## Wednesday, 23 February 2011

### Estimating and testing for center effects in competing risks

Sandrine Katsahian and Christian Boudreau have a new paper in Statistics in Medicine. This develops methods for including frailty terms within a Fine-Gray competing risks model in order to account for clustering, e.g. effects of different centres.

Since the Fine-Gray model is essentially just a standard Cox proportional hazards regression model with additional time dependent weights, based on the censoring distribution, for individuals who have had a competing event, methods appropriate for standard Cox frailty models can be readily adapted.

Katsahian and Boudreau closely follow the approach taken by Ripatti and Palmgren (Biometrics, 2000). They assume a Gaussian frailty. Computation of the likelihood requires integrating out the frailty terms. Here this is performed using a Laplace approximation. A difficulty with the Laplace approximation is that it still requires the modal value of the frailty distribution conditional on the data and current values of the parameters. The authors therefore take a profile likelihood approach in which they fix the frailty variance $\inline \theta$ and maximize the likelihood term with respect to both the regression parameters $\inline \beta$ and the frailty terms,$\inline u$ . Having obtained, $\inline \hat\beta$ and $\inline \hat u$ they can then plug $\inline \hat u$ into the Laplace approximation to get the profile likelihood for $\inline \theta$. The procedure gives a local approximation for $\inline \theta$ which can be used to suggest the updated estimate. Thus the process involves alternating between two Newton-Raphson algorithms until convergence.