Joint modelling of longitudinal outcome and interval-censored competing risk dropout in a schizophrenia clinical trial

Ralitza Gueorguieva, Robert Rosenheck and Haiqun Lin have a new paper in JRSS A. The paper concerns the joint modelling of a longitudinal outcome and an interval censored competing risks outcome that explains drop-out. As is common with these joint longitudinal and survival types of models the two processes are linked via a normally distributed vector of random effects. The novelty of the paper is in the survival part is a competing risks process and the event time is interval censored. The authors adopt a parametric model for the competing risks, using the family of distributions proposed by Sparling et al (Biostatistics, 2006). This makes inference somewhat more straightforward than it would be if a non-parametric baseline cause-specific hazards were used. As recently noted, parametric treatment of competing risks data is surprisingly rare. One problem faced by the authors is that the hazard family of Sparling, while allowing closed form expressions for interval censored univariate survival data, do not result in closed form expressions for interval censored competing risks data (except in special cases). Instead a numerical integral has to be competed. The presence of the overall random effects would mean the likelihood requires nested integration. To avoid this problem the authors adopt an approximation to the true likelihood for competing risks data. If a patient is known to have had a failure of type j in the interval [t0,t1] the authors assume that the patient is censored of all risks except risk j at time t0. It is clear that this approximation will lead to systematic bias as the time at risk from each failure type will be underestimated so the hazards will tend to be overestimated. The amount of bias will depend on the typical length of the intervals [t0,t1].

For the CATIE data example the proposed approximation is probably not an issue. The drop out (competing risks) part of the model is not the primary focus of the inference, and it is really the relative hazards of different types of drop out rather than their absolute values that is important in determining the trajectories of the longitudinal measure without drop out. For instance the estimates for simulated data of a similar type are close to unbiased.
However in extreme cases like current status competing risks data the approximation will do extremely badly.

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

Van den Hout and Matthews have a new paper in JRSS A. This deals with interval censored data from a three-state disease model where subjects may miss scheduled interviews meaning the disease status is not observed. A joint model for the disease state and an observation indicator is developed, being a continuous time generalisation of Cole et al (2005) that allows information from exact death times to be included. Conditional on the disease state and measured covariates, the observation indicator is governed by a logistic model.
In general the method seems promising for dealing with informative observation when the potential observation times are known.

Though not noted, the model can be expressed as a hidden Markov model. The authors state that the logistic model and the three-state Markov model are estimated separately. It is not made clear how this is achieved since the logistic model depends on the unobserved states of the Markov model. In some cases the missing state will in fact be known, for instance if the sequence is 1,-,1 or 2,-,2. However, for sequences like 1,-,2 or 1,-,3 it is not possible to establish the unobserved state.

The main aim of the analysis is to obtain estimates of life expectancy, disease free life expectancy and post-disease life-expectancy. These are complicated functions of the parameter vector as they involve integrals of transition probabilities. In addition to the method of Aalen et al 1997, Van den Hout and Matthews additionally propose to use a Metropolis algorithm to get confidence intervals for the life expectancies. The resulting intervals have a Bayesian interpretation, being the credible intervals from an improper uniform prior, but will not be invariant to changes in parametrisation. In practice, the intervals may give good frequentist coverage, particularly for large samples. However, Van den Hout and Matthews seem to be implying the intervals have exact coverage (apart from Monte-Carlo error through the Metropolis algorithm) which is a substantial misconception. Moreover, no mention of the procedure being Bayesian is given.