Epidemicmodelsandinferenceforthe...

Epidemic models and inference for thetransmission of hospital pathogens

Marie ForresterBachelor of Commerce, Bachelor of Arts

University of Queensland

A thesis submitted for the degree ofDoctor of Philosophy

July 2006

Principal Supervisor: Prof Tony Pettitt

Queensland University of TechnologySchool of Mathematical Sciences

Faculty of ScienceBrisbane, Queensland, 4001, Australia

Thesis examination page

Abstract

The primary objective of this dissertation is to utilise, adapt and extend current stoch-

astic models and statistical inference techniques to describe the transmission of

nosocomial pathogens, i.e. hospital-acquired pathogens, and multiply-resistant or-

ganisms within the hospital setting. The emergence of higher levels of antibiotic

resistance is threatening the long term viability of current treatment options and

placing greater emphasis on the use of infection control procedures. The relative im-

portance and value of various infection control practices is often debated and there

is a lack of quantitative evidence concerning their effectiveness. The methods devel-

oped in this dissertation are applied to data of methicillin-resistant Staphylococcus

aureus occurrence in intensive care units to quantify the effectiveness of infection

control procedures.

Analysis of infectious disease or carriage data is complicated by dependencies within

the data and partial observation of the transmission process. Dependencies within

the data are inherent because the risk of colonisation depends on the number of

other colonised individuals. The colonisation times, chain and duration are often

not visible to the human eye making only partial observation of the transmission

process possible. Within a hospital setting, routine surveillance monitoring permits

knowledge of interval-censored colonisation times. However, consideration needs

to be given to the possibility of false negative outcomes when relying on observations

from routine surveillance monitoring.

SI (Susceptible, Infected) models are commonly used to describe community epi-

demic processes and allow for any inherent dependencies. Statistical inference tech-

niques, such as the expectation-maximisation (EM) algorithm and Markov chain

Monte Carlo (MCMC) can be used to estimate the model parameters when only

partial observation of the epidemic process is possible. These methods appear well

suited for the analysis of hospital infectious disease data but need to be adapted for

short patient stays through migration. This thesis focuses on the use of Bayesian

statistics to explore the posterior distributions of the unknown parameters. MCMC

techniques are introduced to overcome analytical intractability caused by partial ob-

servation of the epidemic process. Statistical issues such as model adequacy and

MCMC convergence assessment are discussed throughout the thesis.

The new methodology allows the quantification of the relative importance of differ-

vi

ent transmission routes and the benefits of hospital practices, in terms of changed

transmission rates. Evidence-based decisions can therefore be made on the impact

of infection control procedures which is otherwise difficult on the basis of clinical

studies alone.

The methods are applied to data describing the occurrence of methicillin-resistant

Staphylococcus aureus within intensive care units in hospitals in Brisbane and Lon-

don

Keywords

Bayesian inference, Markov chain Monte Carlo, reversible jump, transdimensional,

stochastic epidemic model, susceptible-infected model, SI model, generalised lin-

ear model, hospital epidemiology, infectious diseases, infection control, nosoco-

mial infection, hospital-acquired infection, multiply-resistant organisms, antibiotic-

resistant bacteria, Staphylococcus aureus, methicillin-resistant Staphylococcus au-

reus, sensitivity, detectability

Contents

Contents xii

List of Tables xiv

List of Figures xvii

List of Abbreviations xvii

List of Notation xx

1 Overview 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Contribution of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Review of literature 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Epidemic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Deterministic models . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Stochastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Comparison of deterministic and stochastic models . . . . . . . 11

2.3 Statistical inference techniques . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Maximum likelihood (ML-) estimation . . . . . . . . . . . . . . . 11

2.3.2 Martingale techniques . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.3 Bayesian inference using Markov chain Monte Carlo tech-

niques (MCMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Statistical inference techniques applied to stochastic epidemic models 25

2.4.1 Chain binomial and other independent household models . . . 25

2.4.2 Generalised linear models . . . . . . . . . . . . . . . . . . . . . . 26

2.4.3 Stochastic epidemic models . . . . . . . . . . . . . . . . . . . . . 27

2.4.4 Non-transmission models . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Case study: methicillin-resistant Staphylococcus aureus (MRSA) . . . . 31

2.5.1 Staphylococcus aureus and MRSA . . . . . . . . . . . . . . . . . . 32

2.5.2 Transmission dynamics . . . . . . . . . . . . . . . . . . . . . . . 33

x

2.5.3 Infection control procedures . . . . . . . . . . . . . . . . . . . . 34

2.5.4 Epidemic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.5 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Case studies of methicillin resistant Staphylococcus aureus 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Princess Alexandra Hospital (PAH) intensive care unit (ICU) . . . . . . 41


3.2.2 Population and occurrence of MRSA . . . . . . . . . . . . . . . . 43

3.3 ICUs within two London (LON) hospitals . . . . . . . . . . . . . . . . . 47


3.3.2 Analysis of the patient population and extent of MRSA . . . . . 48

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Mechanistic description of the transmission process 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Model definition and assumptions . . . . . . . . . . . . . . . . . . . . . 57

4.3 Deterministic epidemic model . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Stochastic epidemic model . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5.1 Data and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Generalised linear model (GLM) and inference 67

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Model and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3.1 Data and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . 71

5.3.3 Bayesian inference using MCMC techniques . . . . . . . . . . . 72

5.3.4 Model adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.5 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Case study: PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6 Stochastic epidemic model (SEM) and inference 85

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


6.3 Data and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4 Joint likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . 88

6.6 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.6.1 MCMC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.7 Simulated data based on the PAH data . . . . . . . . . . . . . . . . . . . 90


6.7.2 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.8 Case study: PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


6.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xi

7 Stochastic epidemic model extended for imperfect sensitivity (ESEM) and

inference 101

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102



7.4 Joint likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.5 ML- estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


7.6.1 MCMC algorithm and convergence assessment . . . . . . . . . . 105

7.6.2 Model adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.7 Simulated data based on the PAH data . . . . . . . . . . . . . . . . . . . 111



7.8 Case study: PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.8.1 Model assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8 Stochastic epidemic model for an intervention study and inference 133

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133




8.4.1 MCMC algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.4.2 Model adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.5 Simulated data based on the London data and the hazard-phase-

effects (HPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138



8.6 Case study: hazard-phase-effects model, London ICU . . . . . . . . . . 142

8.6.1 Model assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8.7 Simulated data based on the London data and the detection-phase-

effects (DPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9 Conclusions and future work 159

9.1 Summary of methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.2 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A General epidemic model for exponentially distributed infectious periods 167

B Data of the Princess Alexandra Hospital ICU 169

B.1 Sources of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.1.1 Patient data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.1.2 Positive MRSA swab data . . . . . . . . . . . . . . . . . . . . . . . 170

B.1.3 MRSA notification data . . . . . . . . . . . . . . . . . . . . . . . . 171

B.2 Data formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

B.2.1 Discrepancies between data sources . . . . . . . . . . . . . . . . 172

C Antibiotic usage at the London ICU 173

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C.1 Antibiotic usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

C.2 Anti-Staphylococcal properties . . . . . . . . . . . . . . . . . . . . . . . 177

C.3 Gram-negative properties . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C.4 Amenogycide, cephalosporin and quinolone use . . . . . . . . . . . . . 185

C.5 Quantity of antibiotics used . . . . . . . . . . . . . . . . . . . . . . . . . 189

D Source model 193

E Stochastic simulation of the epidemic model 195

F Derivation of the data required for the generalised linear model 197

G Program code for inference of the generalised linear model 201

G.1 ML-estimation using standard statistical software . . . . . . . . . . . . 201

G.2 Bayesian inference using WinBUGS . . . . . . . . . . . . . . . . . . . . . 203

