Robust Inference for Univariate Proportional Hazards Frailty Regression Models

• Published in: The Annals of statistics 2004-08, Vol.32 (4), p.1448-1491
• Main Authors: Kosorok, Michael R., Lee, Bee Leng, Fine, Jason P.
• Format: Article
• Language: English
• Subjects: 60F05; 62B10; 62F40; 62N01; Applications; Consistent estimators; Distribution theory; Empirical process; Estimating techniques; Estimators; Exact sciences and technology; Frailty; Identifiability; implied parameter; Inference; Laplace transform; Linear inference, regression; Linear regression; Mathematical models; Mathematics; Maximum likelihood estimators; Medical sciences; misspecification; Modeling; Non Hodgkin lymphoma; Nonparametric inference; nonparametric maximum likelihood; Parametric models; Probability and statistics; Probability theory and stochastic processes; Regression analysis; Sciences and techniques of general use; semiparametric information bound; Statistics; Studies; Survival Analysis; unobservable heterogeneity
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/009053604000000535

We consider a class of semiparametric regression models which are one-parameter extensions of the Cox [J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] model for right-censored univariate failure times. These models assume that the hazard given the covariates and a random frailty unique to each individual has the proportional hazards form multiplied by the frailty. The frailty is assumed to have mean 1 within a known one-parameter family of distributions. Inference is based on a nonparametric likelihood. The behavior of the likelihood maximizer is studied under general conditions where the fitted model may be misspecified. The joint estimator of the regression and frailty parameters as well as the baseline hazard is shown to be uniformly consistent for the pseudo-value maximizing the asymptotic limit of the likelihood. Appropriately standardized, the estimator converges weakly to a Gaussian process. When the model is correctly specified, the procedure is semiparametric efficient, achieving the semiparametric information bound for all parameter components. It is also proved that the bootstrap gives valid inferences for all parameters, even under misspecification. We demonstrate analytically the importance of the robust inference in several examples. In a randomized clinical trial, a valid test of the treatment effect is possible when other prognostic factors and the frailty distribution are both misspecified. Under certain conditions on the covariates, the ratios of the regression parameters are still identifiable. The practical utility of the procedure is illustrated on a non-Hodgkin's lymphoma dataset.