A semi-competing risks model for data with interval-censoring and informative observation

Jessica Barrett, Fotios Siannis and Vern Farewell have a new paper in Statistics in Medicine. Essentially the paper uses similar methods as in Siannis et al 2006, to investigate informative loss-to-follow-up (LTF) in a study of aging and cognitive function.

LTF, refers to loss-to-follow-up of monitoring cognitive impairment - crucially survival continues to be monitored. LTF (from healthy) is modelled as a separate state in the process with its own transition intensity. Once LTF, the subject experiences different intensities of becoming cognitive impaired or dying. An unidentifiable parameter k determines the relative rate at which people who are lost to follow-up (before becoming cognitively impaired) proceed to the cognitively impaired state rather than the death state, compared to those not lost to follow-up.
k can be varied to see what impact assumptions about those LTF have on overall estimates. In the current study k has quite a large impact on estimates of cumulative incidence of cognitive impairment. This is in contrast to the Whitehall study where these methods were applied to right-censored data, where k had little effect.

A parametric Weibull intensities Markov model is used to model the data. Due to the interval censoring, computation of the likelihood requires numerical integration.

As an informal goodness-of-fit test the authors compare on Cox model of overall survival, with the corresponding survival estimates for the multi-state model with proportional intensity models on each intensity. The authors note that the Cox proportional hazards model for overall survival should be unbiased. Of course, it is only unbiased if the proportional hazards assumption holds on the overall hazard of death. If, however, the covariates are proportional on the individual intensities, as assumed in the multi-state model, the Cox model on overall survival will be biased. This approach is therefore more of a test of robustness to assumptions about the covariates than a goodness-of-fit test because the models aren't nested. The same method was used in Siannis et al 2006 and in Van den Hout et al, 2009.