Modelling time-dependent interaction in a time-varying covariate and its application to rejection episodes and kidney transplant failure

Acute rejection episodes are thought to be prognostic of eventual kidney graft failure. The influence of rejection events on the hazard of transplant failure appears to be a complex function of how long after transplantation the rejection event occurs as well as the time elapsed since the rejection event. To examine the nature of this relationship, we propose a penalized likelihood approach to estimate the parameters of a two-dimensional rectanglewise constant hazard model. The approach appears to be fairly successful at modelling time dependency in a time-varying covariate. The approach is equally applicable for modelling fixed covariates that act in a jointly nonproportional (non-log-linear) and time-dependent manner.