Competing risks and time-dependent covariates

Time-dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors which can not be substituted by time-fixed covariates. The present work briefly recalls the different types of time-dependent covariates, as classified by Kalbfleisch & Prentice (2002), with the intent of clarifying their role and emphasize the limitations in the competing risks setting. If random (internal) time-dependent covariates are to be included in the modelling process, then it is still possible to estimate cause-specific hazards but prediction on the cumulative incidences and survival probabilities based on these is no longer feasible. The present paper aims at providing some possible strategies for dealing with these prediction problems. In a multi-state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen (2007) in order to study cumulative incidences at different sub-intervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause-specific events as final states. In all of those proposals it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplants data is presented in order to compare the different methods.