H Maximum likelihood estimation for the stochastic epidemic model with-

(out) imperfect sensitivity 205

H.1 Unconstrained parameter values . . . . . . . . . . . . . . . . . . . . . . 206

H.1.1 Score function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

H.1.2 Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

H.2 Constrained parameter values . . . . . . . . . . . . . . . . . . . . . . . . 210

H.2.1 Score function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

H.2.2 Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

I Proposal distribution for patient colonisation times in the presence of im-

perfect sensitivity 217

J Convergence diagnostics 221

J.1 Stochastic epidemic model . . . . . . . . . . . . . . . . . . . . . . . . . . 221

J.1.1 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

J.1.2 PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

J.2 Stochastic epidemic model with imperfect sensitivity . . . . . . . . . . 226

J.3 Stochastic epidemic model with imperfect sensitivity for multiple

phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

J.3.1 Hazard-phase-effects model, London ICU . . . . . . . . . . . . . 229

J.3.2 Detection-phase-effects model, simulated data . . . . . . . . . . 232

K Bayesian latent residual method of model assessment 235

Bibliography 239

List of Tables

2. Review of literature

2.1 Gelman and Rubin convergence diagnostic . . . . . . . . . . . . . . . . 18

2.2 Brooks and Guidici convergence diagnostic . . . . . . . . . . . . . . . . 20

3. Case studies of MRSA

3.1 Source of patient admissions to the PAH ICU . . . . . . . . . . . . . . . 43

3.2 Positive swabs per day in the PAH ICU . . . . . . . . . . . . . . . . . . . 44

3.3 Details of admissions to the PAH ICU . . . . . . . . . . . . . . . . . . . . 44

3.4 Time from admission to first swab for detected patients in the PAH ICU 44

3.5 MRSA patients in the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 Key characteristics of the London ICUs . . . . . . . . . . . . . . . . . . . 49

3.7 Number of admissions and transfers to the London ICUs . . . . . . . . 49

3.8 Key characteristics of the 18-bed London ICU . . . . . . . . . . . . . . . 50

3.9 Screening swab patterns and frequency for admissions to the London

ICUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Properties of antibiotics administered to the London ICU patients . . . 54

5. Generalised linear model (GLM)

5.1 MLEs of the GLM parameters for MRSA transmission within the PAH

ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Posterior summaries of the GLM parameters for MRSA transmission

within the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Deviance information criterion for the GLM fitted to the PAH data us-

ing MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Effect of bed occupancy on GLM transmission parameters within the

PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Posterior summaries for parameters of a GLM, that allows annual vari-

ation in transmission, fitted to the PAH ICU data . . . . . . . . . . . . . 80

6. Stochastic epidemic model (SEM)

6.1 Key characteristics of data simulated according to the stochastic epi-

demic model (SEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 SEM MLEs for MRSA transmission based on simulated data . . . . . . . 94

6.3 SEM MCMC sampler efficiency . . . . . . . . . . . . . . . . . . . . . . . 95

xiv

6.4 SEM posterior summaries for MRSA transmission based on simulated

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5 SEM posterior summaries for MRSA transmission within the PAH ICU 96

7. Stochastic epidemic model extended for imperfect sensitivity (ESEM)

7.1 ESEM MLEs for MRSA transmission based on simulated data . . . . . . 113

7.2 Summary statistics of the posterior means of ESEM parameters for

simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Posterior summaries for the ESEM parameters within the PAH ICU . . 117

7.4 Effect of prior swab sensitivity information on the estimated benefit of

isolation, 1 2, in the PAH ICU (based on the ESEM) . . . . . . . . . 127

8. Stochastic epidemic model for an intervention study

8.1 Parameter values used to simulate data according to the hazard-

phase-effects (HPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.2 MLEs for the HPE model parameters based on simulated data . . . . . 139

8.3 Posterior summaries for the HPE model parameters based on simu-

lated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.4 Posterior summaries for the HPE model parameters for the London

ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.5 Parameter values to simulate data according to the detection-phase-

effects (DPE) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.6 MLEs for the DPE model parameters based on simulated data . . . . . 152

8.7 Posterior summaries for the DPE model parameters based on simu-

lated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

9. Conclusions and future work

9.1 Comparison of MRSA parameter estimates . . . . . . . . . . . . . . . . 161

F.1 Calculations to derive data for the generalised linear model . . . . . . . 199

G.1 GLM data variables in S-Plus program code . . . . . . . . . . . . . . . . 202

J.13 Starting values for Markov chains for the HPE model parameters

within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

J.14 Geweke convergence diagnostics for the HPE parameters within the

London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

J.15 Raftery and Lewis convergence diagnostic for the HPE model param-

eters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . . 230

J.16 Heidelberger and Welch convergence diagnostic for the HPE model

parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . 231

J.17 Geweke convergence diagnostics for the DPE parameters within the

London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

J.18 Raftery and Lewis convergence diagnostic for the DPE model param-

eters within the London ICU . . . . . . . . . . . . . . . . . . . . . . . . . 232

J.19 Heidelberger and Welch convergence diagnostic for the DPE model

parameters within the London ICU . . . . . . . . . . . . . . . . . . . . . 233

K.1 Notation to describe the Bayesian latent residual method . . . . . . . . 236

List of Figures

2. Review of literature

2.1 Dynamics of infection and disease states . . . . . . . . . . . . . . . . . . 6

3. Case studies of MRSA

3.1 PAH ICU layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 PAH ICU patient lengths of stay . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Bed occupancy, MRSA prevalence and MRSA incidence at the PAH ICU 46

3.4 Layout of the London ICUs . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Number of patients and positive swabs taken each day in the 18-bed

London ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4. Mechanistic description of the transmission process

4.1 Transmission dynamics of a nosocomial antibiotic resistant bacterium 58

4.2 Stochastic compartmental model of the infection dynamics of noso-

comial antibiotic resistant bacteria . . . . . . . . . . . . . . . . . . . . . 59

4.3 Example clinical time-line for a hospital ward patient colonised with

antibiotic resistant bacteria . . . . . . . . . . . . . . . . . . . . . . . . . 59

5. Generalised linear model (GLM)

5.1 Relationships between data of the generalised linear model . . . . . . . 70

5.2 Directed acyclic graph of the generalised linear model . . . . . . . . . . 74

5.3 Marginal posterior densities of the generalised linear model parame-

ters obtained using MCMC within a Bayesian framework. . . . . . . . . 78

5.4 Correlations for posterior densities of the GLM parameters for the PAH

ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5 Deviance residuals for the posterior means of the GLM parameters for

the PAH ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6. Stochastic epidemic model (SEM)

6.1 Numbers of detected patients in data simulated according to the SEM . 91

6.2 Prevalence and incidence for data simulated using the SEM . . . . . . . 93

6.3 Posterior distributions for the SEM parameters fitted to the PAH ICU . 97

6.4 Weighted latent residuals for the SEM . . . . . . . . . . . . . . . . . . . 98

7. Stochastic epidemic model extended for imperfect sensitivity (ESEM)

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7.1 Allowable Markov chain move types for updating a patients final col-

onisation status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2 Scatter plots showing correlation between the ESEM parameters fitted

to simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.3 Augmented data for the PAH ICU and the ESEM . . . . . . . . . . . . . 117

7.4 Posterior distributions for the ESEM parameters fitted to the PAH ICU . 118

7.5 Correlations for posterior densities of ESEM parameters fitted to PAH

ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.6 Posterior distributions for the relative risks of transmission of the

ESEM in the PAH ICU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.7 Markov chain realisations for the ESEM parameters fitted to the PAH

ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.8 Posterior predictive discrepancies for the ESEM fitted to PAH ICU data 123

7.9 Cross-validation residuals (method 1) for the ESEM fitted to PAH ICU

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.10 Cross-validation residuals (method 2) for the ESEM fitted to PAH ICU

