social Disadvantage (IRSD) in NSW,

categorized by epidemiologically meaningful (i.e. 75th percentile) cut-offs in shades

of colour

As we can see from the three figures above, the results appear different. Use of carefully

justified cut-offs and consistent colour shadings can help with the correct interpretation

of such maps. The choice of colour scheme influences both pattern noting and

interpreting of map details in the context of choropleth maps(129).

Disease maps are used mainly for descriptive purposes. This includes using them as a

tool to identify geographical locations with a heightened or lowered risk of disease. In

particular, areas with heightened risk are identified after sieving out the underlying risk

that is attributable to differences in the population at risk. This is done through

standardization, as we’ve discussed above. The relative risk information, in turn, can be

used by policy makers to aid in resource allocation (i.e. provide more healthcare

39

workers, build more clinics and hospitals, etc in areas which show a heightened risk of

disease).

Classification of areas with high/ low risk can also help researchers design future studies

(perhaps a geographical case-control study) with subjects selected from these regions,

resulting in more targeted subject selection. Areal classification of disease status can

also help with more focused public health education/ promotion. Disease trends, plotted

at the areal level, can also help local policy-makers better understand the impact of

policy changes over time, as well as make changes to priority disease groups, identified

through disease mapping. Increasingly, disease atlases have also been constructed and

made available to the public. These include Atlas of Cancer Mortality(130) in the United

States, which presents deaths from more than 40 types of cancers geographically, and

the Chief Health Report(131) of the New South Wales Department of Health, which

produces a range of disease maps, including cardiovascular deaths by local government

area in the New South Wales area.

However, the use of the SMR (in its crude form) has several disadvantages. Firstly, for

areas which are sparsely populated, the SMR can be imprecise and unstable. This is due

to the variance of the SMR estimate, which becomes hugely inflated because of its

inverse relationship with the expected counts, as shown by the formula below:

i.e. 2

)(i

ii E

OVar .

These sparsely populated areas are often located outside of the cities, and appear

prominent in maps because of their large geographic areas. This heightens the problem

40

of interpretation. The issue is further compounded when studying rare disease outcomes

such as birth defects. Extremely sparse counts can be observed in many of these rural

areas, with some including excessive numbers of zero counts as well. This, together with

dependence of counts in neighbouring areas, results in overdispersion with respect to the

Poisson model, which is a common statistical model used to analyse count data. In a

Poisson model, over-dispersion can be expected when the variance is much greater than

the mean of the distribution. Bayesian smoothing of these SMRs (to be explained in

detail later in the thesis) provides for a robust estimation of point estimates, resulting in

a minimized squared error loss, which is obtained through borrowing information from

the hierarchical structure. These estimates are often termed shrinkage estimators.

4.4. Geographical correlation studies

In addition to mapping and presenting smoothed relative risk estimates, CAR models are

also used for the purpose of examining the occurrence of disease at an areal level, and its

association with independent variables (or risk factors) also measured at the same scale.

These variables can include demographics (such as sex, age and occupation), socio-

economics (such as income or SES), environmental factors (e.g. ambient air pollution,

disinfection by-products in water) and also lifestyle factors (such as smoking and/or

eating habits). The independent variables can be measured at an individual level (e.g.

personal income) or at an areal level (e.g. index of relative social disadvantage) or even

a combination of both. The unit of analysis is often at the areal level. Traditional

regression models are not adequate for analyzing these data, because they fail to account

for geographical correlation, if any, in the data.

41

Correlation in counts of disease between neighbouring areas can arise due to several

reasons. Firstly, the outcome measures themselves may be correlated or clustered. For

instance, the count of cancer in one region can be high, and this region could also be

surrounded by neighbours with a similarly high count of cancer as well. Secondly, there

could be confounders or environmental factors, which themselves are spatially

structured. The CAR model, in this regard, is well suited as an analytical tool, as it

provides for a spatially structured random effect to be specified in the model in addition

to a spatially unstructured random effect term. Furthermore, it allows for the estimation

of more precise risks in sparse areas as described earlier.

4.5. Formulation of the CAR Model

In the context of disease mapping, the CAR model is most commonly formulated as

follows:

Let Yi and Ei, be the observed and expected counts of disease at an area i. Assuming that

the disease is relatively rare and independently distributed in each area,

,,...,1),(~ niEPoissonY iii

where

iiii vx )log(

The xis are covariates measured at the areal-level, with the corresponding parameter

coefficients given by is. The i represents the relative risk in each area. A Conditional

Autoregressive (CAR) prior is placed on the random effect terms vi=(v1,….,vn)

),)]([,0(~ 1 WDNv ni

42

where Nn denotes the n-dimensional normal distribution, D is a nxn diagonal matrix with

diagonal elements mi that denote the number of neighbours of area i, and W is the

adjacency matrix (most commonly denoted by the Queen method of assignment) of the

areal units (i.e. Wii =0, and Wii’ =1 if i’ is adjacent to i and 0 otherwise). -1 is the spatial

dispersion parameter, and is the spatial autocorrelation parameter. The conditional

distribution of vi is given by:

ji i

ji

ji njim

vm

Nijvv~

,,....,1,),1

,(~,,/

where i~j refers to an area i with neighbour j, which are usually defined in terms of

spatial adjacency. The model above reduces to the more commonly used Intrinsic CAR

(ICAR) model when =1. Alternatively, when =0, the rates are smoothed towards the

global value of 1 for standardized rates. An areal random effect formulation containing

both terms (spatially structure and spatially unstructured) is also known as a convolution

prior. For identifiability reasons, either the pair-wise differences in the mean intercept

for each area must be constrained to zero, or the overall intercept must be set to zero. In

the latest version in WinBUGs (the software used to execute and run the Bayesian

analysis for this thesis), this is no longer a problem, as the sum to zero constraint is

automatically implemented.

The variance parameters for the spatially structured term (2v) and the spatially

unstructured random term (2μ) are not comparable as they are not measured on the

same scale (i.e. the former is calculated conditional upon the random effects in

neighbouring areas). For purposes of calculating and comparing the contribution of the

43

variance components, the marginal empirical variance for the spatial random effect term

is used instead, which is calculated from the posterior distribution of the coefficients.

The CAR model helps address the issue of imprecise estimates of the SMR due to the

sparse ‘population at risk’ issue. Essentially, it does this by smoothing the SMR to a

value that is a compromise between the actual value and either a local mean (average of

neighbouring values) or a global mean (often taken as 1 because of the standardization),

depending on the amount of autocorrelation in the risk estimates. The model with both

the spatially structured random effect and a spatially unstructured random effect is often

used in many studies, and this is often termed a convolution prior.

4.6. Comparison of single disease CAR models

There has been a number of CAR models proposed to study single disease outcomes,

and these are summarized in Table 2 below. I make comparisons across these models for

selected studies in a variety of ways. The areal units of the selected studies vary, from as

little as 22 linguistic areas in Italy, to 88 counties in the Ohio state USA, and to 366

communes in Italy. Temporal units ranged from 3 yearly periods to 29 individual years.

It should be noted that most of these outcomes were not rare, except for the study by

Assuncao, where prevalence of Leishmaniasis was relatively rare. In all studies, no

provision was made to the models to account for sparseness. As for model comparison

techniques, the Deviance Information Criteria (DIC) seems the most commonly used,

followed by the Expected Predicted Deviance (EPD) and the cross-validation technique.

We discuss in a later part of this thesis, a comparison of these techniques. Next, in terms

44

of the hyper-prior distribution for the variance component estimates, the Gamma

distribution seems to be the most commonly used (table 3). Specifically, the Gamma

(0.001, 0.001) is the most popular. The other alternative is the chi-square distribution

suggested by Bernardinelli. Recent suggestions have included the uniform prior, and we

discuss in detail the choice of the hyper-prior distribution in a later section of this thesis.

A discussion of the comparative advantages and disadvantages of these models follows.

45

Table 2. Comparison of outcomes used, along with the spatial and temporal level of aggregation and the model comparison

technique of single disease CAR models

Author Outcome Areal Unit Temporal Unit Model Comparison Bernardinelli (132)

Cumulative prevalence of insulin-dependent diabetes mellitus

22 linguistic areas and 366 communes of Sardinia, Italy

4 time periods of 9 years each

Deviance Statistic

Waller (133) Lung cancer deaths in Ohio, USA

88 counties 21 years Expected Predicted Deviance

Held (134) Lung cancer deaths in Ohio, USA

88 counties 21 years Nil

Held (135) Lung cancer deaths in Ohio, USA

88 counties 21 years DIC

Xia (136) Lung cancer deaths in Ohio, USA

88 counties 21 years Expected Predicted Deviance

Dreassi (137)

Lung cancer deaths in males in Tuscany, Italy

287 municipalities 29 years Nil

Dreassi (138)

Lung cancer deaths in males in Tuscany, Italy

287 municipalities 29 years DIC

Catelan (139)

Lung cancer deaths in males in Tuscany, Italy

287 municipalities 6 calender periods DIC

Assuncao (140)

Human visceral Leishmaniasis in Belo Horizonte, Brazil

121 areas 3 years Cross-validation technique

Lawson (141)

Incidence of sudden infant death syndrome in North Carolina, USA

100 counties 5 years DIC

46

Table 3. Comparison of model formulation and hyper-priors used for the variance component of single disease CAR models

Author Year Log-rate function Hyper-prior distribution

Bernardinelli (132) 1995

i~Chi-square(a,b)

Waller (133) 1997

i~Gamma(a,b)

Held (134) 1998

i~Gamma(a,b)

Held (135) 1998 (IBC Conference)

Not stated

Xia (136) 1998

i~Gamma(1,0.01)

Dreassi (137) 2003

i~Gamma(0.001, 0.001)

Dreassi (138) 2005

i~Gamma(0.001, 0.001)

Catelan (139) 2005

Not stated

jijiiij ttv )log(

)()()log( ti

tikjkjijk t vrsrs

iiiktjttijkt vz )log(

itiittit v )log(

it

ikjkjijkt

vt

prsrs

)log(

jljijjiiij xpv ,)log(

l jliljiiiiij Iljxwprzvu *)()()log(

jljijjiiij x ',)log(

47

Assuncao (140) 2001

i~Gamma(0.001, 0.001)

Lawson (141) 2002

Not stated

Table 4. Legend explaining parameters used in single disease CAR models

Variable Description

Bernardinelli (132) ij Relative risk of outcome in ith area and jth time interval

Mean log-rate over all areas i Spatially unstructured random effect

vi Spatially structured random effect

Mean linear time trend i Space-time interaction effect

tj Time interval

i Precision term of random effect

Waller (133) ijkt Relative risk of outcome in ith area, jth and kth subgroup (covariate) and tth time

interval

sj Indicator variable for female gender

rk Indicator variable for non-white race

sjrk Gender and race interaction term

i(t) Spatially unstructured random effect at tth time interval

2)1()1()log( jj iiiij

iiiiii wpvp )1()log(

48

vi(t) Spatially structured random effect at tth time interval

Held (134) ijkt Relative risk of outcome in ith area, jth agegroup and kth subgroup (gender by

race combination) and tth time interval

t Effect of year t

jt Age group j effect at time t

i Spatially unstructured random effect

vi Spatially structured random effect

kt Gender by race effect at time t

zi Centred KT index (a measure of urbanisation)

Held (135) it Relative risk of outcome in ith area and tth time interval

t Temporal term with random walk

t Unstructured temporal term

i Spatially unstructured random effect

vi Spatially structured random effect

it Spatio-temporal interaction term

Xia (136) ijkt Relative risk of outcome in ith area, jth gender and kth race (gender by race

combination) and tth time interval

sj Sex effect

rk Race effect sjrk Sex/ race interaction effect a Overall log-relative risk of all areas

49

vit Spatially structured random effects over time, where the clustering effects are nested within time. These terms are assigned the CAR prior

Fixed time effect pi The true but unobserved smoking proportion in each county

Dreassi (137) ij Relative risk of outcome in ith area and jth time interval

i Spatially unstructured random effect

vi Spatially structured random effect

pj Period (from 0 to 5)

zi Time difference in each region that is spatially unstructured

ri Time difference in each region that is spatially structured

xi,j-l A vector of the material deprivation index for the ith area observed at lagged time

Ijl Indicator matrix

Coefficient for relationship between material deprivation and mortality Dreassi (138) ij Relative risk of outcome in ith area and jth time interval

i Spatially unstructured random effect

vi Spatially structured random effect

pj Effect of jth period, modelled as an autoregressive conditional random term

xi,j-l A vector of the material deprivation index for the ith area observed at lagged time

Vector of indicators, with the associated probabilities, , following a Dirichlet distribution

Catelan (139) ij Relative risk of outcome in ith area and jth cohort

i Spatially unstructured random effect

50

vi Spatially structured random effect

pj Random effect of the jth cohort

j Coefficient of education for each birth cohort

xi, j+l Vector of education scores for the ith area observed at different ages of exposure

j Vector of indicators, with the associated probabilities, , following a Dirichlet distribution

Assuncao (140) ij Relative risk of outcome in ith area and jth time interval

i Logarithm rate of disease in ith area in the first year (assigned a CAR prior)

i Coefficient for linear temporal component (assigned a CAR prior)

i Coefficient for quadratic temporal component (assigned a CAR prior) Lawson (141) i Relative risk of outcome in ith area

i Spatially unstructured random effect

vi Spatially structured random effect

wi Represents discrete jumps in total absolute difference between neighbours.

pi Prior for the mixing probabilities

51

Bernardinelli(132) developed a spatio-temporal model, where both the area-specific

intercepts and temporal trends were modeled as random effects. Although they proposed

extensions to the model to incorporate spatio-temporal effects and allowing for

correlations between the area and time random effects, in the application to their study

data, they did not find these additional terms significant. One limitation of this model is

the use of the linear time trend and the absence of a temporal random effect term.

Inclusion of quadratic and higher order terms should be straightforward. The authors

also included either the spatially unstructured term or the spatially structured term in the

model. An alternative approach would be to include both terms in the model, and let the

data dictate the relative contribution of each term.

In contrast to the Bernardinelli model, Waller(133) et al, proposed a spatio-temporal

model that involves a nested component, where both the spatially structured and

spatially unstructured random effects are nested within time. The authors use vaguely

informative priors for the precision parameters, which were derived from the same data

(i.e. middle period spatial-only model), giving the analysis a slightly empirical Bayes

flavor. Sensitivity analysis to this prior selection was not done. The authors have also

not included age, an important confounder that was available, either as a variable upon

which to standardise the rates, or as a covariate in their model.

Compared to earlier models which made use of the CAR prior for the temporal terms,

Knorr-Held and colleagues(134) allowed for the temporal terms (t) to follow a random

walk with independent Gaussian increments (a type of autoregressive prior distribution).

52

The authors applied block Metropolis steps (specifically updating i and vi in a single

block) to sample from the posterior distribution, mainly because they found the spatially

structured and spatially unstructured parameters to be negatively correlated. However,

the authors did not use any formal model selection procedures to compare competing

models. The models also did not involve more complex space-time interaction effects.

As a follow-up to the limitation discussed in the earlier paper, Knorr-Held(135)

presented a class of models which allow for spatio-temporal interactions. Essentially,

these models assume that the random effects cannot be separated into just temporal and

spatial components. He considered four types of interaction effects: Interaction type 1:

The two unstructured main effects are expected to interact. This can be described in

terms of unobserved covariates in each space-time period that do not have a structure.

Interaction type 2: The temporal effect, which has a random walk prior assigned to it, is

expected to interact with the spatially unstructured term. This can be useful in situations

where we have temporal trends that are different between areal units, but do not have

any structure in space. Interaction type 3: The unstructured temporal term is allowed to

interact with the spatially structured term. This would specify a situation where spatial

patterns are different for the various time points, and there is no temporal trend.

Interaction type 4: This is the result of an interaction between temporal effect with a

random walk assigned, and the spatially structured random effect term, which results in

a rather complex formulation. The prior used was actually a Markov random field. This

type of interaction specification is useful for infectious disease studies. Although the

models look promising, as the author concedes in the paper, the number of parameters in

53

the interaction models is very high, and so some tuning of the Metropolis steps is

needed. Specialised software is also needed to run such models, which puts them out of

reach to most applied statisticians or epidemiologists.

At around the same time, Xia and Carlin (136) presented a spatio-temporal model, which

allows for spatially correlated errors in the observed covariates (in their case smoking).

In their analysis, the authors did not find any significant effect of the spatially

unstructured random effects, and removed it from subsequent analyses. In the paper, the

authors made the assumption that socio-demographic covariates do not interact with

time or space. It is not clear how the results would have varied if this assumption was

relaxed.

Dreassi et al(137) used a modified version of Bernardinelli’s(132) spatio-temporal

model to investigate the time-dependent characteristic of socioeconomic factors, with

lung cancer mortality among males in Tuscany, Italy. The authors assumed that the

effect of material deprivation on lung cancer mortality can be derived from a sum of

different scores, each pertaining to a different temporal lag. Extensions to this basic

assumption should have been explored. In a subsequent follow-up paper, the authors

expanded the methods to also take into account values imputed between the census

years(138). They were able to this in the context of Bayesian hierarchical models. The

material deprivation index was assigned a spatially unstructured random effect and an

autoregressive time random effect. There are several limitations to their approach. The

proposed models do not account for measurement error in the covariates. They also do

54

not take into consideration cohort effects. Equally lacking in the model is a formulation

to incorporate space-time interaction effect.

Coming from a different perspective, Catelan and fellow researchers(139) proposed a

Bayesian hierarchical model with time-dependent covariates, along with space-time

random effects. They considered six birth cohorts (10 year groups from 1905 to 1940).

Using the multiplicative age-cohort model, the observed and expected counts were

aggregated along the diagonals of the Lexis diagram for the six birth cohorts, thus

collapsing over the age dimension. The model was used to evaluate the association

between areal measures of ‘low educational level’ and lung cancer mortality at the

municipality level, among males in Tuscany, Italy. The authors performed a sensitivity

analysis on the choice of the priors for is. However, they have not stated which priors

were used for the precision parameter of the CAR model, and whether any sensitivity

analysis was done regarding the choice of these priors.

In a sharp contrast to traditional areal spatial models used by most researchers, which

took into account the spatial distribution of the response variable and the neighbourhood

structure, Assuncao suggested an innovative method of allowing for the coefficients of

the predictor variables to vary smoothly, based on the area’s geographical location(140).

One main criticism of the usual CAR model is that it tends to over smooth maps,

especially between boundaries that show clear discontinuity (e.g. urban versus rural

areas). To this end, Lawson(141) proposed a new class of mixture models, which allow

55

for the log-relative risk to be split into three additive parts: a spatially unstructured

random effect, and two mixing components (a spatially structured random effect and a

component which models discrete jumps). While the proposed model accounts for

spatial heterogeneity, temporal extensions to the model may prove to be useful,

especially for forecasting purposes.

4.7. Studies that have applied the CAR model

I next examined the hyper-prior choices and other model formulation details of studies

that have applied the single-disease CAR models discussed earlier. The areal units

studied were far more wide-ranging than those studies used to develop the models. For

instance, the CAR models were applied to data including as little as 57 schools in Haiti

to 10,530 wards in the United Kingdom (table 5). The temporal units were equally

diverse, ranging from 91 days in one study, to 36 months in another. The DIC seemed to

be the predominant measure used for model comparison purposes.

The disease outcomes studied using the CAR models, ranged from hospital discharges

for acute myocardial infarction and bronchitis to incidence of prostate cancer and counts

of respiratory episodes. The models have also been used to study infectious diseases

such as prevalence of W. bancrofti infection and incidence of malaria, although it is

questionable whether the models are appropriate here, due to the non-independence of

counts of disease. Once again, it should be noted that these studies can hardly be

classified as rare, and hence the models have used a simpler formulation of the Poisson

distribution, without any extension to incorporate excess zeros in the data. In terms of

56

the hyper-priors for the variance parameters, once again the Gamma distribution seemed

to be the most popular (table 6). However, the scale and shape parameter varied,

depending on each study (e.g. Gamma (0.001, 0.001) to Gamma (0.5, 0.002) to Gamma

(0.5, 0.005)).

57

Table 5. Comparison of outcomes used, along with the spatial and temporal level of aggregation and the model

comparison technique used in the applied studies

Author Outcome Areal Unit Temporal Unit

Model Comparison

Heisterkamp (142) Hospital discharges for acute myocardial infarction and bronchitis in Netherlands

291 postcodes 3 years Expected Predicted Deviance

Johnson (143) Incidence of Prostate cancer

1412 New York state ZIP codes

Nil Mean Deviance

Jarup (144) Incidence of Prostate cancer

10,530 wards in UK Nil DIC

Boyd (145) Prevalence of W. bancrofti infection

57 schools across Leogane Commune in Haiti

Nil Nil

Law (146) Deaths from coronary heart disease

1031 census enumeration districts in Sheffield, UK

Nil DIC

Nobre (148) Incidence of malaria 69 counties in Para, Brazil

36 months DIC

Rodeiro (149) Counts of respiratory episodes

287 census tracts in Boston, USA

91 days DIC

58

Table 6. Comparison of model formulation and hyper-priors used in studies that have applied the single-disease CAR

model

Author Year Log-rate function Hyper-prior distribution

Heisterkamp (142) 2000

i~gamma(0.001, 0.001)

Johnson (143) 2004

i~gamma(0.5, 0.002)

Jarup (144) 2002 Not stated Not stated

Boyd (145) 2005

i~gamma(0.5, 0.005)

Law (146) 2005

i~gamma(0.5, 0.005)

itiitit vx )log(

iii v )log(

ii

iii

v

xx

22110)log(

ii

iii vx

6

10)(logit

59

Nobre (148) 2005

i~gamma(1,1)

Rodeiro (149) 2006

i~gamma(0.5,0.005)

itittij bx )log(

),0(~, 21 Nww tttt

ttt

tit CARb

)log()log(

),(~2

12

2

iiijjij vt )log(

60

Table 7. Legend explaining parameters used in the models that have applied the single-disease CAR model

Variable Description

Heisterkamp (142)

it Relative risk of outcome in ith area and tth time interval

xit Fixed effect covariates (e.g. noise pollution or distance from airport)

vi Spatially structured random effect

it Space-time interaction effects Johnson (143)

i Relative risk of outcome in ith area Mean log-rate over all areas

i Spatially unstructured random effect

vi Spatially structured random effect Jarup (144) Not provided Boyd (145)

i Relative risk of outcome in ith area

vi Spatially structured random effect

0 Baseline risk

i Coefficient for covariates

xi Covariates

Law (146)

61

i Relative risk of outcome in ith area

vi Spatially structured random effect

0 Baseline risk

i Coefficient for covariates such as smoking and Townsend material deprivation index

xi Covariates such as smoking and Townsend material deprivation index

Nobre (148)

it Relative risk of outcome in ith area and tth time interval

bit Spatially structured random effect

t Error term that follows a normal distribution with mean 0 and variance,

Rodeiro (149)

ij Relative risk of outcome in ith area and jth time interval

i Spatially unstructured random effect

vi Spatially structured random effect

tj Temporal trend, which is assigned a random walk prior

ij Spatio-temporal random effect, which is assigned a random walk prior

62

Heisterkamp(142) applied the CAR model to study the relationship between noise and

distance from a national airport in Netherlands, and hospital discharges from bronchitis

and acute myocardial infarction at the postcode level. Based on the Expected Predictive

Deviance, the authors found that the results varied for males and females. Generally, the

CAR model with the interaction term performed best, with the lowest EPD. For males,

addition of the noise index variable did not significantly improve the fit of the model for

acute myocardial infarction, whereas for females it did. As for bronchitis, distance from

airport appeared to fare better for males than for females. It is also unclear whether the

δit term used in the analysis measures the spatio-temporal interaction effects as the

authors also concede in the paper.

In their application of the CAR model, Johnson and colleagues(143) evaluated the

sensitivity of their results to the choice of hyperparameters used for the precision term of

the random effects by considering an alternative parameter. They showed that the

difference in the smoothed standardized incidence ratios hardly changed. The authors

could have improved on their model by examining temporal trends of prostate cancer

incidence in New York, and also perhaps a spatio-temporal extension.

In another project(144), the authors made use of the CAR model to smooth prostate

cancer incidences at the ward level, after accounting for age and areal Carstairs

deprivation index. However, they failed to provide detailed information on how their

model was formulated. Key determinants of the model such as neighbourhood

formulation and parameters for the hyper-priors were also not elaborated upon in the

manuscript. Although the authors claimed to have conducted a small simulation study to

63

evaluate the sensitivity of the model in terms of detecting areas with an elevated risk,

they could have provided details on the study, particularly in terms of choice of hyper-

priors and model selection techniques.

Contrary to other applications involving the CAR model, the authors in this particular

study(145) used a distance-based approach to define neighbourhoods (i.e. all schools

within a distance of 4.35km of a given school were included as neighbours). The authors

also compared the parameter estimates of the covariates with other non-Bayesian spatial

models, including the Generalized Linear Models and the Generalized Linear Mixed

Models. Another research project(146) examined the relationship between smoking and

areal measures of deprivation with coronary heart disease mortality using a CAR

convolution prior. They also performed spatial smoothing of the smoking covariate.

Instead of smoothing the smoking covariate, the authors could have considered using the

errors in covariate model proposed earlier by Bernardinelli(147).

Nobre, et al (148), made use of an amended form of the CAR model to study the

relationship between rainfall and malaria incidence in Brazil. As rainfall is monitored at

fixed locations, and malaria is observed at the areal level (i.e. both measured at disparate

scales), the authors use a gaussian interpolation technique to calculate rainfall at the

county level. For the CAR model, they consider a few variations, including one which

allows for the spatial variability to change over time. Using the DIC, the authors

concluded that the spatial model performed poorly, as compared to the non-spatial

equivalent. The authors did not discuss the reasons for this finding. Also, some sort of

64

sensitivity analysis on the choice of priors should have been performed. Furthermore, it

is unclear if the CAR model is well-suited to study an infectious disease such as Malaria.

Rodeiro and Lawson(149) applied a spatio-temporal CAR model to daily counts of

respiratory episodes in Boston, in order to evaluate it's usefulness as a surveillance tool.

Prior to applying the model on the data, they also perform a simulation study to gauge

the model's performance under 7 different scenarios whereby the risk patterns are

expected to change over time. While the authors should be commended for performing a

simulation study to evaluate their model's utility as a surveillance tool, perhaps they

could have presented the results comparing against several other competing models (as

mentioned briefly in the discussion section of their manuscript).

4.8. Studies that have compared single disease CAR models

There were a few studies which have attempted to compare the performance of

competing CAR models. We discuss these studies briefly. Lawson(150) presented

results of a large simulation study, comparing 154 different models using a variety of

goodness of fit measures, including residual sum of squares, Moran’s I and Bayesian

Information Criteria (BIC). He reported that, in terms of robustness against mis-

specification, the Besag, York and Mollie (BYM) model performed best. The research

also found that mixture models, along with non-parametric smoothing techniques and

linear Bayes methods performed poorly as compared to the BYM model.

In an attempt to investigate the relationship between calcium and magnesium in drinking

water and cerebrovascular mortality, Ferrandiz and colleagues(151) evaluated a number

65

of Bayesian hierarchical models. Specifically, they considered data aggregated in a

number of ways (temporally) and compared models which included/ excluded terms,

including the spatially structured and spatially unstructured random effects. In all, they

compared 11 different models. For model comparison purposes, they used the Pseudo

Bayes Factor and the Deviance Information Criteria. Consistently across the various

ways that data were aggregated, they found that models incorporating the spatially

structured random effect (clustering term) performed better.

Best(152) compared several Bayesian hierarchical models in the context of disease

mapping. Five different univariate spatial models, along with two multivariate models

were compared. The authors used the DIC to compare the performance of the models

across five simulated datasets. They found that the mixture model (153) performed

consistently well (i.e. lower DICs as compared to the other models) amongst the

univariate analysis. In the multivariate analysis, they concluded that the shared

component model(154) had a better ability to detect areas with a true increased risk, as

compared with the multivariate CAR (MCAR) model.

