A multistate Markov chain model for longitudinal, categorical quality-of-life data subject to non-ignorable missingness

Reduced rank multinomial logistic regression in Markov chains with application to cognitive data

Finite Markov chains are useful tools for studying transitions among health states; these chains can be complex consisting of a mix of transient and absorbing states. The transition probabilities, which are often affected by covariates, can be difficult to estimate due to the presence of many covariates and/or a subset of transitions that are rarely observed. The purpose of this article is to show how to estimate the effect of a subset of covariates of interest after adjusting for the presence of multiple other covariates by applying multidimensional dimension reduction to the latter. The case in which transitions within each row of the one-step transition probability matrix are estimated by multinomial logistic regression is discussed in detail. Dimension reduction for the adjustment covariates involves estimating the effect of the covariates by a product of matrices iteratively; at each iteration one matrix in the product is fixed while the second is estimated using either standard software or nonlinear estimation, depending on which of the matrices in the product is fixed. The algorithm is illustrated by an application where the effect of at least one Apolipoprotein-E (APOE) gene ϵ 4 allele on transition probability is estimated in a Markov Chain that includes adjustment for eight covariates and focuses on transitions from normal cognition to several forms of mild cognitive impairment, with possible absorption into dementia. Data were drawn from annual cognitive assessments of 649 participants enrolled in the BRAiNS cohort at the University of Kentucky's Alzheimer's Disease Research Center.

Estimating the number and length of episodes in disability using a Markov chain approach

Background

Markov models are a key tool for calculating expected time spent in a state, such as active life expectancy and disabled life expectancy. In reality, individuals often enter and exit states recurrently, but standard analytical approaches are not able to describe this dynamic. We develop an analytical matrix approach to calculating the expected number and length of episodes spent in a state.

Methods

The approach we propose is based on Markov chains with rewards. It allows us to identify the number of entries into a state and to calculate the average length of episodes as total time in a state divided by the number of entries. For sampling variance estimation, we employ the block bootstrap. Two case studies that are based on published literature illustrate how our methods can provide new insights into disability dynamics.

Results

The first application uses a classic textbook example on prednisone treatment and liver functioning among liver cirrhosis patients. We replicate well-known results of no association between treatment and survival or recovery. Our analysis of the episodes of normal liver functioning delivers the new insight that the treatment reduced the likelihood of relapse and extended episodes of normal liver functioning. The second application assesses frailty and disability among elderly people. We replicate the prior finding that frail individuals have longer life expectancy in disability. As a novel finding, we document that frail individuals experience three times as many episodes of disability that were on average twice as long as the episodes of nonfrail individuals.

Conclusions

We provide a simple analytical approach for calculating the number and length of episodes in Markov chain models. The results allow a description of the transition dynamics that goes beyond the results that can be obtained using standard tools for Markov chains. Empirical applications using published data illustrate how the new method is helpful in unraveling the dynamics of the modeled process.

Flexible parametric multistate modelling of employment history

A multistate model is used to describe employment history. Transition-specific rates are defined using generalized gamma distributions and Gompertz distributions. This flexible parametric modelling of the rate of change is combined with latent classes for unobserved propensity to change jobs. The propensity is described by two latent classes which can be interpreted as consisting of movers and stayers. The modelling is illustrated by analysing longitudinal data from the German Life History Study.

A Latent Transition Analysis Model for Latent-State-Dependent Nonignorable Missingness

Psychologists often use latent transition analysis (LTA) to investigate state-to-state change in discrete latent constructs involving delinquent or risky behaviors. In this setting, latent-state-dependent nonignorable missingness is a potential concern. For some longitudinal models (e.g., growth models), a large literature has addressed extensions to accommodate nonignorable missingness. In contrast, little research has addressed how to extend the LTA to accommodate nonignorable missingness. Here we present a shared parameter LTA that can reduce bias due to latent-state-dependent nonignorable missingness: a parallel-process missing-not-at-random (MNAR-PP) LTA. The MNAR-PP LTA allows outcome process parameters to be interpreted as in the conventional LTA, which facilitates sensitivity analyses assessing changes in estimates between LTA and MNAR-PP LTA. In a sensitivity analysis for our empirical example, previous and current membership in high-delinquency states predicted adolescents' membership in missingness states that had high nonresponse probabilities for some or all items. A conventional LTA overestimated the proportion of adolescents ending up in a low-delinquency state, compared to an MNAR-PP LTA.

