## Friday, 3 December 2010

### Interpretability and importance of functionals in competing risks and multi-state models

Per Kragh Andersen and Niels Keiding have a new paper currently available as a Department of Biostatistics, Copenhagen research report. They argue that three principles should be adhered to when constructing functionals of the transition intensities in competing risks and illness-death models. The principles are:

1. Do not condition on the future.
2. Do not regard individuals at risk after they have died.
3. Stick to this world.

They identify several existing ideas that violate these principles. Unsurprising, the latent failure times model for competing risks rightly comes under fire for violating (3), i.e. to say anything about hypothetical survival distributions in the absence of the other risks requires making untestable assumptions. Semi-competing risks analysis where one seeks the survival distribution for illness in the absence of death has the same problem.

The subdistribution hazard from the Fine-Gray model violates principle 2 because it involves the form $\inline P(X(t + dt) = j | X(t) \neq j)$. Andersen and Keiding say this makes interpretation of regression parameters difficult because they are log(subdistribution hazard ratios). The problem seems to be that many practitioners interpret the coefficients as if they are standard hazard ratios. The authors go on to say that linking covariates directly to cumulative incidence functions is useful. The distinction between this and the Fine-Gray model is rather subtle as in the Fine-Gray model (when covariates are not time dependent): $\inline CIF(t; Z) = 1 - \exp{(-H(t)\exp{(b^{T}Z)})}$ i.e. b is essentially interpreted as a parameter in a cloglog model.
The conditional probability function recently proposed by Allignol et al has similar problems with principle 2.

Principle 1 is violated in the pattern-mixture parametrisation. This is where we consider the distribution of event times conditional on the event type, e.g. the sojourn in state i given the subject moved to state j. This is used for instance in flow-graph Semi-Markov models.

A distinction that is perhaps needed that isn't really made clear in the paper is that there is a difference between violating the principles for mathematical convenience e.g. for model fitting and violating the principles in terms of the actual inferential output. Functionals to be avoided should perhaps be those where no easily interpretable transformation to a sensible measure is available. Thus a pattern-mixture parametrisation for a semi-Markov model without covariates seems unproblematic, since we can retrieve the transition intensities. However, when covariates are present the transition intensities will have complicated relationships to the covariates without an obvious interpretation.

**** UPDATE : The paper is now published in Statistics in Medicine. *****