Statistica Sinica 17(2007), 841-856

A FAST ALGORITHM FOR THE NONPARAMETRIC

MAXIMUM LIKELIHOOD ESTIMATE IN THE COX-GENE

MODEL

I-Shou Chang$^1$, Chi-Chung Wen$^2$, Yuh-Jenn Wu$^3$ and Che-Chi Yang$^4$

$^1$National Health Research Institutes,

$^2$Tamkang University, $^3$ Chung Yuan Christian University and

$^4$ Lunghwa University of Science and Technology

Abstract: The Cox model with the gene effect for age at onset was introduced and studied by Li, Thompson and Wijsman (1998) and Li and Thompson (1997). This paper concerns the numerical performance of the nonparametric maximum likelihood estimate of the environmental effects and the genetic effect in this model. Based on the self-consistency equations derived from the score functions, we propose a fast iterative algorithm for the computations of the nonparametric maximum likelihood estimate and its asymptotic variance. Simulation studies conducted using these algorithms indicate that the profile likelihood-based normal approximations for the estimates are valid with reasonable sample sizes, and the bootstrap methods work well also for smaller sample sizes, and are computationally feasible.

Key words and phrases: Age at onset, bootstrap, frailty, gene effect, profile likelihood, self-consistency equations.