Time to health-related quality of life score deterioration as a modality of longitudinal analysis for health-related quality of life studies in oncology: do we need RECIST for quality of life to achieve standardization?

Purpose

Longitudinal analysis of health-related quality of life (HRQoL) remains unstandardized and compromises comparison of results between trials. In oncology, despite available statistical approaches, results are poorly used to change standards of care, mainly due to lack of standardization and the ability to propose clinical meaningful results. In this context, the time to deterioration (TTD) has been proposed as a modality of longitudinal HRQoL analysis for cancer patients. As for tumor response and progression, we propose to develop RECIST criteria for HRQoL.

Methods

Several definitions of TTD are investigated in this paper. We applied this approach in early breast cancer and metastatic pancreatic cancer with a 5-point minimal clinically important difference. In breast cancer, TTD was defined as compared to the baseline score or to the best previous score. In pancreatic cancer (arm 1: gemcitabine with FOLFIRI.3, arm 2: gemcitabine alone), the time until definitive deterioration (TUDD) was investigated with or without death as event.

Results

In the breast cancer study, 381 women were included. The median TTD was influenced by the choice of the reference score. In pancreatic cancer study, 98 patients were enrolled. Patients in Arm 1 presented longer TUDD than those in Arm 2 for most of HRQoL scores. Results of TUDD were slightly different according to the definition of deterioration applied.

Conclusion

Currently, the international ARCAD group supports the idea of developing RECIST for HRQoL in pancreatic and colorectal cancer with liver metastasis, with a view to using HRQoL as a co-primary endpoint along with a tumor parameter.

Electronic supplementary material

The online version of this article (doi:10.1007/s11136-013-0583-6) contains supplementary material, which is available to authorized users.

Parametric and nonparametric analysis of life courses: an application to family formation patterns

We discuss a unified approach to the description and explanation of life course patterns represented as sequences of states observed in discrete time. In particular, we study life course data collected as part of the Dutch Fertility and Family Surveys (FFS) to learn about the family formation behavior of 1,897 women born between 1953 and 1962. Retrospective monthly data were available on each 18- to 30-year-old woman living either with or without children as single, married, or cohabiting. We first study via a nonparametric approach which factors explain the pairwise dissimilarities observed between life courses. Permutation distribution inference allows for the study of the statistical significance of the effect of a set of covariates of interest. We then develop a parametric model for the sequence-generating process that can be used to describe state transitions and durations conditional on covariates and conditional on having observed an initial segment of the trajectory. Fitting of the proposed model and the corresponding model selection process are based on the observed data likelihood. We discuss the application of the methods to the FFS.

Multi-stage transitional models with random effects and their application to the Einstein aging study

Longitudinal studies of aging often gather repeated observations of cognitive status to describe the development of dementia and to assess the influence of risk factors. Clinical progression to dementia is often conceptualized by a multi-stage model of several transitions that synthesizes time-varying effects. In this study, we assess the influence of risk factors on the transitions among three cognitive status: cognitive stability (normal cognition for age), memory impairment, and clinical dementia. We have developed a shared random effects model that not only links the propensity of transitions and to the probability of informative missingness due to death, but also incorporates heterogeneous transition between subjects. We evaluate four approaches using generalized logit and four using proportional odds models to the first-order Markov transition probabilities as a function of covariates. Random effects were incorporated into these models to account for within-subject correlations. Data from the Einstein Aging Study are used to evaluate the goodness-of-fit of these models using the Akaike information criterion. The best fitting model for each type (generalized logit and proportional odds) is recommended and their results are discussed in more details.

