Book Review of: Competing Risks and Multistate Models with R.

Ross Maller has written a book review of Beyersmann, Schumacher and Allignol's recent Springer book on Competing Risks and Multistate Models with R, published in Australian & NZ Journal of Statistics. This is primarily a rant against the cause-specific hazard approach to modelling competing risks. For instance cause specific hazards "do not explicitly take into account the obvious mixture of distributions inherent in the data." Moreover, the fact that assuming proportionality in cause-specific-hazards (CSHs) can lead to non-proportional, even crossing, relationships for cumulative incidence functions (CIFs), is painted as a terminal weakness to the approach.

Maller's main contribution to survival analysis is through models for cure fractions (see e.g. Maller and Zhou 1995) an approach that he is evidently very taken with. Apparently the correct approach to take in modelling competing risks data is to assume a finite mixture model, such that individuals in a particular class are only at risk of one particular failure rate. Moreover, the problem of covariates is claimed to be entirely solved by allowing proportional hazards within failure types, which Maller says is the approach taken by Fine and Gray (1999).

The entire nature of survival and event history analysis is in modelling the dynamics of the process. In most circumstances it is much more useful to be able to describe the process at time t given no event has occurred by time t than to describe the process conditional on a latent class membership. Moreover, in the vast majority of competing risks data, at least in medical contexts, all patients are at risk of all event types until experiencing an event. A mixture model could therefore only ever be viewed as a mathematical convenience. The fact that in practice a CSH method is actually substantially more convenient, particularly if a non- or semi-parametric approach is to be adopted, hardly aids the case for mixture models.

Maller is also misrepresenting the Fine-Gray approach which does not assume proportional hazards within failure types. The Larson-Dinse (1985) paper that Maller also cites does involves that approach. But that can lead to the same crossing cumulative incidence curves Maller takes issue with in the context of CSH. Fine-Gray assumes proportionality of the sub-distribution hazard for a particular cause. This does allow proportionality for that cause's corresponding CIF but, as a consequence, is unable to provide a covariate model for other CIFs that is guaranteed to lead to a feasible set of CIFs for all covariate values (ie. we can fit a Fine-Gray model to each cause of failure but the resulting models will be contradictory).

Fundamentally, whatever model that is postulated, we can find the implied cause-specific hazards. Assuming proportionality of the cause-specific hazards is obviously only a modelling assumption but in nearly all cases it will be a better starting point than assuming the existence of cure fractions.