Leiden 85+ data binary/binomial responses Data MMSE index ... · MMSE index Modeling issues...

Random effect modelsfor longitudinalbinary/binomial

responses

Marco Alfo

The statistical problem

Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach

Random effect structure

Modeling initial conditions

Data analysis

Missing data

Ignorable dropouts

Non-ignorable dropouts

Data analysis - moreresults

Concluding remarks

Random effect models for longitudinalbinary/binomial responses

Marco Alfo

with A. Spagnoli and J. Houwing-Duistermaat

“Sapienza” Universita di Roma, Leiden Univ. Medical Center

London, April 15, 2009


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Table of contents


Leiden 85+ dataDataMMSE indexModeling issues

Modeling approachRandom effect structureModeling initial conditions

Data analysis

Missing dataIgnorable dropoutsNon-ignorable dropouts

Data analysis - more results

Concluding remarks


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Purpose

I Define a regression model for longitudinal binomialdata to account for dependence on baseline valuesand random effects,

I Adopt an adequate modeling structure to obtainreliable ML parameter estimates;

I Since some data may be missing due to dropout,consider potentially non-ignorable missing data.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Notation

We have a two-stage random sample

yit , i = 1, . . . , n, t = 1, . . . ,T ,

from a binomial distribution f (y | θ) with covariatesX = (xit), and canonical parameter θ.

We will consider binomial longitudinal responses, where iindexes individuals and t indexes time occasions withinindividuals. In this case, T will denote the designedcompletion time.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Intra-cluster Dependence

Primary interest: to allow for apparent and truecontagion.

I Apparent contagion - Individuals are drawn fromheterogeneous populations, each population having aconstant, but different, propensity to the event ofinterest.

I True contagion - Current and future outcomes aredirectly influenced by past ones, which causechanges over time in the corresponding distribution.

We will consider the effect of the baseline outcome onsubsequent responses, with a special focus on randomeffect models for short panel series.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Data

I Between September 1997 and September 1999, 705inhabitants of Leiden (The Netherlands), 85 yearsold, were eligible to participate to the Leiden 85-plusStudy;

I 599 subjects were enrolled, 14 died before they couldbe enrolled and 92 refused to participate(der Wiel etal, 2002);

I Response variable: Mini-Mental State Examination(MMSE) index (Folstein et al, 1975), assessing theglobal cognitive status of older adults (measuredonce a year: 85 up to 90);


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

MMSE index

I Assessed by a standard questionnaire;

I 30 items to be separately answered by each subject;scores on each item are binary (1 for correct answerand 0 otherwise);

I the MMSE index is defined as follows:

Yit =30∑j=1

Yijt Yijt ∈ {0, 1}

where Yijt , i = 1, . . . , n, j = 1, . . . , 30, t = 1, . . . ,Trepresents the MMSE score for the i-th subject onthe j-th item a time t.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Modeling issues

I Investigate the relationship between annual value ofthe cognitive function, the baseline value and a setof covariates.

I Covariates collected at entry time: gender(female=0), educational status classified into twolevels (primary=0 and higher levels);

I A possible approach could be based on a Rasch(1960)-type model:

logit [Pr(Yijt = 1 | xit , ui )] = xTitβ + (ui − γj)

where ui and γj , i = 1, . . . , n, j = 1, . . . , 30,represent individual-specific (random) ability anditem-specific (fixed) complexity.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Model structure

I However, in the following we will consider, asresponse, the MMSE index defined by:

Yit =∑

j

Yijt

i = 1, . . . , n, t = 1, . . . ,T . Thus, we (implicitly)assume:

I local (conditional) independence

I constant (across items and time) item-specificcomplexity, ie γj = γ ⇒ πijt = πit .


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Variance components

Conditional on unobserved characteristics summarized bythe individual-specific vector bi , the responses areassumed to be independent:

Yit | bi ∼ Bin(30, πit)

The canonical model is

θit = xTitβ + zT

itbi

where the random coefficient bi may be associated to asubset of covariates, zit .

The bi may be independent multivariate Gaussian rv’sbi ∼ MVN(0,Σ), or arbitrary with unknown densityg(b | Σ).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Marginal likelihood

The likelihood is

L(β,Σ) =n∏

i=1

∫ [T∏

t=1

f (yit | bi )

]g(bi )dbi .

Under Gaussian assumptions, the likelihood may beapproximated by

L(β,Σ).=

n∏i=1

∑k1,...,km

[T∏

t=1

f (yit | bk1,...,km)

]πk1,...,km ,

where bk1,...,km and πk1,...,km are masspoints and massesfor (K1, . . . ,Km)-point Gaussian quadrature.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Parametric assumptions?

The parametric approach may be implemented using astandard EM algorithm; see Aitkin (1999) for details.

