Incorporating frailty in a multi-state model: application to disease natural history modelling of adenoma-carcinoma in the large bowel

Yen, Chen, Duffy and Chen have a new paper in SMMR. This considers methods for incorporating frailty into a progressive multi-state model under interval censoring (the data resemble current status data being a case-cohort design with known sampling probabilities). They note that the accommodation of frailty is made easier by considering a model where only one transition intensity is subject to heterogeneity. The main deficiency of the paper is the lack of reference to the fairly wide existing literature on frailty and random effects models for panel observed multi-state models. In particular, the tracking model of Satten is directly applicable to their data. In the tracking model, a common frailty affects all transition intensities and results in a likelihood that doesn't require numerical integration. Some comment on the extendability of their model to the case of multiple observations per patient would also have been useful.

Estimation of overall survival in an "illness-death" model with application to vertical transmission of HIV-1

Halina Frydman and Michael Szarek have a new paper in Statistics in Medicine. This presents an additional application to their previous work on non-parametric estimation in a Markov illness-death model under interval censoring. Specifically, they use estimates of survival pre and post disease, to provide insights into the effect on overall survival. Sub-sampling is used to obtain estimates of standard errors (rather than a simple bootstrap).

Analysis of interval-censored disease progression data via multi-state models under a nonignorable inspection process

Chen, Yi and Cook have a new paper in Statistics in Medicine. This covers similar ground to the paper of Sweeting et al as it is also concerned with a nonignorable inspection process for a panel observed progressive Markov multi-state model. Again there are a set of known planned observation times, but unlike Sweeting et al there is no auxiliary data. A MNAR model where the missingness probability depends on the true state at the potential observation time is used. This is similar to the MNAR model in Sweeting et al, except that here piecewise constant transition intensities are used rather than a time homogeneous model. The model is fitted using an EM algorithm.

Both this study and Sweeting et al's depend on there being a known set of planned examination times. In the majority of cases, only the actual examination times are known and the observation process is not fully understood. As Chen et al note, informative observation times could be dealt with by determining a stochastic model for the observation process, but would require assumptions about the process.

Multistate Markov models for disease progression in the presence of informative examination times

Sweeting, Farewell and De Angelis have a new paper in Statistics in Medicine. This deals with the problem of a panel observed disease process where the examination times are generated by a non-ignorable mechanism. This is in general a very difficult problem. The authors consider a special case where, while the disease process is only observed at informative examination times, an auxiliary variable is able to be observed at a full set of planned (or ignorable) times. They consider a model where the disease process is missing at random (MAR) conditional on the values of the auxiliary variable. The likelihood then becomes of the form of a (partially) hidden Markov model. An alternative missing not at random model (MNAR) with missingness dependent on the current state is also considered. However the comparison between the MAR and MNAR models is somewhat unfair. In the simulations and the application, the disease process is only observed at a minority of planned examination times. The auxiliary variable has a fairly strong correlation with the disease process, meaning significant information about the process can be obtained through using the auxiliary variable in a hidden Markov model. Indeed, even without the suspicion of informative missingness the HMM would be worth using (if the strong assumptions about the relationship with the auxiliary variable could be reliably assumed) for improved efficiency. However, the MNAR model does not use the auxiliary variable at all. A better model to compare would be a hybrid of the two where a partially HMM is used in conjunction with the logistic model for the MNAR (although this wouldn't necessarily be consistent under AD-MAR unless the logistic model was correctly specified). This does not seem like a difficult extension.

A further shortcoming of the paper is the lack of any suggestions for model checking. Measurements from the auxiliary variable are assumed to be Normally distributed and independent conditional on the underlying disease states at the examination times. This seems fairly unrealistic and should at least be supported by the data.