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.11 Latent residuals for the ESEM fitted to PAH ICU data . . . . . . . . . . . 126

7.12 Effect of prior swab sensitivity information on the estimated benefit of

isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8. Stochastic epidemic model for an intervention study

8.1 Posterior distributions for the HPE model parameters fitted to simu-

lated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.2 Posterior distributions for the HPE model parameters fitted to the ob-

served London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.3 Markov chain realisations for the HPE model parameters fitted to ob-

served data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.4 Correlations for posterior densities of the HPE model parameters fit-

ted to observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.5 Posterior predictive discrepancies for the HPE model fitted to the Lon-

don ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.6 Cross-validation residuals (method 1) for the HPE model fitted to the

London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.7 Cross-validation residuals (method 2) for the HPE model fitted to the

London ICU data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.8 Latent residuals for the HPE model fitted to the London ICU data . . . 150

8.9 Posterior distributions for the DPE model parameters fitted to simu-

lated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.10 Markov chain realisations for the DPE model parameters fitted to sim-

ulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

C.1 18-bed London ICU patients (a) on antibiotics and (b) not on antibiotics.174

C.2 Positive 18-bed London ICU admissions (split according to antibiotic

usage) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

C.3 18-bed London ICU patients on (a) anti-MRSA, (b) anti-

Staphylococcal, (c) other and (d) no antibiotics. . . . . . . . . . . . . . . 177

C.4 Positive 18-bed London ICU admissions (split according to anti-

Staphylococcal antibiotic usage) . . . . . . . . . . . . . . . . . . . . . . 178

C.5 18-bed London ICU patients on (a) gram-negative, (b) gram-positive

and (c) no antibiotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xvii

C.6 Positive 18-bed London ICU admissions (split according to gram-

negative antibiotic usage) . . . . . . . . . . . . . . . . . . . . . . . . . . 182

C.7 18-bed London ICU patients taking (a) amenogycides, (b) cephalo-

sporins, (c) quinolones and (d) other antibiotics. . . . . . . . . . . . . . 185

C.8 Positive 18-bed London ICU admissions (split according to amenogo-

cycide/quinlone/cephalosporin usage) . . . . . . . . . . . . . . . . . . . 186

C.9 18-bed London ICU patients administered (a) one, (b) two, (c) three or

more and (d) no antibiotics . . . . . . . . . . . . . . . . . . . . . . . . . 189

I.1 Proposal distribution for patient colonisation times . . . . . . . . . . . 217

J.1 Castelloe and Zimmerman convergence diagnostics for Markov

chains of ESEM parameters fitted to PAH ICU data . . . . . . . . . . . . 228

J.2 Castelloe and Zimmerman convergence diagnostics for Markov

chains of the HPE model parameters fitted to the London data . . . . . 231

A list of abbreviations

AIC Akaike information criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

BIC Bayesian information criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

DAG Directed acyclic graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

DIC Deviance information criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

DPE Detection-phase-effects model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

EM Expectation-Maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

ESEM Stochastic epidemic model extended for imperfect sensitivity . . . . 102

GLM Generalised linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

HCW Health-care worker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

HPE Hazard-phase-effects model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

IACF Integrated autocorrelation time function . . . . . . . . . . . . . . . . . . . . . . . . . 21

ICU Intensive care unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

LON London . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

LOS Length of stay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

MCMC Markov chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

ML- Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

MLE Maximum likelihood estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

MRSA Methicillin-resistant Staphylococcus aureus . . . . . . . . . . . . . . . . . . . . . . 31

MSSA Methicillin-susceptible Staphylococcus aureus . . . . . . . . . . . . . . . . . . . 38

NA Not a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

PAH Princess Alexandra Hospital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

RJMCMC Reversible jump MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

SEM Stochastic epidemic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

VRE Vancomycin-resistant enterococci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

CPU Computer processing unit time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A list of symbols and notation

ai Admission time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63a Admission times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63a Parameter 1 of the prior distribution for . . . . . . . . . . . . . . . . . . . . . . . . 72b Parameter 2 of the prior distribution for . . . . . . . . . . . . . . . . . . . . . . . . 72BU Upper bound of the uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . 90BL Lower bound of the uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . 89ci Colonisation time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C Non-isolated (and colonised) patient state . . . . . . . . . . . . . . . . . . . . . . . 57CIx x% credible interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Cj Number of non-isolated patients in the ward at swabtime tj . . . . . 69C(t) Number of susceptible patients in the ward at time t . . . . . . . . . . . . . 58di Bayesian latent residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138dij Time from the j

th swab, which is positive, taken from patient i andnotification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

D() Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22D() Deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Davg() Posterior deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24D Complete data, includes observed and augmented data . . . . . . . . . 12Dd Data emanating from deterministic processes . . . . . . . . . . . . . . . . . . . 64Dobs Observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Dm Latent or missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12ej j

th event time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85e Event times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85f() Density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62F () Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62g Markov chain iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13G Total number of MCMC iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13h(t) Hazard function at time t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61H() Hessian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12jm(s

|s) State proposal probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Kx,K

x Move probabilities defined for x=1, 4 and 5 . . . . . . . . . . . . . . . . . . . . . . 106

l() Loglikelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11L() Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11N(t) Number of patients in the ward at time t . . . . . . . . . . . . . . . . . . . . . . . . . 58NCj Expected number of colonisations in the j

th interval . . . . . . . . . . . . . 76

xxii

NCj Number of colonisation events in the jth interval . . . . . . . . . . . . . . . . 69

NQj Number of isolation events in the jth interval . . . . . . . . . . . . . . . . . . . . 69

NA Number of admissions during the observation period . . . . . . . . . . . 103Ne Number of events during the observation period . . . . . . . . . . . . . . . . 85NFN(D) Number of false negative swabs, which is a function of the data D 103NTP Number of detected patients, or true positive swabs . . . . . . . . . . . . . 89Nsp(s) Number of patients colonised on admission . . . . . . . . . . . . . . . . . . . . . 103P (D|) Likelihood of the data D given the parameters . . . . . . . . . . . . . . . . . 13P (|D) Posterior of the parameters given the data D . . . . . . . . . . . . . . . . . . . 13q() Proposal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14qi Isolation time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63q Isolation times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Q Isolated patient state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Q(t) Number of isolated patients in the ward at time t . . . . . . . . . . . . . . . . 58Qj Number of isolated patients in the ward at swabtime tj . . . . . . . . . . 69ri Discharge time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63r Discharge times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63R Removed patient state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57rDj Deviance residual for the j

th interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75sd Final colonisation status of colonised in ward . . . . . . . . . . . . . . . . . . . 63sp Final colonisation status of colonised on admission . . . . . . . . . . . . . 63ss Final colonisation status of susceptible . . . . . . . . . . . . . . . . . . . . . . . . . . 63S Susceptible patient state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Sj Number of susceptible patients in the ward at time tj . . . . . . . . . . . . 69S() Score function of parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11S(t) Number of susceptible patients at time t . . . . . . . . . . . . . . . . . . . . . . . . . 58tiv1 Last negative swab time for patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86ti Swabtimes of patient i, comprising {ti1, ti2, . . .} . . . . . . . . . . . . . . . . . . . 59ts Swabtimes of all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64tj Time infinitesimally preceding tj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62i : ti ph2 Logical condition that time t is in phase 2 . . . . . . . . . . . . . . . . . . . . . . . . 205vi Positive swab time of patient i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63v Positive swab times for all patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63X() Survivor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62yj Number of patients escaping colonisation in the j