For an arbitrary (nonparametric) assumption, NPMLestimation of the mixing distribution can be achieved in afinite mixture framework using the same algorithm.

Computational details of NPML estimation in the contextof clustered/longitudinal data analysis are discussed inAitkin (1999).

R libraries: lme4 (Bates and Maechler) and npmlreg(Einbeck, Darnell and Hinde).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Dependence on baseline

Unobserved heterogeneity may be (partially) accountedfor by inserting, in the linear predictor, the response valueobserved at t = 1 (baseline value).The corresponding model is, for t > 1,

Yit | Yi1,bi ∼ Bin(30, πit), t = 2, . . . ,T

with

θit = xTitβ + αyi1 + zT

itbi ,

bi ∼ g(bi | Σ).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

ML estimation

In this case, the joint marginal distribution is given by:

f (yi ) =

∫ {T∏

t=2

[f (yit | yi1,bi ) f (yi1 | bi ) g (bi )]

}dbi

However, the term f (yi1 | bi ) is not specified by themodel assumptions; thus, we need to modify the standardVC approach to account for this modeling structure.

This issue can be linked to the initial condition problemin models with both AR and random terms, see Aitkinand Alfo (1998).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

How to handle it...

Various approaches can be adopted. We discuss:

I Conditional Modeling

I Joint modeling of first and subsequent occasions

In the first case, we maximize the likelihood conditionalon the initial conditions yi1; this is not feasible when thebaseline values is included as a covariate in a randomeffect model.In the second case we define a joint model including amodified model for the first occasion, and a standardmodel for subsequent occasions.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Conditional modeling - 1

Since the bi ’s are shared by all subject i ’s responses, thelikelihood function results from the following integral:

L(β,Σ | yi1) =n∏

i=1

∫ {T∏

t=2

[f (yit | yi1,bi )g(bi | yi1)]

}dbi

see Aitkin and Alfo (1998, 2006).However, the random effect distribution has changed dueto conditioning ⇒ parametric assumptions on g(bi ) maynot be valid for g(bi | yi1).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Conditional modeling - 2

The random coefficient vector can be rewritten as:

bi = [b∗i + E (bi | yi1)] = b∗i + Γyi1

where b∗i is independent from Yi1. The canonical model is


it [b∗i + Γyi1]

When random effect models are considered, we have:

θit = xTitβ + αyi1 + b∗i + γyi1 = xT

itβ + αyi1 + b∗i

This means that the estimated α will include the effect ofthe baseline value on the response variable and on therandom effect.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Drawbacks

The standard VC algorithm can be simply adapted toconditional modeling. However,

it discards information about the random effect structurewhich, in the case of short time series, may produceinefficient parameter estimates.

it can not be of any help in the case of random effectsmodels with baseline dependence (in general thecorresponding estimate would be biased).

Solution: Include the term f (yi1 | bi ) in a full modelfor all the data.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Model all the data jointly

As before, the model for t ≥ 2 is defined by:


itbi

while for the first occasion we assume:

θi1 = xTi1β1 + zT

itbi1

with bi1 = Λbi , Λ = diag (λ1, . . . , λm),cov (bi1) = ΛΣΛT.The RE covariance matrix may be different on lateroccasions; the same applies to the fixed parameter vector.

This is handled by defining a dummy variable dt = 1,t ≥ 2, and interacting it with appropriate terms.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Leiden 85+ data, VC model

Variable Coeff. Std. Err.

Cons. 1.3525 0.0783Age -0.1944 0.0066Gender 0.2299 0.1216Educational 1.0240 0.1207

σb 1.3344` -2756AIC 5522

Table: VC model: parameter estimates


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Leiden 85+ data, VC.bas model


Cons. -3.3395 0.0717Age -0.1944 0.0065Gender -0.0133 0.1079Educ. 0.2264 0.0595MMSE85 0.2091 0.0050Cons.bas 1.4313 0.1028Gender.bas 0.0526 0.1311Educ.bas 0.3243 0.0420

σb 0.8705λ 1.1432 0.2221` -2293AIC 4606

Table: VC.bas model: parameter estimates


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Intro

However, 276 of 599 subjects (46.1 %) have incompletesequences; these may occur due to a variety of selectionrules: for example, the individual may refuse to answer tosome questions, or drop out during the study.

This individual behavior may bias the survey design andquestion the representativeness of the observed sample indrawing inference about the general population.

The key question is whether those who dropoutdiffer (in any way relevant to the analysis) fromthose who still remain.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

More on key question

Once other variables in the data have been controlled for,do the missing data depend on the current (unobserved)value of the response variable?

If not, the missing data mechanism is said to be ignorable(ID); otherwise, it is said to be non-ignorable (NID).

