Pseudo-partial likelihood estimators for the Cox regression model with missing covariates

Abstract

By embedding the missing covariate data into a left-truncated and right-censored survival model, we propose a new class of weighted estimating functions for the Cox regression model with missing covariates. The resulting estimators, called the pseudo-partial likelihood estimators, are shown to be consistent and asymptotically normal. A simulation study demonstrates that, compared with the popular inverse-probability weighted estimators, the new estimators perform better when the observation probability is small and improve efficiency of estimating the missing covariate effects. Application to a practical example is reported.

Keywords: Augmented estimator; Biased sampling data; Embedding missing data; Left-truncation; Martingale structure; Right censoring; U-statistic.

Fig. 1.

Selection probabilities π for different simulation settings. Panels (a) and (d) for setting 1, Table 1, (b) and (e) for setting 2, Table 2, (c) and (f) for setting 3, Table 3. In panels (b), (e), (c) and (f), the solid lines are for δ = 1 and the dotted lines are for δ = 0. The curves in panels (c) and (f) are plotted at the mean values of Z^c.

Fig. 2.

Stroke study dataset. Nadaraya–Watson estimates of the selection probabilities with respect to the observed times for (a) different age groups, where the solid line represents the above or equal to median age group and the dotted line represents the below median age group; and (b) different diastolic blood pressure statuses, where the solid line represents the above or equal to median diastolic blood pressure group and the dotted line represents the below median diastolic blood pressure group.