Multi-state Markov models for disease progression in the presence of informative examination times: an application to hepatitis C

In many chronic diseases it is important to understand the rate at which patients progress from infection through a series of defined disease states to a clinical outcome, e.g. cirrhosis in hepatitis C virus (HCV)-infected individuals or AIDS in HIV-infected individuals. Typically data are obtained from longitudinal studies, which often are observational in nature, and where disease state is observed only at selected examinations throughout follow-up. Transition times between disease states are therefore interval censored. Multi-state Markov models are commonly used to analyze such data, but rely on the assumption that the examination times are non-informative, and hence the examination process is ignorable in a likelihood-based analysis. In this paper we develop a Markov model that relaxes this assumption through the premise that the examination process is ignorable only after conditioning on a more regularly observed auxiliary variable. This situation arises in a study of HCV disease progression, where liver biopsies (the examinations) are sparse, irregular, and potentially informative with respect to the transition times. We use additional information on liver function tests (LFTs), commonly collected throughout follow-up, to inform current disease state and to assume an ignorable examination process. The model developed has a similar structure to a hidden Markov model and accommodates both the series of LFT measurements and the partially latent series of disease states. We show through simulation how this model compares with the commonly used ignorable Markov model, and a Markov model that assumes the examination process is non-ignorable.

Estimating stroke-free and total life expectancy in the presence of non-ignorable missing values

A continuous time three-state model with time-dependent transition intensities is formulated to describe transitions between healthy and unhealthy states before death. By using time continuously, known death times can be taken into account. To deal with possible non-ignorable missing states, a selection model is proposed for the joint distribution of both the state and whether or not the state is observed. To estimate total life expectancy and its subdivision into life expectancy in health and ill health, the three-state model is extrapolated beyond the follow-up of the study. Estimation of life expectancies is illustrated by analysing data from a longitudinal study of aging where individuals are in a state of ill health if they have ever experienced a stroke. Results for the selection model are compared with results for a model where states are assumed to be missing at random and with results for a model that ignores missing states.

Estimation of average treatment effect with incompletely observed longitudinal data: application to a smoking cessation study

We study the problem of estimation and inference on the average treatment effect in a smoking cessation trial where an outcome and some auxiliary information were measured longitudinally, and both were subject to missing values. Dynamic generalized linear mixed effects models linking the outcome, the auxiliary information, and the covariates are proposed. The maximum likelihood approach is applied to the estimation and inference on the model parameters. The average treatment effect is estimated by the G-computation approach, and the sensitivity of the treatment effect estimate to the nonignorable missing data mechanisms is investigated through the local sensitivity analysis approach. The proposed approach can handle missing data that form arbitrary missing patterns over time. We applied the proposed method to the analysis of the smoking cessation trial.

Trends in rates of onset of and recovery from disability at older ages: 1982-1994

OBJECTIVES

Although there is substantial evidence of declining prevalence of disability among the older population during the late 1980s and 1990s, evidence on trends in the underlying dynamics of disability is lacking. For this study, we estimated models of transitions between discrete disability and vital states that incorporated simple linear time trends.

METHODS

We analyzed data from the 1982-1994 interviews of the New Haven Established Populations for Epidemiologic Studies of the Elderly study and used three alternative measures of disability status. We estimated separate models of disability prevalence and disability transitions by gender.

RESULTS

Eleven of 12 estimated trends in transition rates were statistically significant. For men and women, and for three alternative disability indicators, we found downward trends in rates of both onset of and recovery from disability among people aged 75 and older. We did not find any consistent pattern of trends in disability among those aging into the 75 and older group during this period.

DISCUSSION

Our findings are consistent with declining population-level disability prevalence only if any downward trend in onset outweighs the downward trend in recovery. These findings are also consistent with a trend toward more severe disability problems among the disabled population.