th interval . . . . . 71yj Observed data used to assess a model(cross-validation) . . . . . . . . . 23yobs Observed data used to assess a model (posterior prediction) . . . . . 22yrep Replicated data used to assess a model . . . . . . . . . . . . . . . . . . . . . . . . . . 22y Data for model fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241x Indicator function taking value 1 if x is true and 0 otherwise . . . . . 72() Acceptance probability for the MCMC algorithm . . . . . . . . . . . . . . . . 140 Background transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 Rate of transmission from non-isolated (colonised) patients . . . . . 612 Rate of transmission from isolated (colonised) patients . . . . . . . . . . 61 Multiplicative effect on the baseline hazard . . . . . . . . . . . . . . . . . . . . . . 134ph(t) Indicator function taking the value if time t occurs in the move

phase and one otherwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Unknown model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Proposed value of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Point estimate of , i.e. posterior mean or MLE . . . . . . . . . . . . . . . . . . 24

xxiii

() Target distribution or full conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13j Constant hazard rate in the interval [tj1, tj) . . . . . . . . . . . . . . . . . . . . . 62j Conditional probability of colonisation in an interval [tj1, tj) . . . 68 Imperfect sensitivity; probability of a false negative swab . . . . . . . . 59c Standard distribution of the proposal distribution for the colon-

isation times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Probability that a patient is colonised on admission to the ward . 58

Statement of Original Authorship

The work contained in this thesis has not been previously submitted for a degree or

diploma at any other higher education institution. To the best of my knowledge and

belief, this thesis contains no material previously published or written by any other

person except where due reference is made.

Signed:

Date:

Acknowledgements

Thank-you to the following people and organisations for support during this degree:

Supervisor, Tony Pettitt expert guidance, time and encouragement. I am especially

grateful for the opportunities provided to collaborate with researchers abroad.

Family and friends camping, capoeira and making life away from this thesis fun

and interesting. Thanks in particular to my parents for milk provisions and Sonia for

assistance to locate glasses, keys, etc.

Collaborators and colleagues to those within the School of Mathematical Sciences,

for making staff room visits a worthwhile distraction, and to those who showed an

interest in this work and offered suggestions or advice. In particular, Gavin Gibson

who assisted with later models of this thesis and Ben Cooper for data and enthusiasm

to assist with and discuss this thesis, facilitate my time in London, and to provide

information on prospective employers.

School of Mathematical Sciences administration and IT support staff rapid assis-

tance to administrative and technical problems encountered.

Princess Alexandra Hospital financial support, data and information regarding the

intensive care unit policies and procedures.

Australian Research Council financial support.

Lindt & Sprungli rewarding the boring and mundane parts of this thesis.

xxviii

CHAPTER 1

Overview

1.1 Introduction

The primary objective of this study is to develop statistical inference techniques

for application to hospital data. The methodology provides a means of quantifying

transmission and the estimated benefit, in terms of changed transmission rates, of

infection control interventions.

Despite a large body of clinical research, there is a growing dissent between practi-

tioners concerning the effectiveness of widely implemented infection control proce-

dures. Clinical studies are generally limited to quasi-experimental designs making

causal relationships difficult to establish. Dependencies within the epidemic pro-

cess make standard statistical tests invalid, making it difficult to rule out stochastic

fluctuation as a cause for any observed changes. Systematic reviews (Cooper et al.,

2003; Loveday et al., 2006) have found major methodological weaknesses in pub-

lished research evaluating the effectiveness of infection control procedures designed

to curb the spread of nosocomial pathogens and in particular, methicillin-resistant

Staphylococcus aureus (MRSA).

Epidemic models and statistical inference are well-equipped to tackle any difficulties

arising from dependencies within the epidemic process. Furthermore, such meth-

ods are able to quantify the estimated effects of infection control interventions, in

terms of changed transmission rates, which is otherwise difficult on the basis of clin-

ical studies.

Patients colonised with nosocomial pathogens such as MRSA may be asymptomatic

so that the transmission process can only be partially observed by routine swabs

testing for colonisation. Markov chain Monte Carlo (MCMC) methods are currently

popular techniques (e.g. Gibson and Renshaw, 1998, 2001; ONeill and Roberts, 1999;

Streftaris and Gibson, 2004b) used to analyse data of partially observed infectious

diseases within the community. The methods appear well suited to routinely col-

lected hospital data but must be adapted to allow for patient turnover. Community

2 Chapter 1. Overview

populations have relatively small turnover and are typically assumed to be closed.

The predominantly short patient stays of hospital ward populations requires the re-

moval of this assumption.

This thesis investigates the use of epidemic models and statistical inference tech-

niques to evaluate the effectiveness of infection control procedures designed to curb

nosocomial transmission of pathogens. Emphasis is placed on the use of MCMC

methods within a Bayesian framework. The methods are adapted for application

to routine surveillance data and extended to allow for imperfect sensitivity of the

surveillance process. The methods are applied to data describing the occurrence of

methicillin-resistant Staphylococcus aureus within a Brisbane and London intensive

care unit.

1.2 Scope

The scope of this thesis is to estimate parameters for stochastic epidemic models

describing transmission of a single nosocomial pathogen that can be carried asymp-

tomatically.

Epidemic models and statistical inference

Stochastic epidemic models to describe the transmission of infectious diseases with-

in a small number of individuals are described and developed. Theoretical proper-

ties of stochastic epidemic models are not investigated. The models are introduced

to infer model parameters describing transmission from data assumed to have been

generated from the defined model. Techniques to infer parameters of the stochas-

tic epidemic model given only partial observation of the transmission process are

developed.

MCMC within a Bayesian framework

MCMC techniques, including reversible jump MCMC, are employed to estimate pa-

rameters of the stochastic epidemic model. Convergence diagnostics and methods

for assessing goodness-of-fit for models fitted to date within a Bayesian framework

are applied. Methods to improve the speed of implementation and convergence are

not a primary focus of this thesis.

Quantification of hospital practices designed to curb nosocomialtransmission

Based on the given epidemic model, assumptions and data describing the occur-

rence of nosocomial pathogens, the rates at which the pathogen is transmitted are

estimated. The effect of control practices are quantified, for example, by including

separate transmission parameters for pre- and post-intervention transmission. The

results of this thesis are not generalisable to different populations, settings or time.

Other than reporting the effects of infection control measures, recommendations for

infection control practices are not within the scope of this thesis.

1.3 Outline of thesis 3

1.3 Outline of thesis

The remaining chapters of this thesis are organised as follows:

Chapter 2 presents an investigation of epidemic models and associated statistical

inference techniques that have been described in the literature. The use of epidemic

models and statistical inference techniques to evaluate the effectiveness of hospi-

tal practices designed to control MRSA and other antibiotic resistant bacteria is dis-

cussed, complimented by a brief introduction to the epidemiology of MRSA and the

main results and limitations of clinical research designed to investigate the effective-

ness of infection control procedures.

Details of admissions, the extent of MRSA and infection control procedures em-

ployed by the Princess Alexandra Hospital (PAH) and three London (LON) intensive

care units (ICUs) are given in Chapter 3.

A stochastic epidemic model to describe the transmission of nosocomial pathogens

that can be carried asymptomatically, e.g. MRSA, is described in Chapter 4. The

model describes the flow of patients through susceptiblecolonisedisolated

removed states according to admission, colonisation, isolation and discharge events.

A critical examination of factors and assumptions which may affect parameter val-

ues is performed. The objectives and data requirements of statistical inference for

the stochastic epidemic model are introduced. Problems posed by dependence and

partial observation of the transmission process are discussed.

In Chapter 5, the stochastic epidemic model is approximated by a generalised linear

model and data is aggregated according to discrete intervals of time. By considering

the data in discrete form, inference techniques are not affected by partial observa-

tion and so standard inference techniques, such as maximum likelihood estimation,

can be used. Bayesian inference using Markov chain Monte Carlo techniques is in-

troduced for comparative and precision purposes. The methods are applied to data

describing MRSA occurrence within the PAH ICU.

