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Journal ArticleDOI

A latent-class mixture model for incomplete longitudinal Gaussian data.

TL;DR: It is argued that analyses valid under MNAR are not well suited for the primary analysis in clinical trials, and one route for sensitivity analysis is to consider, next to selection models, pattern-mixture models or shared-parameter models, a latent-class mixture model.
Abstract: In the analyses of incomplete longitudinal clinical trial data, there has been a shift, away from simple methods that are valid only if the data are missing completely at random, to more principled ignorable analyses, which are valid under the less restrictive missing at random assumption. The availability of the necessary standard statistical software nowadays allows for such analyses in practice. While the possibility of data missing not at random (MNAR) cannot be ruled out, it is argued that analyses valid under MNAR are not well suited for the primary analysis in clinical trials. Rather than either forgetting about or blindly shifting to an MNAR framework, the optimal place for MNAR analyses is within a sensitivity-analysis context. One such route for sensitivity analysis is to consider, next to selection models, pattern-mixture models or shared-parameter models. The latter can also be extended to a latent-class mixture model, the approach taken in this article. The performance of the so-obtained flexible model is assessed through simulations and the model is applied to data from a depression trial.

Summary (3 min read)

1 Introduction

  • Repeated measures are often prone to incompleteness, often taking the dropout form.
  • Since one can never be certain about the dropout mechanism, certain assumptions have to be made.
  • A non-response process is missing completely at random (MCAR) if the missingness is independent of both unobserved and observed data, and missing at random (MAR) if, conditional on the observed data, the missingness is independent of the unobserved measurements.
  • Information from the location and evolution of the response profiles, a selection model concept, and from the dropout patterns, a pattern-mixture idea, is used simultaneously to define latent groups and variables, a shared- 1 parameter feature.
  • Second, apart from providing a more flexible modeling tool, there is room for use as a sensitivity analysis instrument.

2 Latent-Class Mixture Models

  • In principle, one would like to consider the density of the full data f(yi, di|θ,ψ), where the parameter vectors θ and ψ describe the measurement and missingness processes, respectively.
  • The measurement process as well as the dropout process depend on this latent variable, directly and through the subject-specific effects bi.
  • The authors then assume that gij(wij, bi, qik) satisfies logit[gij(wij, bi, qik)] = wijγk + λbi.
  • Not all models that can be formulated in this way are identified, so restrictions might be needed.

3 Likelihood Function and Estimation

  • Estimation of the unknown parameters in the latent-class mixture model described in the previous section will be based on maximum likelihood.
  • Böhning (1999) shows that a mixture of two normals with simultaneously different means and different variances is not identifiable.
  • In line with Böhning (1999) and McLachlan and Peel (2000), one could consider several variations to the target model.
  • Maximizing `(Ω|yo,d, q), the corresponding log-likelihood, is easier than maximizing the loglikelihood `(Ω|yo,d).
  • Denote the expected log-likelihood function, the so-called objective function, by O. The EM algorithm is initiated by means of an initial value Ω(0), after which one oscillates between the E- and M-steps, until convergence.

4 Classification

  • One can also classify the subjects into the different latent subgroups of the fitted model.
  • In certain cases such latent groups can have substantive meaning.
  • A scenario would be that two or more posterior probabilities are almost equal, of which one is the maximum of all posterior probabilities for that particular subject.
  • This makes classification nearly random and misclassification is likely to occur.
  • Therefore, rather than merely considering the classification of subjects into the latent subgroups, it is instructive to inspect the posterior probabilities in full.

5 Simulation Study

  • An advantage of the latent-class mixture model is its flexible structure, making the model a helpful tool for analyzing incomplete longitudinal data.
  • To assess whether this disadvantage counterbalances the advantage of model flexibility, and to assess performance, the authors conduct a simulation study.
  • Section 5.1 describes a simplification of the latent-class mixture model used in the simulations and in the application in Section 6.
  • The design and results of the simulation study are given in Sections 5.2 and 5.3, respectively.