In an attempt to map and study the geographical distribution of deaths from West Nile

Virus infection in the United States from 2003 to 2004, Griffith et al(155) compared and

contrasted six different disease mapping models, including the gaussian spatial

autoregression model, Gaussian spatial filter model, binomial spatial filter model, spatial

filter model and finally the CAR model. They concluded that no one particular model

was superior over the others, and that further information on covariates such as

rural/urban locations, willingness of local populations to accept and fund mosquito

66

control programs and adoptions of these programs could better help describe these

spatial variations. Here, they’ve also used the proper CAR prior instead of the routinely

used improper CAR (ICAR) prior.

In a simulation study encompassing three different patterns of elevated risk (including

single isolated areas with elevated risks, and several larger clusters of adjacent areas

with elevated risks), and three magnitudes for elevated risks (RR=1.5, 2 and 3),

Richardson and colleagues(156) examined the performance of the following three spatial

models specifically in terms of their ability to recover the ‘true’ underlying risk surface:

1) the BYM CAR Model(157), 2) the median-based local smoothing L1-BYM

Model(157) and 3) the MIX Model(153), which allows for discontinuities in the risk

surfaces.

The authors found that generally, the models were conservative in terms of their ability

to detect areas with a raised risk. In other words, the models had poor sensitivity in

terms of detecting areas with a heightened risk. Sensitivity, they argued, improved when

information from the posterior probability was used as well. They found the

performance of the BYM CAR model and the L1-BYM model to be similar, but the

MIX model produced considerably less smoothing, when there was moderate size

expected counts and/ or high true excess risks. They proposed a set of decision rules for

the CAR model involving the posterior probability. This included computing the

probability of the relative risk being above one, with a cut-off between 70% and 80%.

Such a criterion would provide reasonable sensitivity for a range of scenarios having

moderate expected counts and excessive risks in the order of 1.5 to 2. In appendix B of

67

the same article, they also provided a series of loss functions (which involves a trade-off

between sensitivity and specificity) for different weightage given to each component.

4.9. Comparison of multiple disease CAR models

So far, attention has been focused on single disease CAR models. I now examine the

multivariate equivalent. Multivariate CAR models involve the study or 2 or more

diseases simultaneously. In such studies, diseases are selected for the study if there is a

strong possibility that they share a similar geographically varying risk factor. One

advantage of such multivariate modeling is that if similar patterns of geographic

variation of related diseases can be identified in a joint analysis, this may provide more

evidence of real clustering in underlying risk surface than from the analysis of the single

disease separately. I provide a brief summary of currently available spatial CAR models

that have been used to analyse multiple diseases at the same time.

The shared component model for two diseases was originally proposed by Knorr-Held in

2001(154). The main idea of the model was to split the underlying risk surface for each

disease into a disease-specific component and a shared component. The shared

component is thought to be a surrogate for unobserved covariates that display spatial

structure, and are common to both diseases. A scaling factor, , is used to weigh the

contribution of the shared component to the overall relative risk. The introduction of this

scaling factor is needed because of identifiability issues with the model. The two

disease-specific risk components and the shared component are assumed to be

independent, with each having a spatial prior assigned to it. Reversible jump Markov

68

chain Monte Carlo methods were used to run the models. Knorr-Held and colleagues

applied their model to oral cavity and oesophageal cancer mortality data among males in

Germany. They found two large clusters with a large shared component value, and

postulated that it was consistent with the distribution of risk factors in the area. They

also found distinct spatial patterns for each individual disease.

One main limitation of their study was the choice of the prior number of clusters set in

their model. The authors conceded that extensive simulation studies were required to

provide guidelines on choosing the hyperparameters. In their study, Knorr-Held et al

reported some sensitivity analyses on the choice of hyperparameters used for the prior

number of clusters. Also, the cluster models are not available via standard analytical

software (e.g. WinBUGS), but adaption to the model is possible by using a CAR prior

instead.

In another piece of research, Carlin et al(158) formulated and applied the multivariate

CAR model to analyse cancer survival data of 17,146 patients from the 99 counties in

Iowa, USA (including cancers of the colorectal, gallbladder, pancreas, small intestine

and stomach) between the years 1992 to 1998. They extended the model to analyse time-

to-event data and also incorporated temporal components. The use of multivariate

models was justified in the paper by examining the correlations in the frailty terms

between the various cancers (e.g. correlation of 0.528 for pancreas and stomach).

Using the DIC, Carlin and colleagues also compared the single CAR model with several

variations of the MCAR model. They found that the MCAR model, with disease-specific

69

mean, j, performed well with a low DIC of 237. However, the authors acknowledge

problems with using j, since j <1 implied that the mean is a shrunk version of the

average of the neighbouring values.

In contrast to other work on multivariate CAR models, Gelfand et al (159) examined

these models in the context of introducing spatial autoregression parameters for

distributional propriety in the form of a multivariate Gaussian Markov random field. In

particular, they used the proper CAR distribution instead of the improper CAR prior

distribution. Applications to a two-dimensional data (height and weight adjusted for

age) as well as a four-dimensional (population allele frequencies) problem were used to

exemplify the models. Due to the sparseness of allele frequency data at the Resource

Mapping Unit (RMU) level, sparse units with less than five individuals were aggregated

with their closest neighbours. The use of multivariate CAR models was justified in the

analysis by looking at the correlation estimates from the posterior distribution and also

the difference in spatial patterns between the competing models.

Extending his earlier work to include more than 2 diseases, Knorr-Held and colleagues

developed joint models(160) for more than two diseases, and also considered ecological

regression models, where disease rates from a second disease entered as a covariate in

the model, with the first disease being the main one of interest.

They applied their models to study the spatial variation of oral, oesophagus, larynx and

lung cancers in 544 districts in Germany from 1986 to 1990. In particular, they used

70

information on lung cancer rates, known to be related to smoking, as an ecological

surrogate covariate for each of the other three types of cancer, similar to a study done by

Bernardinelli(147). The joint model included a shared component, which was relevant

for both diseases, and an additional part for the residual variation that was relevant for

just one of the disease. The authors used the DIC statistic to compare models,

particularly individual models with the combined ones. They concluded that the shared

component models performed better across all disease groups.

In this paper, Jin and colleagues(161) proposed a flexible class of Generalized

Multivariate Conditionally Autoregressive (GMCAR) models for areal data. The

GMCAR model directly specifies the joint distribution for a multivariate Markov

Random Field (MRF) through the formulation of simpler conditional and marginal

models. The GMCAR distributions were used in the second-stage random effects for the

hierarchical area models. The models allowed for smoothing parameters for the cross-

covariances, as compared to existing MCAR models. The authors then made

comparisons with the multivariate conditional autoregressive (MCAR) model(159) using

simulation studies. The Deviance Information Criteria (DIC) and the Average Mean

Squared Error (AMSE) were used to evaluate the results from the simulation studies.

Using four different studies (100 datasets simulated within each study), the authors

compared five different model formulations (3 GMCAR and 2 MCAR models).

They found that the GMCAR models performed consistently better than the MCAR

models across all the different scenarios. When applied to death rates from lung and

esophagus cancers in Minnesota counties from 1991 to 1998, they found that the

71

GMCAR model performed consistently better than the MCAR (i.e. lower DIC values).

Within competing GMCAR model specifications, they also found that the model

accounting for the bivariate spatial structure in the data performed better than the one

that did not account for this spatial structure. One disadvantage of this model though,

was that the variables being modeled were placed in an arbitrary (causal) order, and this

can lead to different marginal distributions depending on the conditioning sequence.

Although the authors have argued that model selection procedures can be used to select

the optimal model in terms of conditioning order, this is not an established practice. This

has been demonstrated in subsequent simulation studies(162).

Dabney and Wakefield(163) discussed joint mapping of two diseases when the

population at risk was unknown. In particular, they postulated that for a relatively rare

disease, a joint model may better estimate the risk using a more common but related

disease. In addition, the researchers argued that using joint disease mapping tools to

predict areal-level risks is less risky than trying to undertake ecological regression

analysis. The proportional mortality model is applied and compared with the shared

component model(154) and individual disease models, using bladder and lung cancer

incidence data at the Washington state census tract level, from 1996-2000. The

proportional model assumed that the probability of one disease over the other disease is

conditional on the total count of disease. They found similar estimates of shared spatial

effects between the shared component and the proportional mortality models. In terms of

two specific covariates studied, namely income and smoking, the proportional mortality

model found that the relationship between smoking and lung cancer was stronger than

that between smoking and bladder cancer. The converse was true of income, which was

72

another variable they included in their study. The findings were generally similar to the

shared component model. The main advantage was that the proportional mortality model

did not require data on population at risk.

Richardson and colleagues(164) identified a gap in small area analysis; specifically, lack

of useful models to analyse ‘spatio-temporal’ disease counts at a small area level for

multiple disease groups. With such a model, one can borrow strength across space and

time, and also among related disease groups. Such a model can help identify areas with a

heightened shared risk factor, and also point out instances where data collection methods

may not be uniform. The authors proposed four different formulations, of which the

most general form was given by the inclusion of space-time interaction effects common

to both diseases, and the addition of sex-specific heterogeneity terms. For further details

on the model, readers are referred to the article(164). The authors applied the models to

lung cancer incidences among males and females in Yorkshire, England from 1981 to

1999. Data were available at 4 broad periods and 626 wards. Based on the DIC, the

authors concluded that the model with space-time interaction terms common to both

diseases was the most appropriate.

Jin, et al(162), used the linear model of coregionalization (LMC), an approach that was

originally developed for multivariate point-referenced data by Wackernagel(165). The

basic idea was to develop richer spatial association models by transforming simpler

spatial distributions. Once again, the authors performed a simulation study (1000

datasets for each model/ scenario) to compare the new model against existing models.

They applied their model to mortality data from lung, larynx and esophageal cancers

73

between 1990 and 2000 in Minnesota counties, USA. The authors compared 6 different

models (3 MCAR models, 2 GMCAR models and a misspecified geostatistical model

with an exponential covariance function). They found significant correlations between

the different cancers, particularly between lung and esophagus cancers.

They also found that the MCAR model with LMC performed relatively better than the

other models, with a lower DIC and AMSE values. One main advantage of the proposed

model was that they can be used to study higher dimensional problems (i.e. more than 2

diseases) efficiently, as the computation does not involve large matrix calculations. On

the other hand, one limitation was that it did not take into account temporal trends,

although the authors have suggested incorporating multivariate Gaussian AR(1) time

series spatial processes, which have been used in the setting of dynamic models.

Dreassi(166) extended the work done by Dabney(163) specifically by analyzing more

than two diseases with a polytomous logit model, and adopting the shared component

model(154) for the predictor variables. Specifically, Dreassi et al used a single disease as

a reference category and adopted the shared component formula for each predictor. They

applied their model to oral cavity, larynx and lung cancer, with the latter being the

reference category in the polytomous logit model. Dreassi and colleagues concluded that

their method allowed one to study more than 2 diseases at a time, without information

on the population at risk, and also reduced the standard deviation of disease-specific

cluster estimates, when compared with the shared component model.

74

CHAPTER 5. CAR MODELLING ISSUES

5.1. Summary

In this chapter, I provide an overview of issues of importance when one contemplates

the use of the CAR model in an analysis. Firstly, I offer an introduction and discussion

of Bayesian theory, including MCMC analysis and convergence assessment, which are

essential topics in any Bayesian modelling. Next, prior specification of the variance

component in the CAR model is discussed. In particular, I assess the impact of

commonly suggested priors in literature, in terms of a small-scale sensitivity analysis.

Next, the Modifiable Areal Unit Problem (MAUP) and the model selection techniques

available for comparing competing spatial models are discussed. A review is also

undertaken on the topic of boundary analysis, and I argue how the objective of such an

analysis differs from the traditional use of the CAR model. Apart from the sensitivity

analysis of the priors, it should be noted that this chapter does not involve any new

analysis of data. Instead, it consists of a literature review on a few important topics

around the CAR model. Further analysis on these topics is out of scope of this thesis.

Yet, these topics are important for readers, as they provide useful background

information to understand the context in which the rest of the chapters are written.

5.2. Bayesian Theory

There are many advantages to undertaking Bayesian analysis. Bayesian models allow for

more interpretable definitions of the confidence intervals (or credible intervals in

Bayesian sense), provide for confidence intervals on parameters, and also allow one to

incorporate any prior information one may have, in the model. For instance, frequentist

75

interpretation of the 95% confidence interval (as an example) is that if we conduct the

same experiment many many times, in 95% of the time, we will observe the true

parameter within the specified interval. From a Bayesian’s point of view, however, we

say that the confidence interval contains the true parameter with 95% certainty, which is

easily understood by non-statisticians. Advancements in computation speed and

efficiency of computers have also helped to overcome the initial hurdle to using

Bayesian models, which are notorious in taking up large computational memory and

space.

In a Bayesian model, the posterior distribution is computed from the likelihood function

and the prior distribution, as shown below:

)(

)()/(

)(

),()/(

yp

pyp

yp

ypyp

where p(y/) is the likelihood of y under a model and p() is the prior density and p(y)

or the denominator is a fixed normalising factor, which ensures that the posterior

probabilities sum to 1.

More commonly, the posterior distribution is also displayed as:

)()/()/( pyPyp

Posterior distribution (Likelihood x Prior distribution)

It is clear from the above formulation that the estimates from the posterior distribution

are influenced by the prior and the data. The amount of data available, and the type of

prior we specify (e.g. non-informative versus weakly informative or highly informative

priors) will influence the relative influence of the prior. In a spatial modeling setting

76

however, we often use non-informative priors or very weakly informative priors merely

to facilitate the formulation and estimation of the model. This actually addresses one of

the more fundamental criticism of the Bayesian approach, where the use of informative

priors has been denigrated.

5.3. Markov chain Monte Carlo (MCMC)

Instead of obtaining the closed form of the likelihood for a complex spatial model, one

can obtain samples from the posterior distribution of a Bayesian analysis instead. The

sampling techniques or algorithms are known as Markov chain Monte Carlo (MCMC)

methods. These methods include repeated sampling across the entire distribution of the

parameter or parameters we are interested in. This involves sampling parameters and

then updating the latest parameter values(167). Sampling is often done with long and

multiple chains, each chain starting from diverse yet reasonable values. A number of

MCMC methods have been proposed in literature, including the Metropolis-Hastings

algorithm(168), Gibbs sampling(169), Slice sampling(170) and Reversible jump Markov

chain Monte Carlo method. The Gibbs sampler involves sampling each parameter in turn

from the posterior distribution, while keeping the other parameters fixed. The Gibbs

sampler is a special componentwise Metropolis-Hastings algorithm, whereby the

proposal density for updating j is the full conditional p(j|[j]) so that proposals are

accepted with probability 1, in a series of steps shown below(171):

Step 1 ),....,|(~ )()(3

)(211

)1(1

td

ttt f

Step 2 ),....,|(~ )()(3

)1(122

)1(2

td

ttt f …….

Step d ),....,|(~ )1(1

)1(3

)1(1

)1(

td

ttdd

td f

77

The Gibbs algorithm samples from the full conditional distribution of each parameter

and does not need the joint distribution of the parameters. This in turn, eliminates the

need for matrix inversion calculations and makes for computation to be relatively easier.

The slice sampling technique is an alternative to the Metropolis-Hastings algorithm, and

this technique always samples from the exact full conditional distribution, and can be

expected to converge more rapidly(172).

5.4. MCMC Convergence

Because of the nature of MCMC sampling, sampled values are often correlated. Ideally,

one would wish to obtain samples from the posterior when they are thought to arise from

a stationary distribution of the chains. A generally accepted diagnostic tool in Bayesian

analysis is to initiate two or more chains, with each chain consisting of initial values of

parameter estimates starting at diverse values. Convergence of these chains is then

assessed in a number of ways, including the Gelman-Rubin scale reduction factor

(GRSRF) (173) and the Brooks Gelman Rubin (BGR) statistic(174).

Bayesian models that do not show convergence often indicate poorly identified models,

including missing identifiability constraints, over fitting and redundant parameters in the

model. Poor convergence can also be caused by the choice of overly diffuse priors.

The calculation of the GRSRF(171) is shown below:

The variability of samples )(tj within the jth chain is given by:

78

Ts

st

jt

jj T

V1

2)(

1

)( , where j=1,2,….,J, and T is the number of iterations after an initial

burn-in sample of s.

The variability within chains is just the average of Vj. i.e.

J

j

jw J

VV

1

The variability between chain is given by:

J

jjB J

TV

1

2)(1

, where is the average of the j .

W

P

V

VGRSRF , where

1

T

TV

T

VV WB

P

Values of the GRSRF less than 1.2 are indicative of convergence of the chains. In

addition to the diagnostics, one would also wish to examine the Monte Carlo standard

errors of the posterior estimates and ensure that they are relatively smaller than the

standard error of the parameter estimates.

5.5. Specifying the hyperprior distribution

One key issue in Bayesian analysis, in particular the CAR model analysis, is the

formulation of the hyperprior distribution for the variance (or precision) parameter

estimates. In general, there are two ways of specifying the distribution for the precision

term of the spatially structured random effect σs. In the empirical Bayes (EB) approach,

information about σs is provided by the data through maximum likelihood methods or an

initial Bayesian analysis. A disadvantage of doing this is that uncertainty in the

estimation of σs is not taken into account in the model. The fully Bayesian method, on

79

the other hand, allows one to specify a prior distribution on the hyper parameter σs (also

known as hyperprior). This addresses the issue of uncertainty in measuring σs and also

circumvents the issue of providing initial ‘guess’ estimates of σs.

For the CAR model, several guidelines on the choice of the priors for the variance

parameters have been proposed. Congdon reviews common ways priors have been

specified(171). In the first instance, the variance from the crude relative risks is used in

the computation as described below. This was the method suggested by Mollie(175).

)(

1

rVarr , where ri= Oi/Ei, and Oi and Ei are the observed and expected counts of

disease in an area i. The priors on the non-spatial and spatial precision terms are then

specified as follows:

),(~ ccGamma r and

),(~ cN

cGamma r , where c is a small constant and N is the average number of

neighbours.

The other alternative suggestion by Bernardinelli(176) makes the assumption that the

marginal standard deviation of the spatial effects is proportional to the conditional

standard deviation, as shown below:

i

s

NsSD

7.0)(

, where SD(s) is the empirical estimate of the marginal standard deviation of the spatial

effect, which is estimated during a MCMC run. Ni denotes the average number of

80

adjacencies. In addition, the authors recommend the following chi-square distributions

for the spatially unstructured precision term and spatially structured precision term

as shown below.

21~

VS

21~

VS

, where Sμ and Sν denote the scale factors and Vμ and Vν denote degrees of freedom

respectively. The prior means are given by Vμ/ Sμ and Vν/ Sν and variances 2Vμ/ (Sμ)2

and 2Vν/ (Sν)2 respectively.

The authors have, via a simulation study, examined the sensitivity of relative risk

estimates from a BYM model, to the choice of the hyperprior distribution(176). For ease

of interpretability, they expressed the hyperprior in a more understandable figure (i.e.

ratio of high versus low rate ratios). They found that their model results (i.e. relative risk

estimates) were insensitive to the choice of priors. Possible reasons for their finding

include their large size of data (which seems to suggest that the data overwhelms the

information provided by the prior), and the small simulation study performed.

In addition to the chi-square distribution, the gamma prior has also been widely used for

the precision terms (also termed inverse gamma prior for the variance terms). This

conveniently produces a marginal distribution for the residual relative risks in a closed

form (177). The investigators propose that the choice of hyperprior parameter selected

revolve around some statement about the postulated range of residual relative risks. The

81

authors allow for the specification of p, the proportion of total variation due to spatial

structure, and assign a Beta prior, Beta (c,d). The parameters c and d are, in turn,

formulated from some postulated information on correlation decay in relative risks over

a distance. In addition, the authors criticize the choice of Gamma (0.001, 0.001) prior, as

this does not allow for all reasonable levels of variability in the residual relative risks, in

particular small values. In their study, they did not find any significant variation in the

spatial parameters according to the different priors used. It is unclear, however, how the

priors could have affected the overall relative risk estimates, as these have not been

reported in the study.

The gamma prior is often chosen because it produces appropriate estimate for the

outcome, which is always positive, demonstrates conjugacy (i.e. distribution of posterior

is also similar to that of the data, i.e. Poisson) and it is flexible in modeling skewness in

the data. The choice of the gamma prior for the precision term of the CAR prior has

been criticized by Gelman(178). In his paper, comparisons were made between the

uniform prior (on the standard deviation parameter) and the inverse gamma prior (on the

variance parameter). Gelman recommended noninformative but proper uniform prior

(useful for modeling in WinBUGs, which requires a proper distribution to be specified).

5.6. Conjugate priors and improper priors

A family of prior distributions ρ() is conditionally conjugate for if the conditional

posterior distribution, ρ(\y) is also in that class(178). Computationally, this means that

we will be able to draw from the posterior distribution using the Gibbs sampler if it is

82

possible to draw from this class of prior distribution. Conditional conjugacy also

allows the prior distribution to be interpreted in terms of equivalent data.

Improper priors can, but do not necessarily, lead to proper posterior distributions. Two

commonly used improper priors are the uniform (0, x), x→∞, and inverse gamma(y, y),

y→0. As compared to the uniform prior, Gelman argues that the inverse gamma prior

does not have any proper limiting posterior distribution.

5.7. Sensitivity analysis on priors

I performed a small-scale analysis, looking at the effect of various prior specifications

on the amount of smoothing performed by the CAR model. 10 different priors were

compared, as indicated in table 8 below. I calculated standardised expected counts for

each of seven birth defects: spina bifida (SB). After discarding the first 50,000 samples

as burn-in, I ran a further 30,000 iterations which were used in the calculation of the

posterior estimates. Two different chains were started from diverse initial values.

Estimates for the mean smoothed relative risk, ratio of relative risk associated with

shared component, fraction of variation in relative risk of outcome explained by the

shared component and their corresponding 95% credible intervals were derived from the

posterior distribution.

The chi-squared residual sum of squares (RSS) was used to determine the amount by

which the estimated counts of birth defects differed from the actual counts. This is the

sum of the squared differences between the observed and estimated number of birth

83

defects standardised by the estimated number of births. A higher RSS indicates a greater

amount of smoothing. Third, I categorised the estimated relative risks in each area using

epidemiologically meaningful cut-offs based on the 25th and 75th percentiles of the crude

relative risks. The kappa statistic was then used to assess the consistency between the

modelled and crude relative risks, thereby assessing the degree of smoothing that

occurred. A positive value of kappa approaching +1 indicated strong consistency and

thus little smoothing, a positive value approaching zero indicated more smoothing.

Table 8.Comparison of smoothed risks and standard deviation by various priors

Number Spatially unstructured prior Spatially structured prior Mean

RR SD

Observed relative risk 1.165 2.258

1 G(0.0125, 0.001) G(0.0042, 0.001) 1.030 0.271

2 G(0.0125, 0.001) G(0.0025, 0.001) 1.034 0.272

3 G(0.0125, 0.001) G(0.0018, 0.001) 1.028 0.273

4 G(0.001, 0.001) G(0.001, 0.001) 1.032 0.271

5 G(0.5, 0.005) G(0.5, 0.005) 1.026 0.2666 σ~ Uniform(0.1, 20) σ~ Uniform(0.1, 20) 1.025 0.2787 σ~ Uniform(0.001, 1000) σ~ Uniform(0.001, 1000) 1.039 0.272

8 G(0.25, 0.000125) G(0.25, 0.00025) 1.023 0.269

9 G(1, 0.02) G(1, 0.04) 1.020 0.264

10 G(5, 1.25) G(5, 2.5) 1.030 0.333

84

Table 9. Influence of type of prior on fraction of random variation attributable to spatial effects Number Spatially unstructured prior Spatially structured prior Fraction 95% CI

1 G(0.0125, 0.001) G(0.0042, 0.001) 92.1 (64.4, 99.7)

2 G(0.0125, 0.001) G(0.0025, 0.001) 93.4 (65.3, 99.7)

3 G(0.0125, 0.001) G(0.0018, 0.001) 94.1 (70.6, 99.7)

4 G(0.001, 0.001) G(0.001, 0.001) 92.1 (63.4, 99.7)

5 G(0.5, 0.005) G(0.5, 0.005) 91.8 (67.2, 99.0) 6 σ~ Uniform(0.1, 20) σ~ Uniform(0.1, 20) 83.5 (56.7, 95.2) 7 σ~ Uniform(0.001, 1000) σ~ Uniform(0.001, 1000) 91.9 (64.4, 100.0)

8 G(0.25, 0.000125) G(0.25, 0.00025) 95.4 (73.7, 100.0)

9 G(1, 0.02) G(1, 0.04) 86.5 (60.4, 97.5)

10 G(5, 1.25) G(5, 2.5) 61.5 (43.6, 76.4)

Table 10. Residual sum of squares and kappa values by different prior specifications

Number Spatially unstructured prior Spatially structured prior RSS Kappa

1 G(0.0125, 0.001) G(0.0042, 0.001) 0.879 0.190

2 G(0.0125, 0.001) G(0.0025, 0.001) 0.882 0.190

3 G(0.0125, 0.001) G(0.0018, 0.001) 0.884 0.200

4 G(0.001, 0.001) G(0.001, 0.001) 0.877 0.190

5 G(0.5, 0.005) G(0.5, 0.005) 0.893 0.190 6 σ~ Uniform(0.1, 20) σ~ Uniform(0.1, 20) 0.833 0.250 7 σ~ Uniform(0.001, 1000) σ~ Uniform(0.001, 1000) 0.864 0.230

8 G(0.25, 0.000125) G(0.25, 0.00025) 0.899 0.190

9 G(1, 0.02) G(1, 0.04) 0.883 0.210

10 G(5, 1.25) G(5, 2.5) 0.633 0.350

85

As can be seen from table 8 above, regardless of which type of prior was specified for

the variance parameters, the Bayesian models always provided smoothed estimates of

the relative risks, which were closer to 1, and which had a lower standard deviation.

However, when I examined the fraction of variation in relative risk that was attributed to

spatial differences, differences were found across models (table 9 above). In particular, I

found that the Gamma (0.25, 0.00025) prior provided the largest variation (95.4%) and

the Gamma (5, 2.5) with the lowest (61.5%). While it should be noted that the

specification of the priors implicitly determines the fraction of variation, models that use

the same prior specification should not face a problem when evaluating the contribution

of additional covariates, in terms of their ability to reduce this fraction. The Kappa

values from table 10 indicate that the uniform priors exhibit similar amount of

smoothing across epidemiologically relevant cut-offs as compared to the Gamma priors.

5.8. Model selection techniques for spatial models

In Bayesian analysis, it is often possible that one has several competing models to select

the ‘best’ model from. Questions often arise as to which is the most appropriate and

parsimonious model to apply for a particular dataset. In addition, researchers may also

want to know whether the inclusion of a covariate in a regression model helps to

improve the fit of the model. In the frequentist approach, there are several model fit

criterions, such as the Akaike Information Criteria (AIC) or the adjusted R-square.

These measures offer a trade-off between measure of fit (some deviance statistic) and

complexity (number of free parameters in the model). The Bayesian version of this is

called the Bayesian Information Criteria (BIC). To this end, Spiegelhalter has suggested

86

the Deviance Information Criteria (DIC)(179) which consists of the adequacy (goodness

of fit) and complexity (penalty for increasing model complexity) parameters, as

described below. The DIC has been widely used in many complex hierarchical spatial

studies.

DpDDIC 2)( ,

The pD parameter measures the effective model size and D() measures the deviance.

DICs themselves have no meaning, so differences between models are reported and

interpreted instead.

In the case of the usual disease mapping study, the observed counts following a Poisson

distribution is formulated as follows:

i

iiiiiii EyEyyD }])exp({})exp(/log{[2)( , and yi, Ei and i refer to the

observed, expected counts and relative risk in each area 1.

The Bayesian Information Criteria (BIC)(180), which is shown below, is another

proposed alternative to the DIC. However, the BIC is not adequate for Bayesian models

that are complex in nature, with parameters often outnumbering the observations. In

addition, Spiegelhalter mentioned that this may only be useful in situations where one

believes that only one of the models is the correct one(179).

