A reanalysis of a longitudinal scleroderma clinical trial using non-ignorable missingness models

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Abstract

When analyzing incomplete longitudinal clinical trial data, it is often inappropriate to assume that the occurrence of missingness is at random, especially in cases where visits are entirely missed. We present a framework that simultaneously models multivariate incomplete longitudinal data and a non-ignorable missingness mechanism using a Bayesian approach. A criterion measure is presented for comparing models. We demonstrate the feasibility of the methodology through reanalysis of two of the longitudinal measures from a clinical trial of penicillamine treatment for scleroderma patients. We compare the results for univariate and bivariate, ignorable and non-ignorable missingness models.

Introduction

Contemporary clinical trial data commonly consist of multivariate longitudinal measurements of primary and secondary outcomes. These measures tend to be incompletely recorded due to missed values within a visit, intermittent missed visits, or dropout of subjects. If the missingness occurs at random (MAR), as described in Little and Rubin (2002), then the mixed effect linear models approach (e.g. Laird and Ware, 1982) is a common analysis technique and will give sensible inference when the mean and covariance models are appropriate. However, it is well known that the MAR mechanism, especially for dropout, is often strongly violated. Much recent research has gone into developing inferential techniques for non-ignorable missingness models and especially non-ignorable dropout models.

Likelihood based methods for non-ignorable missing data problems require specification of the joint distribution of the response outcome and the missing data mechanism. Current methods for handling non-ignorable missing data include selection models, pattern mixture models, and shared parameter models.

Selection models (Diggle and Kenward, 1994) factor the joint distribution of response and missing into the marginal distribution of response times the conditional distribution of missingness given response. The term selection model comes from the econometric literature and it can be seen that a subject's missing data are selected through the probability model, given their measurements, whether observed or not. An appealing feature of selection models is that they directly model the marginal distribution of response, which is of interest with complete data. However, selection models are potentially sensitive to distributional assumptions about the missing data, which are not testable given the observed data. Recent references on selection models include Ibrahim et al. (2001), Carpenter et al. (2002), Scharfstein et al. (2003), Molenberghs et al. (2001), Molenberghs and Thijs (2003), Troxel et al. (1998), and Albert and Follman (2003).

In pattern mixture models (Little, 1993), the population is stratified by the pattern of missingness. Subjects with the same pattern of missingness are assumed to share a common response distribution. The observed data is a mixture of these response models with weights equal to the probability of the missingness patterns. Pattern mixture models avoid strong assumptions about the mechanism of missingness and do not suffer from the sensitivity to distributional assumptions seen in selection models. However, in clinical studies, the number of missing data patterns might be large and some patterns may have very few subjects requiring restrictive assumptions for parameter estimation. Extensive reviews of pattern mixture models are given by Little (1995) and Verbeke and Molenberghs (2000). Recent articles discussing pattern mixture models for longitudinal data include Daniels and Hogan (2000), Demirtas and Schafer (2003), Fitzmaurice and Laird (2001), Guo et al. (2004), Hedeker and Gibbons (1997), Lin et al. (2004), Roy (2003), and Thijs et al. (2002). Articles by Glynn et al. (1986) and Michiels et al. (2002) contrast selection and pattern-mixture models.

Shared parameter models (Wu and Carroll, 1988, Wu and Bailey, 1988, De Gruttola and Tu, 1994, Follman and Wu, 1995, Ten Have et al., 1998) assume the probability of dropout depends on the underlying trajectory of response. The longitudinal outcomes are assumed to be independent of the dropout process conditional on the random effects.

In this paper, we will focus our attention on selection models as they have both strengths and weaknesses relative to pattern mixture models, and, in our experience, are particularly interpretable and understandable to our scientific collaborators. The approach described in this paper is relatively straightforward to generalize to shared parameter models, but we will not discuss this in what follows. In the context of univariate continuous longitudinal data, Diggle and Kenward (1994) proposed a selection model for non-ignorable dropout with the missingness mechanism modeled as a logistic regression of dropout indicator on previous values and current (possibly unobserved) value. They used an EM algorithm to obtain maximum likelihood estimates. Many authors have since extended this methodology to the Bayesian setting; Carpenter et al. (2002) is particularly noteworthy as it includes WinBUGS (Spiegelhalter et al., 1995) code to implement their model.

In Section 2, we extend this methodology to simultaneously model multivariate incomplete longitudinal data and the non-ignorable missingness mechanism. In order to compare models, we derive a model selection criterion in Section 3. The methodology is illustrated through reanalysis of clinical trial data from a study of penicillamine treatment for scleroderma patients (Clements et al., 1999) in Section 4. We employ two of the longitudinal measures (the primary outcome, total skin score, and the secondary outcome, Health Assessment Questionnaire disability index) for assessing the outcome of the study. We compare the results for univariate and bivariate non-ignorable missingness models, summarize our results, and present a sensitivity analysis investigating the effect of prior distribution choices on inference. Conclusions and directions for future work are given in Section 5.

Section snippets

Selection models for analyzing incomplete multivariate longitudinal data

We assume the following general selection model for the hypothetical complete set of q measurement vectors (each of length t) on the ith subject, yi=(yi1T,,yiqT)Tyi|β,αi,ΣNtq(Xiβ+Ziαi,Σy),αiNs(0,Σα),p(β,Σy,Σα)=p(β)p(Σy)p(Σα),where i=1,,n indexes subjects, β are “fixed” effects with prior distribution p(β), αi are s-vectors of “random” effects, Σy is the variance–covariance matrix of the data discrepancies from the subject-specific means with prior distribution p(Σy), and Σα is the

Full-sample log score

A variety of model fit measures have been developed for Bayesian inference, including Bayes factors, the Deviance Information Criterion (DIC), and the log score. A recent article by Draper and Krnjajic (2005) discusses the relationship between these three approaches. They define the cross-validated log score for a model M as LSCV(M|d)=1ni=1nlogp(di|d-i,M),where d=(d1,,dn) are the (observed) data on n subjects, d-i is d with the ith subject's data removed, and p(di|d-i,M) is the posterior

Data description

Clements et al. (1999) conducted a randomized controlled trial of high-dose (1000 mg daily) versus low-dose (125 mg every other day) penicillamine for the treatment of patients with early diffuse scleroderma (SSc). Early diffuse SSc was defined as cutaneous induration (i.e. hardening of the skin) proximal, as well as distal, to the elbows and knees, with or without face involvement. This trial was motivated because of continuing uncertainty about penicillamine's effectiveness and toxicity in SSc.

Discussion and future directions

We have presented a model for multivariate longitudinal data with a non-ignorable missingness mechanism. We have also generalized the full-sample log-score measure to this situation as an aid in choosing an appropriate model, which is especially important in applied work as many alternative models for the mean structure, random effects structure, and missingness structure must be evaluated. A reanalysis of bivariate data from a longitudinal penicillamine trial in scleroderma patients showed

Acknowledgments

All three authors were partially supported by a grant award from the Scleroderma Foundation. WJB was also partially supported by NIH Grants MH60213, NS30308, and AI28697, and WKW by GM72876. The authors would like to give special thanks to Philip Clements, MD in the UCLA School of Medicine for allowing us to use the data in Section 4.

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