5.1 A Simplification of the Latent-Class Mixture Model

  • The matrices Σ (k) i usually depend on i only through the dimension of the response vector for subject i, while parameters are common to all.
  • The authors further simplify the model in two steps.
  • First, it is assumed that there is only one subject-specific effect bi, a shared intercept, influencing the measurement process, not the dropout process.
  • Second, the measurement process is assumed to depend on the latent variable, not in a direct way, but only through the shared intercept.

5.2 Design of the Simulation Study

  • The authors simulated 250 datasets, each containing measurements and covariate information of 100 subjects.
  • In line with Section 5.1, the 9 variances of these two normal distributions are assumed to be equal and are denoted by d2.
  • While only the measurement error variance is increased in the second setting, σ = 0.75, both variance parameters are increased in the third setting, d = 3.5 and σ = 1.00.
  • Up to the third setting, the chosen parameters result in a bimodal, well-separated mixture distribution.
  • Finally, in the dropout model, the logistic regression is based on an intercept only, which differs for both latent classes, namely, γ1 = −2.5 and γ2 = −1.25, respectively, with corresponding probabilities 0.73 and 0.45 of completing the study.

5.3 Results of the Simulation Study

  • Table 1 contains the results of the simulation study.
  • So, the authors can conclude the model fits the data well, even with a larger within-subject variability.
  • In the penultimate simulation setting, not only the measurement error variance is increased, but also the variance in the mixture components.
  • Bias and MSE values remain small, the order of magnitude not exceeding 10−1 and 10−3, respectively.
  • Thus the latent-class mixture model does fit the simulated data well.

6 Analysis of Depression Trial Data

  • The authors apply the latent-class mixture model to a depression trial, arising from a randomized, double-blind psychiatric clinical trial, conducted in the United States.
  • The primary objective of this trial was to compare the efficacy of an experimental anti-depressant with a nonexperimental one.
  • In these retrospective analyses, data from 170 patients are considered.
  • The Hamilton Depression Rating Scale (HAMD17) is used to measure the depression status of the patients.
  • In the two subsequent sections, a latent-class mixture model is fitted to the depression trial and a sensitivity analysis performed.

6.1 Formulating a Latent-Class Mixture Model

  • The latent-class mixture model framework is used to analyze the depression trial, assuming the patients can be split into g latent subgroups.
  • Table 2 shows that when assuming dropout model (6), AIC opts for the model with two 13 latent subgroups (Model 2), whereas BIC gives preference to the shared-parameter model (Model 1).
  • Parameter estimates with corresponding standard errors and p-values of the two-component latent-class mixture model are shown in Table 3.
  • A more formal comparison of both latent groups regarding their change of HAMD17 score versus baseline confirms this association between the classification and the profile over time.
  • The first latent group mainly contains patients who complete the study, 62 in total.

6.2 A Sensitivity Analysis

  • In addition to the two-component latent-class mixture model shown in Section 6.1, a classical shared-parameter model will be fitted to the depression trial, as well as a pattern-mixture model, and two selection models, based on the selection models introduced by Diggle and Kenward (1994).
  • Next, the Diggle-Kenward (DK) model combines a multivariate normal model for the measurement process with a logistic regression model for the dropout process.
  • Since the main interest of the depression trial was in the treatment effect at the last visit, Table 5 shows the estimates, standard errors, and p-values for this effect under the five fitted 17 models.
  • Note that using both the two-component latent-class mixture model and the classical shared-parameter model, the standard error is reduced by 0.3 units, compared to either selection model, or pattern-mixture model, resulting in a more accurate confidence interval for the treatment effect at the last visit.
  • The p-values are clearly moving around the significance level of 0.05.

7 Concluding Remarks

  • Through its structure, the model captures unobserved heterogeneity between latent subgroups of the population.
  • As shown in the simulation study, the flexibility of such latent-class mixture models outweighs the expected modelling complexity.
  • Of course, care has to be taken when interpreting latent classes, since in some applications they may merely be artifacts, without any substantive grounds.
  • This is a tricky but well documented problem (McLachlan and Peel 2000).
  • Details on starting value selection are embedded in an electronically available companion manual.