)log()}\(log{2 npypBIC

, where p is the number of free parameters to be

estimated, n is the sample size and )\(

yp is the maximized value of the likelihood.

87

Generally, smaller values of the DIC or BIC indicate a better fit and hence better model.

It has been suggested that models with DIC within 1-2 of the best deserve consideration,

and 3-7 deserve considerably less support (179). The DIC allows us to compare several

competing models and choose the best, but it does not tell us if the models are

appropriate for the data. In other words, all the models can be bad in the first place.

The model selection procedure described earlier is different from the procedure for

determining whether a particular model fits the data well. The latter can be examined

through cross-validation predictive checks (i.e. leave data from some area out of analysis

and predicting the values using the remaining data). These measures of fit are less

commonly used, and have been discussed elsewhere (181, 182).

5.9. Modifiable Areal Unit Problem (MAUP)

The Modifiable Areal Unit Problem (MAUP) is concerned with the effects of scale

change on models and their parameters. In the context of disease mapping, one would be

concerned with the differential amount of smoothing across the various scales of

analysis. In ecological regression models, concerns would be raised if we have different

regression coefficient estimates for the various levels of aggregation.

In the analysis of areal-level data, decisions on the choice of the appropriate scale of

analysis need to be made prior to analysis. In Australia for instance, a wide range of

social and economic data are provided at the various Australian Standard Geographical

Classification (ASGC) levels. The ASGC provides a hierarchy of geographic area codes.

88

As an example, the Statistical Local Areas (SLAs) aggregate to form Statistical

Subdivisions (SSDs) which in turn aggregate to form Statistical Divisions (SDs) and the

SDs aggregate to form States. All these levels cover the whole of Australia without gaps

or overlaps. Analysis of data at the most basic and smallest geographical level would

seem the most logical approach. However, significant challenges arise from the sheer

amount of data needed to process, along with the associated expensive computation

time, and changes to boundaries over time. Some software, such as WinBUGs also have

restrictions on the maximum amount of observations that can be handled (i.e. 50,000).

Counts of disease/ outcome become sparse at a finer areal level, and more sophisticated

statistical methods are needed to address such data. As a partial solution to these

problems, one often aggregates data up into the next higher available hierarchy, and a

certain level of compromise is made between fineness of areal units and computational

efficiency.

The choice of areal scale of analysis is also governed by practical issues such as

usefulness of the study results. In NSW, health services are spread across 8 area health

services, covering more than 200 public hospitals and 500 community, family and

children’s health centres. In the latest NSW Chief Health Officer Report, various disease

indicators such as cardiovascular disease deaths, hospital separations and avoidable

deaths have been reported at the 148 Local Government Area (LGA) in terms of

Standardised Mortality Ratios (SMRs)(183), as this has been decided to be most

applicable for local planning.

89

5.10. Boundary Analysis

In spatial studies of areal-level data, data are often available at some aggregate level,

which often corresponds to some administrative boundaries (e.g. Statistical Local Area

or Local Government Area). Sometimes, boundaries are drawn up purely for

administrative reasons (e.g. falling within jurisdiction of the local government), and

have no meanings and relations with occurrence of health outcomes such as birth

defects.

Occasionally, interest in disease mapping studies lies in identifying boundaries which

separate the risk of disease into areas with high/ row risk. The boundaries are known as

difference boundaries, and these separate areal units with vastly different observed

responses. Traditionally, researchers assign a boundary likelihood value (BLV) to each

boundary. The BLVs are calculated using some sort of dissimilarity statistics (using

functions such as the Euclidean distance metric)(184). Generally, such analysis is also

known as wombling, edge detection or barrier analysis. Such methods of differentiation

can help improve means of allocating healthcare resources. Creation of these new

boundaries can also help researchers further develop study plans to characterize the

reasons for difference in risks across the regions (for example by examining the

population distribution or areal-level characteristics such as socio-economic status).

Lu and colleagues presented a model-based approach to areal wombling, where they

incorporated spatial and non-spatial random effects in a Bayesian hierarchical

framework to determine boundaries which separate areas with distinctly different risk

90

estimates(184). In a two-stage analysis, they first performed Bayesian smoothing of

areal-level risks using the CAR model. Subsequently, based on the smoothed relative

risks, they performed boundary analysis using the commercial software

BoundarySeer(185). For the first stage analysis, they considered various formulations of

the CAR model (including intrinsic and proper CAR model and the MIX model(141)).

Based on the DIC, they selected the intrinsic CAR model, which gave the lowest DIC in

their analysis. In addition to the traditional method of wombling based on the relative

risk estimates, the authors also proposed wombling the spatial residuals. This, they

argued, provided further information on missing covariates, thus lending evidence to

policy makers on the issue of environmental justice.

In a follow-up study to their earlier work, Lu et al, extended their spatial hieararchical

areal wombling method to allow for the choice of the neighbourhood structure to be

determined by the value of the process in each region, and by variables determining the

similarity of two regions(186). Traditional methods of wombling can be divided into two

types: crisp wombling which encompasses a binary situation where every edge can be

classified as being a boundary element or not, and fuzzy wombling, where partial

membership in the boundary is allowed(186). Here, the authors allowed for weight

matrix in the CAR models to have crisp or fuzzy boundaries. Based on simulation

studies, the authors compared their method with their earlier model (which incorporated

spatial elements but with a fixed adjacency), and showed that their model performed

better than the other models, with greater agreement between true and wombled

boundaries.

91

Extensions to the areal boundary techniques in terms of handling multivariate disease

outcomes have also been developed(187). This model allows for the identification of a

unified set of boundaries for a number of diseases, while making provisions for

correlations across diseases and locations. Here, the authors considered various

competing multivariate models and showed that the model which incorporated the

shared component joint model(154) provided the best fit, with the lowest DIC(187).

One major draw-back of boundary analysis (or wombling) is that, in practice, they often

produce maps with collections of disconnected segments. This makes the interpretation

of the maps difficult for the users. Figure 4 (below) shows as example of an application

of the wombling technique in studying breast cancer late detection ratios in Minnesota,

USA (186).

Figure 4. Fuzzy wombled map based on Standardised Late Detection Ratio (SLDR)

of breast cancer in Minnesota, USA, after adjusting for age and cancer incidence

rates

Source of map: Lu H, Reilly CS, Banerjee S, et al. Bayesian areal wombling via adjacency modeling. Environmental and Ecological Statistics 2007;14: pg 448.

92

In the map (figure 4), counties (blue arrow) may have boundaries that are disconnected

with their neighbours. It remains unclear how such information can be used by local

policy makers or epidemiologists, as it shows that the county is different from the

adjacent neighbours, but this difference is not consistent across all neighbours. This

limitation was also pointed out by the authors in the same manuscript. Although they

suggested adding a second-level Conditional Autoregressive effect into the model, it

remains to be proven whether such a solution will indeed solve the problem. In addition,

current wombling techniques are restricted to spatial-only data. Extending these models

to spatio-temporal data would provide greater flexibility in terms of modeling and

application.

I have discussed issues revolving around boundary analysis, which is very different from

the seemingly related concept of edge effect. There are two types of edge effects:

sampling bias in which the probability of observing an event depends on the size and

shape of the area, and secondly censoring effects, where we are prevented from

observing the full extent of an object that lies partially within a window. Techniques to

address edge effects include providing a buffer area around the borders of a map. This in

turn, provides information on neighbours, but themselves are not included in the

analysis. Various weighting schemes for proximity to edges have also been proposed

(188). Rodeiro and Lawson compared and contrasted the various spatial models which

can tackle this issue. Through a simulation study, they also showed that the BYM

convolution model best addressed this issue(189).

93

CHAPTER 6: NEIGHBOURHOOD WEIGHT MATRIX

SPECIFICATION

Evaluating the effect of neighbourhood weight matrices on smoothing properties of

Conditional Autoregressive (CAR) models

Arul Earnest1§, Geoff Morgan1,2, Kerrie Mengersen3, Louise Ryan4, Richard

Summerhayes1,5, John Beard1,5,6

1 Northern Rivers University Department of Rural Health, The University of Sydney, New South Wales, Australia 2 Population Health & Planning, North Coast Area Health Service, New South Wales, Australia 3 Faculty of Science, Queensland University of Technology, Queensland, Australia 4 Department of Biostatistics, Harvard School of Public Health, Boston, USA 5 Graduate Research College, Southern Cross University, New South Wales, Australia 6 Centre for Urban Epidemiologic Studies. New York Academy of Medicine, New York, USA §Corresponding author Email addresses: AE: arul_earnest@hotmail.com GM: geoff.morgan@ncahs.health.nsw.gov.au KM: k.mengersen@qut.edu.au LR: lryan@hsph.harvard.edu RS: rsumm@doh.health.nsw.gov.au JB: jbeard@nyam.org

This research was published in the International Journal of Health Geographics,

November 2007, Volume 29;6: pp 54-65.

94

The authors listed below have certified that: 1. they meet the criteria for authorship in that they have participated in the conception,

execution, or interpretation, of at least that part of the publication in their field of expertise;

2. they take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

3. there are no other authors of the publication according to these criteria;

4. potential conflicts of interest have been disclosed to (a) granting bodies, (b) the editor or publisher of journals or other publications, and (c) the head of the responsible academic unit, and

5. they agree to the use of the publication in the student’s thesis and its publication on the

Australasian Digital Thesis database consistent with any limitations set by publisher requirements.

In the case of this chapter:

Contributor Statement of contribution

Arul Earnest Conceptualised the entire study, performed data management, wrote computer codes for the analysis, performed literature review and statistical analysis and wrote the manuscript for publication

26 February 2010

Geoff Morgan

Contributed to writing of manuscript and study design

Kerrie Mengersen

Contributed to writing of manuscript and study design

Louise Ryan

Contributed to writing of manuscript

Richard Summerhayes

Contributed to writing of manuscript

John Beard

Contributed to writing of manuscript and study design

Principal Supervisor Confirmation I have sighted email or other correspondence from all Co-authors confirming their certifying authorship. _______________________ ____________________ ______________________ Name Signature Date

95

Abstract

Background

The Conditional Autoregressive (CAR) model is widely used in many small-area

ecological studies to analyse outcomes measured at an areal level. There has been little

evaluation of the influence of different neighbourhood weight matrix structures on the

amount of smoothing performed by the CAR model. We examined this issue in detail.

Methods

We created several neighbourhood weight matrices and applied them to a large dataset

of births and birth defects in New South Wales (NSW), Australia within 198 Statistical

Local Areas. Between the years 1995-2003, there were 17,595 geocoded birth defects

and 770,638 geocoded birth records with available data. Spatio-temporal models were

developed with data from 1995-2000 and their fit evaluated within the following time

period: 2001-2003.

Results

We were able to create four adjacency-based weight matrices, seven distance-based

weight matrices and one matrix based on similarity in terms of a key covariate (i.e.

maternal age). In terms of agreement between observed and predicted relative risks,

categorised in epidemiologically relevant groups, generally the distance-based matrices

performed better than the adjacency-based neighbourhoods. In terms of recovering the

underlying risk structure, the weight-7 model (smoothing by maternal-age 'Covariate

model') was able to correctly classify 35/47 high-risk areas (sensitivity 74%) with a

specificity of 47%, and the 'Gravity' model had sensitivity and specificity values of 74%

and 39% respectively.

96

Conclusion

We found considerable differences in the smoothing properties of the CAR model,

depending on the type of neighbours specified. This in turn had an effect on the models'

ability to recover the observed risk in an area. Prior to risk mapping or ecological

modelling, an exploratory analysis of the neighbourhood weight matrix to guide the

choice of a suitable weight matrix is recommended. Alternatively, the weight matrix can

be chosen a priori based on decision-theoretic considerations including loss, cost and

inferential aims.

97

Background

The Conditional Autoregressive (CAR) model is widely used in small-area ecological

studies to map outcomes measured at some areal level and to examine associations with

covariates. Most of these applications are in the field of disease mapping (See Elliott for

a list of studies [1]). The advantages of using the CAR model instead of presenting crude

relative risks are well-described in the literature. One component of the CAR analysis is

the use of a Bayesian model to include spatial association between observations. This

approach offers a trade off between bias and variance reduction of the estimates, and has

been shown to produce a set of point estimates that have improved properties in terms of

minimising squared error loss, particularly in cases where the sample size is small [2].

When there is geographical correlation inherent in the data, ignoring such correlation

can also lead to biased and inefficient inference, as the observations are strictly not

independent. Elliott provides a summary of the various applications in disease mapping

studies and some current methodological issues [1]. Lawson also highlights some of the

applications of CAR models in disease mapping studies [3].

The CAR model was originally suggested by Besag [4] in the context of image analysis

and is also known as the intrinsic CAR model with a convolution prior, or the Besag,

York and Mollie (BYM) model. The original BYM model applied to continuous data

that could be assumed to be normally distribution. In disease mapping studies, this has

been adapted to incorporate normally distributed spatially correlated random effects into

Poisson models for disease counts. The BYM model allows for the smoothing of relative

98

risk estimate in each region towards the mean risk in the neighbouring areas. This

provides for a more precise or reliable estimate of both mean and variance compared to

using the crude rate. This is especially so, as the variance for the estimate of the raw rate

with a small expected count can be large and unreliable. Risks are also smoothed

towards the global mean to account for overdispersion. This ‘shrinkage’ of estimates

towards the mean can be shown mathematically to be optimal if the aim is to minimise

the squared-error loss, in a decision theory framework [5].

When undertaking CAR modelling of data at an areal level, it is necessary to define a,

so–called, adjacency matrix that characterizes the neighbourhood structure of the data

being analysed. There are several approaches to doing this, including defining

neighbours according to the distances between centroids, declaring two regions to be

neighbours if they share a boundary, and so on. It may also be necessary to specify the

level of aggregation of data if small area data is available. Other influences include the

choice of hyperprior distribution used for the precision estimates (e.g. gamma versus

uniform), and the nature and sparseness of data. In this analysis, we are primarily

concerned with the influence of different neighbourhood weight matrices on the amount

of smoothing.

The CAR model has been used in many published studies, but only a few have provided

justification on the choice of neighbourhood weight matrix structure. One exception is

an ecological study to investigate the relationship between benzene emissions and the

incidence of childhood leukaemia in Greater London, in which the authors considered

three alternative levels of data aggregation in their analysis, and examined adjacency

99

versus distance-based neighbourhood spatial weights for each of analysis [6]. For both

the grid-level and ward-level analyses, they found that the adjacency based

neighbourhood structure provided a better fit of the data, based on DIC (Deviance

Information Criteria) comparisons. They also found a significant difference in the

estimates of the spatially structured random effects between the distance-based and

adjacency-based neighbourhood structures for the ward-level analysis for the model

without any covariates. In another paper looking at the issue of spatial priors and single

versus joint-disease models, the authors concluded that sensitivity to structural

assumptions as well as hyperprior specification should be explored as part of any

disease-mapping study [7].

A recent study of prostate cancer incidence in New York, recently published in the

International Journal of Health Geographics, applied the CAR spatial model to obtain

smoothed risk estimates at the ZIP-code level, and highlighted the use of distance-based

weight functions in the model formulation [8]. While this paper included some

sensitivity analysis on the choice of hyperpriors, the issue of neighbourhood weights was

not addressed.

Other authors have used various specifications of the neighbourhood weight matrix.

Wall [9] looked at spatial structure in the US SAT college entry exam results by state,

having from one neighbour up to eight neighbours. A similar approached was used by

Rasmussen [10] with Scottish lip cancer data and by MacNab [11] with chronic lung

disease in neonatal intensive care units. Bell [12] used first-order neighbours for a

spatial neighbour matrix in a CAR model describing the association of intrauterine

100

growth restriction and area level covariates. English [13] mapped low birth weight to

0.5km grids and used a CAR model to assess and adjust for residual spatial correlation

using four neighbours for each cell (north, south, east and west) to describe spatial

dependence. Kousa [14] geocoded acute myocardial infarctions in Finland to 10km grids

to examine spatial variation associated with geochemistry of ground water. The

neighbours were defined using eight neighbours (side and corner) for each cell.

Abrial [15] used 23km hexagonal grids to map cases of Bovine spongiform

encephalopathy (BSE) Johnson [16] used adjacent zipcodes in a study on prostate cancer

as the basis for geographical weighting. Other methods to define neighbours have

included Euclidean distance [17], geographic distance and population size in prostate

cancer [18].

In Australia, New South Wales (NSW) Health [19] recently used CAR models to

produce smoothed maps of selected health indicators from 1999/2000 to 2003/2004 for

166 Local Government Areas in NSW. Similarly, the Cancer Council of NSW has

provided on its website [20] maps of cancer incidence and mortality across New South

Wales by Local Government Area (LGA) for the period 1998 to 2002. They used

Bayesian methodology to smooth the maps of standardised incidence and mortality

ratios.

The overall crude birth defect rate in NSW has decreased from 22.7/1,000 births in 1998

to 20.5 per 1,000 births in 2003 [21]. The only increases over this period were cases of

chromosomal abnormalities which increased from 4.2 to 5.3, Cleft Palate (from 0.8 to

101

1.0 per 1000 births) and Down Syndrome which increased from 2.2 to 2.6 per 1,000

births. The spatial distribution of birth defects in NSW has only been published by very

large spatial units (8 Area Health Services (AHS)) standardised for maternal age in the

NSW population. The same report found that the Greater Southern AHS had the lowest

rate of birth defects (15.7 per 1,000 births) and Hunter New England AHS had the

highest (24.2 per 1,000 births) across the period 1998-2004. Spatial analysis at a smaller

spatial unit than the 8 AHS within NSW will provide more information about the

geographical distribution of the birth defects, and allow the development of hypotheses

to explain this spatial variation.

More extensive assessment of the strengths and weaknesses of these, and other, possible

approaches to neighbourhood weighting is urgently needed, since the results of analysis

may vary substantially depending on the model chosen. Lawson [22] talks briefly about

possible weighting schemes in the context of edge effects, including distance functions

and surrogate measures derived from the perimeter of shared borders between

neighbours. He also mentions the need to conduct sensitivity analysis on the choice of

weights. However, no quantitative case-studies are shown to highlight this point. Best

and colleagues [23], looked at the use of adjacency versus distance-based weights to

define the neighbourhood structure for the residuals. However, their study examined just

two specific weights (the rook adjacency and a distance decay function) and their results

were based on a small dataset which included simulated risk structures. Model

comparison was also done on the same data.

102

In another related study, Conlon [24] et al examined the effect of three different

neighbourhood weight structures (namely fixed weights based on adjacency, parametric

distance-based weights and distance-based covariances). For the limited data (again the

Scottish lip cancer data) that they worked on, they found that the adjacency and variable

covariance models seemed to provide better fit as compared to the variable distance

model. In other words, they found differences according to the type of neighbours

defined. Our study aims to look at a more comprehensive list of weight matrices, and the

use of innovative measures to compare competing models.

Aims

The main aims of our study were two-fold. Firstly, to explore any differences in the

smoothing properties between the contiguity (adjacency) and distance-based methods of

defining spatial weights. We also studied whether there were differences between the

type and order of neighbours included within the contiguity method of neighbourhood

definition. Secondly, for the distance-based method, we assessed the impact of including

various formulations of the weight matrix. We performed external validation of the

models by applying them to birth defects data in New South Wales.

Methods

We obtained data on birth defects from the NSW Midwives Data Collection (MDC) and

Birth Defects Register (BDR) databases from 1995 to 2003. We calculated standardised

expected counts of total birth defects. For instance, the expected count of birth defects in

each areal unit at a particular time period was defined as

103

Ei=(Birthsi/Totalbirths)*Totalbirthdefects, where Birthsi refers to total births in the ith

SLA, and Totalbirths and Totalbirthdefects refer to the overall number of births and

birth defects in the NSW study region for that particular time-period. Analyses were

carried out at the SLA level, for which there were 198 SLAs defined within the NSW

study area. These represent administrative districts that relate to local government

jurisdictions. Statistical Local Area (SLA)-specific relative risk estimates were

calculated as the ratio of the observed and expected counts for each area. Because a few

of the SLA had no births recorded during a particular study period, we added a small

constant (10-5) to both the numerator and denominator to ensure that the relative risks

were well-defined. This constant is absorbed by the smoothing of these extreme relative

risk estimates in the subsequent analysis. The data were grouped into 3 equal three-year

long time periods: 1995-1997, 1998-2000 and 2001-2003. After confirming that these

three time periods were similar with respect to relevant measures, we used the first two

time-periods to build the model coefficients and then assessed model fit using data from

2001-2003. We excluded cases that had missing data for year of birth and maternal age.

Year of birth was needed to assign the cases to each of the three time-periods, whereas

maternal age was required to create one of the weight matrices. Also, to ensure that only

good quality geocoded addresses was used, we excluded indeterminate geocodes,

geocodes resulting in many addresses, many streets, many localities and those without

any matches.

The New South Wales (NSW) Midwives Data Collection is a population-based

surveillance system covering all births in NSW public and private hospitals, as well as

home births. The information for each birth is recorded by either the attending midwife

104

or medical practitioner. It encompasses all live births and stillbirths of at least 20 weeks

gestation or at least 400 grams birth weight. The MDC receives notifications of women

whose usual place of residence is outside NSW but who give birth in NSW. However,

the MDC does not receive notifications of births outside NSW to women usually

resident in NSW [21].

The New South Wales BDR is a population-based surveillance system established to

monitor birth defects detected during pregnancy or at birth, or diagnosed in infants up to

one year of age. The BDR was established in 1990 and, under the NSW Public Health

Act 1991, from 1 January 1998 doctors, hospitals, and laboratories have been required to

notify birth defects detected during pregnancy, at birth, or up to one year of life [21]. For

the purposes of this statistical methodological study we considered all birth defects

together.

The MDC and BDR data for 1990 to 2003 have recently been geocoded using software

developed by NSW Health and the Australian National University and the geocoding

process is described in detail elsewhere [25].

Ethical approval was obtained for the use of the NSW Midwives Data Collection and the

NSW Births Defects Registry data from the NSW Department of Health Ethics

Committee, and for the study itself from the University of Sydney Ethics Committee.

The formulation of the CAR model used in our analyses is shown below:

105

)(~ ikik PoiO

221 **)log()log( kikiiiikik ttvuE

where Oik and Eik are the observed and expected birth defects for a SLA in the ith region

and jth time period, ui is a spatially structured random effect and vi is a spatially

unstructured random effect. We also added a quadratic temporal random effect term to

capture time trends. This model is an amended version from Bernardinelli [26]. The

main difference lies in the exclusion of the space-time interaction random effect term;

because the main focus of this paper is a comparison of the prior imposed on the spatial

random effect, the inclusion of an interaction term would potentially blur this

comparison and increase the computational cost substantially. Possible spatial

correlation was accommodated in the model by introducing a conditional autoregressive

(CAR) prior for the spatial random effects, as shown below (from Lawson AB 2003:

Disease Mapping with WinBUGS and MLwiN).

),(~],|[ 22, iiuji uNjiuu

jijj

jij

i wuw

u1

jij

ui w

22

As we can see from the above equations, estimation of the risk in any area is conditional

on risks in neighbouring areas. Subscripts i and j refer to an SLA and it’s neighbour

respectively, and j ε Ni where Ni represents the set of neighbours of region i. Besides

the identification of neighbours, the assigned weights also affect the risk estimation. The

weights for the adjacency and distance models are given by weightsij (wij)=1 if i,j are

106

adjacent, and 0 otherwise. For the other distance-based models, various formulations of

the weights (described in detail below) were used.

We created four different neighbourhood adjacent weight matrices that are commonly

used in spatial regression, namely Queen-1, Queen-2, Rook-1 and Rook-2. The numbers

reflect the order of contiguity, and the main difference between the Queen and Rook

method of assigning neighbours is that the latter uses only common boundaries to define

neighbors, while the former includes all common points (boundaries and vertices).

Please see Figure 1 for more details. For instance, the Queen-1 neighbourhood matrix

for a SLA would include all its immediate neighbours that share common points with

that area, while a Queen-2 matrix would include the immediate neighbours of the

neighbours as well. All neighbours for the adjacency weight matrix contribute equal

weights.

We also computed seven distance-based matrices. The simple distance-based matrix

included all SLAs as neighbours and assigned them equal weights. Matrices Weight-1 to

Weight-7 also include all SLAs as neighbours, but the weights were assigned differently.

For Weights 1-3, the following formulations were used: wij=1/distij , wij=1/dist2ij and

wij=1/dist3ij respectively. The weight matrix for the “Gravity” model was defined by

wij=eiej/distij. The corresponding weight matrix for the “Entropy” model was defined by

wij=exp(-10*distij), and for the “Density” model: wij=(1/distij)*densityi*densityj, with

distij being the distance (decimal degrees) between the two SLAs, ei and ej being the

standardised counts of births for an SLA and its respective neighbour, and densityi and

densityj being the respective standardised birth population densities. A distance decay

107

parameter of 10 was chosen, based on a preliminary exploratory examination of the

correlogram of the relative risks over distance.

The weights for the “Gravity” and “Density” models were standardised against their

mean and standard deviation for the purpose of comparability. Finally, for the Weight-7

model, the weights were wij=1/(distij*absolute(maternalagei-maternalagej +0.0001)),

with maternalagei and maternalagej being the mean maternal ages in SLAs i and its

neighbour j respectively. A small constant was needed to ensure that weights were

defined for those pairs with identical values.

Our selection of the 7 distance-based models provides a variety of scenarios whereby the

relative risks, in reality, are spatially correlated. The first three models (Weights 1-3)

consider only distance in the weight function, placing greater weights on SLAs that are

closer together. The “Gravity” model was used to examine whether placing greater

weights on neighbours which themselves were relatively more populated, made a

difference in smoothing. This was to examine the hypothesis that sparsely populated

neighbours provide little information. The “Entropy” model was designed to provide a

scenario whereby immediate neighbours were assigned most of the weights, and the

weights were reduced drastically for those that were further away. This was done to

exemplify a situation where one could expect localised environmental hazards to be

present. The “Density” model was similar to the “Gravity” model, except that we

weighted the neighbours according to the population density, instead of just the

population.

108

For the covariate model, we chose maternal age mainly because this has been previously

reported to be associated with birth defects in our study population (i.e. incidence of

birth defects found to be increasing with maternal age in NSW[21]), and also because

maternal age was less likely to be subject to recall bias as compared with other

covariates, such as smoking.

The priors for the means were set to a normal distribution, with standard deviation set to

cover a wide range of values, whereas the priors for the standard deviations of the

precision estimates were set to a uniform distribution [27] with a wide yet plausible

interval (i.e. range from 0.00001 to 20). This range was selected from initial exploratory

analysis of the data.

We ran 12 different CAR models for the various adjacencies described above, using

WinBUGS (version 1.4.1, Imperial College and Medical Research Council, UK). The

models were run through Stata using a customised ado program file (written by Dr John

Thompson, Department of Health Sciences, University of Leicester 2006). We discarded

the first 40,000 samples as burn-in and ran a further 20,000 iterations which were used

in the calculation of the posterior estimates. We ran two different chains, starting from

diverse initial values and convergence was assessed using the Gelman-Rubin

convergence statistic, as modified by Brooks and Gelman [28].

Estimates for the smoothed relative risk, posterior probability of relative risk greater

than one, spatially structured random effect, spatially unstructured random effect and

their corresponding 95% credible intervals were derived from the posterior distribution.

109

We also computed the fraction of total random variation explained by the model as a

ratio of the empirical variance of the spatial component against the total variance. This

fraction provides us with a means to explore how much of the spatial variation in

relative risk is explained by the model. The formulas are given below:

)1(/)()( 2 nuuuVari i

)1(/)()( 2 nvvvVari i

Fraction ))()(/()( vVaruVaruVar

ui is a spatially structured random effect and vi is a spatially unstructured random effect,

with i ranging from 1 to n=198.

We compared the different models in several ways. Firstly, we used the Deviance

Information Criterion (DIC) developed by Spiegelhalter [29] to assess the complexity

and fit of the models. The DIC is computed as the sum of the posterior mean deviance

and estimated effective number of parameters.

,DpDDIC

with D and pD being the sum of the posterior mean deviance and estimate of the

effective number of parameters.