Chapter 6 describes a Bayesian framework using MCMC techniques to infer the pa-

rameters of the stochastic epidemic model. The use of MCMC avoids any problems

associated with partial observation of the epidemic process. In this chapter, the

swabs allowing for partial observation of asymptomatic pathogen transmission are

assumed to have perfect sensitivity. Knowledge of a patients colonisation status on

admission is also assumed. The methods are illustrated using the PAH data.

In Chapter 7, techniques are introduced to infer parameters of a stochastic epidemic

model allowing for imperfect sensitivity. Reversible jump MCMC is used to facilitate

inference. A by-product of the methodology is that knowledge of a patients colon-

isation status is no longer required nor assumed. The methodology is used to in-

fer the rates of cross-colonisation, spontaneous acquisition, importation probability

and screening swab sensitivity within the PAH ICU.

Chapter 8 is motivated by an intervention study which took place at an 18-bed ICU in

London. The stochastic epidemic model and techniques for inference are extended

to estimate pre- and post-intervention model parameters.

An overview and discussion of the methodology and results are given in Chapter 9.

Possible extensions to the research presented in this thesis are indicated.

4 Chapter 1. Overview

1.4 Contribution of thesis

The main purpose of this thesis is to develop statistical methodology to enable esti-

mation of parameters for stochastic epidemic models that describe transmission of

single nosocomial pathogens that can be carried asymptomatically. The techniques

are applied to hospital surveillance data. This thesis provides the following:

Methodological contributions

adapts methods for quantitative infectious disease epidemiology to hospitalepidemiology

provides a tool for estimating parameters that describe the transmission ofasymptomatic pathogens in small populations with high turnover

develops reversible jump MCMC methodology to allow for imperfect sensitiv-ity

Applied contributions

provides insight into the underlying transmission process for the spread of asingle nosocomial pathogen that can be carried asymptomatically

quantifies the relative importance of cross-colonisation verse spontaneous ac-quisition (as measured by background transmission)

develops a method for analysing routine surveillance data

quantifies the effect of infection control measures

Publications

Forrester, M., Pettitt, A., and Gibson, G. (2006). Bayesian inference of hospi-tal transmission and control given imperfect surveillance data. Biostatistics.

Accepted for publication.

Forrester, M. and Pettitt, A. N. (2005). Use of stochastic epidemic modeling toquantify transmission rates of colonization with methicillin-resistant Staphyl-

ococcus aureus in an intensive care unit. Infect Control Hosp Epidemiol, 26(7):

598606.

CHAPTER 2

Review of literature

2.1 Introduction

Analysis of infectious disease data provides insight into underlying epidemic pro-

cesses, a means of assessing proposed interventions aimed at reducing transmission

and a summary of the data (Becker, 1989). By summarising the data according to

key parameters, the disease can be compared between different areas and times and

with other diseases. The task of the analysis is essentially one of statistical inference:

given an observation of a process, which is believed to be governed by a

known epidemic model, what range of model parameters could plausibly

explain the observations (Gibson and Renshaw, 1998).

The behaviour of an epidemic model is characterised by the model parameters.

Both epidemic modelling and the analysis of infectious disease data are complicated

by dependencies within the epidemic process (Becker, 1989; Becker and Britton,

1999; Andersson and Britton, 2000; ONeill, 2002). Dependencies arise because the

risk of acquisition depends on the number of individuals who are infected. If for

example, the occurrence of the disease did not require the presence of infected indi-

viduals, data could be analysed using survival analysis (Cox and Oakes, 1984; Collett,

1994) techniques. Such techniques require the specification of the hazard function

to describe the risk of an individual becoming infected. The hazard function may be

specific to each individual and/or contain a random parameter correlated between

related individuals, which is known as a frailty model. Coxs proportional hazard is a

well known hazard function.

Partial observation of the epidemic process further complicates the task of infec-

tious disease data analysis (Becker and Britton, 1999; Andersson and Britton, 2000;

ONeill, 2002). For example, it is usually unknown who infects whom, the time of

infection and the duration of infectiousness. Poor quality data and low and fluctu-

ating reporting rates are often further obstacles to statistical inference (Becker and

6 Chapter 2. Review of literature

Britton, 1999). Becker and Britton (1999) highlight the importance of the data and a

need to consider the type and quantity of data required to make objective decisions

concerning infectious disease control procedures.

The focus of this thesis lies with statistical inference techniques for disease data.

Epidemic models form a basis for statistical inference and as such, some general

epidemic models (Section 2.2) are described below, followed by a discussion of sta-

tistical inference techniques (Section 2.3).

2.2 Epidemic models

Epidemic models describe the dynamics of infectiousness as opposed to the dynam-

ics of disease (see Figure 2.1) (Halloran, 1998). The infectious process can be repre-

FIGURE 2.1. Dynamics of infection and disease states.

sented by a succession of states or compartments. Such a representation is referred

to as a compartmental model (Jacquez, 1996). For example, diseases characterised

by Figure 2.1 can be represented by a SEIR compartmental model which consists

of four compartments; susceptible (S), exposed (E), infected (I) and removed (R).

Individuals are initially considered susceptible. Upon contact with an infectious in-

dividual, the individual is exposed. The individual remains exposed for a duration of

time, the latent period, until becoming infectious. Upon termination of infectious-

ness, the individual is considered removed from the infectious cycle. Removal can

correspond to immunity or death. There are many well-known variants to the SEIR

model (Hethcote, 1994), for example, using the same notation, possible models are

SI, SIS, SEI, SEIS, SIR, SIRS, SEIR, and SEIRS.

The SEIR model and variants are approximations to the epidemic process; a com-

plete epidemic model would also describe the infectious load in each member of the

host population (Dietz and Schenzle, 1985). A detailed model describing the num-

ber of parasites in each member of the host population was attempted by Rvachev

in 1967 (Dietz and Schenzle, 1985), however most models ignore the number of par-

asites per host or more generally, the infectious load.

The compartmental model can be described by either deterministic or stochastic

mathematical equations. In a deterministic model, the number of infections in a

short time interval can be assumed proportional to the number of susceptible and
hallaThis figure is not available online. Please consult the hardcopy thesis available from the QUT Library

2.2 Epidemic models 7

infectious individuals and to the time interval. In a stochastic model, the probability

of one new case in a short interval is equivalently proportional to the same quantity.

SIR models involve only a minor variation to predator-prey models such as the deter-

ministic Lotka-Volterra and the stochastic Volterra models applied in ecology (Ren-

shaw, 1999).

Mathematical models can be applied to both epidemic and endemic situations. The

term epidemic refers to outbreaks which can usually be attributed to a point source.

Mathematical models are used to describe biological and transmission mechanisms,

threshold densities and to predict the course of epidemics (Bailey, 1975). In partic-

ular, they are used to predict the initial conditions which lead to an epidemic, the

shape of the epidemic curve, the number of cases at the peak of the epidemic, the

duration of the total epidemic and the total number of cases (Dietz and Schenzle,

1985).

The term endemic is used to refer to randomly occurring low numbers of cases

punctuated by the intermittent re-introduction of the disease (Thompson, 2004).

In endemic situations, mathematical models are important tools for evaluating and

comparing control interventions (Bailey, 1975; Dietz and Schenzle, 1985). For exam-

ple, they can be used to answer questions concerning the control effort required to

make the positive equilibrium unstable and zero incidence stable. For viruses, they

can estimate the required vaccination coverage and for vector transmitted patho-

gens, the minimum reduction in vector density required.