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Made available by Hasselt University Library in https://documentserver.uhasselt.be
A latent-class mixture model for incomplete longitudinal Gaussian data
Peer-reviewed author version
BEUNCKENS, Caroline; MOLENBERGHS, Geert; VERBEKE, Geert & Mallinckrodt,
Craig (2008) A latent-class mixture model for incomplete longitudinal Gaussian data.
In: BIOMETRICS, 64(1). p. 96-105.
DOI: 10.1111/j.1541-0420.2007.00837.x
Handle: http://hdl.handle.net/1942/9518

A Latent-Class Mixture Model For
Incomplete Longitudinal Gaussian Data
Caroline Beunckens,
1,
Geert Molenberghs,
1
Geert Verbeke,
2
and Craig Mallinckrodt
3
1
Center for Statistics, Hasselt University, Agoralaan 1, 3590 Diepenbeek, Belgium.
2
Biostatistical Centre, Catholic University of Leuven,
Kapucijnenvoer 35, 3000 Leuven, Belgium.
3
Eli Lilly & Company, Lilly Corporate Center, Indianapolis, IN 46285, U.S.A.
email: caroline.beunckens@uhasselt.be
Summary. In the analyses of incomplete longitudinal clinical trial data, there
has been a shift, away from simple methods that are valid only if the data are
missing completely at random (MCAR), to more p rincipled ignorable analyses,
which are valid under the less restrictive missing at random (MAR) assumption.
The availability of the necessary standard statistical software nowadays allows
for such analyses in practice. While the possibility of data missing not at random
(MNAR) cannot be ruled out, it is argued that analyses valid under MNAR are
not well suited for the primary analysis in clinical trials. Rather than either
forgetting about or blindly shifting to an MNAR framework, th e optimal place
for MNAR analyses is within a sensitivity analysis context. One such route
for sensitivity analysis is to consider, next to selection models, pattern-mixture
models or shared-parameter models. The latter can also be extended to a latent-
class mixture model, the route taken in this paper. The so-obtained flexible model
is submitted to the test in simulations and applied to data from a depression trial.
Key Words: Latent class, Nonrandom missingness, Random effect, S hared
parameter
0

1 Introduction
Repeated measures are often prone to incompleteness, often taking the dropout form. The
nature of the dropout mechanism can affect inference. Since one can never be certain about
the dropout mechanism, certain assumptions have to be made. We will use the terminol-
ogy introduced by Rubin (1976). A non-response process is missing completely at random
(MCAR) if the missingness is independent of both unobserved and observed data, and miss-
ing at random (MAR) if, conditional on the observed data, the missingness is independent
of the unobserved measurement s. A process that is neither MCAR nor MAR is non-random
(MNAR). For likelihood inference, and when the parameters describing the measurement
process are functionally independent of the p arameters describing the missingness process,
MCAR and MAR are ignorable, in which case the missingness process can be ignored when
interest is in inference for the longitudinal pro cess only. For frequentist inference, the stronger
condition of MCAR is required for ignorability.
Recently, there has been a move away from simple methods, such as complete case analysis
and last observation carried forward analysis, towards such likelihood-based methods as
mixed models (Molenberghs et al., 2004; Jansen et al., 2006). Of course, since non-random
methods allow the missingness to depend on the unobserved or missing values, it is clear that
the MNAR assumption is not verifiable (Laird, 1994; Molenberghs, Kenward and Lesaffre,
1997). Therefore, fitting a single MNAR model will not be trustworthy, which does not mean
we should ignore such models but rather frame them within a sensitivity analysis. So far, we
have used selection-model concepts. Alternatively, pattern-mixture and shared-parameter
models can be used. They are introduced in the next section.
We propose a latent-class mixture model, bringing together features of the selection, pattern-
mixture, and shared-parameter model frameworks. Information from the location and evo-
lution of the response profiles, a selection model concept, and from the dropout patterns, a
pattern-mixture idea, is used simultaneously to define latent groups and variables, a shared-
1

