  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.
