Bias reduction methods in the hierarchical likelihood approach for shared frailty model of clustered failure time data.

Shared frailty models are widely used for the analyses of multivariate survival data. A hierarchical likelihood (h-likelihood) approach has been developed for estimating the regression parameters and frailty variates, in which the latent frailties are treated as 'parameters' and estimated jointly with other parameters of interest. The h-likelihood estimators generally perform well in various frailty models. However, they are known to be biased for non-normal random effects. Existing modifications to the h-likelihood employ the total derivative and second-order Laplace approximation, which is computationally intensive with complicated mathematical derivations. In this work, we propose two effective bias correction methods for the h-likelihood estimators under the shared gamma frailty models. The first method modifies the adjusted profile likelihood by adding a logarithmic transformation of the variance of the frailty parameter to avoid zero estimates in the frailty parameter. The second approach modifies the score function of the adjusted profile likelihood. Thus, in the two modifications, we avoid the use of the total derivative and second-order Laplace approximation. Simulation studies show that the proposed approaches reduce the bias in the h-likelihood estimators, especially for the estimate of the frailty parameter. Applications of both methods are illustrated using recurrent kidney infection data.