A hidden Markov model for informative dropout in longitudinal response data with crisis states

Abstract

We adopt a hidden state approach for the analysis of longitudinal data subject to dropout. Motivated by two applied studies, we assume that subjects can move between three states: stable, crisis, dropout. Dropout is observed but the other two states are not. During a possibly transient crisis state both the longitudinal response distribution and the probability of dropout can differ from those for the stable state. We adopt a linear mixed effects model with subject-specific trajectories during stable periods and additional random jumps during crises. We place the model in the context of Rubin’s taxonomy and develop the associated likelihood. The methods are illustrated using the two motivating examples.

Introduction

Longitudinal studies often suffer attrition, in that individuals drop out of the study before the scheduled completion time, and thus present incomplete data. A variety of methods have by now been developed for dealing with the possibility that dropout is related to responses (Hogan et al., 2004, Molenberghs et al., 2004, Philipson et al., 2008, Tsiatis and Davidian, 2004), though caution in using such methods is always needed (Molenberghs et al., 2004, Molenberghs et al., 2008).

Recently le Cessie et al. (2009) recognised that longitudinal data analysis can be complicated by the fact that during follow-up, subjects can change condition or state, examples being remission, relapse and death for cancer patients. Both longitudinal responses and the dropout probability can depend on the current state and this needs to be accounted for in analysis. The methods developed by le Cessie et al. (2009) are appropriate when the underlying state is observed. If a state is defined by the level of a response variable but obscured by measurement error, then the hidden Markov methods of Satten and Longini (1996) or Guihenneuc-Jouyaux et al. (2000) can form a basis for analysis. But, as argued by Liestøl and Andersen (2002), there are circumstances where a subject’s state is either hidden or vaguely defined. For the liver cirrhosis application considered by Liestøl and Andersen (2002) for example, some subjects experienced apparent “crises”, marked by a sudden change in response values. These crises could be transient or could indicate a terminal disease stage.

In this work we build on the ideas of le Cessie et al. (2009) and Liestøl and Andersen (2002) and develop a hidden state modelling approach for longitudinal data subject to dropout. We assume that during follow-up subjects can experience different states, which we will think of as a stable state, a crisis state and dropout. The first two states are transient and reversible, while the third, dropout, is an absorbing state. The crisis state can be defined as an intermediate phase where significant changes of the response values can be observed and where the probability of dropout is increased. We assume that the longitudinal response is associated with the underlying state but we assume that states other than dropout are not directly observed, and perhaps not precisely defined. We exclude situations where the state is defined by the response, such as for AIDS when the CD4 T-cell first reaches a given level, and leukaemia relapse when an residual leukaemic cell count is over a defined threshold (De Lorenzo et al., 2005).

In Section 2 we provide brief details of two applications which motivated our work. Our model is introduced and the estimation outlined in Section 3, where we also argue for the merits of proper treatment of time ordering when considering missingness mechanisms for longitudinal data. Section 4 includes summaries of our analyses of the motivating data sets, and some brief comments in Section 5 conclude the paper.