2.2.1 Deterministic models

Deterministic models are generally based on the mass-action principle which states

that the course of an epidemic depends on the number of susceptible individuals

and the contact rate between susceptible and infectious individuals. The mass ac-

tion principle was formulated in discrete time by Hamer in 1906 and then in continu-

ous time by Ross in 1908 (Andersson and Britton, 2000). In a deterministic model, the

future state of the epidemic process can be determined from the initial numbers of

susceptible and infectious individuals, together with the infection-, recovery-, birth-

and death-rates (Bailey, 1975).

The first mathematical model is generally attributed to Kermack and McKendrick

(Kermack and McKendrick, 1927). The set of equations were however first published

by Ross and Hudson in 1917 (Diekmann et al., 1995; Dietz and Schenzle, 1985). The

mathematical equations describe the dynamics of a SIR disease in continuous time.

It is assumed that the disease being modelled occurs in a large, closed, homoge-

nous and uniformly mixing population of equally susceptible individuals, contacts

are made according to the law of mass action and infection triggers an autonomous

process within the host (Diekmann and Heesterbeek, 2000). These assumptions are

summarised by the equation

S(t) =dS

dt= S(t)

0A()S(t ) d, (2.1)

where S(t) is the spatial density of susceptibles, i.e. the number of susceptibles perunit area, at time t, S(t) is the incidence, i.e. the number of infection events ina unit of time, at time t and A() is the expected infectivity of an individual thatbecame infected time units ago (Diekmann et al., 1995).


Equation 2.1, referred to as the Kermack and McKendrick model, is usually expressed

for the specific case in which infectivity has an exponential distribution. If is therate at which an infectious individual has contact (sufficient for transmission) with

susceptible members and is the removal rate, then the Kermack and McKendrickmodel with exponential infectivity is

S(t) = S(t)I(t), (2.2a)I (t) = S(t)I(t) I(t). (2.2b)

The derivation is obtained by defining A() = e and I(t) = 10 A()S

(t )d = 1

0 A(t )S()d and differentiating. The proof is included in Ap-

pendix A.

The Kermack and McKendrick model is often referred to as the general epidemic

model. To quote Bailey (1975), the term general is used in the sense that the model

is not confined to infection only, i.e. the possibility of removal is also considered.

Hethcote (1994) provides a brief outline of possible generalisations.

Deterministic models approximate actual state changes by considering that integer

valued variables are varying continuously. When numbers of susceptible and infec-

tive individuals are both large and mixing is reasonably homogeneous, the deter-

ministic model is likely to be satisfactory as a first approximation (Bailey, 1975). The

approximation will not be good when any of the integer-valued variables become

small such that the population becomes close to extinction (Mollison et al., 1994;

Daley and Gani, 1999; Andersson and Britton, 2000). Deterministic epidemic mod-

els are discussed in detail by Bailey (1975), Anderson and May (1991) and Daley and

Gani (1999).

2.2.2 Stochastic models

In a stochastic model, probability distributions of the numbers of susceptible and in-

fectious individuals occurring at any instant replace the real-values of deterministic

treatments. The majority of stochastic models are based on variants of two classical

models, namely the general epidemic model and the chain binomial model (Lefevre,

1988). Both classical models are special cases of a more general SIR model of a closed

and homogeneous mixing population in which pairs of individuals make indepen-

dent contact. The infectious periods are assumed to be independent and identically

distributed. Assumptions concerning the distribution of the infectious periods differ

between the two classical models. For example, within the general epidemic model

the infectious period is assumed to be exponential and in the chain binomial model,

the infectious period is assumed to be of a pre-determined fixed length. The general

epidemic model is usually described in continuous time dynamics and the chain bi-

nomial model in discrete time dynamics.

Stochastic epidemic models are discussed in detail by Bailey (1975), Lefevre (1988),

Daley and Gani (1999) and Andersson and Britton (2000). The references provide

an overview of analyses of stochastic epidemic models. For example, asymptotic

and exact distributions of the final size of the epidemic, the total area under the tra-

jectory of infective individuals, and approximations to the epidemic model are dis-

cussed. In addition, generalisations of the stochastic epidemic model to allow for

several classes of susceptible and infected individuals are made and the phenomena

of recurrence and competition and spatial aspects are considered.

2.2 Epidemic models 9

The general stochastic epidemic model was first studied by McKendrick in 1926

and then ignored until 1949 when analysed by Bartlett. Deterministic general epi-

demic models assume that the actual number of new infections in a time interval

is proportional to the product of the susceptible and infective population sizes and

the time interval. Stochastic general epidemic models, on the other hand, assume

that the probability of a new infection in a short interval is proportional to this same

amount, i.e. the product of the susceptible and infective population sizes and the

time interval.

The stochastic version of Equation 2.2 expresses the probability of infection and re-

moval, respectively, as

Pr{S(t + t) = s 1, I(t + t) = i + 1|S(t) = s, I(t) = i} = sit + o(t) (2.3a)

and

Pr{S(t + t) = s, I(t + t) = i 1|S(t) = s, I(t) = i} = it + o(t). (2.3b)

The force-of-infection is defined as the rate at which an individual is infected, i.e.

I(t). This formulation for the force-of-infection is known as the pseudo mass-action assumption (de Jong et al., 1995). It is used when the number of effective

transmissions is expected to remain the same regardless of the population size. An-

other approach, true mass-action assumes that the probability of contact decreases

as population size increases. In this case, should be divided by the population size.The force-of-infection at a time t is sometimes referred to as the hazard h(t). For themodel described by the system of equations (2.3), the hazard at time t is

h(t) = I(t). (2.4)

The system of equations (2.3) define an infection process in which contacts between

uniformly mixing susceptible and infectious individuals occur at times given by

points of a homogeneous Poisson process with intensity . An implicit assumptionis that the infectious periods are exponentially distributed with mean 1. It is onlynecessary to know the times and nature of all events occurring in the course of an

epidemic; the individuals to which each event applies do not affect the next event. A

further implication is that the conditional distribution of an infection time in givenprevious infection times i0, i1, . . . in1, depends only on the most recent infectiontime in1. Infectious or latent periods that are modelled as random variables froman exponential distribution are said to be Markovian.

The Selke construction (Selke, 1983) provides a biological interpretation of the gen-

eral stochastic epidemic model. Each susceptible individual is considered to have a

threshold to infection which is exponentially distributed with unit mean. Once the

infection pressure reaches this level, the susceptible individual becomes infected.

The total infection pressure exerted on a susceptible up to time t is the integral ofthe hazard from exposure to time t.

Stochastic models that do not assume homogenous mixing have received consider-

able attention. Such models include multitype models (Ball, 1986; Ball et al., 1997;

Andersson and Britton, 2000), which divide the population into homogenous sub-

populations, and their extensions (Ball and Neal, 2003), social cluster models (Schi-

nazi, 2002) and random network models (Andersson, 1999).


Generalised linear models can be used to model epidemics in the community

(Becker, 1983, 1986). Equation 2.3a of the general stochastic epidemic model implies

that the probability of a given susceptible escaping infection in the interval (t, t + 1),conditional on the individual being susceptible at time t, is

exp{

t+1

tI(s)ds

}

. (2.5)

If the time unit is chosen so that not many events, or infections, occur in one unit of

time then the conditional probability in Equation 2.5 can be approximated by

exp{I(t)}.

If Ni(t) is the number of individuals infected in time (t, t + 1) and individuals areinfected independently during a given time unit, the number of infections is bino-

mially distributed,

Ni(t) Bin(

S(t), 1 exp{I(t)})

.

Chain binomial models are suitable for diseases with a short infectious period and

a longer constant latent period. Long latent periods form a means of separating dis-

ease progression into stages or generations. Specifically, the latent period is used to

model the time between generations. With short infectious periods, it follows that

infectives remain infectious for one generation only. The number of infections in

any generation therefore depends on the state of the epidemic in the previous gen-

eration only.

