Ignorability for general longitudinal data

Randomized Trials With Repeatedly Measured Outcomes: Handling Irregular and Potentially Informative Assessment Times

Randomized trials are often designed to collect outcomes at fixed points in time after randomization. In practice, the number and timing of outcome assessments can vary among participants (i.e., irregular assessment). In fact, the timing of assessments may be associated with the outcome of interest (i.e., informative assessment). For example, in a trial evaluating the effectiveness of treatments for major depressive disorder, not only did the timings of outcome assessments vary among participants but symptom scores were associated with assessment frequency. This type of informative observation requires appropriate statistical analysis. Although analytic methods have been developed, they are rarely used. In this article, we review the literature on irregular assessments with a view toward developing recommendations for analyzing trials with irregular and potentially informative assessment times. We show how the choice of analytic approach hinges on assumptions about the relationship between the assessment and outcome processes. We argue that irregular assessment should be treated with the same care as missing data, and we propose that trialists adopt strategies to minimize the extent of irregularity; describe the extent of irregularity in assessment times; make their assumptions about the relationships between assessment times and outcomes explicit; adopt analytic techniques that are appropriate to their assumptions; and assess the sensitivity of trial results to their assumptions.

Missing at random: a stochastic process perspective

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing-data framework, we give a novel characterization of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness-at-random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration. These measurability conditions ensure the usual factorization of likelihood ratios. We illustrate how the theory can be extended easily to incorporate explanatory variables, to describe longitudinal data in continuous time, and to admit more general coarsening of observations.

Summarizing the extent of visit irregularity in longitudinal data

Background

Observational longitudinal data often feature irregular, informative visit times. We propose descriptive measures to quantify the extent of irregularity to select an appropriate analytic outcome approach.

Methods

We divided the study period into bins and calculated the mean proportions of individuals with 0, 1, and > 1 visits per bin. Perfect repeated measures features everyone with 1 visit per bin. Missingness leads to individuals with 0 visits per bin while irregularity leads to individuals with > 1 visit per bin. We applied these methods to: 1) the TARGet Kids! study, which invites participation at ages 2, 4, 6, 9, 12, 15, 18, 24 months, and 2) the childhood-onset Systemic Lupus Erythematosus (cSLE) study which recommended at least 1 visit every 6 months.

Results

The mean proportions of 0 and > 1 visits per bin were above 0.67 and below 0.03 respectively in the TARGet Kids! study, suggesting repeated measures with missingness. For the cSLE study, bin widths of 6 months yielded mean proportions of 1 and > 1 visits per bin of 0.39, suggesting irregular visits.

Conclusions

Our methods describe the extent of irregularity and help distinguish between protocol-driven visits and irregular visits. This is an important step in choosing an analytic strategy for the outcome.

Independence conditions and the analysis of life history studies with intermittent observation

Multistate models provide a powerful framework for the analysis of life history processes when the goal is to characterize transition intensities, transition probabilities, state occupancy probabilities, and covariate effects thereon. Data on such processes are often only available at random visit times occurring over a finite period. We formulate a joint multistate model for the life history process, the recurrent visit process, and a random loss to follow-up time at which the visit process terminates. This joint model is helpful when discussing the independence conditions necessary to justify the use of standard likelihoods involving the life history model alone and provides a basis for analyses that accommodate dependence. We consider settings with disease-driven visits and routinely scheduled visits and develop likelihoods that accommodate partial information on the types of visits. Simulation studies suggest that suitably constructed joint models can yield consistent estimates of parameters of interest even under dependent visit processes, providing the models are correctly specified; identifiability and estimability issues are also discussed. An application is given to a cohort of individuals attending a rheumatology clinic where interest lies in progression of joint damage.

Dealing with death when studying disease or physiological marker: the stochastic system approach to causality

The stochastic system approach to causality is applied to situations where the risk of death is not negligible. This approach grounds causality on physical laws, distinguishes system and observation and represents the system by multivariate stochastic processes. The particular role of death is highlighted, and it is shown that local influences must be defined on the random horizon of time of death. We particularly study the problem of estimating the effect of a factor V on a process of interest Y, taking death into account. We unify the cases where Y is a counting process (describing an event) and the case where Y is quantitative; we examine the case of observations in continuous and discrete time and we study the issue of whether the mechanism leading to incomplete data can be ignored. Finally, we give an example of a situation where we are interested in estimating the effect of a factor (blood pressure) on cognitive ability in elderly.