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.

翻译：比例率模型是分析计数过程率函数的最常用方法之一。尽管该模型能够直接解释协变量效应的率比，但比例率假设要求协变量不改变率函数的形状。当该假设不成立时，我们提出通过两个指标——形状指数与尺度指数——来描述率函数与协变量之间的关系。形状指数允许协变量灵活地影响率函数的形态，而尺度指数则保留了协变量效应在率函数量级上的可解释性。为克服同时估计两组参数的挑战，我们提出条件伪似然方法在形状估计中消除尺度参数，以及事件计数投影方法用于尺度估计。所提出的估计量具有渐近正态性，收敛速度为根号n。通过模拟实验和基于SEER-Medicare数据的重复住院数据分析，我们验证了所提方法的有效性。