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Robust inference for univariate proportional hazards frailty regression models. (English) Zbl 1047.62090

Summary: We consider a class of semiparametric regression models which are one-parameter extensions of the Cox model [D. R. Cox, J. R. Stat. Soc., Ser. B 34, 187–220 (1972; Zbl 0243.62041)] 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 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 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.

MSC:

62N02 Estimation in survival analysis and censored data
62P10 Applications of statistics to biology and medical sciences; meta analysis
62G08 Nonparametric regression and quantile regression
62N01 Censored data models
60F05 Central limit and other weak theorems
62F35 Robustness and adaptive procedures (parametric inference)
62G35 Nonparametric robustness

Citations:

Zbl 0243.62041
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References:

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