Imputing missing covariate values for the Cox model

Abstract

Multiple imputation is commonly used to impute missing data, and is typically more efficient than complete cases analysis in regression analysis when covariates have missing values. Imputation may be performed using a regression model for the incomplete covariates on other covariates and, importantly, on the outcome. With a survival outcome, it is a common practice to use the event indicator D and the log of the observed event or censoring time T in the imputation model, but the rationale is not clear.We assume that the survival outcome follows a proportional hazards model given covariates X and Z. We show that a suitable model for imputing binary or Normal X is a logistic or linear regression on the event indicator D, the cumulative baseline hazard H(0)(T), and the other covariates Z. This result is exact in the case of a single binary covariate; in other cases, it is approximately valid for small covariate effects and/or small cumulative incidence. If we do not know H(0)(T), we approximate it by the Nelson-Aalen estimator of H(T) or estimate it by Cox regression.We compare the methods using simulation studies. We find that using logT biases covariate-outcome associations towards the null, while the new methods have lower bias. Overall, we recommend including the event indicator and the Nelson-Aalen estimator of H(T) in the imputation model.

Figure 1

Smoothed mean and SD of X|T, D with β_X = 0.7, h₀(t) = 1.