Frailties in multi-state models: Are they identifiable? Do we need them?

Hein Putter and Hans van Houwelingen have a new paper in Statistical Methods in Medical Research. This reviews the use of frailties within multi-state models, concentrating on the case of models observed up to right-censoring and primarily looking at models where the semi-Markov (Markov renewal) assumption is made. The paper clarifies some potential confusion over when frailty models are identifiable. For instance, for simple survival data with no covariates, if a non-parametric form is taken for the hazard then no frailty distribution is identifiable. A similar situation is true for competing risks data. Things begin to improve once we can observe multiple events for each patient (e.g. in illness-death models), although clearly if the baseline hazard is non-parametric, some parametric assumptions will be required for the frailty distribution. When covariates are present in the model, the frailty term has a dual role of modelling dependence between transition times but also soaking up lack of fit in the covariate (e.g. proportional hazards) model.

The authors consider two main approaches to fitting a frailty; assuming a shared Gamma frailty which acts as a multiplier on all the intensities of the multi-state model and a two-point mixture frailty, where there are two mixture components with a corresponding set of multiplier terms for the intensities for each component. Both approaches admit a closed form expression for the marginal transition intensities and so are reasonably straightforward to fit, but the mixture approach has the advantage of permitting a greater range of dependencies, e.g. it can allow negative as well as positive correlations between sojourn times.

In the data example the authors consider the predictive power (via K-fold cross-validation) of a series of models on a forward (progressive) multi-state model. In particular they compare models which allow sojourn times in later states to depend on the time to first event, with frailty models and find the frailty model performs a little better. Essentially, these two models do the same thing and as the authors note, a more flexible model for the effect of time of first event on the second sojourn time may well give a better fit than the frailty model.

Use of frailties in models with truly recurrent events seems relatively uncontroversial. However for models where few intermediate events are possible their use, rather than some non-homogeneous model allowing dependence both on time since entry to the state and time since initiation, the choice is largely related to either what is more interpretable in terms of the application at hand or possibly what is more computationally convenient.