Inference in generalized linear regression models with a censored covariate

The problem of estimating the parameters in a generalized linear model when a covariate is subject to censoring is studied. A new method based on an estimating function approach is proposed. The method does not assume a parametric form for the distribution of the response given the regressors and is computationally simple. In the linear regression case, the proposed approach implies the use of mean imputation of the censored regressor. The use of flexible parametric models for the distribution of the covariate is employed. When survival time is considered as the covariate subject to censoring, the use of the generalized gamma distribution is explored, since it is considered as a platform distribution covering a wide variety of hazard rate shapes. The method can be further robustified by considering models of nonparametric nature typically used in survival analysis such as the logspline for the censored covariate. For models involving additional, fully observed, covariates the use of a generalized gamma accelerated failure time regression model is explored. In this setting, no parametric family assumption for the extra covariates is needed. The proposed approach is broader than likelihood based multiple imputation techniques. Moreover, even in cases with a known parametric form for the response distribution, the method can be considered a feasible alternative to likelihood based estimation. Simulation studies are conducted for continuous, binary and count data to evaluate the performance of the proposed method and to compare the estimates to standard ones. An application using a well known data set of a randomized placebo controlled trial of the drug D-penicillamine (DPCA) for the treatment of primary biliary cirrhosis (PBC) conducted at the Mayo Clinic is presented. Possible extensions of the method regarding the robustness as well as the type of censoring are also discussed.