parameter feature. This brings several appealing features. First, one uses information in a
more symmetric, elegant way. Second, apart from providing a more flexible modeling tool,
there is room for use as a sensitivity analysis instrument. Third, a strong advantage over
existing methods is that we are able to classify subjects into latent groups. If done with
due caut ion, it can enhance substantive knowledge and generate hypotheses. Fourth, while
computational burden increases, fitting the proposed method is remarkably stable and ac-
ceptable in terms of computation time for the settings considered here. Clearly, neither the
proposed model nor any other alternative can be seen as a tool to definitively test for MAR
versus MNAR, as amply documented in the sensitivity analysis literature. This is why the
method’s use predominantly lies within the sensitivity analysis context. Such a sensitivity
analysis is of use both when it modifies the results of a simpler analysis, for further scrutiny,
as well as when it confirms these.
The general latent-class mixture model is presented in Section 2. A simulation study is
reported in Section 5. In Section 6, data from a depression clinical trial are analyzed using
a latent-class mixture model within a sensitivity analysis.
2 Latent-Class Mixture Models
Let Y
ij
denote the response for the ith individual, designed to be measured at time t
ij
,
i = 1, . . . , N, j = 1, . . . , n
i
. Group the outcomes into a vector Y
i
= (Y
i1
, . . . , Y
in
i
)
0
. Define
a dropout indicator D
i
for the occasion at which dropout occurs, with the convention that
D
i
= n
i
+ 1 for a complete sequence. Split the vector Y
i
into observed (Y
o
i
) and missing
(Y
m
i
) components, respectively.
In principle, one would like to consider the density of the full data f(y
i
, d
i
|θ, ψ), where
the parameter vectors θ and ψ describe the measurement and missingness processes, respec-
tively. Covariates are allowed but suppressed from notation. This full density function can be
factorized in different ways. The selection model framework is based on (Rubin, 1976; Little
2

and Rubin, 1987): f(y
i
, d
i
|θ, ψ) = f(y
i
|θ)f(d
i
|y
i
, ψ). The first factor is the marginal den-
sity of the measurement process and the second one is the density of the missingness process,
conditional on the outcomes. Alternatively, one can consider pattern-mixture models (Little,
1993, 1994) using the reversed factorization f(y
i
, d
i
|θ, ψ) = f(y
i
|d
i
, θ)f(d
i
|ψ). This can be
seen as a mixture of different populations, characterized by the observed pattern of missing-
ness. Instead of using the selection modelling or pattern-mixture modelling framework, the
measurement and the dropout process can be jointly modeled by using a shared-parameter
model (Wu and Carrol, 1988; Wu and Bailey, 1989; Ten Have et al., 1998). These methods
assume there exists a vector of random effects b
i
, conditional upon which the measure-
ment and dropout processes are independent: f (y
i
, d
i
|b
i
, θ, ψ) = f(y
i
|b
i
, θ)f(d
i
|b
i
, ψ). We
propose an extension, capturing possible heterogeneity between the subjects not measured
through covariates but rather through a latent variable. We call this model a latent-class
mixture model. Next to one or more so-called shared parameters, b
i
, the model contains
a latent variable, Q
i
, dividing the population into g subgroups. This latent variable is a
vector of group indicators Q
i
= (Q
i1
, . . . , Q
ig
), defined as Q
ik
= 1, if subject i belongs to
group k, and 0 otherwise. The measurement process as well as the dropout process depend
on this latent variable, directly and through the subject-specific effects b
i
. The distribution
of Q
i
is multinomial and defined by P (Q
ik
= 1) = π
k
(k = 1, . . . , g), where π
k
denotes the
group or component probability, also termed prior probabilities of the components. These
are restricted through
P
g
k=1
π
k
= 1.
The measurement process will be modeled by a heterogeneity linear mixed model (Verbeke
and Lesaffre, 1996; Verbeke and Molenberghs, 2000): Y
i
|q
ik
= 1, b
i
N (X
i
β
k
+Z
i
b
i
, Σ
(k)
i
),
where X
i
and Z
i
are design matrices, β
k
component-dependent fixed effects, b
i
denote the
shared parameters, following a mixture of g normal distributions with mean vectors µ
k
and
covariance matrices D
k
, i.e., b
i
|q
ik
= 1 N (µ
k
, D
k
) and thus b
i
P
g
k=1
π
k
N(µ
k
, D
k
).
The measurement error terms ε
i
follow a normal distribution with mean zero and covariance
3

