Methods for Contrasting Gap Time Hazard Functions: Application to Repeat Liver Transplantation

In studies featuring a sequence of ordered events, gap times between successive events are often of interest. Despite the rich literature in this area, very few methods for comparing gap times have been developed. We propose methods for estimating a hazard ratio connecting the first and second gap times. Specifically, a two-stage procedure is developed based on estimating equations. At the first stage, a proportional hazards model is fitted for the first gap time. Weighted estimating equations are then solved at the second stage to estimate the hazard ratio between the first and second gap times. The proposed estimator has a closed form and, being analogous to a standardized mortality ratio, is easy to interpret. Large sample properties of the proposed estimators are derived, with simulation studies used to evaluate finite sample characteristics. Extension of the approach to accommodate a piecewise constant hazard ratio is considered. The proposed methods are applied to contrast repeat (second) versus primary (first) liver transplants with respect to graft failure, based on national registry data.