In the first case, one may use standard ML methods forconsistent estimation. Otherwise, one has to take intoaccount the missing data mechanism to obtain consistentmodel parameter estimates.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Some notation

Let us denote by Rit a missing data indicator, defined as:

I Rit = 1 if the i-th unit drops out at any time∈ (t − 1, t), t = 1, . . . ,T ,

I Rit = 0 otherwise.

The number of available responses for the i-th subject is:

Si = T −T∑

t=1

Rit

We focus on a special case, assuming that, once a persondrops out, he/she is out forever (attrition is an absorbingstate).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Statistically speaking...

If the mechanism is ignorable, the joint density is:

f (yOi , ri | bi , φi ) =

∫f (yO

i , yMi | bi )h(ri | yi , φi )dyM

i =

=

∫f (yO

i , yMi | bi )h(ri | yO

i , φi )dyMi =

= f (yOi | bi )h(ri | yO

i , φi )

If bi⊥φi , the likelihood function can be factorized andML estimates for β can be derived from the first term,using standard approaches.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

RCBDM

When dropout is non-ignorable, we may choose aparameterization to describe association between theprimary outcome and the missing data mechanism.

In clinical trials, unobserved disease status could influenceboth the primary response and the dropout due e.g. tonon-compliance.

The dependence between the outcome and the missingdata indicator may thus arise because they are jointlydetermined by shared (or correlated) latent effects.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

On the model side...

Assuming conditional independence given the randomcoefficients (bi , φi ), we may introduce an explicitdiscrete-time model for the dropout process:

h(ri | φi , vi , yi ) = h(ri | vi , φi )

which leads to the following joint density:

f (yi , ri ) =

∫f (yi | Xi ,bi )h(ri | Vi , φi )dG (bi , φi )

The previous density may be specified using differentapproaches.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Modeling choice

Little and Rubin (2002) discussed two classes of modelsto handle non-ignorable missing data:

I Pattern mixture models: the observed sample isstratified according to the observed patterns ofdropout. See e.g. Alfo and Aitkin (2000), Verbekeand Molenberghs (2000), Roy (2003),Wilkins andFitzmaurice (2007).

I Selection models: a complete-data model is definedfor the primary response, augmented by a modeldescribing the missing-data mechanism conditionalon the complete data. See e.g. Verzilli andCarpenter (2002), Gao (2004), Lin et al.(2004),Rizopoulos et al. (2008).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Leiden 85+ data, Pattern Mixture model


Cons. -3.2729 0.0718Age -0.1994 0.0067Gender -0.0201 0.1080Educ. 0.2065 0.0688MMSE85 0.2044 0.0051S 0.0542 0.0160Cons.bas 1.4305 0.1029Gender.bas -0.0208 0.0484Educ.bas 0.1314 0.0524S.bas 0.0158 0.0114

σb 0.8622λ 1.1429 0.2223` -2287AIC 4598

Table: Pattern mixture model: parameter estimates


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Leiden 85+ data, Selection model

Response Variable Coeff. Std. Err.

Cons. -3.3343 0.0711Age -0.1893 0.0086Gender 0.0924 0.1027

Yit Educ. 0.1867 0.0476MMSE85 0.2012 0.0069Cons.bas 1.4311 0.1032Gender.bas 0.1006 0.1277Educ.bas 0.0674 0.0454

Age.surv -0.1064 0.00921− Rit Gender.surv -1.0755 0.2604

Educ.surv 0.3600 0.1604MMSE85.surv 0.1008 0.0191

σb 0.9322λ 1.1543 0.1929σφ 2.4102ρ 0.6923` -2775AIC 5584

Table: Selection model: parameter estimates


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Findings...

I Once we control for the baseline value, the effect ofeducational drops significantly;

I however, the effect remains significant in both theVC-bas and the PM model;

I when we move to the selection model, educational isnot significant in the baseline model, while it is insubsequent occasions as well as in the discrete timesurvival model

I these findings are, obviously, just a starting point,they need to be validated but point out the need forproper handling of missing data.


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

Next steps?

I From discrete-time to continuous-time survivalmodels (event times available);

I Genetic factors (in particular APOE - apolipoproteinE) have to be considered;

I From random coefficient based to response drivendropout models; Bayesian approach?

I Perform sensitivity analysis: do model assumptionsinfluence results? Explore the impact of potentialdeviations from non-ignorability (see eg Troxel, Maand Heitjan, 2004);

I From static to dynamic mixing: eg HMM (Cappe etal., 2005).


responses

Marco Alfo


Leiden 85+ data

Data

MMSE index

Modeling issues

Modeling approach



Data analysis

Missing data

Ignorable dropouts



Concluding remarks

THANK YOU!

Leiden 85+ data binary/binomial responses Data MMSE index ... · MMSE index Modeling issues...

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