Statistical Inference for Counting Processes Under Shape Heterogeneity

Proportional rate models are among the most popular methods for analyzing recurrent event data. 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 the proportionality assumption fails to hold, we propose to characterize covariate effects on the rate function through two types of parameters: the shape parameters and the size parameters. The former allows the covariates to flexibly affect the shape of the rate function, and the latter 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, followed by an event count projection approach for size estimation. The proposed estimators are asymptotically normal with a root- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> convergence rate. Simulation studies and an analysis of recurrent hospitalizations using SEER-Medicare data are conducted to illustrate the proposed methods.