The first chain-binomial model was anticipated by Enko in 1889 (Dietz and Schen-

zle, 1985). However, the most well-known chain-binomial model was put forward by

Reed and Frost in lectures in 1928. Another well-known variant is the Greenwood

model which was formulated independently by Greenwood and published in 1931.

If is the probability of no infection by any single infective and does not vary be-tween generations, the chain-binomial model can be expressed for each of the de-

scribed variants,

P (Sj+1 = sj+1|Sj = sj, Ij = sj) =

Bin(

sj ,{

1 ijN1}k)

Enko,

Bin(sj , ij ) Reed-Frost,

Bin(sj , ) Greenwood,

(2.6a)

ij = sj1 sj . (2.6b)

The Enko model assumes a fixed number of contacts k during one time interval,whereas the Reed-Frost model does not specify the distribution of contacts (Dietz

and Schenzle, 1985). In the Reed-Frost model, the probability of escaping infection

when exposed simultaneously to i infective individuals in one generation is equiv-alent to the probability of being exposed to a single infective in each of i separategenerations. It is applicable to diseases which are primarily transmitted by close

contact between individuals (Becker, 1989). The Greenwood model assumes that si-

multaneous exposure to two or more infectives in one generation is equivalent to

being exposed to just one infective. It is applicable if the household environment

is saturated with infectious material (Becker, 1989), i.e. if the risk of infection of

a susceptible depends only on the presence of viruses in the environment, not on

the number of infectives. The Greenwood model is therefore applicable to airborne

spread of infection (Dietz and Schenzle, 1985).

2.3 Statistical inference techniques 11

2.2.3 Comparison of deterministic and stochastic models

The stochastic versus deterministic model debate is often centred around model

simplicity and realism (Andersson and Britton, 2000). Deterministic models are sim-

pler to analyse than their stochastic counterparts (Mollison et al., 1994; Bailey, 1975).

For a stochastic model to be mathematically manageable, a simpler and not entirely

realistic model is required. Although algebraic determination of stochastic model

properties is difficult, approximations can provide information about the intrinsic

variability of the system (Mollison et al., 1994).

Deterministic models are unsuitable for small populations. For larger populations,

the mean number of infectives in a stochastic model may not always be approx-

imated satisfactorily by the equivalent deterministic model (Mollison et al., 1994;

Daley and Gani, 1999; Andersson and Britton, 2000). According to Renshaw (1999),

it should always be assumed that stochastic effects play an important role in any

given process unless proven otherwise.

2.3 Statistical inference techniques

The models described in Section 2.2 are used to describe the mechanism by which

observed data are generated. Statistical inference is then used to estimate the pa-

rameters of the model. Although deterministic models can be fitted to observa-

tions, for example, by least-squares minimisation, the subsequent discussion con-

cerns statistical inference for data said to arise from a stochastic model. Methods of

inference for the stochastic models described in the preceding section include max-

imum likelihood estimation, Bayesian inference using Markov chain Monte Carlo

and Martingale techniques. An introduction to statistical inference techniques for

non-transmission models is given towards the end of the section.

2.3.1 Maximum likelihood (ML-) estimation

For data D generated from a stochastic epidemic model defined by parameters ,

a maximum likelihood estimate (MLE) of is the value that maximises the likeli-

hood L(;D), or equivalently, the loglikelihood l(;D). If the likelihood is differen-tiable, unimodal and bounded above (Tanner, 1996), the MLE is unique and found

by setting the score function S() = l(;D) to zero and solving for . Numerical ap-proaches can be used when the maximum of the likelihood can not be determined

analytically. The Newton-Raphson and the expectation-maximisation (EM) algo-

rithms are discussed below; other numerical approaches include the Nelder-Mead

simplex algorithm, quadratic optimisation and the quasi-Newton method.

Newton-Raphson

The Newton-Raphson algorithm is based on the Taylor series expansion of f(x) aboutx[k1] for scalar x:

f(x) = f(x[k1]) + f (x[k1])(x x[k1]) + f(x[k1])(x x[k1])2

2!+ . . .

f(x[k1]) + f (x[k1])(x x[k1]). (2.7)


The stationary points of the linear approximation in Equation 2.7 are found by set-

ting Equation 2.7 to zero and solving for x. Doing so implies that

x = x[k1] f(x[k1])

f (x[k1]). (2.8)

Equation 2.8 can be applied to the vector of score functions S() to obtain an itera-tive solution of the MLE. If f(x) = S() is the score function and f (x) = H() is the

Hessian matrix with ijth element 2L(;D)ij

, an updated estimate, [k], of a current es-

timate, [k1], is given by [k] = [k1]H1([k1])S([k1]). Given an initial startingvalue [0], the sequence [0],[1], . . . converges to a root of S().

Expectation-Maximisation (EM) algorithm

The expectation-maximisation (EM) algorithm, first named so by Dempster et al.

(1977), can simplify ML-estimation of the parameter vector by considering a more

complete and hypothetical data set. The complete data set D is formed by augment-

ing the observed data Dobs with fictitious data Dm (referred to as latent or missing

data). The latent data should be chosen such that the loglikelihood of the complete

data is relatively straightforward. The algorithm is an iterative method which con-

sists of two steps: the E-step (expectation step) and the M-step (maximisation step).

Once an initial parameter choice [0] is chosen, the E-step and M-step are performed

repeatedly until convergence occurs, that is until the difference between successive

iterates is negligible.

E-step. The E-step consists of computing the expected value of the complete data

loglikelihood conditional on the observed data and current estimate,

EDm|[i],Dobs l(;D

obsDm).

M-Step. The M-step requires maximising the expectation calculated in the E-step

with respect to to obtain the next iterate [i].

Iterates obtained using the EM algorithm converge to a turning point of the likeli-

hood. Readers are referred to Hastie et al. (2001) for an explanation as to why the EM

algorithm works. A numerical example is provided in Tanner (1996).

2.3.2 Martingale techniques

Martingale methods can be used to infer parameters of a stochastic epidemic model

given partial observation of the epidemic process. A drawback of using martin-

gale techniques for inference is the necessity of the Markov assumption (ONeill,

2002). The literature on Martingales requires a sound foundation in measure the-

ory, among other topics, and will not be described further. For an introduction to

Martingales without the imposition of mathematical rigour, readers are referred to

Lan and Lachin (2005).


2.3.3 Bayesian inference using Markov chain Monte Carlo techniques(MCMC)

Bayesian inference (Gelman et al., 2000; Congdon, 2001) is the process of fitting a

model to data and summarising the results by a probability distribution, known as

the posterior distribution, on the parameters and unobserved quantities. The treat-

ment of model parameters as random and data as fixed is in contrast to the frequen-

tist framework in which model parameters are treated as fixed and uncertainty is

expressed in terms of potential replicates of the data. The posterior distributions

obtained within a Bayesian framework provide information about parameter uncer-

tainty and permit the formulation of direct probability statements, appropriate for

small size samples, about parameters. Frequentist approaches, on the other hand,

estimate only the standard errors rather than full probability distributions. Proba-

bility statements within a frequentist approach rely on indirect statements based on

confidence intervals and p-values. Calculation of the frequentist confidence levels

may require development of appropriate theoretical results and the usual conditions

that require asymptotic normality of maximum likelihood estimators are often vio-

lated (ONeill, 2002).

A Bayesian framework incorporates prior information on the model parameters in

the form of a prior distribution P (). This prior distribution along with the likelihoodof the data D, P (D|) = L(;D), defines the joint posterior distribution, denotedP (|D). The posterior distribution is given by

P (|D) = P ()P (D|)

P ()P (D|) d . (2.9)

The posterior is the distribution of the model parameters conditional on the data.