Generally, smaller values of DIC are preferred. We used the model-selection decision

criteria suggested by Best [7], to suggest that models with DIC values within 1 or 2 of

the ‘best’ model are also strongly supported, values within 3 and 7, weakly supported,

and models with a DIC greater than 7 are substantially inferior.

110

For further model comparison, we also calculated the chi-squared residual sum of

squares (RSS) to determine the amount by which the estimated counts of birth defects

differed from the actual counts for the third time-period [3]. This is computed as the sum

of the squared differences between the observed and estimated number of birth defects

standardised by the estimated number of births:

,ˆ

)ˆ( 2

i

iiORSS

with Oi and i being the observed and estimated number of birth defects respectively.

In order to determine the magnitude of smoothing in relation to epidemiologically

meaningful cut-offs in the relative risks, we tabulated risk estimates into 3 groups, based

on the 25th and 75th percentiles (Low: RR<0.65, Neutral: 0.65<RR<1.15, High:

RR>1.15) and cross tabulated the observed with the predicted relative risk estimates.

Using percentiles was a reasonable way to ensure enough numbers in each group to

access sensitivity and specificity. To quantify the extent of change, we calculated the

Kappa statistic [30], as a means to compare across models.

),1/()(ˆ eeo ppp

with op and ep being the observed and expected proportion of agreement respectively.

In addition, we computed and mapped the probability of a relative risk (RR) more than

one. Although a cut-off of 0.7 has been shown to provide reasonable sensitivity to detect

areas with an elevated risk [31] for a range of scenarios having moderate expected

counts and excess risks of about 1.5, we estimated our own optimal cut-offs using the

receiver-operating characteristic (ROC) curve.

111

Data extraction, management, analysis and diagnostics were done in Stata (version 9.2,

Stata Corp, College Station, USA) and the maps were produced in Stata as well as

ArcMap version 9.0 (ESRI, USA). All the weight matrices were created using the

GeoDA software (version 0.9.5-I, University of Illinois, USA) and Stata.

Results

The number of birth defects recorded during the time periods 1995-1997, 1998-2000 and

2001-2003 that we used in the analysis were 5924, 6161 and 5510 respectively. The total

number of births in the corresponding periods was 257353, 258147 and 255138

respectively. The mean number of first-order neighbours is shown in Table 1. For the

Queen method of assignment, a mean of 5 neighbours were identified; for second-order

assignment, the mean number of neighbours increased by about three-fold. There was

little difference in the number of neighbours assigned by the Queen and Rook method

due to the irregularity of the SLA areal units. The distance-based method of assignment

resulted in all SLAs having 197 neighbours (i.e. all areas were considered neighbours).

Next, we compared the characteristics of the neighbourhood types in terms of various

measures (Table 2). Generally, there was greater agreement between observed and

predicted relative risks, categorised in quartiles, using the distance-based matrices

compared to the adjacency-based neighbourhoods. The Queen-1 model had a low Kappa

value of 0.05, indicating a larger amount of smoothing. Among the distance-based

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models, the ‘Gravity’ model (Weight-4) performed better with a Kappa of 0.15. The

‘Covariate’ model fell in between the adjacency and distance-based models.

In terms of fraction of random effect due to ‘spatially structured random effects’, the

adjacency-based models, generally, had lower values as compared with the distance-

based models. For instance, when we used the Queen-1 matrix, only 22% of the

variation in the relative risks could be attributed to spatial effects. Among the distance-

based models, the ‘Gravity’ model had the highest value (fraction=99%), followed by

Weight-2 (fraction=98%) and ‘Density’ models (fraction=97%).

When we examined the DIC as a basis of model-selection, we found that the ‘Gravity’

model had the lowest DIC of 2202, followed by Weight-2 (DIC=2204) and ‘Density’

model (DIC=2208). Generally, the adjacency-based matrices had higher DICs.

Using the decision tool suggested by Richardson et al [31], we also compared the

models in terms of their ability to detect areas with an elevated risk (i.e. RR>1.15).

When we used the cut-off of 0.7 as suggested by their study, we found that all the

models performed well in terms of specificity, but poorly in terms of sensitivity (data not

shown).

The ROC analysis indicated that the posterior probabilities from the distance-based

models had a similar discriminatory ability (i.e. same area under the curve) compared to

the adjacency-based models (table 2). However, the sensitivities can be improved by

choosing a different threshold. When we used a threshold of 0.33 as determined by our

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ROC analysis to optimise sensitivity, we found some improvements in the models. For

instance, the Queen-1 model was now able to correctly classify 36/47 high-risk areas

(sensitivity 77%) with a specificity of 44%. The distance-based neighbourhood matrices

had similar ranges of sensitivities and specificities. As with all such models, the choice

of threshold depends on the inferential aims of the analysis, including the loss or cost

associated with making a wrong positive or negative decision.

Finally, we compared the amount of smoothing performed by the various models. The

distance-based models generally had lower RSS values as compared with adjacency-

based models, indicating a lower amount of smoothing, and hence a better ability to

predict the observed risk in an area. Most of the smoothing occurred in areas with low

expected counts (table 3) as expected.

Figure 2 depicts the crude (observed) standardised relative risk of birth defects by SLA

regions. There appear to be pockets of areas with an elevated risk, and these areas seem

to be surrounded by regions with similar risk values. Figures 3 and 4 features the

predicted relative risk of birth defects using the Queen-1 adjacency neighbourhood

matrix and the ‘Covariate’ method of assigning weights. It is apparent that the latter

seems to perform better in recovering the true underlying relative risk.

In terms of diagnostics, the Gelman and Rubin plots indicated convergence after about

20,000 iterations for the posterior estimates of the regression coefficients, 3 randomly

selected relative risk estimates and 3 randomly selected posterior probability estimates.

The ‘fraction’ parameters took a longer time to converge (around 40,000 iterations); thus

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for consistency, we discarded the first 40,000 iterations for all the parameters. Due to the

complex weight structures, the models were computationally intensive, and a further

20,000 iterations provided us with a reasonable Monte Carlo standard error (less than

1% of the standard deviation) for the estimates.

Discussion

We found considerable differences in the smoothing properties of the CAR model,

depending on the type of neighbours specified. This in turn had an effect on the models’

ability to predict the observed risk in an area. These results have significant implications

for all researchers using CAR models, since the neighbourhood weight matrices chosen

may markedly influence a study’s findings.

For instance, if one were primarily concerned with classifying areas into low/ high risk

of birth defects, the distance-based models appear to perform better than the adjacency-

based ones. These models also have a higher Kappa, indicating low levels of changes in

relative risk estimates across epidemiologically relevant thresholds. While we used

Kappa with equal weights for the different categories, it is possible to use a weighted

Kappa instead and assign higher weights for more important categories (e.g. smoothing

a high relative risk).

If the aim were to explain away the spatial relationship in the relative risk estimates,

then again distance-based models, such as ‘Gravity’ or Weight-2 models might be

useful. Conversely, if one wants to preserve the spatial structure of the relative risks and

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examine the relationship between covariates that were spatial in nature, then one might

prefer a model that has a low fraction of random effect due to spatially structured

random effects to begin with.

The DIC appears to reflect the choice of models based on the level and nature of

smoothing performed. For instance, the model with the lowest DIC, ‘Gravity’, had a

high fraction (99%) and a high Kappa (0.15) as well. The Weight-2 model also seems to

be ‘strongly supported’. However, reliance on just the DIC alone fails to account for the

nature of spatial relationship inherent in the model.

In terms of detecting areas with an elevated risk, we found that using information from

the posterior probability had a higher sensitivity than looking at the smoothed relative

risks alone. For instance, in the ‘Covariate’ model, using the smoothed (predicted)

relative risk allowed us to detect 11 (22%) of the 50 SLAs with an elevated risk, as

compared to 74% if we used information from the posterior probability instead.

As illustrated by the ROC analysis, the choice of threshold in the posterior probabilities

involves a trade-off between sensitivity and specificity. Depending on the aims of the

study, one should choose an appropriate cut-off point, using the ROC analysis. If for

instance, one were to undertake exploratory spatial analysis to generate hypothesis on

possible explanatory factors of elevated birth defect rates at an areal level, one could

choose a low sensitivity but with a high specificity, so that the false negatives are

minimised.

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Our finding that distance-based neighbourhood models perform better than adjacency

models is not surprising. This can be attributed to the highly irregular shapes and sizes

of the SLAs (see Figure 1). The adjacency models can be expected to perform better

with regular shaped areas such as grids.

The Kappa values in our study ranged from 0.05 to 0.14. While these may seem to

afford modest values of agreement, we do note that these were ecological models and

the models were evaluated on an ‘external’ time-period.

It is also worth mentioning the relatively good performance of the ‘Gravity’ model,

which had the highest Kappa, the highest fraction of random effect due to spatially

structured random effects and the lowest DIC. One current limitation of the CAR model

is that it ‘borrows strength’ from neighbours even if they themselves are sparsely

populated. The ‘Gravity’ model weighs neighbours according to the size of their

population, and we have shown that this improves precision. The importance will be

more marked for sparsely populated maps.

In our analysis, we used the simple spatial formulation of the CAR model comprising of

spatially structured and spatially unstructured random effects and a temporal term. The

purpose was not to make comparisons with the other formulations of the CAR model

(e.g. mixture model, spatio-temporal interaction models, multivariate CAR models, etc),

but further research might examine if these results reported here can be replicated in

those models as well. This is also not the first time this particular space-time model has

been applied to study epidemiological data. Assuncao and colleagues have used a similar

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model to map and project the rates of visceral Leishmaniasis in Belo Horizonte, Brazil

across 117 study areas and 3 time points [32]. For our CAR models, we also used the

Uniform priors on the standard deviations, instead of a Gamma prior on the precision

estimates, as recent research has identified problems associated with the Inverse-Gamma

prior [27], in particular the poor performance of the prior in terms of being non-

informative.

The thresholds used in our definition of epidemiologically relevant cut-offs for the

relative risks in our study may seem arbitrarily selected (based on the 25th and 75th

percentiles). However, the same thresholds were used for all models, thus ensuring

comparability, and we do not believe that the choice of alternative cut-offs would affect

inferences.

It was also not the aim of this paper to undertake studies of association or ecological

regression models, although covariates and interactions can easily be included in the

existing models. Our research group is currently undertaking work to examine the

impact of socio-economic status, demographic and environmental risk factors of adverse

birth outcomes at an areal-level. A complete understanding of the structural form of

CAR models is needed before this process of ecological modelling is pursued. Future

work on incorporating landscape features and spatial smoothing based on additional

covariates (e.g. multivariate similarity index) may improve the performance of the CAR

models in disease mapping studies.

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Conclusion

Disease mapping studies that make use of the CAR model to smooth relative risks at

some areal level need to take into account structural forms of the model specified.

Depending on the aims of the study, various forms of the neighbourhood weight

matrices provide differential levels of smoothing. Prior to risk mapping or ecological

modelling, the weight matrix should be chosen according to the inferential and decision-

theoretic aims of the study or through an exploratory analysis of the nature and degree of

spatial correlation. In addition, a sensitivity analysis on the choice of neighbourhood

weight matrix should be performed.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgements

The authors acknowledge the support of the Australian Research Council Linkage

Grant LP0348628, the North Coast Area Health Service, the NSW Department of

Health, the Commonwealth Department of Health and Ageing, Dr Lee Taylor (NSW

Health), Douglas Lincoln, Paul Houlder, Therese Dunn and Danielle Taylor.

119

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Figures

Figure 1 - Neighbourhood assignment based on adjacency

Note: For Rook method, only neighbours 2,4,6 and 8 assigned to SLA(i) For Queen method, all neighbours (i.e. 1-8) are assigned to SLA(i)

Figure 2 - Observed relative risk of birth defects: 2001-2003

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Figure 3 - Predicted relative risk of birth defects: 2001-2003

Queen -1 Model

Figure 4 - Predicted relative risk of birth defects: 2001-2003

Weights- 7 Covariate Model

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Tables Table 1. Characteristics of neighbourhood weight matrices Neighbourhood Type Mean Median Min Max SD Sum Queen-1 5 5 1 12 2 960 Queen-2 15 15 3 29 6 3018 Rook-1 5 5 1 11 2 956 Rook-2 15 15 3 29 6 3010 Distance 197 197 197 197 NA 39006

Table 2. Comparison of model fit and sensitivity of detecting areas with an elevated risk

Neighbourhood type Kappa Fraction DIC AUC*

Sensitivity of detecting SLAs with elevated risks (PP cut-

off=0.33) SpecificityQueen-1 0.05 22% 2283 0.61 77% 41% Queen-2 0.05 51% 2282 0.62 78% 43% Rook-1 0.05 14% 2282 0.62 76% 41% Rook-2 0.05 41% 2282 0.62 78% 39%

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Distance 0.07 41% 2274 0.62 80% 30% Weight-1 (1/distance) 0.07 41% 2213 0.62 80% 30% Weight-2 (1/distance2) 0.12 98% 2204 0.61 75% 44% Weight-3 (1/distance3) 0.08 93% 2218 0.59 75% 46% Weight 4 (Gravity) 0.15 99% 2202 0.60 74% 39% Weight 5 (Entropy) 0.08 91% 2227 0.60 74% 47% Weight 6 (Density) 0.14 97% 2208 0.60 74% 37% Weight 7 (Covariate) 0.10 95% 2210 0.62 74% 47%

* Area under curve from ROC analysis Table 3. Comparison of amount of smoothing performed, stratified by size of population

Neighbourhood type RSS

RSS (Areas with low expected

count)*

RSS (Areas with high expected

count)

Queen-1 91427 91254 173 Queen-2 91349 91176 174

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Rook-1 91150 90976 174 Rook-2 90975 90802 173 Distance 88987 88816 171 Weight-1 (1/distance) 88987 88816 171

Weight-2 (1/distance2) 89612 89436 176

Weight-3 (1/distance3) 92666 92490 176 Weight 4 (Gravity) 87510 87330 179 Weight 5 (Entropy) 90550 90378 172 Weight 6 (Density) 87336 87162 174 Weight 7 (Covariate) 79478 79292 185

* Expected count less than median value of 9

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CHAPTER 7: MODELLING SPARSE DISEASE COUNTS Small Area Estimation of Sparse Disease Counts using Shared Component Models- Application to Birth Defect Registry Data in New South Wales, Australia. Arul Earnesta*, John Beardb, Geoff Morgana,c, Douglas Lincolna, Richard Summerhayesd, Deborah Donoghuea, Therese Dunna,c, David Muscatelloe, Kerrie Mengersenf aNorthern Rivers University Department of Rural Health, New South Wales, Australia b Department of Ageing and Lifecourse, World Health Organisation, Geneva, Switzerland cNorth Coast Area Health Service, New South Wales, Australia dGraduate Research College, Southern Cross University, New South Wales, Australia, eCentre for Epidemiology and Research, New South Wales Department of Health, New South Wales, Australia fFaculty of Science, Queensland University of Technology, Queensland, Australia

* Corresponding author: Arul Earnest, Northern Rivers University Department of Rural Health, 55 Uralba Street, Lismore, NSW, Australia 2480, Tel: 61 2 6620 7570, Fax: 61 2 6620 7270, E-mail: arul_earnest@hotmail.com

Accepted for publication in Health and Place Journal on 23 Feb 2010

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In the case of this chapter:

Contributor Statement of contribution

Arul Earnest Conceptualised the entire study, performed data management, wrote computer codes for the analysis, performed literature review and statistical analysis and wrote the manuscript for publication

23 February 2010

Geoff Morgan

Contributed to writing of manuscript

Kerrie Mengersen

Contributed to writing of manuscript

Douglas Lincoln

Contributed to writing of manuscript

Deborah Donoghue

Contributed to writing of manuscript

Therese Dunn

Contributed to writing of manuscript

David Muscatello

Contributed to writing of manuscript

Richard Summerhayes

Contributed to writing of manuscript

John Beard

Contributed to writing of manuscript

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Abstract

In the field of disease mapping, little has been done to address the issue of analysing

sparse health datasets. We hypothesised that by modelling two outcomes

simultaneously, one would be able to better estimate the outcome with a sparse count.

We tested this hypothesis utilising Bayesian models, studying both birth defects and

caesarean sections using data from two large, linked birth registries in New South Wales

from 1990 to 2004. We compared four spatial models across seven birth defects: spina

bifida, ventricular septal defect, OS-atrial septal defect, patent ductus arteriosus, cleft lip

and or palate, trisomy 21 and hypospadias. For three of the birth defects, the shared

component model with a zero-inflated Poisson (ZIP) extension performed better than

other simpler models, having a lower Deviance Information Criteria (DIC). With spina

bifida, the ratio of relative risk associated with the shared component was 2.82 (95% CI:

1.46-5.67). We found that shared component models are potentially beneficial, but only

if there is a reasonably strong spatial correlation in effects for the study and referent

outcomes.

Word count: 173

Keywords : CAR; Sparse; Spatial; Defects

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Introduction

In recent years, there has been considerable interest in the development and application

of spatial models to analyse areal-level data. Most of these applications have been in the

field of disease mapping. Bayesian methods, in particular, have been used to calculate

smoothed relative risks of a particular disease at some areal level. While much attention

has been given to developing temporal extensions to spatial models and methods for

simultaneously analysing multiple outcomes, very little has been done to address the

issue of analysing sparse datasets, where there could be an abundance of zero counts or

large number of areas with extremely low expected counts of the disease.

The Conditional Autoregressive (CAR) model provides estimates of disease risk that

borrow strength from neighbouring areas. One advantage of this approach is to provide

more precise estimates of disease risk in areas with a small population, thus helping to

overcome the problem of larger uncertainty associated with a smaller disease count.

However, the performance of the model is questionable when neighbouring areas

themselves are sparsely populated, as there is little information to borrow. This is

particularly true of studies involving rare disease outcomes.

In epidemiological research, several outcomes can share the same risk factor (e.g. lung

cancer, chronic obstructive pulmonary disease (COPD) with smoking)(Division of Pop.

Health, 2001). We hypothesized that Bayesian modelling which examined outcomes

with shared risk factors simultaneously, may be a useful means of overcoming the

problems of small area estimation of sparse counts. We also hypothesised that modelling

two outcomes simultaneously should improve the estimation of the outcome with the

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sparse count, provided that both share a common spatially varying covariate, which need

not be measured. We then tested this hypothesis in models incorporating birth defects

and caesarean section rates. Birth defects are relatively rare events, but community

perceptions of clusters of defects cause great concern. Therefore, they are a useful health

outcome for evaluating methods for estimating spatially varying risks.

Relatively few studies have examined the spatial variation in risks of specific birth

defects, with neural tube defects being most commonly found to be spatially correlated

(Meyer and Siega-Riz, 2002, Tuncbilek et al., 1999, Ericson et al., 1988, Borman and

Cryer, 1993, Frey and Hauser, 2003, Rankin et al., 2005), followed by oral clefts (Saxen,

1975, Poletta et al., 2007). In anophthalmia and microphthalmia (Dolk et al., 1998)

prevalence was found to be higher in rural versus urban areas, as was diaphragmatic

hernia and gastroschisis (Rankin et al., 2005). Proximity of maternal residence to landfill

sites was associated with certain birth defects such as neural tube defects, hypospadias

and epispadias and abdominal wall defects (Elliott et al., 2001). Other studies have

examined spatial distribution of birth defects in general (Rushton and Lolonis, 1996,

Kuehl and Loffredo, 2006, Rushton et al., 1996). Caesarean sections are a common

procedure and spatial variation in the distribution of caesarean section rates has also

been well established (Taffel, 1994, Magadi et al., 2001, Baicker et al., 2006, Clarke and

Taffel, 1996). One study found a fourfold variation between low and high use areas

(Baicker et al., 2006).

Maternal age is commonly studied as a risk factor for both birth defects and caesarean

counts, and is a readily available demographic variable in most birth defect registries.

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The effect of maternal age on the occurrence of birth defects is not uniform, with both

very young maternal age and old maternal age associated with a different range of birth

defects. For defects like Down Syndrome (Trisomy 21), several studies (Gaulden, 1992,

Hsieh et al., 1995, Reefhuis et al., 1999) have found a positive association between

advanced maternal age and the risk of having babies with Down Syndrome. An analysis

of two large birth registries combined together, for instance, showed that that mothers

aged 40 years and above were 4.96 times (95% CI: 3.44-7.16) more like than those aged

below 40 to give birth to a baby with Down Syndrome, and this relationship was

statistically significant (p<0.001)(Reefhuis et al., 1999). Further, maternal age has been

associated with caesarean section in a number of studies (Padmadas et al., 2000, Taffel,

1994, Seshadri and Mukherjee, 2005, Sims et al., 2000, Maslow and Sweeny, 2000,

Witter et al., 1995, Parrish et al., 1994, Peipert and Bracken, 1993, Gordon et al., 1991).

We use maternal age here as an example. We would like to emphasise that our

subsequent model is not restricted to just maternal age, as it accommodates a number of

variables, which can be latent, but varying spatially.

We evaluated the shared component Bayesian modelling approach by examining both

birth defect and caesarean section counts simultaneously. We hypothesised that by

modelling two related outcomes simultaneously, one should be able to better estimate

the outcome with a sparse count, provided both share a common spatially varying

covariate, which need not be measured.

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Methods

The shared component model was developed by Knorr-Held and Best (Knorr-Held and

Best, 2001) and applied to the investigation of oral and oesophageal cancer mortality

data for males in Germany. The model was initially used to separate the underlying risk

surface for each outcome into a shared and outcome-specific component. The shared

component was to be interpreted as a surrogate for unobserved covariates that display

spatial structure and are common to both outcomes. The two outcome-specific risk

components and the shared component are assumed to be independent, each with a

spatial prior. The authors found two large clusters with a large shared component value,

and they postulated that this was consistent with the distribution of risk factors in the

neighbourhood. They also found distinct spatial patterns for each individual outcome.

We obtained data on birth anomalies in the state of New South Wales (NSW), Australia

for the period 1990 to 2003 (inclusive) from the NSW Midwives Data Collection (MDC)

and Birth Defects Register (BDR) databases. These mandatory registers of all births in

the state are completed by clinicians at the time of the birth. In 1998, a 2% sample of

the Midwives Data Collection records was validated against hospital records. The

excellent quality of this database is reflected in high correlations and low missing data

for almost all covariates (Centre for Epidemiology and Research, 2007b). Further details

on the two registries can be found elsewhere (Centre for Epidemiology and Research,

2007a). Data from both registers was geocoded to the mother’s usual address using

software developed by NSW Health and the Australian National University using a

previously described process (Summerhayes et al., 2006).

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We calculated standardised expected counts for each of seven birth defects: spina bifida

(SB), ventricular septal defect (VSD), ostium secundum atrial septal defect (ASD),

patent ductus arteriosus (PDA), cleft lip and or palate (CLP), trisomy 21 (T21) and

hypospadias (HPS). For example, the expected count of hypospadias in each Statistical

Local Area (SLA) was defined as Ei = (Birthsi/Totalbirths) * Totalhypospadias, where

Birthsi refers to total births in the ith SLA, and Totalbirths and Totalhypospadias refer to

the overall number of births and hypospadias defects in the NSW study region for that

particular time-period. These birth defects were studied for several reasons. They were

the more common defects reported in Australia, and they covered a spectrum of body

systems. They were also correlated with caesarean rates at varying strengths, which is

important to allow us to quantify the performance of the shared component model,

described in detail below.

We also calculated caesarean section counts for use in the shared component model.

Analyses were undertaken at the Statistical Local Area (SLA) level, for which there

were 198 SLAs available. The median size of the SLAs in our study was 2069 km2

(Interquartile range 181 to 4103 km2), while the median annual number of births in each

SLA was 193 (Interquartile range 66 to 670). SLA-specific relative risk estimates were

calculated as the ratio of the observed and expected counts for each area. Caesarean

section counts were obtained from the Midwives Data Collection registry, and we

included both emergency and elective caesarean counts from 1990 to 2003.

Ethical approval was obtained for the use of the NSW Midwives Data Collection and the

NSW Births Defects Registry data from the NSW Population and Health Services

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Research Ethics Committee, and for the study itself from the University of Sydney

Ethics Committee.

We compared the following four models in our analysis: a simple CAR model, a CAR

model with a zero-inflated Poisson (ZIP) extension to model the excess zeros, a shared

component model and finally a shared component with a ZIP extension.

The simple CAR model consisted of both a spatially structured prior and spatially

unstructured prior, which is sometimes known as the convolution prior or the Besag,

York and Mollie (BYM) model(Besag et al., 1991). The BYM model allows for the

smoothing of relative risk estimate in each region towards the mean risk in the

neighbouring areas. This provides for a more precise or reliable estimate of both mean

and variance compared to using the crude rate. Risks are also smoothed towards the

global mean to account for overdispersion.

The basic shared component model was formulated as follows:

)(~ 111 iiii EPoiO

)(~ 2

1

22 iiii EPoiO

where O1i and O2i are the observed counts of birth defects and caesarean counts in the ith

SLA respectively, and E1i and E2i are the expected counts of birth defects and caesarean

counts in the ith SLA respectively. i is the shared component, and 1i and 2i refer to

the components specific to birth defects and caesarean counts respectively. δ refers to

the scaling parameter that allows for a different ‘risk gradient’ to be associated with this

136

component for each outcome. We extended the model to incorporate a ZIP component,

in the form of a mixture model, where the proportion of zeros and non-zeros are

assigned different distributions and priors accordingly (see appendix 1 for WinBUGs

code and appendix 2 for model details). CAR priors were also incorporated for the three

components described earlier. When undertaking CAR modelling of data at an areal

level, it is necessary to define a neighbourhood matrix that characterizes the

neighbourhood structure of the data being analysed. A previous study of neighbourhood

matrices led us to choose the Queen contiguous adjacency method of assigning

neighbours to preserve the underlying spatial distribution in the relative risks (Earnest et

al., 2007). This method of neighbourhood assignment selects neighbours which share

both common boundaries and vortices. However, each neighbour is assigned equal

weight.

Reversible jump, Markov chain Monte Carlo methods were used to run the models in the

initial analysis (Knorr-Held and Best, 2001). As these cluster models were not available

through standard analytical software, we used an adapted version of a model that made

use of convolution priors for shared and outcome-specific random effects instead. We

discarded the first 50,000 samples as burn-in and ran a further 30,000 iterations which

were used in the calculation of the posterior estimates. We then ran two different chains,

starting from diverse initial values, and convergence was assessed using the Gelman-

Rubin convergence statistic, as modified by Brooks and Gelman (Brooks and Gelman,

1998). Estimates for the mean smoothed relative risk, ratio of relative risk associated

with shared component, fraction of variation in relative risk of outcome explained by the

shared component and their corresponding 95% credible intervals were derived from the

137

posterior distribution. The fraction is computed as the ratio of variance due to the shared

component over the total variance, and it’s importance in this study lies in quantifying

the magnitude of shared risk factor between the birth defect and caesarean rates.

Models were compared in three ways. Firstly, we assessed model fit using the deviance

information criterion (DIC) developed by Spiegelhalter (Spiegelhalter et al., 2002). This

consisted of the posterior mean deviance and a term (pD) that penalised for excessive

parameters in the model. Smaller DIC values indicate a better model, and a high pD

value will indicate presence of excessive parameters in the model. Criteria suggested by

Best (Best et al., 2005) were used to interpret the DIC models, with DIC values within 1

or 2 of the ‘best’ model also strongly supported, values within 3 and 7, weakly

supported, and models with a DIC greater than 7 interpreted as substantially inferior.

Second, we calculated the chi-squared residual sum of squares (RSS) to determine the

amount by which the estimated counts of birth defects differed from the actual counts

(Lawson et al., 2000). This is the sum of the squared differences between the observed

and estimated number of birth defects standardised by the estimated number of births. A

higher RSS indicates a greater amount of smoothing. RSS was also calculated by SLAs

stratified by the 75th percentile (high expected count) and 25th percentile (low expected

count) of the expected counts of each defect.

