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.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000053
