Statistical inference for counting processes under shape heterogeneity

Proportional rate models are among the most popular methods for analyzing the rate function of counting processes. Although providing a straightforward rate-ratio interpretation of covariate effects, the proportional rate assumption implies that covariates do not modify the shape of the rate function. When such an assumption does not hold, we propose describing the relationship between the rate function and covariates through two indices: the shape index and the size index. The shape index allows the covariates to flexibly affect the shape of the rate function, and the size index retains the interpretability of covariate effects on the magnitude of the rate function. To overcome the challenges in simultaneously estimating the two sets of parameters, we propose a conditional pseudolikelihood approach to eliminate the size parameters in shape estimation and an event count projection approach for size estimation. The proposed estimators are asymptotically normal with a root-$n$ convergence rate. Simulation studies and an analysis of recurrent hospitalizations using SEER-Medicare data are conducted to illustrate the proposed methods.