Data extraction, management and analysis were performed in Stata (version 9.2, Stata

Corp, College Station, USA). The Bayesian models were run in WinBUGs (version

1.4.3, Imperial College and Medical Research Council, UK) through Stata, using a

138

customised ado program file (Thompson et al., 2006). All the weight matrices were

created using the GeoDA software (version 0.9.5-I, University of Illinois, USA). Maps

were produced in ArcMap version 9.0 (ESRI, USA). A copy of the WinBUGS code we

used for the analysis is provided in appendix 1.

Results

Table 1 briefly describes the seven birth defects and their British Paediatric Association

International Classification of Diseases, 10th Revision (BPA-ICD10) codes. In the 198

NSW SLAs from 1990 to 2004, there were 1,287,460 births, including 590 births with

spina bifida, 2594 with ventricular septal defect, 2377 with atrial septal defect, 1557

with patent ductus arteriosus, 2043 with cleft lip/ palate, 1973 with trisomy 21 and 2260

with hypospadias included in our analysis. Any birth with multiple types of the same

defect (e.g. spina bifida subtypes) was only counted once. Caesarean section was around

100-fold more common than the selected anomalies, with 263,199 counts across the

same region. Depending on the type of defect, the proportion of SLAs without the birth

defects ranged from 15% (29/198) for hypospadias to 39% (78/198) for spina bifida. As

expected, the average expected count for each of the birth defects was low: 3 for spina

bifida, 13 for ventricular septal defect, 12 for OS- atrial septal defect, 8 for patent ductus

arteriosus, 10 for cleft lip/ palate, 10 for trisomy 21 and 11 for hypospadias.

An exploratory analysis examining the correlation in the crude relative risk estimates of

birth defects with caesarean showed that there were moderate but significant correlations

ranging from 0.153 for spina bifida (p=0.031) to 0.250 for trisomy 21 (p<0.001). For

139

patent ductus arteriosus, the relationship was marginally significant (rho 0.136,

p=0.057).

Table 2 compares the smoothed relative risks for each birth defect for each of the four

models with the mean crude relative risks. In the CAR models, the relative risks are

smoothed towards 1. More importantly, because of the shrinkage property of these

models, the standard deviations for the smoothed models are much smaller than for the

crude estimates. This is an expected finding, as the CAR model produces smoother,

more stable estimates in areas with small counts. This method of comparison suggests

little difference in standard deviation between the models.

Using the DIC, we compared model fit for each of the 28 models in our analysis (Table

3). For spina bifida, patent ductus arteriosus and trisomy 21, the shared component ZIP

model clearly performed better than the other three models, with a much lower DIC. For

cleft lip/ palate, the CAR ZIP model was clearly superior. For ventricular septal defect,

OS-atrial septal defect and hypospadias, the shared component ZIP model and the CAR

ZIP model had similar DIC scores. Even after penalising for the excess parameters in the

model, the DIC showed that the shared component ZIP model performed better in some

instances.

Table 4 details the model parameters from the shared component model. Here, we only

discuss the results with respect to Spina Bifida, Patent Ductus Arteriosus and Trisomy

21, for which the shared component ZIP model performed best. For Spina Bifida, the

ratio of relative risk associated with the shared component was found to be 2.82 (95%

140

CI: 1.46-5.67), indicating that the unobserved risk factors common to both outcomes

(i.e. Spina Bifida and Caesarean) have a stronger association with Spina Bifida. The

corresponding ratio of relative risk for Patent Ductus Arteriosus and Trisomy 21 are

similar at 2.60 (95% CI: 1.37-4.96) and 2.03 (95% CI: 1.21-3.18) respectively. The

model also shows that 27% of the between-SLA variation in relative risk of Spina Bifida

is explained by the shared component. For Patent Ductus Arteriosus and Trisomy 21,

this figure is higher at 42% and 57%. The results strongly indicate the strong presence of

a shared covariate that is common to both the defect and caesarean rates.

Next, we examined the residual sum of squares for all outcomes, and stratified by areas

of low or high expected counts, defined by the 25% and 75% percentile of the expected

counts (Table 5). For spina bifida, all four models smoothed the relative risks by roughly

the same amount. For the other birth defects, the shared component ZIP model generally

smoothed the crude relative risks by a smaller amount. The stratified analysis indicates a

greater amount of smoothing occurred at areas with smaller expected counts, as we

would expect.

Finally, we plotted the smoothed relative risk estimates of the seven birth defects and

compared them across the four competing models. Some differences in risk estimates

between competing models were observed. The difference between the shared

component models and the simple CAR models could be attributed to the ‘borrowing

strength’ from the spatial pattern of the caesarean risk estimates through the shared

component. Figure 1 shows the crude relative risk of trisomy 21 across NSW. Outside of

Sydney, there appear to be large pockets of areas with a low risk. These are actually

141

areas with small or zero counts. As expected, after smoothing these unstable risks, we

note that most of them are now closer to 1 (Figure 2). Similarly within Sydney, a small

number of areas which appear to have a higher risk get smoothed closer to one. The

relative risk maps for caesarean rates appear to have a strong spatial correlation (Figure

3). There is a striking similarity in the maps between caesarean rates and maternal age,

particularly in the eastern Sydney region (Figure 4).

Discussion

The common challenge of low or absent areal counts in spatial analysis of uncommon

outcomes motivated us to explore the benefit of shared component extensions to

frequently used CAR models. We examined these approaches in comprehensive datasets

of birth defects and caesarean sections, based on the assumption that maternal age is a

shared risk factor between these outcomes. We found that the shared component model

worked well for some outcomes (e.g., spina bifida and trisomy 21), resulting in models

with better fit and smaller standard deviations. However for others, (e.g. hypospadias,

ventricular septral defect and cleft lip/ palate) this approach was of less benefit.

A likely explanation of these findings is that shared component models are dependent on

the strength of the relationship each outcome has with the shared risk factor. Advanced

maternal age is a well known risk factor for caesarean rates (Sims et al., 2000, Peipert

and Bracken, 1993, Gordon et al., 1991, Parrish et al., 1994, Padmadas et al., 2000). In

NSW, the overall risk of having a baby with a birth defect also increases after 40 (Centre

for Epidemiology and Research, 2007a). However the association between age and birth

defects varies markedly by defect type. Older maternal age is strongly associated with

142

trisomy 21 (Reefhuis et al., 1999, Hsieh et al., 1995). The relationship between maternal

age and spina bifida has also been shown in a number of studies (Meyer and Siega-Riz,

2002, Nili and Jahangiri, 2006, Strassburg et al., 1983, Tuncbilek et al., 1999). On the

other hand the association of age with the other defects is unclear. Besides maternal age,

other factors such as maternal diabetes or socio-economic status could have contributed

to our findings.

The shared component model we used in the analysis assumes that the shared and

outcome-specific component is independent, which is reasonable for a non-

communicable disease such as birth defects. In the context of birth defects and caesarean

rates, we had no reason to believe that the outcome-specific effects and the shared

components could be correlated. In addition, we also acknowledge the presence of

ecological bias in this study, in particular the challenges of inferring that areal-level

association is the same as an individual subject level association.

While we have chosen to present the application of the proposed models on real-life

data, it would be useful if future research can make use of large simulation studies to

examine if our results still hold under more general conditions. The closest large-scale

simulation study we have come across is from Best and colleagues(Best et al., 2005), but

extensions to the models using the ZIP component have not been considered. The

challenge in those simulations studies would lie in setting up computer codes to sample

from a correlated but over-dispersed bivariate Poisson distribution.

143

For computational reasons, we restricted our analysis to spatial-only models. Extensions

of these models/ ideas to spatio-temporal models should be fairly straightforward, and

this has been considered in another study by Richardson, et al(Richardson et al., 2006).

One main concern with the CAR models that we used is that they may tend to over-

smooth maps. In our analyses, the residual sum of squares shows that this is not the case.

Disease maps are often provided in age standardised form, so that meaningful

comparisons can be made between regions after accounting for differences in these

important confounders. In our study, age standardising the expected counts that were

used in the models would have precluded inclusion of maternal age as a variable in our

analysis. While we did not directly include age in the model, we effectively included the

latent effect of age in the model, through the shared component. The key advantage of

this is that the shared component model does not only account for age but also other key

covariates that share a similar geographic relationship with both caesarean rates and

birth defects. It is not the aim of this paper to discuss the aetiology or risk factors for the

various defects. In a related issue, we chose to model all caesareans (as compared to just

emergency caesareans), as data on type of caesarean delivery was only available prior to

1998. Considering just emergency caesareans would have reduced the effective sample

size in our study, as well as restrict the generalizability of the study results to the period

1990 and 1997.

It should be noted that there are other multivariate areal spatial models available, and it

is not the aim of this paper to make comparisons across these models. A multivariate

Bayesian model has been used to estimate cancer rate in one particular region, while

144

borrowing information from other cancer sites(Assuncao and Castro, 2004), but it is

unclear how this model would handle excessive zeros in the outcome, or incorporate

unmeasured covariates. The model also does not allow for geographical spatial

dependency. We also preferred the shared component model over the alternative

multivariate CAR model(Gelfand and Vounatsou, 2003), as the latter makes a (rather

strong) assumption that the outcomes are correlated. On the other hand, the shared

component stratifies and includes a shared component (i.e. latent risk factor) which is

more likely to reflect the relationship between caesarean rates and birth defects.

The shared component model has been used recently in a few studies. In one such study,

researchers studied the joint prevalence of fever and diarrhoea and their association with

individual, familial and community risk factors(Kazembe et al., 2009). The authors

found evidence of common variation of prevalence of childhood fever and diarrhoea

suggestive of a common/ overlapping risk factor. The models used may not be suitable

to study birth defects though, due to the lower prevalence and excessive zeros.

Extensions to more than 2 disease outcomes have also been performed using the shared

component model (Downing et al., 2008). In that particular study, the authors performed

a joint analysis on six cancer sites with 3 shared components, and showed the superiority

of joint modelling over individual modelling via reductions in the DIC. However, once

again, the models were used to study chronic diseases that are not rare (i.e. oesophagus,

stomach, pancreas, lung, kidney and bladder cancers), and it is unclear how the models

would fare for a sparse outcome such as birth defects.

145

Conclusion

We found that the shared component models are potentially beneficial, but only if there

is a reasonably strong spatial correlation in effects for the study and referent outcomes.

In our study, we found this to be particularly true for spina bifida and trisomy 21. This

finding can be explained by shared risk factors, which are spatially correlated, for

example maternal age. Disease maps for sparse outcomes, such as birth defects, can

benefit from simultaneous modelling with a correlated outcome, such as caesarean

section counts. Prior to modelling, we recommend exploratory analyses focussing on the

correlation in risks between the two, to choose a suitable referent outcome.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgements

The authors acknowledge the support of the Australian Research Council Linkage

Grant LP0348628, the North Coast Area Health Service, the NSW Department of

Health, the Commonwealth Department of Health and Ageing, Dr Lee Taylor, Paul

Houlder, and Danielle Taylor.

146

Table 1. Description of birth defects and their associated ICD codes and abbreviations

Birth defect

Abbrevia

tion

Affected

organ Description*

ICD10-

AM

Code

Spina Bifida SB Spinal cord Defect in which part of the vertebral column

is incomplete

741

Ventricular Septal

Defect

VSD Heart A hole in the wall separating the ventricles

in the heart

745.4

Ostium Secundum

Atrial Septal Defect

ASD Heart An abnormally large opening in the atrial

septum at the site of the foramen ovale and

the ostium secundum.

745.5

Patent Ductus

Arteriosus

PDA Heart Failure for the ductus arteriosus, an arterial

shunt in fetal life, to close on schedule.

747.0

Cleft lip/ Palate CLP Oro-facial Cleft lip is a physical split or separation of

the two sides of the upper lip and appears as

a narrow opening or gap in the skin of the

upper lip. Cleft palate is a split or opening in

the roof of the mouth.

749,

749.0,

749.1,

749.2

Trisomy 21 (“Downs

Syndrome”)

T21 Multiple organs A defect caused by having an extra

chromosome 21

758.0

Hypospadias HPS Reproductive

organ (penis)

Defect in males, where the urethra opens on

the undersurface of the penis

752.6,

752.60

* Source: The Harper Collins Illustrated Medical Dictionary (4th Ed, 2001, Harper Collins publisher, NY)

and http://www.medicinenet.com. Assessed 08 May 2008

BPA-ICD10: British Paediatric Association International Classification of Diseases, 10th Revision

147

Table 2. Comparison between mean crude and smoothed relative risk estimates and their associated

standard deviations

Crude RR CAR model

CAR ZIP

model

Shared

Component

model

Shared Component

ZIP model

RR SD RR SD RR SD RR SD RR SD

Spina Bifida 1.17 2.26 1.03 0.28 1.35 0.35 0.94 0.28 1.68 0.47

Ventricular Septal Defect 1.88 11.55 0.97 0.16 1.03 0.11 0.91 0.18 1.05 0.16

OS Atrial Septal Defect 1.03 1.09 0.97 0.18 1.04 0.15 0.9 0.19 1.05 0.18

Patent Ductus Arteriosus 2.54 19.11 0.99 0.2 1.11 0.19 0.92 0.22 1.18 0.24

Cleft lip and or Palate 1.35 3.13 1.03 0.07 1.07 0.08 1.06 0.11 1.16 0.13

Trisomy 21 1.03 1.71 0.92 0.2 1.02 0.16 0.82 0.20 1.00 0.19

Hypospadias 1.09 0.92 1.06 0.11 1.11 0.13 1.11 0.15 1.21 0.18

SD denotes standard deviation and RR Relative Risk

Table 3. Summary of Deviance Information Criteria (DIC) values for the four competing models

CAR model CAR ZIP model

Shared Component

model

Shared Component

ZIP model

DIC pD DIC pD DIC pD DIC pD

Spina Bifida 593 37 493 30 568 32 488 34

Vent. Septal Defect 965 53 889 46 946 54 887 56

OS Atrial Septal Defect 923 54 845 47 904 53 844 56

Pat. Ductus Arteriosus 838 51 758 45 814 48 753 51

Cleft lip/ Palate 873 31 826 31 869 39 837 47

Trisomy 21 836 46 746 42 812 43 740 48

148

Hypospadias 890 42 840 42 871 45 841 53

Caesarean section 1971 178 1949 168

Note: Model with lowest DIC (+/- 3) highlighted in bold. Two bold values in the same row indicate similar fit

DICs for shared component models have been discounted for the contribution by "Caesarean only" model to

ensure comparability

Table 4. Summary of model parameters

Shared Component Model Shared Component ZIP Model

RR 95% Credible Interval RR 95% Credible Interval

Spina Bifida

Ratio - RR assoc. with shared component 3.27 1.51, 6.16 2.82 1.46 5.67

Fraction of variation in RR of defect explained

by shared component 0.34 0.11, 0.73 0.27 0.10 0.69

Fraction of variation in RR of c. section

explained by shared component 0.24 0.10, 0.44 0.25 0.10 0.45

Vent. Septal Defect

Ratio - RR assoc. with shared component 2.32 1.21, 4.03 2.02 1.10 3.69

Fraction of variation in RR of defect explained

by shared component 0.48 0.22, 0.75 0.48 0.26 0.71

Fraction of variation in RR of c. section

explained by shared component 0.28 0.14, 0.49 0.28 0.14 0.48

OS Atrial Septal Defect

Ratio - RR assoc. with shared component 2.15 1.14, 3.87 2.22 1.13 3.88

Fraction of variation in RR of defect explained

by shared component 0.39 0.16, 0.70 0.48 0.20 0.73

149

Fraction of variation in RR of c. section

explained by shared component 0.29 0.14, 0.48 0.26 0.14 0.45

Pat. Ductus Arteriosus

Ratio - RR assoc. with shared component 2.77 1.30, 5.37 2.60 1.37 4.96

Fraction of variation in RR of defect explained

by shared component 0.41 0.15, 0.77 0.42 0.17 0.76

Fraction of variation in RR of c. section

explained by shared component 0.26 0.11, 0.47 0.23 0.11 0.39

Cleft lip/ Palate

Ratio - RR assoc. with shared component 1.62 0.90, 2.74 1.61 0.94 2.53

Fraction of variation in RR of defect explained

by shared component 0.46 0.27, 0.66 0.46 0.26 0.66

Fraction of variation in RR of c. section

explained by shared component 0.29 0.15, 0.47 0.29 0.16 0.48

Trisomy 21

Ratio - RR assoc. with shared component 2.15 1.30, 3.37 2.03 1.21 3.18

Fraction of variation in RR of defect explained

by shared component 0.57 0.26, 0.80 0.57 0.29 0.78

Fraction of variation in RR of c. section

explained by shared component 0.39 0.20, 0.57 0.36 0.21 0.56

Hypospadias

Ratio - RR assoc. with shared component 1.90 1.06, 3.20 1.92 1.11 3.13

Fraction of variation in RR of defect explained

by shared component 0.47 0.26, 0.70 0.47 0.26 0.67

Fraction of variation in RR of c. section

explained by shared component 0.27 0.14, 0.47 0.25 0.14 0.39

150

Table 5. Residual sum of squares for the overall sample, and for areas with low expected counts and

high expected counts

Models

Spina

Bifida

Vent.

Septal

Defect

OS Atrial

Septal

Defect

Pat. Ductus

Arteriosus

Cleft lip/

Palate Trisomy 21

Hypo-

spadias

Overall

CAR Model 0.83 2.04 1.15 2.19 1.37 0.92 0.83

CAR ZIP Model 0.82 2.08 1.19 2.08 1.32 0.89 0.80

Shared Component Model 0.83 2.06 1.19 2.22 1.24 1.13 0.72

Shared Component ZIP Model 0.89 1.88 1.08 1.85 1.18 0.83 0.75

High expected count

(4) (16) (15) (10) (12) (12) (14)

CAR Model 0.45 0.20 0.17 0.28 0.44 0.35 0.41

CAR ZIP Model 0.69 0.27 0.24 0.35 0.43 0.42 0.42

Shared Component Model 0.51 0.31 0.33 0.33 0.35 0.79 0.33

Shared Component ZIP Model 1.04 0.18 0.17 0.35 0.45 0.33 0.53

Low expected count

(0.35) (1.55) (1.42) (0.93) (1.22) (1.18) (1.35)

CAR Model 1.29 5.91 2.48 6.28 3.05 1.60 1.13

CAR ZIP Model 0.98 5.75 2.37 5.59 2.87 1.25 1.07

Shared Component Model 1.29 6.03 2.52 6.54 2.78 1.82 1.06

Shared Component ZIP Model 0.74 5.32 2.15 4.97 2.45 1.27 0.97

Note: High and low expected count for each birth defect was calculated from the areas within the 75th

and 25th percentile of expected count of the birth defect respectively, and the values are reflected in

parenthesis

151

Figures

/

125 0 125 250 375 50062.5

Kilometers

RR

<0.75

0.75 - 1.00

1.01 - 1.50

>1.50

Figure 1. Crude relative risk estimates of Trisomy 21

Sydney

NSW

/

125 0 125 250 375 50062.5

Kilometers

RR

<0.75

0.75 - 1.00

1.01 - 1.50

>1.50

Figure 2. Smoothed relative risk estimates of Trisomy 21

(Shared component ZIP model)

NSW

Sydney

152

/

125 0 125 250 375 50062.5

Kilometers

RR

<0.75

0.75 - 1.00

1.01 - 1.50

>1.50

Figure 3. Crude relative risk estimates of caesarean rates

NSW

Sydney

/

125 0 125 250 375 50062.5

Kilometers

Proportion aged 30 and above

<37.5

37.5 - 41.7

41.7 - 49.8

>49.8

Figure 4. Proportion of births among mothers aged 30 and above

NSW

Sydney

153

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Appendix 1. Copy of WinBUGS code for the shared component ZIP model model { # Likelihood for (i in 1:N) { for (k in 1:2) { zeros[i,k] <- 0 zeros[i,k] ~ dpois(mu[i,k]) mu[i,k]<-(1-step(O[i,k]))*(-log(1-p[i,k] + p[i,k]*exp(-lambda[i,k]))) + (step(O[i,k]-1))*(-(log(p[i,k])-lambda[i,k] + O[i,k]*log(lambda[i,k])-logfact(O[i,k]))) logit(p[i,k]) <- alpha0[k] log(lambda[i,k])<-log(E[i,k]) + eta[i, k] } } for(i in 1:N) { # Define log relative risk in terms of disease-specific (psi) and shared (phi) random effects eta[i,1] <- phi[i] *delta + psi[1, i] # changed order of k and i index for psi eta[i,2] <- phi[i] /delta + psi[2, i] # (needed because car.normal assumes right hand index is areas) } # Spatial priors (BYM) for the disease-specific random effects for (k in 1:2) { for (i in 1:N) { psi[k, i] <- bind[k, i] + bspat[k, i] # convolution prior = sum of unstructured and spatial effects # unstructured disease-specific random effects bind[k, i] ~ dnorm(alpha[k], vind[k]) } # spatial disease-specific effects bspat[k,1:N] ~ car.normal(adj[], weights[], num[], vspat[k]) } # Spatial priors (BYM) for the shared random effects for (i in 1:N) { phi[i] <- Ush[i] + Ssh[i] # convolution prior = sum of unstructured and spatial effects # unstructured shared random effects Ush[i] ~ dnorm(0, vunstr) } # spatial shared random effects Ssh[1:N] ~ car.normal(adj[], weights[], num[], vstr) # Weights for(k in 1:sumnumneigh) { weights[k]<-1 } # Other priors for (k in 1:2) { alpha[k] ~ dflat() alpha0[k] ~ dnorm(0,0.0001) vind[k] <-1/(sigmaind[k]*sigmaind[k]) vspat[k] <-1/(sigmaspat[k]*sigmaspat[k]) sigmaind[k]~dunif(0.1,20) sigmaspat[k]~dunif(0.1,20) } vunstr <-1/(sigmaunstr*sigmaunstr) vstr <-1/(sigmastr*sigmastr) sigmaunstr~dunif(0.1,20) sigmastr~dunif(0.1,20) # scaling factor for relative strength of shared component for each disease logdelta ~ dnorm(0, 5.9)

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# (prior assumes 95% probability that delta^2 is between 1/5 and 5; lognormal assumption is invariant to #which disease is labelled 1 and which is labelled 2) delta <- exp(logdelta) # ratio (relative risk of disease 1 associated with shared component) to (relative risk of disease 2 # associated with shared component)- RRratio <- pow(delta, 2) # Output for (i in 1:N) { SMR1[i] <- O[i,1]/E[i,1] # SMR for disease 1 SMR2[i] <- O[i,2]/E[i,2] # SMR for disease 2 RR1[i] <- exp(alpha[1] + eta[i,1]) # overall RR of disease 1 in area i RR2[i] <- exp(alpha[2] + eta[i,2]) # overall RR of disease 2 in area i residualRR1[i]<- exp(psi[1,i]) # residual RR specific to disease 1 residualRR2[i]<- exp(psi[2,i]) # residual RR specific to disease 2 sharedRR[i] <- exp(phi[i]) # shared component of risk common to both diseases # Note that this needs to be scaled by delta or 1/delta if the absolute magnitude of shared RR for each disease # is of interest logsharedRR1[i] <- phi[i]*delta logsharedRR2[i] <- phi[i]/delta } # empirical variance of shared effects (scaled for disease 1) varshared[1] <- sd(logsharedRR1[])*sd(logsharedRR1[]) # empirical variance of shared effects (scaled for disease 2) varshared[2] <- sd(logsharedRR2[])*sd(logsharedRR2[]) # empirical variance of disease 1 specific effects varspecific[1] <- sd(psi[1,])*sd(psi[1,]) # empirical variance of disease 2 specific effects varspecific[2] <- sd(psi[2,])*sd(psi[2,]) # fraction of total variation in relative risks for each disease that is explained by the shared component fracshared[1] <- varshared[1] / (varshared[1] + varspecific[1]) fracshared[2] <- varshared[2] / (varshared[2] + varspecific[2]) }

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Appendix 2. Shared Component ZIP Model Assuming that y1i and y2i are observed counts of 2 diseases (here we use birth defects and caesarean counts) and e1i and e2i expected counts in the ith region, the responses yi1 and yi2 are assumed to be conditionally independent Poisson random variables with means as shown below:

)(~ 111 iiii ePoiy

)(~ 2/1

22 iiii ePoiy ,

where θi is the shared component and Øid, d=1,2 is the disease-specific component. δ is the scaling parameter that allows for a different ‘risk gradient’ in terms of the contribution of the shared component to the overall relative risk, for each disease. The mixture model we used in our analysis is defined below:

,.....2,1,!

)())(exp()|Pr(

)),(exp((1)|0Pr(

yy

zzpzyY

zppzYy

iiii

iiii

For both diseases k=1,2, λ(zik), is expressed as a function of the expected counts ei,k and a random effects ηi,k through a log transformation as shown below. log(λ(zik) = log(ei,k) + ηi,k A logit function is used for the pi,k as shown below: logit(pi,k) = αk with a diffuse normal prior as seen below: αk ~ Normal(0, 10000) The log relative risk for each disease is then formulated as follows: For each disease, the log relative risk is split into a disease-specific (Øi1, Øi2) and shared (θi ) random effects as shown below: ηi1 = θi *δ + Øi1 ηi2 = θi /δ + Øi2

The shared component random effect is assigned a CAR normal spatial prior with a conditional mean and variance, as shown below:

),(~ 2siCARi Normal

The standard deviation parameter is assigned a diffuse but plausible uniform distribution: σs ~Uniform(0.1, 20) The disease-specific random effects are in turn split into two components: a spatially structured random effects (Si1 and Si2) and an unstructured random effects (Ui1 and Ui2). This is also known as a convolution prior: Øi1= Si1 + Ui1

Øi2= Si2 + Ui2 The spatially structured terms are again assigned a CAR normal prior;

),(~ 2111 siCARi NormalS

),(~ 2222 siCARi NormalS

σs1 ~Uniform(0.1, 20) σs2 ~Uniform(0.1, 20) The unstructured terms are assigned a non-informative normal prior distribution as shown below:

),(~ 2111 ui uNormalU

),(~ 2222 ui uNormalU

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σu1 ~Uniform(0.1, 20) σu2 ~Uniform(0.1, 20) u1 and u2 are assigned flat non-informative hyper-priors. The scaling factor, δ, is assigned a log-normal distribution, as shown below. The prior assumes 95% probability that δ2 is between 1/5 and 5, invariant to whichever disease is labelled 1 or 2: log (δ) ~ Normal(0, 0.169) δ is scaling parameter that is reciprocated for the second disease, as this provides an attractive invariance feature. The logarithm of the scaling parameter, δ, has a normal prior with mean 0 and variance τ2. Since the prior for δ is symmetric around zero on a log-scale, any value δo is as ‘equally likely’ as the reciprocal value 1/δo a priori. P(δl≤ δ ≤ δu) = P(1/δu≤ δ ≤ 1/δl), for any positive values δl < δu If we switch the indices for the two diseases (i.e. birth defects and caesarean counts), we obtain exactly the same posterior distribution for the joint and specific components, and the posterior of δ will change to the reciprocal distribution. This will ensure that the posterior distribution of the relative risk for each disease will be exactly the same.

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CHAPTER 8: STRATEGIES FOR COMBINING AREAL WITH INDIVIDUAL DATA Sampling and sample size strategies for including individual with areal-level covariates in the spatial analysis of a sparse disease outcome Arul Earnesta,*,, John Beardb, Geoff Morgana, c, Deborah Donoghuea, Therese Dunna, c, David Muscatellod, Danielle Taylore, Kerrie Mengersenf. a Northern Rivers University Department of Rural Health, The University of Sydney, b

Department of Ageing and Lifecourse, World Health Organisation, Geneva, Switzerland, c Population Health & Planning, North Coast Area Health Service, New South Wales, d Centre for Epidemiology and Research, New South Wales Department of Health, e University of Adelaide, f Faculty of Science, Queensland University of Technology, Queensland.

