A semiparametric additive rates model for recurrent event data

Abstract

Recurrent event data often arise in biomedical studies, with examples including hospitalizations, infections, and treatment failures. In observational studies, it is often of interest to estimate the effects of covariates on the marginal recurrent event rate. The majority of existing rate regression methods assume multiplicative covariate effects. We propose a semiparametric model for the marginal recurrent event rate, wherein the covariates are assumed to add to the unspecified baseline rate. Covariate effects are summarized by rate differences, meaning that the absolute effect on the rate function can be determined from the regression coefficient alone. We describe modifications of the proposed method to accommodate a terminating event (e.g., death). Proposed estimators of the regression parameters and baseline rate are shown to be consistent and asymptotically Gaussian. Simulation studies demonstrate that the asymptotic approximations are accurate in finite samples. The proposed methods are applied to a state-wide kidney transplant data set.

Appendix

Proof of Theorem 1

To begin the proof of strong consistency, we write:

$$ \begin{aligned} \widehat{\varvec{\theta}}-\varvec{\theta}_0&=\widehat{\bf A}^{-1}(\widehat{\bf U}-\widehat{\bf A}\varvec{\theta}_0)\\ &=\widehat{\bf A}^{-1} \left[ n^{-1}\sum_{i=1}^n \int_0^\tau \{{\bf Z}_i(s)-\overline{\bf Z}(s)\}\{\hbox{d}N_i(s)-I(C_i> s){\bf Z}_i^T(s)\varvec{\theta}_0\,\hbox{d}s \} \right]\\ &=\widehat{\bf A}^{-1} n^{-1}\sum_{i=1}^n \int_0^\tau \{{\bf Z}_i(s)-\overline{\bf Z}(s)\}\hbox{d}M_i(s;\varvec{\theta}_0). \end{aligned} $$

Through repeated applications of the Strong Law of Large Numbers (van der Vaart 2000), this quantity can be shown to converge almost surely to 0. With respect to asymptotic normality,

$$ n^{1/2}(\widehat{\varvec{\theta}}-\varvec{\theta}_0)= \widehat{\bf A}^{-1}n^{-1/2}\sum_{i=1}^n\int_0^\tau\{{\bf Z}_i(s)-\overline{\bf Z}(s)\}\hbox{d}M_i(s;\varvec{\theta}_0), $$

(15)

which can be shown to be asymptotically equivalent to

$$ {\bf A}^{-1}n^{-1/2}\sum_{i=1}^n \int_0^\tau\{{\bf Z}_i(s)-\overline{\bf z}(s)\}\hbox{d}M_i(s;\varvec{\theta}_0), $$

(16)

using the strong convergence of $\widehat{\bf A}$ to A and using results from empirical processes (Pollard 1990; Bilias et al. 1997) to demonstrate the convergence of $n^{-1/2}\sum_{i=1}^n\int_0^\tau\{\overline{\bf Z}(s)-\overline{\bf z}(s)\}\hbox{d}M_i(s;\varvec{\theta}_0)$ to 0 in probability. The proof of convergence to a zero-mean random vector is complete upon applying the Multivariate Central Limit Theorem (Sen and Singer 1993) to (16).

Proof of Theorem 2

Set $\widehat{\xi}(t)=\widehat{\xi}_1(t)+\widehat{\xi}_2(t)$, where

$$ \begin{aligned} \widehat{\xi}_1(t)&=\widehat{\mu}_0(t;\widehat{\varvec{\theta}})-\widehat{\mu}_0(t;\varvec{\theta}_0),\\ \widehat{\xi}_2(t)&=\widehat{\mu}_0(t;\varvec{\theta}_0)-\mu_0(t). \end{aligned} $$

By the Triangle Inequality, $|\widehat{\xi}(t)|\leq |\widehat{\xi}_1(t)| + |\widehat{\xi}_2(t)|$. Now,

$$ \begin{aligned} \widehat{\xi}_1(t)&=-n^{-1}\sum_{i=1}^n \int_0^t\widehat{\pi}(s)^{-1}I(C_i> s){\bf Z}_i(s)^T \hbox{d}s\ (\widehat{\varvec{\theta}}-\varvec{\theta}_0),\\ \widehat{\xi}_2(t)&=n^{-1}\sum_{i=1}^n \int_0^t\widehat{\pi}(s)^{-1}\hbox{d}M_i(s;\varvec{\theta}_0). \end{aligned} $$

Using the Uniform SLLN (Pollard 1990) and the almost sure convergence of $\widehat{\varvec{\theta}}$ to $\varvec{\theta}_0$, $\widehat{\xi}_1(t)$ converges almost surely to 0 uniformly in $t\in[0,\tau]$. In a similar fashion, $\widehat{\xi}_2(t)$ can be shown to converge strongly to 0 uniformly.

With respect to the process $n^{1/2}\widehat{\xi}_1(t)$, we begin with

$$ n^{1/2}\widehat{\xi}_1(t)=-n^{-1} \sum_{i=1}^n \int_0^t\widehat{\pi}(s)^{-1}I(C_i> s){\bf Z}_i(s)^T\,\hbox{d}s\ n^{1/2}(\widehat{\varvec{\theta}}-\varvec{\theta}_0). $$

Substituting in (15),

$$ n^{1/2}\widehat{\xi}_1(t)=-\int_0^t\overline{\bf Z}(s)^T\,\hbox{d}s\ \widehat{\bf A}^{-1}n^{-1/2}\sum_{i=1}^n\left\{\int_0^\tau\{{\bf Z}_i(s)-\overline{\bf Z}(s)\}\hbox{d}M_i(s;\varvec{\theta}_0)\right\}, $$

then using the strong convergence of $\widehat{\bf A}$ to A along with various empirical process results gives

$$ n^{1/2}\widehat{\xi}_1(t)=-\int_0^t\overline{\bf z}(s)^T \hbox{d}s\ {\bf A}^{-1}n^{-1/2}\sum_{i=1}^n{\bf U}_i +o_p(1), $$

(17)

which behaves asymptotically as sum of independent and identically distributed mean-zero variates. Similarly, it can be shown that

$$ n^{1/2}\widehat{\xi}_2(t)= n^{-1/2}\sum_{i=1}^n \int_0^t\pi(s)^{-1}\,\hbox{d}M_i(s;\varvec{\theta}_0)+o_p(1). $$

(18)

Combining (17) and (18),

$$ n^{1/2}\widehat{\xi}(t)=n^{-1/2}\sum_{i=1}^n \phi_i(t) + o_p(1), $$

which has finite distributions which converge to those of a zero-mean normal variate. Tightness can be shown by demonstrating the manageability of the components of $\phi_i(t)$, thus proving that $n^{1/2}\widehat{\xi}(t)$ converges to a mean-zero Gaussian process with covariance function $E[\phi_1(s)\phi_1(t)]$.