Citations
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Journal ArticleDOI
01 May 2009-Test
TL;DR: Elements of taxonomy include: missing data patterns, mechanisms, and modeling frameworks; inferential paradigms; and sensitivity analysis frameworks; andensitivity analysis frameworks are described in detail.
Abstract: Incomplete data are quite common in biomedical and other types of research, especially in longitudinal studies. During the last three decades, a vast amount of work has been done in the area. This has led, on the one hand, to a rich taxonomy of missing-data concepts, issues, and methods and, on the other hand, to a variety of data-analytic tools. Elements of taxonomy include: missing data patterns, mechanisms, and modeling frameworks; inferential paradigms; and sensitivity analysis frameworks. These are described in detail. A variety of concrete modeling devices is presented. To make matters concrete, two case studies are considered. The first one concerns quality of life among breast cancer patients, while the second one examines data from the Muscatine children’s obesity study.

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Abstract: Most statistical developments in the joint modelling area have focused on the shared random-effect models that include characteristics of the longitudinal marker as predictors in the model for the time-to-event. A less well-known approach is the joint latent class model which consists in assuming that a latent class structure entirely captures the correlation between the longitudinal marker trajectory and the risk of the event. Owing to its flexibility in modelling the dependency between the longitudinal marker and the event time, as well as its ability to include covariates, the joint latent class model may be particularly suited for prediction problems. This article aims at giving an overview of joint latent class modelling, especially in the prediction context. The authors introduce the model, discuss estimation and goodness-of-fit, and compare it with the shared random-effect model. Then, dynamic predictive tools derived from joint latent class models, as well as measures to evaluate their dynamic pre...

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Journal ArticleDOI
TL;DR: 2 classic MNAR modeling approaches for longitudinal data are described: the selection model and the pattern mixture model, which are now quite easy to estimate in popular structural equation modeling programs, particularly Mplus.
Abstract: The past decade has seen a noticeable shift in missing data handling techniques that assume a missing at random (MAR) mechanism, where the propensity for missing data on an outcome is related to other analysis variables. Although MAR is often reasonable, there are situations where this assumption is unlikely to hold, leading to biased parameter estimates. One such example is a longitudinal study of substance use where participants with the highest frequency of use also have the highest likelihood of attrition, even after controlling for other correlates of missingness. There is a large body of literature on missing not at random (MNAR) analysis models for longitudinal data, particularly in the field of biostatistics. Because these methods allow for a relationship between the outcome variable and the propensity for missing data, they require a weaker assumption about the missing data mechanism. This article describes 2 classic MNAR modeling approaches for longitudinal data: the selection model and the pattern mixture model. To date, these models have been slow to migrate to the social sciences, in part because they required complicated custom computer programs. These models are now quite easy to estimate in popular structural equation modeling programs, particularly Mplus. The purpose of this article is to describe these MNAR modeling frameworks and to illustrate their application on a real data set. Despite their potential advantages, MNAR-based analyses are not without problems and also rely on untestable assumptions. This article offers practical advice for implementing and choosing among different longitudinal models.

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Abstract: This article uses a general latent variable framework to study a series of models for nonignorable missingness due to dropout. Nonignorable missing data modeling acknowledges that missingness may depend not only on covariates and observed outcomes at previous time points as with the standard missing at random assumption, but also on latent variables such as values that would have been observed (missing outcomes), developmental trends (growth factors), and qualitatively different types of development (latent trajectory classes). These alternative predictors of missing data can be explored in a general latent variable framework with the Mplus program. A flexible new model uses an extended pattern-mixture approach where missingness is a function of latent dropout classes in combination with growth mixture modeling. A new selection model not only allows an influence of the outcomes on missingness but allows this influence to vary across classes. Model selection is discussed. The missing data models are applied to longitudinal data from the Sequenced Treatment Alternatives to Relieve Depression (STAR*D) study, the largest antidepressant clinical trial in the United States to date. Despite the importance of this trial, STAR*D growth model analyses using nonignorable missing data techniques have not been explored until now. The STAR*D data are shown to feature distinct trajectory classes, including a low class corresponding to substantial improvement in depression, a minority class with a U-shaped curve corresponding to transient improvement, and a high class corresponding to no improvement. The analyses provide a new way to assess drug efficiency in the presence of dropout.

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This paper proposed a latent-class mixture model, which combines features of the selection, pattern-mixture, and shared-parameter model frameworks.