Statistica Sinica 19 (2009), 819-842

USING PROFILE LIKELIHOOD FOR SEMIPARAMETRIC

MODEL SELECTION WITH APPLICATION TO

PROPORTIONAL HAZARDS MIXED MODELS

Ronghui Xu$^1$, Florin Vaida$^1$ and David P. Harrington$^{2}$

$^1$University of California, San Diego and

$^2$Dana-Farber Cancer Institute and Harvard School of Public Health

Abstract: We consider selection of nested and non-nested semiparametric models. Using profile likelihood we can define both a likelihood ratio statistic and an Akaike information for models with nuisance parameters. Asymptotic quadratic expansion of the log profile likelihood allows derivation of the asymptotic null distribution of the likelihood ratio statistic including the boundary cases, as well as unbiased estimation of the Akaike information by an Akaike information criterion. Our work was motivated by the proportional hazards mixed effects model (PHMM), which incorporates general random effects of arbitrary covariates and includes the frailty model as a special case. The asymptotic properties of its parameter estimate has recently been established, which enables the quadratic expansion of the log profile likelihood. For computation of the (profile) likelihood under PHMM we apply three algorithms: Laplace approximation, reciprocal importance sampling, and bridge sampling. We compare the three algorithms under different data structures, and apply the methods to a multi-center lung cancer clinical trial.

Key words and phrases: Akaike information, bridge sampling, Laplace approximation, likelihood ratio test, profile likelihood, reciprocal importance sampling, testing on the boundary.