Statistical Inference for Counting Processes Under Shape Heterogeneity

Abstract

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- $n$ convergence rate. Simulation studies and an analysis of recurrent hospitalizations using SEER-Medicare data are conducted to illustrate the proposed methods.

Keywords: dimension reduction; kernel smoothing; recurrent event process; single index model.

FIGURE 1 |

Shape and size decomposition of common recurrent event models. For mathematical definitions of shape and size, please refer to Section 2. ${\mathit{\alpha }}_0$ is a vector of regression parameters; ${\mu }_0\left(t\right)$ and $m_0\left(t\right)={\int }_0^t{\mu }_0\left(u\right)du$ are the baseline rate and cumulative rate functions, respectively. Let $[0,\tau ]$ be the study period of interest. For the transformation model, $H$ is an increasing function, and $\dot{H}$ is its first order derivative. The right column shows how covariates affect the shape function under a special case where $\mathit{Z}$ is one dimensional, ${\mu }_0\left(t\right)=\text{sin}\left(t\right)+1.5$, ${\alpha }_0=1$, $\tau =2\pi$, and $H$ is the cumulative distribution function of the Weibull distribution with shape parameter 1.5 and scale parameter 5.

FIGURE 2 ∣

Contour plot of the estimated cumulative shape function. Color indicates the value of ${\widehat{F}}_h(t,x,\tilde{\mathit{\beta }})$. The x-axis corresponds to time $\left(t\right)$, and the $y$-axis corresponds to the shape index $\left(x\right)$.