Data Missing Problem in Cox Model with Time-dependent Covariates and Some Reasonable and Convenient Methods to Address It

To elucidate the association between outcomes and covariates in long-term follow-up data, a time-to-event analysis (survival analysis) that incorporates time-dependent covariates (longitudinal data) is required. However, in real data applications, Cox models with time-dependent covariates are almost always forced to deal with incomplete data due to the missing of time-dependent covariates necessary for parameter estimation. The purpose of this article is to highlight this data missing problem and to introduce some relatively reasonable and convenient ways to address it. Multiple imputation method provides a persuasive framework in which uncertainty of imputed values for the missing time-dependent covariates is considered in a straightforward manner. We propose the usage of Gaussian process regression technique in the imputation step. This enables a flexible fitting to various longitudinal data easily and is compatible with the multiple imputation framework. In addition, Gaussian process regression can readily handle irregularly observed multiple time series data.