*Correspondence to: Arul Earnest, Northern Rivers University Department of Rural Health, 55 Uralba Street, Lismore, NSW, Australia 2480. E-mail: arul_earnest@hotmail.com Keywords: Ecological bias, CAR model, Disease mapping, Spatial analysis, Birth defects Research Manuscript submitted to Statistics in Medicine Journal on 1 Oct 2009

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Contributor Statement of contribution

Arul Earnest Conceptualised the entire study, performed data management, wrote computer codes for the analysis, performed literature review and statistical analysis and wrote the manuscript for publication

23 February 2010

Geoff Morgan

Contributed to writing of manuscript

Kerrie Mengersen

Contributed to writing of manuscript

Deborah Donoghue

Contributed to writing of manuscript

Therese Dunn

Contributed to writing of manuscript

David Muscatello

Contributed to writing of manuscript

John Beard

Contributed to writing of manuscript

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SUMMARY

Aim

Our study explored the optimal model for examining individual level and real level risk

factors for sparse outcomes. Our example uses individual maternal age and hospital

status and area level socioeconomic deprivation to examine birth defects.

Methods

We performed a series of simulation analyses with 13 different scenarios, each covering

different aspects of sampling design and sample size considerations. Bayesian hybrid

models were used to examine the effect of both individual and areal-covariates. The

models also took into account possible spatial correlation and over-dispersion in the

data. We applied the models to a large linked registry of births and birth defects in New

South Wales (NSW) from 1990 to 2003.

Results

We found that a simple case-control study design best selects individual samples for

inclusion in a hybrid model, as compared to proportional sampling and various other

methods. It is clear that stratifying the sampling frame first by case-control status, and

then selecting the samples in each stratum is a much better strategy for a sparse disease

outcome. Further stratification by other known risk factor provides only marginal

improvement.

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Conclusion

Our findings will help future researchers address the computational difficulties

encountered when typically analysing large datasets, through careful sampling of the

cases and controls. Specifically, sampling according to case-control status is a good

strategy for diseases that are rare.

Word count 220

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1. INTRODUCTION

Birth defects, though rare, are a major public health concern as they are a leading cause

of neonatal and infant mortality, and a key cause of morbidity in later life. Risk factors

for birth defects can operate at various levels. This includes individual-level factors such

as maternal age, maternal smoking during pregnancy, maternal socio-economic status

and areal-level factors such as socio-economic status of the neighbourhood and area of

maternal residence. In the analysis of such data, one needs to account for the different

scale upon which the data operates, the prevalence (in particular sparseness) of the

outcome variable, and possible correlation in the outcome variable. Unfortunately,

models for the analysis of such data are not readily available. The motivation for this

research came from our wish to examine risk factors for birth defects, specifically key

individual- level covariates such as maternal age and hospital type, and an areal-level

measure such as indices of socioeconomic disadvantage.

We reviewed the birth defects literature for associations with socio-economic status

(both individual and areal-level measures) and maternal age. Among the various

individual-level socio-economic indicators, occupation[1-8] seems to be the most

commonly studied proxy measure, followed by education[1, 2, 9-13], race[14-19],

income[1, 14, 15] and insurance status[12, 22]. In contrast, areal-level measures of

socio-economic status have been evaluated in a number of studies of occurrences of

birth defects[2, 16-23]. Two studies evaluated both individually measured socio-

economic status along with areal measures, with mixed results. One study found a

significant effect of lower individual socio-economic status and residence in a SES-

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lower neighbourhood on the occurrence of neural tube defects[21]. An increased risk of

neural tube and facial clefts with household SES index, but not individual SES measures

was found in another study [29].

Maternal age is clearly associated with birth defects, and younger mothers have been

shown to have a higher risk of giving birth to babies with gastroschisis[24-28],

omphalocele[27] , chromosomal abnormalities[28, 29], cystic hygroma, monogenic

disorders, ventricular septal defects[28], anencephaly, all ear defects and female genital

defects[27]. Older mothers, on the other hand, have a higher risk of having babies with

various types of atresia[23, 33], eye deformities[30], heart defects, male genital defects

and craniosynostosis,[27], chromosomal abnormalities[28, 31] including trisomies 18

and 21[32], hip dysplasia, abnormalities of the pancreas and cleft lip or palate[9].

A few studies have examined the spatial distribution of birth defects and their

association with possible spatially varying risk factors such as high altitude. The study

by Castilla[33] found that adjusted relative risks were significantly higher among those

living in highlands, specifically for cleft lip, microtia, preauricular (branchial arch

anomaly complex, constriction band complex and anal atresia). On the other hand, risks

were lower for spina bifida, anencephaly and hydrocephaly. The study by Lopez-

Camelo also linked altitude with the risk of microtia[34], whilst Poletta and

colleagues[35] found higher cleft lip/ palate birth prevalence rates.

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1.1. Ecological bias

Ecological bias is an umbrella term describing a number of different types of biases that

operate when we undertake ecological analysis (i.e. analysis of aggregated data). They

can be broadly grouped into specification bias, effect modification and confounding.

Specification bias occurs when the aggregated model does not account for any non-

linear relationship inherent in the data between the outcome and covariate. Confounding

in ecological analysis may happen when there is background variation in rate of disease

across covariate groups. This confounds baseline risk for groups, creating interactions

and effect modifications. The lack of mutual standardisation between the outcome and

exposure variables can also create bias[36], which typically occurs when the exposure is

correlated with the variable used for standardisation. The issue of ecological bias has

been discussed in detail by Morgenstern[37, 38] while mathematical derivation of the

biases described above are also available[39].

Inclusion of individual-level data on areal-level models would help alleviate the

potential ecological bias described above. An analysis of both individual and aggregate-

level data of the same construct would allow us to differentiate between individual-level

risks as compared to areal-level risks. In areal-level models, the generalisability of

results to the range of values that exposure variables take is restricted, as often the range

of values taken by the covariate becomes narrower as we aggregate the data.

Conversely, inclusion of areal-level data in individual-level models can help to improve

the power of the study by reducing the mean square error of parameter estimates. There

are several advantages to the use of ecological data, including the use of readily

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available routinely collected outcome, covariate and confounder data, and greater

exposure contrasts provided by the large geographical areas covered[40]. Areal studies

can also be useful to estimate exposure effects that may be difficult to detect within any

one group of individuals, and group-level summaries, in certain instances, can provide

more correct estimates than individual measurements[41].

1.2. Addressing ecological bias

Various options are proposed to tackle ecological bias issues. For instance, Wakefield et

al, suggest an aggregate-level model when there is within-area variability in the

exposure and the exposures are measured with error[42], or by constructing a model

from the underlying continuous risk surface[40]. However, this approach can be

computationally challenging.

The aggregate data study design proposed by Prentice and colleagues[43] is derived by

collating individual-level models within groups, and using individual covariate data

from a random sub-sample. Although the approach sounds appealing, it does not address

the issue of possible spatial dependency in the disease rates. This approach was further

developed to allow for residual spatial correlation in disease rates through a Bayesian

hierarchical model, but the customised software required for analysis is not easily

available[41]. Glynn and colleagues highlight an application of combining ecological

data with sub-samples in a linear ecological model but it is unclear how the models

would work in non-linear situations[44]. Haneuse proposed an alternative method of

combining ecological data with individual case-only data[45], and also considered

model extensions with spatially unstructured random effects[46] and spatially structured

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random effects[47] between areas. Salway et al, proposed a new model that reduces

within-area variability bias when only small samples of individual covariate data are

available, but it is not clear how the models would work if one were to incorporate

spatial extensions[48].

Jackson provides a neat framework of supplementing aggregate-level data with

individual-level information and performed a comprehensive simulation study to

examine the utility of such approach under various scenarios (e.g. availability of within-

group variability, correlated exposures, interactions, measurement errors and unobserved

confounding)[49]. He also extended the framework to handle more complex situations,

including models with several explanatory variables measured at both the areal and

individual-level[50].

1.3. Sampling techniques and sample size

After describing strategies of including individual-level data in aggregate-level models

in the previous section, we note that little attention has been paid to the issue of

sampling techniques along with the number of samples needed in such a hybrid model to

reduce ecological bias. Suggested options for the size of sample in each group include:

inversely proportional to the expected number of disease, proportional to a large

‘between to within-group variation’ in exposure distribution and also greater asymmetry

in the within-group exposure distribution[41]. Other proposals have included probability

sampling[43], specifying parametric assumptions on the exposures and confounders[51],

and complex non-probability case-control two-stage methods[40].

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Glynn proposes an optimal subsampling design that is conditional on the ecological data

(through stratified random sampling within confounders), and argues that a combined

estimate almost always has less error than the estimate based solely on ecological

data[44]. However, Wakefield suggests a two-stage sampling method that stratifies on a

combination of case/control status and then in the second stage, the exposure or even

confounding variables[52]. Through simulations, Wakefield highlights the efficiency

gained by stratifying on variables which are known to be confounders.

There is currently no consensus on the sample size needed for each areal unit. Based on

simulation studies under a variety of scenarios, Guthrie[41] proposes that a 1% sample

provides a relatively unbiased estimate of the exposure effect. Wakefield suggests at

least 100 in each area, with more required if the within-area distribution of exposure

variable or confounders is skewed[42]. Building on the work of the Prentice and

Sheppard[43], Guthrie performed simulation studies to compare sampled covariate data

(a 10% sub-sample) with the individual and complete aggregate data models[53]. A

sample size of 100 individual observations has also been proposed for the aggregate data

model, with reasonable efficiency[54]. Jackson, et al, makes use of 5-10 individuals

within areas with around 5000 people[49].

In the two-stage sampling design used by Wakefield, equal sample sizes or proportional

sampling have been suggested, with case sample size of 500 and 200 shown to be

reasonably good [52]. The authors also argue that the choice of variables used to stratify

the study in the first stage is more important than the specific sample size or method of

allocating samples. Through simulation studies, Haneuse shows that comparable

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estimates of regression parameters can be obtained regardless of varying sample sizes

within the case-control samples (n=20, 40 and 100)[47]. The variance component

estimates from the hybrid models are also shown to be independent of the sample size.

However, the authors demonstrate that a hybrid scheme that samples case-control

numbers in proportion to the overall numbers in the region provides estimates that are

closer to the individual-level analysis.

Our key aim is to identify the optimal sampling and sample size selection regime

required to choose individual-level data to include with areal-level measures in the

analysis of sparse health outcomes (such as birth defects) and covariates. The objectives

are to reduce the bias associated with sampling, and also to mitigate the computational

problems associated with such a large dataset. This is particularly so, with the increasing

availability of routinely collected large population-based health data, and the

development of health record linkage systems. This large dataset (often millions of

records) requires sampling strategies for computational efficiency, as current software

has limitations in terms of size of dataset and computation time.

This was done in the context of a hybrid model (to be described later) which took into

consideration possible spatial correlation. More importantly, the resulting hybrid model

has been shown to reduce ecological bias. Section 1.1 in this manuscript describes the

issue of ecological bias and the need for combining areal with individual-level

information. Section 2.1 describes the data we used in our analysis and 2.2 outlines the

statistical model we used, with model comparison techniques explained in 2.3. The

simulation studies are described in section 2.4, while the application of the models to

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specific birth defects is shown in section 2.5. Results can be found in section 3,

discussions in section 4 and finally, we provide the concluding remarks in section 5.

2. METHODS

2.1. Data

We analysed birth defect data, grouped in major categories, obtained from

probabilistically linked databases of the New South Wales (NSW) Mothers Data

Collection (MDC) and Birth Defects Register (BDR)[55]. These data have been

geocoded and assigned Statistical Local Area (SLA) groupings using software

developed by NSW Health and the Australian National University and the geocoding

process is described in detail elsewhere [56].

Ethical approval was obtained for the use of the NSW Births Defects Registry data from

the NSW Population and Health Services Research Ethics Committee, and for the study

itself from the University of Sydney Ethics Committee.

The period of study was from 1990 to 2003. We calculated observed and standardised

expected counts of birth defects for the following birth defect categories: nervous

system, cardiovascular system, respiratory system, gastro-intestinal system and genito-

urinary system. Analysis of defects at a category level is useful for policy-makers, who

need such macro-level data for planning purposes. The data was aggregated and

analysed at the SLA level, for which there were 198 SLAs within the study area, NSW.

For the individual data analysis, observations with missing data for the variables

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maternal age, type of hospital admitted to, maternal smoking and maternal diabetes

status were excluded. Also, to ensure that only good quality geocoded addresses were

used in the analysis, we only included address geocoded to exact street addresses, streets

or suburbs.

We used socioeconomic index data from the Australian Bureau of Statistics (ABS) to

describe the level of social and economic well-being in areal-levels of NSW. We used

SLA-level Index of Relative Socioeconomic Disadvantage (IRSD) in our analysis [57].

Higher values reflect less disadvantage and the variables that were used to compute this

index include income, educational attainment, unemployment, and dwellings without

motor vehicles. The data was standardised to have a mean of 1000 units and a standard

deviation of 100. The IRSD data was not available for one SLA, and we imputed the

average value for the entire study region in that SLA.

2.2. Statistical model

A brief outline of the models used in our analysis is provided below. The basic

individual-level model we used was a logistic regression model:

iijijiij zyxit 321)(log , where

ij is the probability of a particular birth defect category for the jth mother in the ith SLA.

i represents the spatially unstructured random effect, representing between-SLA

heterogeneity effects. The covariates xij and yij refer to births in public hospitals and

173

maternal age respectively. Finally, the covariate zi refers to SLA-level values of IRSD.

i is assigned a normal prior distribution with the mean set to a flat hyper-prior and the

variance component (i.e. standard deviation) set to a uniform hyper-prior with a wide

range specified. The coefficients 1 to 3 are assigned weak normal priors, with

precision estimates set, such that odds ratios were expected to vary in the range of 0.2 to

5. An ecological model that incorporates the within-SLA variation in maternal age was

built up as follows. The number of people in a SLA with the particular birth defect (Oi)

was modelled as a Poisson variable

)(~ ii muPoissonO

Before individual-level exposures were measured and conditioned on, each individual in

SLAi had an identical marginal probability of outcome, and this probability is the

integral of the individual’s conditional outcome probability over all exposures with joint

distribution in the ith SLA. Further details of these calculations are explained clearly in

the article by Jackson, et al[49]. We extended the basic model proposed by Jackson to

incorporate the Poisson, instead of the Binomial distribution, and we also created a Zero-

Inflated Poisson (ZIP) extension to account for the excess zeros in the outcome and

possible overdispersion (see Appendix 1 for details). The risk of disease is obtained by

integrating firstly over the binary exposure public hospitals, and then over the

continuous variable mean maternal age, as shown below:

174

iiiii xxmu 21 )1(

iiiiii tau

zbaseE2

)log()log(22

232

iiiiii tau

zbaseE2

)log()log(22

2131

2

1

ii AgeSD

tau

where i refers to the mean maternal age in SLA i, AgeSDi refers to the within-SLA

estimates of the standard deviation of maternal age, 1 to 3 refer to the coefficients of

the three variables, similar to the individual data described above. Ei is the expected

count of birth defects in the corresponding SLA, and the random effect estimate basei is

split into a spatially unstructured and a spatially structured random effect term as shown

below:

iii vbase

where i is formulated as before, and vi is assigned a conditional autoregressive (CAR)

prior for the spatial random effects. The models are applied to both the ecological data

alone and individual data with the ecological data.

),(~],|[ 22, iiuji vNjivv

jijj

jij

i wvw

v1

175

jij

ui w

22

Estimation of the risk in any area is conditional on risks in neighbouring areas.

Subscripts i and j refer to an SLA and it’s neighbour respectively, and j ε Ni where Ni

represents the set of neighbours of region i. There are various methods of assigning

neighbours, and a previous project by the authors has looked at the comparisons[58]. We

chose the Queen adjacency method of assigning neighbours, with weights given by

weightsij (wij)=1 if i,j are adjacent, and 0 otherwise, which allowed us to preserve the

underlying spatial variation in the risks.

2.3. Model comparison

Model comparison was done in several ways. Firstly, we calculated the average mean

squared error (AMSE) for each parameter, using the estimate from all observations as

the referent number[59], along with the associated Monte Carlo standard error of the

AMSE estimate.

N

ri

n

i

riNn

AMSE1

2

1

)( )ˆ(1 ,

N

ri

n

i

ri AMSE

NnNnAMSEse

1

22

1

)( ])ˆ[()1(

1)( ,

Where ri is the log odds parameter estimate from the posterior distribution for the ith

iteration and the rth dataset (r=1 to 20). We also calculated the change in AMSE from

the best model, with the best model defined as the one with the lowest AMSE.

176

2.4. Simulations

We performed a series of simulation studies, which is described in detail below. We

sampled individual-level data, including presence of cardiovascular birth defects

(yes/no), maternal age (in years) and public hospital status (yes/no) according to the

following thirteen different scenarios:

Sampling Technique

Scenario 1- Simple random sample of 20% of SLAs, followed by a random sample of 50

observations within each selected SLA.

Scenario 2- Simple random sample of 40% of SLAs, followed by a random sample of 50

observations within each selected SLA.

Scenario 3- Simple random sample of 60% of SLAs, followed by a random sample of 50

observations within each selected SLA.

Scenario 4- Simple random sample of 80% of SLAs, followed by a random sample of 50

observations within each selected SLA.

Scenario 5- Simple random sample of 80% of SLAs, followed by a random sample of

observations selected proportional to the overall sample size in each SLA, with total

sample size restricted to 8000.

Sample Size

Scenario 6- Simple random sample of 20% of SLAs, followed by a random sample of

100 observations within each selected SLA.

Scenario 7- Simple random sample of 20% of SLAs, followed by a random sample of

150 observations within each selected SLA.

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Scenario 8- Simple random sample of 20% of SLAs, followed by a random sample of

200 observations within each selected SLA.

Case-control

Scenario 9- Simple random sample of 20% of SLAs, followed by all cases in each SLA

and an equal number of controls selected at random (1:1).

Scenario 10- Simple random sample of 20% of SLAs, followed by all cases in each SLA

and twice as many controls selected at random (1:2).

Scenario 11- Simple random sample of 20% of SLAs, followed by all cases in each SLA

and three times as many controls selected at random (1:3).

Scenario 12- Simple random sample of 20% of SLAs, followed by all cases in each SLA

and an equal number of controls selected at random, stratified by maternal smoking

status.

Scenario 13- Simple random sample of 20% of SLAs, followed by all cases in each SLA

and an equal number of controls selected at random, stratified by maternal diabetes

status.

In the event that there are fewer observations than the required sample size in a SLA, all

the observations will be included in the analysis. For each scenario, we created 20

random datasets. For each dataset, we ran a combined ecological and individual

combined model. We ran 10,000 iterations as burn-in and a further 20,000 iterations that

were used in the computation of the estimates from the posterior distribution. The mean

coefficient estimates were calculated for maternal age and public hospital status.

178

2.5. Example

Based on the results from the simulation study, we created models using the optimal

sampling strategy and applied the models to the five different birth defect system-based

categories of nervous, cardiovascular, respiratory, gastrointestinal and genito-urinary

systems.

After discarding the first 10,000 samples as burn-in, we ran 100,000 iterations, which

were used in the calculation of the posterior estimates. Initial analysis showed that the

posterior estimates were correlated, and we used thinning to include every fifth

observation. We then ran two different chains, starting from diverse initial values, and

convergence was assessed using the Gelman-Rubin convergence statistic, as modified by

Brooks and Gelman[60]. Here, posterior estimates were derived for maternal age, public

hospital status, IRSD and fraction of variation explained by the spatially structured

random effects. This fraction is defined as the proportion of total variation in relative

risk that is explained by the spatial random effect.

The Bayesian models were run in WinBUGS (version 1.4.3, Imperial College and

Medical Research Council, UK) through Stata, using a customised ado program file[61].

Data extraction, management, analysis and diagnostics were done in Stata (version 9.2,

Stata Corp, College Station, USA), and all the weight matrices were created using the

GeoDA software (version 0.9.5-I, University of Illinois, USA).

179

3. RESULTS

Table 1 lists the British Paediatric Association International Classification of Diseases,

10th Revision (BPA-ICD10) code for birth defects included in our study, their associated

numbers and the covariates studied. Of the 1,210,850 births recorded and geocoded from

1990-2003, 7922 were dropped because of poor geocoding, and a further 302,190

records excluded due to missing information for maternal, hospital type, smoking status

and maternal diabetes status. Genito-Urinary defects were the most common in the study

period, with 4,265 cases across the 198 SLAs in NSW over the 14 year study period,

followed by Cardiovascular defects (4041) and Gastro-intestinal defects (2736). On the

other hand, there were fewer cases of Respiratory birth defects (357). The crude relative

risks were slightly less than 1, indicating a number of areas with zero counts, and the

large standard deviations indicate possible over-dispersion. The mean individual-level

maternal age was similar to the mean age aggregated at the areal-level (29.5 vs 29.1

years respectively). However, we note that the variation was reduced considerably when

data was aggregated (standard deviation decreasing from 5.5 to 1.5). The percentage of

mothers giving birth at public hospitals was 81%, while at the SLA-level this figure was

88%.

The results of our simulation studies are summarised in Tables 2, 3 and 4. Keeping the

sample size constant per SLA and varying the proportion of SLAs initially sampled,

made little difference to the estimates for both maternal age and public hospital, as

shown in Table 2. Increasing the number of SLAs sampled from 40 (i.e. 20%) to 160

(i.e. 80%) resulted in a change in AMSE from 0.349 to 0.449 for maternal age. Sampling

180

in proportion to the total number of births in each SLA also made little difference, with

an AMSE value of 21.8.

Table 3 gives the results of the next stage of our simulation study, where the proportion

of SLAs sampled in the first step was fixed at 20%, and the sample size was varied in

increments of 50. There were little changes in the AMSE estimates across the various

sample sizes. Increasing the sample size by 4-fold from 50 to 200 resulted in a change in

AMSE value between 0.349 and 0.402 for maternal age, and between 20.7 and 27.9 for

public hospital.

Table 4 shows the results of using a case-control study design, along with a two-stage

step, where we stratify the subjects based on case-control status as well as known

confounders such as smoking and diabetes status. Among all the sampling strategies we

considered, the 1:1 case-control study performed the best for a sparse outcome such as

ours, with the lowest AMSE value of 0.000142 for maternal age and 0.155 for public

hospital. Increasing the number of controls (i.e. 1:2 and 1:3 design) did not materially

improve the performance of the model.

Stratifying the sampling frame further by maternal smoking and diabetes status showed

some interesting results (Table 4). Stratifying by smoking showed no improvement over

the basic 1:1 case-control study design, with an AMSE of 0.000154 and 0.000142 for

maternal age, and 0.157 and 0.155 for public hospital respectively. Stratifying by

diabetes status showed better results, with AMSE values similar to the basic 1:1 case-

181

control study design. The AMSE for maternal age were 0.000149 and 0.146, and this

will be discussed later in the manuscript.

Although the 1:1 case-control study and the design with further stratification by diabetes

status performed similarly well, for reasons of parsimony, we chose to use the basic

case-control design and apply it to the five birth defect categories.

Table 5 shows the relative risks and 95% Confidence Intervals for the results of the

various defect categories. For maternal age and IRSD, we did not find any significant

relationship with the various birth defect categories. Mothers who gave birth in public

hospital had a higher likelihood of having a baby with respiratory birth defect (RR=1.94,

95%CI: 1.15-3.14) as compared to births in private hospital. For respiratory defects,

52% of the total variation in relative risk was due to spatial random effects, as compared

to 93% for cardiovascular system defects.

4. DISCUSSION

We have examined several strategies for incorporating samples of individual-level data

to examine the relationship between key covariates measured on an areal-level such as

socio-economic status, and those measured at an individual-level such as maternal age

on the occurrence of birth defects.

Regression analysis of ecologic data is problematic, especially in instances where there

is limited variation in the covariate[62]. Hence the development of our complex model is

necessary, and a simpler alternative such as aggregating the individual factors to the

182

areal-level is simplistic. A hybrid model incorporating both aggregate and individual-

level data can overcome this problem. This is apparent in our study data when we

aggregated and examined maternal age, for instance, at the SLA-level. We observed that

the interquartile range was between 28.1 and 29.9 years, compared to an individual age

interquartile range of 25.7 to 33.3 years. This also highlights the inadequacy of

standardising for maternal age, which is routinely done in disease mapping studies, as

there is very little variability in maternal age at the aggregate-level.

With the availability of complete data at the individual-level, our reason for sampling

was for computational efficiency as well as to reduce bias associated with sampling. A

typical model with about 7000 individuals takes 3.5 hours to run and that number

increases exponentially if one were to incorporate say, 1 million observations, which is

typically the number of records in a birth registry. The problem is compounded by

including covariates in the model. Also, limitations in the number of observations that

can be handled by the current software inhibit the use of the complete number of

observations available. We postulate that spatial studies involving large registries will

become increasingly common in future, and methods to deal with the issue of

computational efficiency will become increasingly important.

The magnitude of crude relative risks for the birth defects we observed in the study was

very small (most interquartile ranges were between 0.8 and 1.2). This also justified the

combination of areal data with small samples of individual-level data to improve the

power of the study[40]. The use of a hybrid ecological study design in our case was

183

necessitated by the rare but clinically important outcome birth defect, and the

availability of routinely collected data from two large registries.

In this study, we did not examine spatio-temporal models, multivariate CAR models, or

other extensions of the basic spatial CAR model. It has been argued that in terms of bias

in estimates of association between risk of disease and environmental factor, biases

because of data anomalies and confounding take precedence over spatial dependency

issues[40].

In Australia, very little research has been done on the effect of areal measures of socio-

economic status on health. In related work using similar spatial models, our group have

shown the effect of area-level socioeconomic disadvantage on an individual’s risk of

acute coronary syndrome[63], and the birthweight[64].

Simulation studies have shown that the bias in an areal-individual study increases as

between-to within area variance ratio of exposure decreases (poor if ratio less than 0.5,

and ideally should be more than 1), and small sample size of individual

measurements[48]. In our data, the intraclass correlation coefficient of age at the SLA-

level was 0.08 (95% CI: 0.05 to 0.10), justifying the necessity of using individual-level

measurements of age. In contrast, the intraclass correlation for IRSD was a moderate

0.393 (95% CI: 0.320 to 0.466) indicating the suitability of IRSD as an appropriate areal

measure.

184

Our finding that a simple case-control study design to select individual samples for

inclusion in a hybrid model, warrants further investigation. It is intuitively clear that

stratifying the sampling frame first by case-control status, and then selecting the samples

in each strata is a much better strategy than the simple random sample approach, as our

dataset involves a sparse disease outcome. With simple random sampling, the births with

defects will stand a lower chance of being selected for inclusion. Further stratification by

potential confounders only worked for some variables like diabetes status, but not for

others like smoking. We explored the reasons for this by analysing the relationship

between diabetes and smoking with cardiovascular birth defects in our dataset. We

found that smoking was not related to the outcome, with an odds ratio of 1.0004 (95%

CI: 0.92-1.08). Maternal diabetes status, on the other hand, was strongly associated with

cardiovascular birth defects in our cohort, with an odds ratio of 3.97 (95% CI: 3.13-

5.03). We argue that further stratification by variables is only appropriate if the variable

is a strong confounder (known risk factor) in the dataset. Even then, the effect is still

marginal, as most of the benefit is derived from the originally splitting the sampling

frame using the 1:1 case-control study design. In addition, stratifying by a confounder

means we will not be able to study the effects of the variable in the model.

A key limitation of our analysis would be the use of individual type of hospital admitted

to (i.e. public versus private). Although this would be the closest proxy measure to

individual socio-economic status available to us in the dataset, it’s actual meaning is

unclear. The assumption that all women who give birth at public hospitals don’t have

private health insurance is problematic[65]. Women with private insurance and a baby

prenatally diagnosed with a major birth defect may choose to deliver their baby at a

185

public hospital with specialist facilities for treating their child. This could partly explain

the increased risk for public hospital births that we found in our study.

Further research is needed to tease out the intricacies of this variable’s meaning. The

role of private health insurance on the healthcare system of other countries may also be

very different, and this will affect the generalizability of the results. In our study though,

we feel the comparisons across the competing models still hold, regardless of how we

choose to interpret the variable in question, as the same variable was used for comparing

across models.

5. CONCLUSION

We found that an ecological , careful sampling of subgroups via a case-control sampling

design can help us overcome the computational issues of spatial modelling of large

datasets, while reducing bias in coefficient estimates to a reasonable amount.

