Estimation of the Asymptotic Variance of Semiparametric Maximum Likelihood Estimators in the Cox Model with a Missing Time-Dependent Covariate
Abstract The relationship between a time-to-event and a time-dependent or time-independent covariate is usually assessed using the Cox model. A frequently encountered problem however is occurrence of missing covariate values. A recent approach for estimating the Cox model with a missing covariate jointly models the time-to-event and covariate. In the case of a time-dependent covariate, Dupuy and Mesbah [Dupuy, J.-F., Mesbah, M. (2002). Joint modeling of event time and nonignorable missing longitudinal data. Lifetime D. Anal. 8:99–115] have proposed a joint model and have obtained a semiparametric maximum likelihood estimator of the regression parameter of the Cox model that is consistent and asymptotically normal. Furthermore, an explicit expression was obtained for the asymptotic variance of this estimator. In this paper, we examine the problem of estimating this variance. We propose a computationally simple estimator and we show its consistency. We illustrate the approach by applications to real data sets.