Because the denominator of Equation 2.9 is not a function of , and since integration

is with respect to , the posterior distribution is proportional to the product of the

prior and likelihood distributions,

P (|D) P ()P (D|). (2.10)

When making inference about the parameters, one is usually concerned with point

and interval summaries, such as the mean, variance or quantiles, of the posterior

distribution. Point and interval summaries are expressed in terms of their expect-

ation of a function of the unknown parameters,

E(

f()|D)

=

f()P ()P (D|) d

P ()P (D|) d .

Except in the simplest cases, the integrals cannot be evaluated analytically. Given

a realisation of a Markov chain {[g]}, g = 1, 2, . . . (obtained using a MCMC algo-rithm) whose stationary distribution is the joint posterior distribution, the integral

E(

f()|D)

can be estimated by Monte Carlo integration,

E(

f()|D)

1G

G

g=1

f([g]).

Markov chain realisations can be obtained using MCMC techniques (Geyer, 1992;

Gilks et al., 1996b; Brooks, 1998) which iteratively generate samples from some tar-

get distribution () that is known only up to proportionality. Within a Bayesian


framework, the target distribution is the joint posterior distribution of the model pa-

rameters (see Equations 2.9 and 2.10). The drawn samples form the Markov chain.

The process is continued until the chain has converged to its stationary distribution

(). Once converged and after discarding initial samples (to remove dependence ofthe simulated chain on its starting location), the Markov chain can be used to esti-

mate functions of the target distribution (Kass et al., 1998).

The transition kernel P([g+1]|[g]) is the probability that the next state of the chainlies within some set, given that that the chain is currently in state [g] (Brooks, 1998).

If the transition kernel satisfies the detailed balance condition,

([g])p([g],[g+1]) = ([g+1])p([g],[g+1]),

then the Markov chain will have a stationary, distribution (Brooks, 1998; Tierney,1994). Markov chains with transition kernels that satisfy the detailed balance equa-

tion are said to be reversible (p.230 Robert and Casella, 2004; Chib and Greenberg,

1995).

For convergence of the Markov chain to the stationary, regularity conditions of ir-

reducibility, aperiodicity and positive recurrence are required (Roberts, 1996). An

irreducible Markov chain will reach any non-empty set with positive probability in

some number of iterations. The aperiodic condition prevents the Markov chain from

oscillating between different states in a regular periodic fashion. If a Markov chain

is positive recurrent and an initial value is sampled from a distribution () then allsubsequent iterations will also be distributed according to ().

The Metropolis-Hastings algorithm (Metropolis et al., 1953; Hastings, 1970; Chib and

Greenberg, 1995) is a Markov chain simulation method useful for drawing samples

from Bayesian posterior distributions (Gelman et al., 2000). The Metropolis sampler

(Metropolis et al., 1953) and the Gibbs sampler (Geman and Geman, 1984; Gelfand

and Smith, 1990; Tanner and Wong, 1987; Casella and George, 1992) are special cases

of the Metropolis-Hastings algorithm.

The Metropolis-Hastings algorithm is as follows:

start with an initial value [0]

obtain a realisation [1],[2], . . . from a Markov chain by repeating the followingsteps for g = 0, 1, . . .:

1. sample a point from some proposal distribution q(|[g]),2. evaluate

([g],) = min(

1,()q([g]|)([g])q(|[g])

)

, (2.11)

3. set [g+1] equal to with probability ([g],), otherwise set [g+1] to [g].

The transition kernel for the Metropolis-Hastings algorithm,

P([g+1]|[g]) ={

q([g+1]|[g])([g],[g+1]) if is accepted such that [g+1] = ,1

q(|[g])([g],)d if is rejected such that [g+1] = [g],

satisfies the detailed balance equation,

([g])P([g+1]|[g]) = ([g+1])P([g]|[g+1]), (2.12)


(Roberts, 1996; Robert and Casella, 2004, p.272). This implies that is the stationarydistribution of the Markov chain (Brooks, 1998; Tierney, 1994; Chib and Greenberg,

1995).

The irreducible, aperiodic and positive recurrent conditions, which regulate conver-

gence to the stationary distribution, will be satisfied by Markov chains generated us-

ing the Metropolis-Hastings algorithm if the proposal distribution provides full sup-

port for (Robert and Casella, 2004, p.272). It should be noted that Markov chainsthat are irreducible with stationary distribution will be positive recurrent (Roberts,1996).

It is often more computationally efficient to update the components of one by

one rather than all at once using a single component Metropolis-Hastings algorithm

(Gilks et al., 1996a). For [g] = {[g]1 , [g]2 , . . . ,

[g]h } at iteration g of the Markov chain let

[g]i = {

[g+1]1 , . . .

[g+1]i1 ,

[g]i+1, . . . ,

[g]h }. In place of steps 1 to 3 above, the algorithm is

to instead, for each component of ,

1. sample a point i from some proposal distribution q(i |

[g]i ,

[g]i),

2. evaluate ([g]i,

[g]i ,

i ) = min

(

1,(i |

[g]i)qi(

[g]i |

i ,

[g]i)

([g]i |

[g]i)qi(

i |

[g]i ,

[g]i)

)

,

3. set [g+1]i equal to

i with probability (

[g], i ), otherwise set [g+1]i to

[g]i .

Note that (i|i) is known as the full conditional; it is the distribution of the ithcomponent of conditioning on all remaining components,

(i|i) =()

()di.

If the proposal distribution is symmetric, i.e. if q(|) = q(|), then a special caseof the Metropolis-Hastings algorithm, referred to as the Metropolis algorithm, can be

used. Given a symmetric proposal distribution, the candidate value is accepted as

the next value with probability = min(

1, ()

()

)

.

Gibbs sampling is a special case of the single-component Metropolis-Hastings algo-

rithm in which the proposal distribution for updating the ith component of at the

gth iteration of the Markov chain is same as the full conditional. If q(i |[g]i ,

[g]i) =

(i |[g]i),

= min(

1,(i |i)qi(

[g]i |i ,

[g]i)

(i|i)qi(i |[g]i ,

[g]i)

)

= min(

1,(i |i)(i|i)(i|i)(i |i)

)

=1,

such that the proposal candidate is always accepted.

A Markov chain defined by the Metropolis-Hastings algorithm will satisfy the irre-

ducible, aperiodic and positive recurrent conditions required for convergence to a

stationary distribution as long as the proposal distribution ensures that the support

of may be fully explored (Robert and Casella, 2004, pp.272-276).


A number of frameworks exist that allow parameter subspaces of differing dimen-

sionality (Green, 1995, 2003). These include jump diffusion (Grenander and Miller,

1994; Phillips and Smith, 1996), point processes (Preston, 1977; Ripley, 1977; Geyer

and Mller, 1994; Stephens, 2000), product space formulations (Carlin and Chib,

1995; Godsill, 2001, 2003) and reversible jump MCMC (RJMCMC) (Green, 1995).

When making transdimensional proposals from a state [g] of dimension dm to

of dimension dm, Green (1995, 2003) shows that detailed balance (2.12) will be pre-served if the new state is accepted with probability

([g],) = min(

1,()jm(

)gm(u)

([g])jm([g])gm(u)

(,u)

([g],u)

)

. (2.13)

Here jm([g]) is the probability of choosing a move m which takes [g] of dimension

dm to of dimension dm, gm(u) is the probability of generating u of dimension rm,

u is of dimension rm and dm+rm = dm+r

m. Retaining detailed balance ensures that

the target distribution is the stationary distribution provided the chain is irreducible

and aperiodic (Dellaportas et al., 2002). Given the current state of the Markov chain

[g] of dimension dm, the reversible jump MCMC (RJMCMC) algorithm (Green, 1995,2003) is to repeat the following four steps:

1. propose a move m which takes [g] of dimension dm to of dimension dm with

probability jm([g]),

2. generate u from gm(u),

3. set = h(,u),

4. accept with probability ([g],

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