ACKNOWLEDGEMENTS

The authors acknowledge the support of the Australian Research Council Linkage

Grant LP0348628, the North Coast Area Health Service, the NSW Department of

Health, the Commonwealth Department of Health and Ageing, Dr Lee Taylor and

Douglas Lincoln, Paul Houlder and Richard Summerhayes.

186

APPENDIX 1: WINBUGS CODE FOR MODEL (INCORPORATING AGGREGATE WITH INDIVIDUAL-LEVEL DATA)

model { ## ecological data for (i in 1:198) { ## The number of individuals with the outcome at sla-level is modelled as a Poisson: O[i] ~ dpois(mu[i]) ## This risk is obtained by integrating the individual-level risk, firstly over the binary exposure: public hospital. see Jackson, page 6. mu[i] <- public_hospital[i] * pni[i] + ( 1 – public_hospital[i] ) * pi[i] ## and secondly over the continuous variable, mean age. log(pi[i]) <- ( log(E[i]) + mu.base[i] + a.irsd*irsd[i] + a.age*meanage[i] + (pow(a.age, 2) / (tau.x[i]*2))) log(pni[i]) <- (log(E[i]) + mu.base[i] + a.irsd*irsd[i] + a.public_hospital + a.age*meanage[i] + (pow(a.age, 2) / (tau.x[i]*2))) tau.x[i] <- 1 / pow(agesd[i], 2) ## The risk includes a random area-specific effect mu.base[i], to account for spatial heterogeneity between areas due to unobserved factors. mu.base[i] <- U[i] + V[i] } U[1:198] ~ car.normal(adj[], weights[], num[], tau.u) for(i in 1:198) { V[i]~dnorm(alpha,tau.v) } ## Weights for(k in 1:960) { weights[k]<-1 } ## Model for outcome in the individual-level data is just a logistic regression with the same coefficients for (i in 1:m) { outcome[i] ~ dbern(pzy[i]) logit(pzy[i]) <- mu.base[sla[i]] + a.irsd*irsd[sla[i]] + a.public_hospital * public_hospital[i] + a.age *age[i] } ## Log odds ratios for area-level irsd, public hospital and age a.irsd ~ dnorm(0, 1.48) ## 95% prior belief that OR(a.public_hospital) is between 1/5 and 5

187

a.public_hospital ~ dnorm(0, 1.48) a.age ~ dnorm(0, 1.48) ## Odds ratios for area-level irsd, public hospital and age or.public_hospital <- exp(a.public_hospital) or.age <- exp(a.age) or.irsd <- exp(a.irsd) alpha~dflat() tau.u<-sigma.u*sigma.u sigma.u~dunif(0.1, 20) tau.v<-sigma.v*sigma.v sigma.v~dunif(0.1, 20) variance.U<-sd(U[])*sd(U[]) # variance spatial component variance.V <- sd(V[])*sd(V[]) # variance non-spatial component fraction<- variance.U/(variance.U + variance.V) # fraction of total variation in logRR due to spatial effects }

188

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Table I. Descriptive summary of birth defect categories and covariates (1990-2003)

Birth Defect Category Number of cases

Crude RR

(SD)

BPA-ICD10

Code

Nervous System 1332 0.93 (0.92) 740-742999

Cardiovascular System 4041 0.95 (0.51) 745-747999

Respiratory System 357 0.89 (1.34) 748-748999

Gastro-intestinal System 2736 0.93 (0.63) 749-751999

Genito-Urinary System 4265 0.94 (0.58) 752-753999

Covariates

Mean individual age (SD) 29.5 years (5.5)

Mean age at SLA-level (SD) 29.1 years (1.5)

Mothers giving birth at a public hospital (n (%)) 728,970 (80.9%)

Mothers giving birth at a public hospitals at the

SLA-level (mean percent) 88%

Mean IRSD at SLA-level (SD) 997 (54)

BPA-ICD10: British Paediatric Association International Classification of Diseases, 10th Revision

193

Table II. Results of simulation study comparing different proportion of SLAs sampled

Sampling

technique

Sample size per

SLA Covariate AMSE

Std. Error

(AMSE) Change

SRS 20% 50 Maternal age 3.49E-01 6.56E-05 3.49E-01

Public hospital 2.07E+01 1.02E-02 2.05E+01

SRS 40% 50 Maternal age 4.02E-01 6.34E-05 4.02E-01

Public hospital 2.73E+01 8.59E-03 2.72E+01

SRS 60% 50 Maternal age 4.32E-01 6.36E-05 4.32E-01

Public hospital 2.88E+01 7.27E-03 2.86E+01

SRS 80% 50 Maternal age 4.49E-01 4.76E-05 4.49E-01

Public hospital 2.90E+01 5.80E-03 2.88E+01

SRS 80% In proportion* Maternal age 5.25E-01 1.34E-04 5.25E-01

Public hospital 2.18E+01 6.63E-03 2.17E+01

* In proportion to total births in each SLA

194

Table III. Results of simulation study comparing different sample sizes per SLA while keeping the

proportion of SLAs sampled fixed

Sampling

technique

Sample size per

SLA Covariate AMSE

Std. Error

(AMSE) % Change

SRS 20% 50 Maternal age 3.49E-01 6.56E-05 3.49E-01

Public hospital 2.07E+01 1.02E-02 2.05E+01

SRS 20% 100 Maternal age 3.69E-01 1.36E-04 3.69E-01

Public hospital 2.56E+01 9.07E-03 2.55E+01

SRS 20% 150 Maternal age 3.88E-01 1.34E-04 3.88E-01

Public hospital 2.76E+01 9.77E-03 2.74E+01

SRS 20% 200 Maternal age 4.02E-01 1.36E-04 4.02E-01

Public hospital 2.79E+01 8.72E-03 2.77E+01

195

Table IV. Results of simulation study comparing case-control study design along with

stratification by key covariates

Sampling

technique

Sample size per

SLA Covariate AMSE

Std. Error

(AMSE) % Change

SRS 20%** 1:1 case-control Maternal age 1.42E-04 3.34E-07 0.00E+00

Public hospital 1.55E-01 1.63E-04 0.00E+00

SRS 20% 1:2 case-control Maternal age 3.04E-04 7.53E-07 1.62E-04

Public hospital 3.13E-01 3.26E-04 1.58E-01

SRS 20% 1:3 case-control Maternal age 5.16E-04 1.11E-06 3.74E-04

Public hospital 5.25E-01 4.95E-04 3.70E-01

SRS 20%

1:1 case-control and

stratified by

smoking status Maternal age 1.54E-04 4.27E-07 1.24E-05

Public hospital 1.57E-01 1.81E-04 2.35E-03

SRS 20%

1:1 case-control and

stratified by

diabetes status Maternal age 1.49E-04 3.04E-07 7.10E-06

Public hospital 1.46E-01 1.71E-04 -8.57E-03

** Best model with the lowest AMSE

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Table V. Application of optimal model to birth defect categories

System Covariate RR 95% CI

Cardiovascular Age 1.01 0.99 1.03

IRSD 1.00 1.00 1.00

Public hospital 1.16 0.94 1.43

Fraction 0.93 0.73 0.99

Gastrointestinal Age 1.01 0.99 1.03

IRSD 1.00 1.00 1.00

Public hospital 0.90 0.71 1.18

Fraction 0.56 0.04 0.96

Genitourinary Age 1.00 0.98 1.01

IRSD 1.00 1.00 1.00

Public hospital 1.20 0.97 1.46

Fraction 0.74 0.06 0.98

Nervous Age 0.99 0.96 1.02

IRSD 1.00 1.00 1.00

Public hospital 1.32 0.92 1.87

Fraction 0.83 0.10 0.99

Respiratory Age 1.03 0.98 1.07

IRSD 1.00 1.00 1.00

Public hospital 1.94 1.15 3.14

Fraction 0.52 0.05 0.98

197

CHAPTER 9: FORECASTING BIRTH DEFECTS AT THE SMALL AREA LEVEL

Forecasting birth defects at the small-area level in New South Wales, incorporating spatial correlation and changes in demography Arul Earnesta, Kerrie Mengersenb, Geoff Morgana,c, John Beardd. a Northern Rivers University Department of Rural Health, The University of Sydney, b

Faculty of Science, Queensland University of Technology, Queensland, c Population Health & Planning, North Coast Area Health Service, New South Wales, d Department of Ageing and Lifecourse, World Health Organisation, Geneva, Switzerland.

Correspondence: Arul Earnest, Northern Rivers University Department Of Rural Health, 55 Uralba Street, Lismore, NSW, Australia 2480. E-mail: arul_earnest@hotmail.com Keywords : Forecasting, CAR model, Disease mapping, Spatial analysis, Birth defects Research Manuscript submitted to BMC Health Services Research Journal on 1 Oct 2009

198

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execution, or interpretation, of at least that part of the publication in their field of expertise;

2. they take public responsibility for their part of the publication, except for the responsible author who accepts overall responsibility for the publication;

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In the case of this chapter:

Contributor Statement of contribution

Arul Earnest Conceptualised the entire study, performed data management, wrote computer codes for the analysis, performed literature review and statistical analysis and wrote the manuscript for publication

23 February 2010

Geoff Morgan

Contributed to writing of manuscript

Kerrie Mengersen

Contributed to writing of manuscript

John Beard

Contributed to writing of manuscript

Principal Supervisor Confirmation I have sighted email or other correspondence from all Co-authors confirming their certifying authorship. _______________________ ____________________ ______________________ Name Signature Date

199

Abstract

Background

There are several complications in forecasting birth defects at a small-area level. One is

the sparseness and excess zeros in the outcome variable. Possible spatial correlation in

the risk between neighbouring regions, as well as demographic changes over time needs

to be considered.

Methods

We used a bayesian spatio-temporal model, along with population forecasts to provide

predictions of birth defect categories from the years 2001 to 2030. The models were

amended to incorporate sparseness in the data and applied to a sample of sixteen

randomly selected SLAs, representing the various levels of socio-economic status and

levels of remoteness in NSW.

Results

We found considerable spatial variation in the projected counts of birth defects in NSW,

Australia. Even in regions which showed negligible risk compared to other areas (e.g.

SLA 11), we projected the actual numbers of cardiovascular defects to increase from 29

in the year 2001 to 48 in the year 2031. Similar patterns were seen for gastrointestinal

defects. We also highlighted how our model projected more realistic forecasts for birth

defects as compared to traditional methods of forecasting (e.g. Autoregressive

Integrative Moving Average (ARIMA) models). Sensitivity analyses around the

200

assumptions for the population forecasts showed that our projections were robust across

a range of scenarios.

Conclusion

We have proposed an innovative method of forecasting birth defects at a small area

level, which addresses several inefficiencies of earlier models. This method will prove

useful for policy-makers in terms of planning purposes.

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Introduction

Statistical models to forecast occurrence of disease counts, at a small area level, can be

useful tools for public health professionals and policy makers. In particular, these

models can help with healthcare resource distribution and planning for healthcare

workers and health-related facilities like hospitals and clinics at a regional level. There

are several foreseeable challenges to developing such a model, especially for sparse

disease counts such as birth defects. Firstly, the disease is rare, making the process of

forecasting challenging because of the small numbers encountered. Specifically,

forecasting rare outcomes often provide imprecise estimates with wide prediction

intervals. This problem is exacerbated when forecasting is done in finer small area

levels, further reducing the numbers. These statistical models for areal-level data should

also take into account correlations in the outcome variable between neighbouring

regions, so as to strengthen the predictive ability of the models, by borrowing strength

from their neighbours.

Generally, statistical models that have previously been developed and used to forecast

diseases fall into two broad categories. In the first category, historical trends and

correlations in the disease counts are often used to extrapolate what we would expect to

see in the future. One common class of models would be the Autoregressive Integrative

Moving Average (ARIMA) models. The ARIMA models require the specification of

the autoregressive terms (e.g. how the observation at time t and time (t-1) are related, in

addition to the moving average term, and whether the series needs to be differenced.

Examples when they have been used in this field include the assessment of seasonal

variation in selected medical conditions[1], and as a surveillance tool for outbreak

202

detection[2], and to forecast attendance at accident and emergency departments in the

United Kingdom. Researchers have shown that the forecasting methodology can be

improved by incorporating the ARIMA method[3]. More generally, time series analysis

has been used for assessing predictive performance in terms of modelling respiratory,

gastrointestinal and neurological complaints in emergency department visits, specifically

in terms of indicating subsequent outbreaks[4]. Time series analysis within the Bayesian

framework has also been performed. One such analysis makes use of bayesian analysis

to demonstrate that Klebsiella pneumonia is related to third generation antibiotic use

(cephalosporin) in hospital, with a lag of three months[5]. Others include a bayesian

hierarchical time series model to detect outbreaks of Rubella and Salmonella

infections[6] and a two-component model to incorporate both seasonal and epidemic

characteristics of notifiable infectious diseases[7].

One major limitation of time series models are that they provide reasonable forecasts for

only short periods of time. For long-term projections, they often converge towards the

mean values of past observations. Confidence intervals for the predictions also get large

early in the predictions, and more importantly time series models do not take account of

changes in population demographics or spatial autocorrelation.

The second category of models used for disease projections includes some form of

covariate data (e.g. population counts, age-sex stratified population, projections of risk

factors, etc) in the calculations and projections. These models require some knowledge

on the future changes in health systems and other population dynamics. One of the more

common classes of models includes the Age-Period-Cohort analysis[8-10], where the

203

rates of disease are broken down into age, period and birth cohort effects. The traditional

age-period-cohort models may well improve fit of the models, but often result in erratic

projections because of the large random variation in the parameter estimates[11] and

also require a more restrictive linear trend assumption. More complex polynomial age-

period-cohort models have been shown to produce wide prediction intervals, which

render them ineffective[12].

The Functional Data Analysis (FDA) approach, which has traditionally been used to

forecast cancer mortality, has been used recently in Australia to forecast expected

number of new major cancer (incidence) cases from 2002 to 2011[13]. This method is

known to be better than previous approaches, which includes projecting age-specific

rates using separate models for each age-group, as the latter is more volatile for small

counts of disease. Other methods include the combination of incidence data (using

Poisson models) and a prevalent component (using the Markov model) such as in the

projection of end-stage renal disease projections for Canada between 1995 and

2005[14].

Dyba and colleagues compared a model based on the observed number of cases (using a

Poisson model) with models based on age-adjusted and age-specific rates. Using

simulation studies, they found that the Poisson method provided the smallest coverage

error for the prediction interval, along with high precision[12]. Variations of this Poisson

model were subsequently used by researchers in Netherlands to predict counts of skin

cancer up to the year 2015[15], and by researchers in Northern Ireland to study cancer

death rates from 1984 to 2004[11].

204

Aim

Building upon our earlier research, we aim to provide a Bayesian method of forecasting

disease counts to address some of the limitations of current methods. A comprehensive

comparison of the various alternative prediction models is beyond the scope of this

paper.

Methods

Data

We analysed birth defect data, grouped in major categories, obtained from the

probabilistically linked New South Wales (NSW) Midwives Data Collection (MDC) and

Birth Defects Register (BDR) databases. Details of the registries can be found here[16].

The births and birth defect data have been geocoded using software developed by NSW

Health and the Australian National University and the geocoding process is described in

detail elsewhere[17]. Ethical approval was obtained for the use of the data from the

NSW Population and Health Services Research Ethics Committee, and for the study

itself from the University of Sydney Ethics Committee.

We calculated standardised expected counts for two main classes of birth defects

organised by body systems, namely cardiovascular defects and gastrointestinal defects.

For example, the expected count of cardiovascular defects in each Statistical Local Area

(SLA) was defined as Ei = (Birthsi/Totalbirths) * Totalcardios, where Birthsi refers to

total births in the ith SLA, and Totalbirths and Totalcardios refer to the overall number

of births and cardiovascular defects in the NSW study region for that particular time-

period. These birth defects were studied because they were the more common defects

205

reported in Australia, and they also covered a spectrum of body systems. In addition,

policy-makers are usually interested in natural groupings of the birth defects rather

studying them individually. The ratio of the observed over the expected values provides

us with estimates of the relative risk of the particular categories of birth defects.

Births and birth defects within the period 1990 to 2003 were included in the analysis.

The data was aggregated and analysed at the Statistical Local Area (SLA) level, for

which there were 198 SLAs defined within the NSW study area. Also, to ensure that

good quality geocoded addresses were used, we only incorporated address geocoded to

exact street addresses, streets or suburbs.

We performed spatio-temporal modelling on all the SLAs, but for ease of presentation,

we forecasted birth defects for 16 selected SLAs, representing a range of socio-

economic status and remoteness background. The socioeconomic index data from the

Australian Bureau of Statistics (ABS) was used to describe the level of social and

economic well-being in areal levels of NSW. This data, along with the information paper

is readily available from the ABS website[18]. We used the Index of Disadvantage

(IRSD) in our analysis. Higher values reflect lack of disadvantage and the variables that

were used to compute this index include income, educational attainment, unemployment,

and dwellings without motor vehicles.

We made use of the ARIA plus (Accessibility/ Remoteness Index of Australia) index,

which was developed by researchers from the National Centre for Social Applications of

Geographic Information Systems (GISCA), University of Adelaide. The index is

206

calculated from road distances from populated localities to the nearest service centres in

five categories. The values were then spatially interpolated at 1 square km grids, which

allowed for mean levels to be calculated for the Collection District (CD) levels, and

hence for each of the other higher hierarchical levels of the ASGC. ARIA+ values

ranged from 0 (high accessibility) to 15 (high remoteness), and we made use of the

following classification system used by the ABS[19].

Based on a cross-tabulation of the IRSD and ARIA+ levels, we selected SLAs from a

range of diverse socio-economic and remoteness classification (see table 1). The aim is

to project birth defects for a sample of SLAs which are representative of NSW in terms

of socio-economic status and level of remoteness. We used the 5-year interval forecasts

of women for each SLA, which was available from the NSW Department of

Planning[20]. The methods used for projecting populations in SLAs within metropolitan

NSW was different from that used to project SLAs outside of metropolitan NSW, but

both methods used a common starting point of estimated resident population (ERP) as at

30 June 2004, stratified by single years of age and sex. We used the projections that

were done by the Department of Planning, which are preferred by NSW state agencies.

Detailed information on the methods used in the calculation can be found in the

following document[21].

We calculated birth defect forecasts in each SLA, for the years 2001 to 2031 in the

following stages:

Step 1: Perform Bayesian spatio-temporal smoothing of relative risk of defects, with

data split into 2 year intervals from 1990 to 2003.

207

Step 2: Tabulate the projected number of women in each SLA, in 5 year breaks, from

2001 to 2031.

Step 3: Calculate the projected number of births in each SLA, in 5 year breaks, using the

proportion of births in each SLA in 2001.

Step 4: Use the smoothed relative risk estimates to calculate the number of defects in

each SLA in 2001.

Step 5: Calculate the projected increase in number of defects from 2001 to 2031, using

the rate of yearly increase in births in each SLA as reference.

Step 6: Perform a sensitivity analysis for NSW as a whole, based on different population

scenarios. The scenarios make certain assumptions about net fertility rate, life

expectancy, internal and overseas migration, etc.

For ease of presentation, we highlight forecasts for 16 randomly selected SLAs

representing the various levels of ARIA-plus and IRSD levels.

Statistical Model

The formulation of the CAR model used in our analyses is shown below:

)(~ ikik PoiO

221 **)log()log( kikiiiikik ttvuE

where Oik and Eik are the observed and expected birth defects for a SLA in the ith region,

i=1,2, …,198 and kth time period, k=1,2,…,7, ui is a spatially structured random effect

and vi is a spatially unstructured random effect and a quadratic temporal random effect

term to capture time trends. This model is an amended version from Bernardinelli[22],

with the main difference being the exclusion of the space-time interaction random effect

208

term. In addition to the basic Poisson model, we added a component to capture excessive

zero counts of defects in the data (see appendix 1 for details). The spatially structured

random effect was assigned a CAR prior, as shown below:

),(~],|[ 22, iiuji vNjivv

jijj

jij

i wvw

v1

jij

ui w

22

Estimation of the risk in any area is conditional on risks in neighbouring areas.

Subscripts i and j refer to an SLA and it’s neighbour respectively, and j ε Ni where Ni

represents the set of neighbours of region i. There are various methods of assigning

neighbours, and we compared alternative methods in a previous project[23]. We chose

the covariate method of assigning neighbours. Specifically, we provided for more

weights when two SLAs had similar mean maternal age values, as this method seems to

explain away the spatial variation in relative risks better than the other schemes of

neighbourhood assignment[23]. Alternate weight matrices such as socio-economic status

of the SLAs can also be considered, but this is not within the scope of this manuscript.

The priors for the means were set to a normal distribution, with standard deviation set to

cover a wide range of values, whereas the priors for the standard deviations of the

precision estimates were set to a uniform distribution with a wide yet plausible interval.

209

We discarded the first 40,000 samples as burn-in. In initial analyses, we noticed that the

samples from the posterior distribution were highly correlated, and so we ran a further

200,000 iterations and selected every fifth sample, which were used in the calculation of

the posterior estimates. The iterations resulted in Monte Carlo standard errors of less

than 1% of the posterior standard deviation for the parameters that we were interested in.

Estimates for the smoothed relative risk, temporal trend, spatially structured random

effect, spatially unstructured random effect and their corresponding 95% credible

intervals were derived from the posterior distribution.

We compared the Bayesian approach described above with an alternative approach using

ARIMA models to predict future birth defects. We used selected ARMIA models of

annual cardiovascular defects for SLA `X’ from 1990 to 2003, and used the ARIMA

model results to calculate yearly projections up till the year 2031. The basic ARIMA

model has the following three parameters: 1) Autoregressive (AR) term, 2) Moving

average (MA) term and 3) Differencing (D) term. The AR term relates the observation

made at year t to the previous year (t-1) or earlier years. The MA term relates the error

(defined as the difference between observed and predicted defects) at year t to the

previous year (t-1) or earlier year. The D term allows one to model the differenced series

(i.e. t minus 1t ) in the event of non-stationarity in the time series.

Data extraction, management and analysis were performed in Stata (version 9.2, Stata

Corp, College Station, USA). The Bayesian models were run in WinBUGs (version

1.4.3, Imperial College and Medical Research Council, UK) through Stata, using a

210

customised ado program file[24]. All the weight matrices were created using the GeoDA

software (version 0.9.5-I, University of Illinois, USA). The ARIMA models were run

through Stata as well. A copy of the WinBUGS code we used for the analysis is

provided in appendix 1.

Results

Of the 1,306,679 births and 28,844 births with at least one birth defect in our database,

1,287,460 births and 27,899 births with defects were assigned an ASGC unit. After

excluding imprecisely mapped addresses and data falling outside of our study period, we

were left with a total of 1,202,928 births and a total of 6263 cardiovascular and 4153

gastrointestinal defects in our study period. The percentage of SLAs with no

cardiovascular defects ranged from 30% in 1996-1997 to 38% in 2002-2003 (table 2).

As for gastrointestinal defects, the percentage with no defects was higher (ranging from

34% to 45%). The mean expected count for gastrointestinal defects was also

considerably lower than cardiovascular defects.

We found considerable spatial variation in the smoothed relative risk of birth defects

across the various SLAs. The smoothed relative risk of cardiovascular defects for ‘SLA

4’ was 6.42, but in terms of absolute numbers, the model projected a constant 2 new

cases every five years from 2001 to 2031 (table 3). On the other hand, ‘SLA 11’ had

negligible risk compared to overall regions, but in terms of absolute numbers, we

projected an increase in incident numbers between 29 in 2001 to 48 in 2031. Similarly,

for gastrointestinal defects, our model projects certain regions such as ‘SLA 11’, ‘SLA

13’ and ‘SLA 14’ to show greater increase in absolute numbers of defects, as compared

211

to other SLAs (table 4). Figures 1 and 2 highlight the projection of cardiovascular and

gastrointestinal defects for selected regions. Table 5 lists the associated uncertainties in

the projection estimates for these selected regions.

Figure 3 shows the temporal trend in the relative risk estimates for cardiovascular

defects for the entire study region. As we can see, there is spatial variation in the

temporal trends. The relative risk of cardiovascular defects increased by a factor of 1.11

in Inverell SLA for every 2 year increase, while for Hawkesbury, we see the risk of

defects decrease by a factor of 0.981 for every 2 year increase.

Our sensitivity analysis for the whole of NSW (figure 4) showed that the projections

were robust across a range of assumptions (i.e. 19 different scenarios). The various

scenarios considered relaxation of the assumptions made in the model. For instance,

scenario D assumed replacement fertility and made the assumption that total fertility for

New South Wales increases to 2.11 children per women by 2014, then remains constant.

Under this scenario, we found that the projected number of defects was highest with

16216 cardiovascular defects by 2031. On the other hand, if one were to consider zero

‘all migration’ (scenario S), where all migration flows, overseas, interstate and intrastate

are set to zero, the model predicts a more conservative 11,780 cardiovascular defects in

New South Wales. Table 4 in appendix 2 provides details on the assumptions made

under the various scenarios.

As a simple form of comparison, we evaluated how the projections would look like if we

ignored all the spatial correlations and demographic changes and examined just the

212

serial trends within each SLA, using the traditional ARIMA model. When we ran 5

different ARIMA models for ‘SLA 11’, we found that the predictions generally

converged to a mean value that was dependent on the previous observations (see figure

5). This finding is discussed later in the manuscript.

Discussion

In this study, we have found that demographic changes at the SLA level is going to be a

major contributor of increased number of selected birth defects, after accounting for

spatial correlation and sparseness in the data. There is spatial variation in the absolute

number of birth defects across the SLAs studied. Certain SLAs are going to be faced

with a significantly greater increase in the number of birth defects as compared to other

SLAs. Our sensitivity analysis also shows that the findings are generally robust to many

of the assumptions upon which the demographic projections are based upon.

Our method of using population projections in conjunction with rates of disease to

develop forecasts of disease is not something new altogether. The prevalence of diabetes

cases has been projected in the United States from 2000 to 2050 using such a

technique[25]. In Australia, population projections, stratified by sex/ age-group, were

used alongside two different statistical models: demographic change model and the trend

model to examine future bed-day utilisation in Australia from 2005 to 2050. The trend

model used was based on a power function[26].

Bayesian spatio-temporal models have been extensively used in the disease mapping

literature[27-31], but few have extended it’s application to forecasting disease counts.

213

Most of the published studies have used non-spatial models and have focused on

relatively non-rare disease outcomes like cancer. Our contention that the spatio-temporal

model is favourable to other existing forecasting models, follows from a recent

study[32] that specifically compared four different models: 1) a quadratic time series

model, 2) a state-space method, 3) piece-wise linear regression method, and 4) semi-

parametric Dirichlet process method against a spatio-temporal model. In that particular

paper, the authors used a spline-based approximation for spatial and temporal

autocorrelation, instead of the usual Conditional Autoregressive (CAR) model. Several

reasons have been postulated to explain why the spatio-temporal model performed better

than conventional models, including its ability to borrow strength from regional

neighbours.

In our study, we have examined ARIMA models as an alternative method of forecasting

disease counts at a small area level, and shown their deficiency in terms of long-term

forecasting. In particular, the ARIMA models we used (regardless of the autoregressive

and moving average terms specified) projected an unrealistic static value of 12 for the

SLA that we studied. The ARIMA model also fails to account for demographic changes

as well as utilise information on the spatial correlation in rates between neighbouring

SLAs.

In NSW, the provision of health services is overseen by the department of health, but

administratively run by eight Area Health Services (AHS) responsible for health care

delivery in a wide range of settings. Four of the AHS are within the metropolitan region,

whilst the other four cover the rural and remote regions of NSW. The organisation of

214

healthcare in geographical regions points to the need for regional (as opposed to state-

wide or national) projections of disease counts such as birth defects. This can help local

health authorities to better respond to regional variations in disease with geographically

targeted health service delivery and programs.

