Bayesian Latent Class Models for the Multiple Imputation of Cross-Sectional, Multilevel and

149 Pages·2017·0.79 MB·English

BAYESIAN LATENT CLASS MODELS FOR
THE MULTIPLE IMPUTATION OF
CROSS-SECTIONAL, MULTILEVEL AND
LONGITUDINAL CATEGORICAL DATA
Davide Vidotto
Tilburg University
©2017DavideVidotto. AllRightsReserved.
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BAYESIAN LATENT CLASS MODELS FOR
THE MULTIPLE IMPUTATION OF
CROSS-SECTIONAL, MULTILEVEL AND
LONGITUDINAL CATEGORICAL DATA
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan Tilburg University op
gezag van de rector magnificus, prof. dr. E.H.L. Aarts, in het
openbaar te verdedigen ten overstaan van een door het college voor
promoties aangewezen commissie in de aula van de Universiteit
2 2018 1400
op vrijdag maart om . uur
door
Davide Vidotto
21 1988
geboren op april te Treviso, Italië
Promotor: Prof.dr. J.K.Vermunt
Copromotor: Dr. K.VanDeun
OverigeledenvandePromotiecommissie: Prof.dr. S.vanBuuren
Prof.dr. F.Bassi
Dr. A.O.J.Cramer
Prof.dr. L.A.vanderArk
Allamiafamiglia
Tomyfamily
Voormijnfamilie

TABLE OF CONTENTS
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 MultipleImputationofMissingCategoricalDatausingLatentClassModels:
StateoftheArt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 LatentClassmodelsandMultipleImputation . . . . . . . . . . . . . 10
2.3 FourDifferentImplementationsofLatentClassMultipleImputation 15
2.4 Real-dataExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Appendixa BayesianTools . . . . . . . . . . . . . . . . . . . . . . . . . 31
Appendixb BayesianMultipleImputationviaMixtureModeling . . . 33
Appendixc GeneratingtheExtraMissingnessfortheReal-dataExample 37
3 Bayesian Latent Class Models for the Multiple Imputation of Categorical
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 BayesianLatentClassImputation . . . . . . . . . . . . . . . . . . . . 42
3.3 SimulationStudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Real-dataStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Bayesian Multilevel Latent Class Models for the Multiple Imputation of
NestedCategoricalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 TheBayesianMultilevelLatentClassModelforMultipleImputation 68
4.3 Study1: SimulationStudy . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Study2: Real-datacase . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Multiple Imputation of longitudinal categorical data through Bayesian mix-
turelatentMarkovmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 TheBayesianmixtureLatentMarkovModelforMultipleImputation 97
5.3 SimulationStudies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Real-dataStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Appendixa Settingthepriordistribution . . . . . . . . . . . . . . . . . 119
Appendixb BMLMmodelestimation . . . . . . . . . . . . . . . . . . . 120
6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
1
INTRODUCTION
Thisdissertationdealswiththemultipleimputation(MI;Rubin(1987))ofcategorical
data coming from different types of data collection and data analysis designs. In
particular, the use of latent class (LC) models (Lazarsfeld, 1950) for the MI of data
coming from cross-sectional study designs (as it was first proposed by Vermunt,
VanGinkel,VanderArkandSijtsma(2008))willserveasastartingpointtoobtain
imputation models that can deal with more complex designs, such as multilevel
(i.e., when multiple individuals are nested within a group) and longitudinal (i.e.,
whenmultipleobservationsforeachindividualareobservedacrosstime)designs.
LatentClassmodelsforMultipleImputation
LCmodelsareknownamonganalystsandmethodologistsfortheirsubstantiveuse,
in which the estimates provided by the model are used to define latent types (or
profiles, clusters) of units. These profiles differ from each other for some character-
istics,identifiedbythedistributionofthescoresontheindicatorvariables(usually
categorical variables). Within each LC, the joint distribution of these features is
describedbyaproductoflocallyindependentcategorical(e.g.,Multinomial)distri-
butions by means of the local independence assumption. Local independence makes
the model easily interpretable, and allows to take into account a large number of
indicators for a specific theoretical construct. A graphical representation of the LC
model is given in Figure 1.1, in which X represents the LC variable and the Y’s
representthe J indicators.
However,LCmodels-whicharemembersofthefamilyofmixturemodels-can
be used in contexts other than latent groups identification. That is, since mixture
2 1 INTRODUCTION
X
y1 y2 ··· yJ
Figure1.1:LCmodel,graphicalrepresentation. X: latentclassvariable;Y’s: indicators(J in
total).
models can correctly pick up unobserved heterogeneity and relevant relationships
inthedataifthenumberofspecifiedLCsislargeenough(McLachlan&Peel,2000),
LC models can be used as a density estimation tool. In density estimation, the
goal is to estimate and describe the joint distribution of the variables present in
a dataset, retrieving all possible associations which tie the variables to each other.
Under this framework, interpretation of the model parameters is of little interest,
and the main focus is on the predictions the model provides by means of these
parameters. Furthermore, models used for density estimation are likely to require
the estimation of a tremendous number of parameters, which would make them
very hard to interpret. Thus, the model parameters are merely a device used to
obtainpredictionsand/oranoveralldescriptionofthejointdistributionofthedata.
Vermuntetal.(2008)exploitedthisfeatureofLCmodels,andproposedthemfor
application in MI. In MI, the missing data of a dataset are replaced (or imputed,
predicted) M > 1 times by different sets of values, the distribution of which is esti-
mated with the imputation model. In particular, the task of the imputation model
is to provide values sampled from Pr(Dmis|Dobs), that is, the distribution of the
missing data given the observed data. When the missing data mechanism is ignor-
1
able , MI can retrieve the correct distribution of the data (for some analysis model
of interest), leading to proper substantive inferences. Furthermore, by obtaining
M different imputations it becomes possible to quantify the uncertainty about the
imputed missing values at the analysis stage. More specifically, substantive analy-
ses are performed on each of the M imputed datasets, where for correct statistical
inferencestheresultsarepooledusingRubin(1987)’srules.
LC models require a very easy model specification (the number of LCs), which
makesthemflexibleandautomatic,sincerelevantassociationsinthedataneednot
to be specified a-priori. Concerning the model selection issue, in MI selecting a
model that overfits the data (i.e., a model that capture sample-specific features) is
1 Thatis,themissingdatageneratingmechanismisindependentoftheunobserveddataanditsparameter
isdistinctfromtheonesoftheassumeddatageneratingmodel(Rubin,1976).