In our analysis, we grouped data into 2 year time periods from 1990 to 2003. This

resulted in a total of 7 temporal units. We chose a 2 year time period instead of annual

periods to ensure that there were enough cases to provide for a robust prediction model,

and also for computational efficiency, as WinBUGs software takes a longer time to run

for larger datasets. We also postulate that temporal trends for birth defects do not change

drastically over a period of one year. The other point we would like to discuss is keeping

identity of SLAs anonymous, in the forecasting of birth defects. The main reason for

doing so was not to create any anxiety in interpretation of results for those SLAs that

show a larger projected increase. It is also not the aim of this thesis to produce any form

of comparison between SLAs (e.g. in the form of league tables).

Conclusion

We have presented an innovative method of providing long-term forecasts of rare

disease outcome outcomes at a small-area level. This involves modifying the usually

applied Conditional Autoregressive Model to incorporate excessive zeros in the

outcome, applying a weighting scheme (via modelling maternal age as weights) to

maximise the model’s ability to explain spatial variation in the relative risk, and

subsequently incorporating future projections of demographic changes. We used the

model to project selected birth defect categories in NSW at the SLA level up to the year

215

2031. These projections will help policy-makers immensely in their healthcare planning

work, as currently, there is limited information in this area.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgements

The authors acknowledge the support of the Australian Research Council Linkage

Grant LP0348628, the North Coast Area Health Service, the NSW Department of

Health, the Commonwealth Department of Health and Ageing, Douglas Lincoln, Paul

Houlder, Deborah Donoghue, Therese Dunn, Richard Summerhayes, Danielle Taylor,

Dr Lee Taylor and Dr David Muscatello.

Legend of tables

Table 1. List of regions selected by remoteness and IRSD status

Table 2. Descriptive statistics on birth defects across the study period

Table 3. Projected number of cardiovascular defects up till the year 2031

Table 4. Projected number of gastrointestinal defects up till the year 2031

Table 5. Projected birth defects and associated credible intervals for the estimation

216

Legend of figures

Figure 1. Projected number of cardiovascular defects for selected regions

Figure 2. Projected number of gastrointestinal defects for selected regions

Figure 3. Temporal trend of relative risk for cardiovascular defects from 1990 to 2003

Figure 4. Cumulative projection of cardiovascular defects (Scenarios A to S)

Figure 5. Observed and forecasted cardiovascular defects for SLA ‘X’ using ARIMA

models

217

Appendix 1. WinBUGs Code for model used in the analysis

model { for (i in 1:N) { for (k in 1:7) { # Poisson likelihood for observed counts with Zero Inflated Poisson extension zeros[i,k] <- 0 zeros[i,k] ~ dpois(mu[i,k]) mu[i,k]<-(1-step(O[i,k]))*(-log(1-p[i,k] + p[i,k]*exp(-lambda[i,k]))) + (step(O[i,k]-1))*(-(log(p[i,k])-lambda[i,k] + O[i,k]*log(lambda[i,k])-logfact(O[i,k]))) logit(p[i,k]) <- alpha0[k] log(lambda[i,k])<-log(E[i,k])+b.spat[i]+b.ind[i] +beta1*t[k]+ beta2*t[k]*t[k] +delta[i]*t[k] # Relative Risk in each area and period of time RR[i,k]<-exp(b.spat[i]+b.ind[i] +beta1*t[k]+ beta2*t[k]*t[k] +delta[i]*t[k]) } RRarea[i]<-exp(b.spat[i]+b.ind[i]+beta1*t[7]+ beta2*t[7]*t[7] +delta[i]*t[7]) TT[i]<-exp(beta1 + beta2 +delta[i]) PP1[i]<- step(RRarea[i]-1+eps) PPT[i]<- step(TT[i]-1+eps) } eps<-1.0E-3 # CAR prior distribution for spatial correlated random effects: b.spat[1:N]~car.normal(adj[],weights[],num[],v.spat) delta[1:N]~car.normal(adj[],weights[],num[],v.delta) # Normal prior distribution for uncorrelated random effects for(i in 1:N){ b.ind[i]~dnorm(alpha,v.ind)} alpha0[1] ~ dnorm(alpha,0.0001) alpha0[2] ~ dnorm(alpha,0.0001) alpha0[3] ~ dnorm(alpha,0.0001) alpha0[4] ~ dnorm(alpha,0.0001) alpha0[5] ~ dnorm(alpha,0.0001) alpha0[6] ~ dnorm(alpha,0.0001) alpha0[7] ~ dnorm(alpha,0.0001) for(i in 1:N){ O[i,7]~dnorm(alpha, 0.0001) } alpha~dflat() # Hyperprior distributions on inverse variance parameter: beta1~dnorm(alpha,1.0E-5) beta2~dnorm(alpha,1.0E-5) v.ind<-1/(sigma.ind*sigma.ind) v.spat<-1/(sigma.spat*sigma.spat) v.delta<-1/(sigma.delta*sigma.delta) sigma.ind~dunif(1.0E-5, 10) sigma.spat~dunif(1.0E-5, 10) sigma.delta~dunif(1.0E-5, 10) sigma2.ind<-sd(b.ind[])*sd(b.ind[]) # variance sigma2.spat <- sd(b.spat[])*sd(b.spat[]) # variance sigma2.delta <- sd(delta[])*sd(delta[]) # variance fraction<-sigma2.spat/(sigma2.spat + sigma2.ind) # fraction of total variation in logRR due to spatial effects }

218

Appendix 2. Description of the various assumptions for scenarios A to S

Scenario Assumptions

A - Preferred series

Total fertility rate declines to 1.66 by 2014, then constant. Life expectancy at birth increases to 88.0 years for males and 91.3 years for females by 2051. Net internal migration in the long term of about minus 16,000 per year. Net overseas migration of 42,000 per year to New South Wales (100,000 to Australia).

B - Constant fertility Total fertility rate for New South Wales remains constant at 1.77 children per woman

C - Low fertilityTotal fertility rate for New South Wales declines to 1.4 children per woman by 2014, then remains constant.

D - Replacement fertility Total fertility rate for New South Wales increases to 2.11 children per woman by 2014, then remains constant.

E - High life expectancy Life expectancy at birth in New South Wales increases to 91.9 years for males and 95.0 years for females in 2051.

F - Low life expectancy Life expectancy at birth in New South Wales increases to 84.0 years for males and 87.6 years for females in 2051.

G - Constant life expectancy Life expectancy at birth in New South Wales remains constant at 77.2 years for males and 82.7 years for females.

H - Low overseas migration Net overseas migration to New South Wales of 29,400 per year (70,000 to Australia).

I - High overseas migration Net overseas migration to New South Wales of 52,500 per year (125,000 to Australia).

J - Very high overseas migration Net overseas migration to New South Wales of 63,000 per year (125,000 to Australia).

K - Zero overseas migration Zero overseas arrivals to and departures from New South Wales and Australia.

L - High interstate migration loss Net interstate migration loss to New South Wales of about minus 22,000 per year.

M - Low interstate migration loss Net interstate migration loss to New South Wales of about minus 10,000 per year.

N - Zero interstate migration Zero interstate arrivals to and departures from New South Wales.

O - High intrastate migration to Sydney

10% increase in rate of people moving from other regions to Sydney, and a 10% decrease in the rate of people moving out from Sydney to other regions.

P - Low intrastate migration to Sydney

10% decrease in rate of people moving from other regions to Sydney, and a 10% increase in the rate of people moving out from Sydney to other regions.

Q - Zero intrastate migration Zero movement between regions in New South Wales, and to and from the State.

R - Zero internal migration Zero movement between regions in New South Wales. S - Zero all migration All migration flows, overseas, interstate and intrastate. Set to zero. Source: 'Department of Planning NSW State and Regional Population Projections - 2005 Release. (Comparison of scenarios)'.

219

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Table 1. List of regions selected by remoteness and IRSD status

Statistical Local Area

2001 Women

Population * ARIA-plus IRSD

Liverpool (C) 78907 Cities First quartile Penrith (C) 88927 Cities Second quartile Parramatta (C) 73975 Cities Third quartile North Sydney (A) 30745 Cities Fourth quartile Lismore (C) - Pt A 16084 Inner region First quartile Dubbo (C) - Pt A 18130 Inner region Second quartile Dungog (A) 4138 Inner region Third quartile Wagga Wagga (C) - Pt B 2227 Inner region Fourth quartile Kyogle (A) 4780 Outer region First quartile Griffith (C) 12143 Outer region Second quartile Berrigan (A) 4011 Outer region Third quartile Coolamon (A) 2020 Outer region Fourth quartile Walgett (A) 3655 Remote First quartile Bogan (A) 1528 Remote Second quartile Central Darling (A) 1055 Very remote First quartile Bourke (A) 1868 Very remote Second quartile Note: ARIAP; <0.20 Cities, 0.20-2.40 Inner region, 2.40-5.92 Outer region, 5.92-10.53 Remote, >10.53 Very remote IRSD; <961 First quartile, 961-988 Second quartile, 988-1025 Third quartile, >1025 Fourth quartile * Estimated resident women population based on the 2001 Census by the Australian Bureau of Statistics

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Table 2. Descriptive statistics on birth defects across the study period Study period Cardiovascular defects Gastrointestinal defects

SLAs with no

defects

Mean expected

count SLAs with no

defects

Mean expected

count 1990 to 1991 69/198 (35%) 4.0 71/ 198 (36%) 2.7 1992 to 1993 60/198 (30%) 5.3 90/198 (45%) 3.3 1994 to 1995 63/198 (32%) 4.8 75/198 (38%) 3.2 1996 to 1997 59/198 (30%) 4.8 70/198 (35%) 3.0 1998 to 1999 62/198 (32%) 4.7 68/198 (34%) 3.2 2000 to 2001 67/198 (34%) 4.3 87/198 (44%) 2.9 2002 to 2003 75/198 (38%) 4.1 77/198 (39%) 2.8

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Table 3. Projected number of cardiovascular defects up till the year 2031

SLA Expected Smoothed risk 2001 2006 2011 2016 2021 2026 2031

SLA 1 0.26 2.37 1 1 1 1 1 1 1 SLA 2 0.48 4.08 2 2 2 2 2 2 1 SLA 3 0.88 1.27 1 1 1 1 1 1 1 SLA 4 0.37 6.42 2 2 2 2 2 2 2SLA 5 0.51 2.90 1 1 1 1 1 1 2 SLA 6 5.94 0.96 6 6 6 6 7 7 7 SLA 7 0.88 1.44 1 1 1 1 1 1 2 SLA 8 4.01 0.79 3 3 3 3 4 4 4 SLA 9 0.93 1.01 1 1 1 1 1 1 1 SLA 10 3.85 0.68 3 3 3 3 3 3 3 SLA 11 28.91 1.00 29 31 35 39 41 45 48 SLA 12 6.79 0.98 7 7 7 7 8 8 8 SLA 13 21.85 1.00 22 23 24 25 26 27 28 SLA 14 28.71 1.00 29 29 30 32 34 35 35 SLA 15 0.49 1.69 1 1 1 1 1 1 1 SLA 16 1.22 1.97 2 2 2 2 2 2 2

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Table 4. Projected number of gastrointestinal defects up till the year 2031

SLA Expected Smoothed risk 2001 2006 2011 2016 2021 2026 2031

SLA 1 0.17 1.56 0 0 0 0 0 0 0 SLA 2 0.32 1.78 1 1 1 0 0 0 0 SLA 3 0.59 2.46 1 1 1 1 1 1 1 SLA 4 0.25 6.67 2 2 2 1 1 1 1 SLA 5 0.34 1.55 1 1 1 1 1 1 1 SLA 6 4.00 1.38 6 6 6 6 6 7 7 SLA 7 0.59 1.18 1 1 1 1 1 1 1 SLA 8 2.70 1.19 3 3 3 4 4 4 4 SLA 9 0.62 1.29 1 1 1 1 1 1 1 SLA 10 2.59 1.03 3 3 3 3 3 3 3 SLA 11 19.46 1.12 22 24 26 29 31 34 37 SLA 12 4.57 1.06 5 5 5 5 6 6 6 SLA 13 14.70 0.93 14 14 15 15 16 17 17 SLA 14 19.32 0.97 19 19 20 21 22 23 23 SLA 15 0.33 1.40 0 0 0 0 0 0 0 SLA 16 0.82 1.49 1 1 1 1 1 1 1

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Table 5. Projected birth defects and associated credible intervals for the estimation

Year 2001 2006 2011 2016 2021 2026 2031

Cardiovascular defects SLA A 29(24-35) 31(26-38) 35(29-42) 39(32-47) 41(34-50) 45(37-54) 48(40-58) SLA B 7(5-8) 7(6-8) 7(6-9) 7(6-9) 8(6-10) 8(7-10) 8(7-10) SLA C 22(18-26) 23(19-27) 24(20-28) 25(21-29) 26(21-31) 27(22-32) 28(23-33) SLA D 29(24-35) 29(24-35) 30(25-36) 32(26-38) 34(28-40) 35(29-42) 35(29-42) Gastrointestinal defects SLA A 22(18-26) 24(20-28) 26(22-31) 29(25-35) 31(26-37) 34(29-40) 37(31-43) SLA B 5(4-6) 5(4-6) 5(4-6) 5(4-7) 6(5-7) 6(5-7) 6(5-8) SLA C 14(11-16) 14(12-17) 15(12-18) 15(13-18) 16(13-19) 17(14-20) 17(14-20) SLA D 19(16-22) 19(16-22) 20(17-23) 21(17-24) 22(18-26) 23(19-27) 23(19-27) Note: numbers in parenthesis refer to estimates based on the 2.5% and 97.5% of the posterior distribution

227

Figure 1. Projected number of cardio defects for selected regions

0

10

20

30

40

50

60

70

80

90

100

2001 2006 2011 2016 2021 2026 2031

No

. o

f c

ard

io d

efe

cts

SLA A

SLA B

SLA C

SLA D

228

Figure 2. Projected number of gastro defects for selected regions

0

50

100

2001 2006 2011 2016 2021 2026 2031

No

. o

f g

astr

o d

efec

ts

SLA A

SLA B

SLA C

SLA D

229

/

< 1.01

1.01 - 1.20

1.20 - 1.40

>1.40

Figure 3. Temporal trend of relative risk for cardiovascular defects from 1990 to 2003

230

Figure 4. Cumulative projection of cardiovascular defects (Scenarios A to S)

0

5000

10000

15000

20000

2001-2006 2006-2011 2011-2016 2016-2021 2021-2026 2026-2031

A B C D E F G H

I J K L M N O P

Q R S

231

Figure 5. Observed and forecasted cardiovascular defects for SLA ‘X’ using

ARIMA models 5

10

15

20

1990 2000 2010 2020 2030Year

ARIMA1 ARIMA2ARIMA3 ARIMA4ARIMA5 Observed

Note: ARIMA 1 Model: AR(1), ARIMA 2 Model: AR(2), ARIMA 3 Model: AR(3), ARIMA 4 Model: AR(2) MA(1), ARIMA 5 Model: AR(2) MA(2)

232

CHAPTER 10: CONCLUSION

10.1. Summary of results

Using a large disease registry of births and birth defects in NSW, I accomplished the

following tasks. Firstly, a series of neighbourhood weight matrices was examined, and I

showed how specifying a weight matrix according to similarity in terms of a key

confounder instead of a conventionally used Queen method of assignment, helped

improve the model’s ability to recover the underlying relative risk. Next, to tackle the

sparseness and excess zeros commonly encountered in the analysis of rare outcomes

such as birth defects, I compared a few models, including an extension of the usual

Poisson model to encompass excess zeros in the data. This was achieved via a mixture

model. Using the Deviance Information Criteria (DIC), I showed how the proposed

model performed better than the usual models, but only when both outcomes shared a

strong spatial correlation. With the aid of extensive simulation studies, I next evaluated

thirteen different sampling schemes along with variations in sample size. This was done

in the context of an ecological regression model that incorporated spatial correlation in

the outcomes, as well as accommodating both individual and areal measures of

covariates. Using the Average Mean Squared Error (AMSE), I showed how a simple

random sample of 20% of the SLAs, followed by selecting all cases in the SLAs chosen,

along with an equal number of controls, provided the lowest AMSE. For the final

objective, I combined the improved spatio-temporal CAR model with population (i.e.

women) forecasts, to provide 30-year annual estimates of birth defects at the Statistical

Local Area (SLA) level in New South Wales, Australia. A further discussion about the

233

implications of this work, along with the limitations and directions for future work will

be held below.

10.2. Implications of research

This project examined the commonly used Conditional Autoregressive (CAR) model in

disease mapping, and fine-tuned several aspects of the model (including the

neighbourhood weight matrix, and the model’s ability to incorporate sparseness and

overdispersion in the data) and applied it to a large registry of birth defect data at the

Statistical Local Area (SLA) in NSW, in order to make projections from 2001 to 2031.

Health policy-makers will find this information useful for several reasons.

Sparse outcomes are difficult to study for the many reasons outlined earlier in this thesis.

On the other hand, too much data also presents challenges, specifically in terms of

computational efficiency. I examined this issue in detail in chapter 8 of this thesis.

Through extensive simulation studies, I have shown that a careful sampling strategy

based on a case-control design, and further stratification by a key confounder, can help

reduce the dimension of the data substantially. This is an important finding for

researchers and policy-makers, as often data for disease mapping and ecological

regression studies comes from large registries. This large amount of data currently poses

problems for commonly used software such as WinBUGs, which only handles 50,000

observations.

234

In all my analyses, the commonly used Bayesian software, WinBUGS, was used to run

the CAR models. The CAR priors are programmed within the software. In addition to

WinBUGs, I also used the Stata (V9.2, College Station, Tx, USA) statistical software to

perform data management and run many of the models simultaneously and automatically

through a custom-written set of commands, developed by John Thompson from the

University of Leicester, United Kingdom. I wrote extensive Stata program files (also

called ado files) to manage data, manipulate data into a format suitable for analysis in

WinBUGs, write WinBUGs script files, invoke and run the models within WinBUGs,

save the output files and perform the analysis and model diagnostics, all within Stata. In

this research, I have shown that it is possible and efficient to run the models from within

Stata. The computer codes will be made available for other researchers who are

interested to perform these spatial analyses from within Stata.

Disease maps of birth defects, by themselves, are of limited use to policy-makers. For

instance, such a map may show that an SLA ‘X’ has a higher risk of birth defect when

compared to other regions. It may be difficult to develop health services for defects or

undertake studies to identify causes of defects, at such a short time to address the

increased risk that the particular region faces. Instead, it is probably useful to know

which local area actually shows an elevated risk some time down the road, whereby

planning for increased health services can be actualised. It is also possible for a certain

SLA to show lower or no risk of birth defect in a cross-sectional analysis, but have an

increased risk over time instead. As an example, we have seen that areas like Liverpool

and Penrith started with low counts of defects, but showed around 1.5 times increase in

counts of defects across a 30-year period.

235

In this research project, I applied the models to analyse birth defect data in New South

Wales. The models can be used to study other sparse outcomes as well. This includes

childhood cancers or other rare conditions such as blood cells cancer or pancreatic

cancer. For instance, in the 2004-2005 National Health Survey in Australia, self-reported

medical conditions in the general population ranged from 104.1 (per thousand people)

for otitis media, to 206.2 (per thousand people) for glaucoma to 2100.7 (per thousand

people) for hypertensive disease(190). When further stratified by age-group (as was

done in the same report), it was reported that the rates had a relative standard error of

more than 50%, which was considered too unreliable for general use. This was

particularly true for sparse outcomes like otitis media. Further splitting the data by

geographical regions only makes the problem more complex, and statistical models (like

the ones we’ve proposed in this thesis) are urgently needed to address this sparse data

problem.

There are a number of statistical contributions resulting from this thesis. Firstly, I have

shown that in a CAR neighbourhood weight structure, applying weights according to

similarity in key covariate (i.e. maternal age) confers much benefit in terms of

recovering the underlying risk estimates. This is a new method. I have also extended the

shared component model to incorporate excess zero counts in the data, via a mixture

model. The comparison between the various models is also something that has not been

done before. Next, I performed an extensive simulation study, to examine the

appropriate sampling regime, as well as sample size required, when selecting individual

cases in a hybrid spatial model that was designed to examine the effect of both

individual and areal covariates simultaneously in the study of a sparse outcome such as

236

birth defect, in relation to the sampling strategy for computational efficiency. Finally,

based on the earlier results, I proposed a method of combining areal smoothed estimates

of the relative risk along with population forecasts to provide 30-year annual estimates

of birth defects at the SLA level in NSW, Australia.

10.2. Limitations

There are several limitations in this research. Firstly, I used the mother’s residential

address at the time of birth for geocoding purposes, and assigned the areal units

accordingly. The fact that some of the mothers could have moved in/out of a particular

SLA just before their delivery cannot be ruled out. At the macro level, there is also the

possibility that the women could have moved inter-state prior to delivery. The MDC

registry actually receives notification of births in NSW from mothers who reside outside

of NSW, but not from mothers who live in NSW and give birth outside of NSW. In my

analysis, I assumed that these events are small in number, and that the effects are non-

differential across the SLAs studied (i.e. the numbers moving in/ out of each SLA are

constant). On a similar issue, I used the ASGC classification based on the 2001 census.

It is possible that over the study period, the boundaries of some CCDs may have

changed, came into existence, removed or even amalgamated with other areas. It is

possible that these changes could have affected the study results if I had performed the

analysis at some other level (e.g. CCD), but I believe that aggregating the data at the

broader SLA level minimizes this bias, as the accuracy of the geocoding process is not

affected. These boundary changes are also done for a range of reasons not related to

health status/outcomes of the population, including 1) to maintain the size of the

237

population so that it would not be too large for a collector's workload, 2) to maintain

confidentiality a population size of at least 100 is required, and 3) the CDs must not

overlap existing ASGC boundaries.

In my analysis, I made extensive use of data from the births and birth defect registries in

NSW, Australia. While the NSW Birth Defects Register (BDR) is a population-based

surveillance system, it was only in 1998 that doctors, hospitals, and laboratories have

been required by law to notify birth defects detected during pregnancy, at birth, or up to

one year of life. It is possible that the level of reporting may actually be different from

1998 onwards. In a spatial-only analysis, this may not be a problem. However, for

spatio-temporal analysis, one needs to careful in terms of not grouping data across the

year 1998, as the level of reporting may be very different before and after this year.

It is unclear how much of the findings are attributable to actual differences in risks

between regions, and how much due to differential levels of reporting between the

regions. One possible source of explanation of regional variations in reported birth

defects within NSW could be the way cases are diagnosed and classified in different

regions. For instance, mild forms of some defects like hypospadias, microphthalmia and

microtia may be missed out in the reporting by some localities. Variations in diagnostic

practice also contribute to the variation in prevalence of birth defects geographically. For

instance, karyotyping rates and indications for karyotyping, may affect accurate

reporting of chromosomal anomalies(191) (e.g. Trisomy 13 or 18 and Down Syndrome),

while autopsy rates for stillbirths and neonatal deaths will affect the diagnosis of certain

non-obvious defects, such as hypoplastic left heart syndrome. It is possible that

238

geographical variation in risk factors can contribute to the differences in risk between

regions.

Regional variations in births with multiple defects also contribute to differences in

reported rates. Practice could vary according to whether the name of the syndrome only,

or all of its component defects are recorded in the Registry. Defects that are a

consequence of other defects (e.g. hydrocephaly with spina bifida) are also generally not

recorded separately, and inconsistencies in recording across regions can contribute to the

variation in rates geographically. Prenatal screening policies, as well as facilities and

resources for prenatal screening, can vary spatially, and this can also contribute to the

variations in termination of pregnancies. The practice of terminating pregnancies due to

detection of birth defects also varies geographically(191). For deadly conditions like

anencephaly, terminations are more common, as opposed to conditions like spina bifida,

which are more variable in terms of severity.

The difficulty of using the birth defect registry to understand geographical variations in

risk has been highlighted in one paper, which suggested that the higher prevalence of

birth defects in the Metropolitan areas of Western Australia as compared to rural areas

can be attributed to better ascertainment(192). It is also probable that infants with birth

defects born to mothers who live near borders, can be transferred inter-state for their

routine care, and hence not reported to the registry.

The other main limitation of my analysis revolves around the coding and grouping of

birth defects in our analysis. I used the classification from the British Paediatric

239

Association (BPA) Classification of Diseases, which is primarily organised by body

systems. This is the system used by the NSW Health in their classification and reporting

of birth defects in their annual NSW Mothers and Babies Report. It is possible to group

birth defects with different etiologies together in the same category, using this

classification system. While it was not the main aim of this thesis to examine the risk

factors (and hence etiology) of selected birth defects, I do acknowledge that more

research needs to be done in the area of classification of birth defects, which will have

more meaningful epidemiological utility. One might perhaps classify defects according

to known main risk factors, or defects that are embryologically and pathogenetically

similar, rather than use administrative classifications, which are primarily based on the

body organ system.

10.3. Directions for future research

In this thesis, I examined in detail, various neighbourhood weight matrix formulations of

the CAR model, and concluded that smoothing by a key covariate such as maternal age,

helped to explain the spatial variation in the relative risks of sparse outcomes such as

birth defects. It would be interesting to see whether this can be extended to more than

one variable, perhaps a multivariate ‘smoothing similarity index’, consisting of maternal

age and socio-economic index, for instance. Smoothing with just one covariate may not

be adequate, as one may miss out on important correlations between the various

confounding variables. On the other hand, including too many covariates may provide a

case of over-fitting the data, or present challenges in terms of missing data. A

compromise is obviously needed. Smoothing by a covariate is advantageous over

240

including the variable as a confounder in the model, as the former provides more

information (e.g. neighbouring regions’ values as well). Future research can examine

this issue of multivariate smoothing by covariates in detail.

To address the issue of sparseness and over-dispersion in the data, I incorporated a zero-

inflated Poisson distribution through a mixture model. I also used a shared component

model to borrow strength from an outcome which shared similar risk factors which

varied spatially. This concept can be extended to model more than two disease groups

simultaneously. Further extensive simulation studies are needed to make comparisons

across these models, and evaluate their utility in analysing sparse outcomes.

In my forecasting model, I came up with a strategy to project birth defect categories at

the small area level. This strategy included spatio-temporal smoothing of the relative

risk of birth defects at the Statistical Local Area (SLA), followed by calculation of future

counts of defects using population projections. Further research can examine how the

two steps can be merged and done together. I foresee that there could be computational

challenges for the Bayesian models (given the large number of local areas involved in

forecasting), and also problems in model formulation (i.e. how would we handle the

large number of autoregressive terms for each local area), and data source (is there

enough historical data to inform future projections).

In this thesis, I also discussed briefly the issue of the Modifiable Areal Unit Problem

(MAUP). The choice of the areal unit of analysis is often made along practical

considerations. For instance, healthcare services may be coordinated within certain

241

administrative boundaries, and it would make sense to produce disease maps and

regression models at these corresponding areal levels. Sometimes, population or other

covariate data is made available at a different aggregate unit, and a compromise is then

made in terms of analysing all data at a more common areal unit. When one has an

option of choosing between various levels of aggregation in areal units as we do in NSW

(with census Collection Districts (CDs) aggregated up to Statistical Local Areas (SLAs)

and Local Government Areas (LGAs)), the most obvious choice would be to select the

smallest unit of aggregation. However, this often presents computational challenges

because of the large amount of areal units, and excessive amounts of areas with no

observations, which renders the analysis stage difficult. Future research could perhaps

look at simultaneously including all the various hierarchical levels of aggregation in the

same model, analysing the data at a more aggregate level, and then weighting the results

(perhaps providing weights in relation to importance of the particular scale) to provide

an overall measure of effect size.

In terms of data quality, I have discussed some of the concerns with regards to

interpreting the spatial variation in risk of defects within NSW, Australia. This problem

is compounded when we attempt to make comparisons across the states at the national

level. Before such an analysis can take place, several improvements in the way data is

collected need to be instituted. Firstly, a minimum dataset needs to be defined and

collected across the states. The clinical definitions for congenital anomalies and how

they are classified into groups need to be standardised. The reporting of congenital

anomalies must be fixed for a period of up to one year of birth for all states. Termination

242

of pregnancies needs to be included, regardless of gestational age. These suggestions

were made in a recent review paper(193) by the National Perinatal Statistics Unit.

243

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