Methods for Estimating Center Effects on Recurrent Events

Abstract

In this article, we develop methods for quantifying center effects with respect to recurrent event data. In the models of interest, center effects are assumed to act multiplicatively on the recurrent event rate function. When the number of centers is large, traditional estimation methods that treat centers as categorical variables have many parameters and are sometimes not feasible to implement, especially with large numbers of distinct recurrent event times. We propose a new estimation method for center effects which avoids including indicator variables for centers. We then show that center effects can be consistently estimated by the center-specific ratio of observed to expected cumulative numbers of events. We also consider the case where the recurrent event sequence can be stopped permanently by a terminating event. Large-sample results are developed for the proposed estimators. We assess the finite-sample properties of the proposed estimators through simulation studies. The methods are then applied to national hospital admissions data for end stage renal disease patients.

Appendix

1.1 Consistency of $\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}},t)$

Let $\phi_{0}(t)=\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}},t) -\mu_{0}(t)=\phi_{1}(t)+\phi_{2}(t)$, where $\phi_{1}(t)=\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}},t) -\widehat{\mu}_{0}(\boldsymbol{\beta},t)$ and $\phi_{2}(t)=\widehat{\mu}_{0}(\boldsymbol{\beta}_{0},t) -\mu_{0}(t)$. By the triangle inequality,

(13)

A Taylor series expansion of

$$\phi_1(t)=n^{-1}\sum_{i=1}^n \sum_{k=1}^K w_k \biggl\{\int _0^t \frac{dN_{ik}(s)}{\mathbf{S}_k^{(0)}(\widehat{\boldsymbol{\beta}}, s)}-\int _0^t \frac{dN_{ik}(s)}{\mathbf{S}_k^{(0)}(\boldsymbol{\beta}_0, s)} \biggr\} $$

yields

$$\phi_1(t)=H\bigl(\boldsymbol{\beta}^{\dagger}, t\bigr)^T \bigl( \widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_0\bigr)+o_p\bigl(n^{-1/2} \bigr), $$

where β ^† lies in the line segment between $\widehat{\boldsymbol{\beta}}$ and β ₀, and $H(\boldsymbol{\beta}, t)=n^{-1}\sum_{i=1}^{n} H_{i}(\boldsymbol{\beta}, t)$ and

$$H_i(\boldsymbol{\beta}, t)= -\sum_{k=1}^K w_k\int_0^t\frac{\overline{\mathbf{Z}}_k(\boldsymbol{\beta},s)}{\mathbf{S}_k^{(0)}(\boldsymbol{\beta},s)} \,dN_{ik}(s). $$

Since N _ik (s), $\mathbf{S}_{k}^{(1)}(\boldsymbol{\beta}, s)$ are bounded and $\mathbf{S}_{k}^{(0)}(\boldsymbol{\beta}, s)$ is bounded away from 0 for $\boldsymbol{\beta}\in \mathcal{B}$ and s∈[0,τ] as n→∞, it follows that $H(\boldsymbol{\beta}, t)=n^{-1}\sum_{i=1}^{n} H_{i}(\boldsymbol{\beta}, t)$ is bounded for sufficiently large n. Since $\widehat{\boldsymbol{\beta}} \stackrel{\mathrm{a.s.}}{\longrightarrow} \boldsymbol{\beta}_{0}$ as n→∞, it follows that

(14)

Further, since $\sum_{k=1}^{K} w_{k} \theta_{k}=1$, we find

Since $\mathbf{S}_{k}^{(0)}(\boldsymbol{\beta}_{0}, s)$ is bounded away from 0 and $n^{-1}\sum_{i=1}^{n} \sum_{k=1}^{K} M_{ik}(\boldsymbol{\beta}_{0}, s)\stackrel{\mathrm{a.s.}}{\longrightarrow} 0$ by the Strong Law of Large Numbers (SLLN) as n→∞ for s∈[0,τ], we have

(15)

The uniform consistency of $\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}}, t)$ follows from (13), (14), and (15).

1.2 Weak Convergence of $\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}},t)$

We now consider the process n ^1/2 ϕ ₀(t)=n ^1/2 ϕ ₁(t)+n ^1/2 ϕ ₂(t). We see that

where $\widehat{\varPsi}_{2i}(\boldsymbol{\beta}_{0}, t)=\sum_{k=1}^{K} w_{k} \int_{0}^{t} \mathbf{S}_{k}^{(0)}(\boldsymbol{\beta}_{0}, s)^{-1}\,dM_{ik}(\boldsymbol{\beta}_{0}, s)$.

Under conditions (a)–(g), it follows that $\sup_{t \in [0, \tau]} |\widehat{\mu}_{0k}(\widehat{\boldsymbol{\beta}}, t) -\mu_{0k}(t)|\stackrel{\mathrm{a.s.}}{\longrightarrow} 0$, k=1,…,K, and $H(\boldsymbol{\beta}_{0}, t) \stackrel{\mathrm{a.s.}}{\longrightarrow} h(\boldsymbol{\beta}_{0}, t)$ for t∈[0,τ].

The partial likelihood score equation (6) can be written as

Under condition (e), $\widehat{\varPsi}_{1i}(\boldsymbol{\beta})$ converges to Ψ _1i (β) for i=1,…,n. A Taylor series expansion of the score equation at $\boldsymbol{\beta}=\widehat{\boldsymbol{\beta}}$ around β ₀ yields

As n→∞, $n^{-1}\widehat{I}(\boldsymbol{\beta}_{0})\stackrel{\mathrm{a.s.}}{\longrightarrow} \mathbf{A}$, where A is the positive definite matrix defined in condition (h). We can see that

(16)

Under condition (e), $\widehat{\varPsi}_{2i}(\boldsymbol{\beta}_{0}, t)$ converges to Ψ _2i (β ₀,t) for i=1,…,n. Hence

(17)

Combining (16) and (17), we see that $n^{1/2}\phi_{0}(t)=n^{-1/2} \sum_{i=1}^{n} \varPhi_{i}(\boldsymbol{\beta}_{0}, t)+o_{p}(1)$, which converges weakly to a Gaussian Process with covariance function ξ.

1.3 Consistency of $\widehat{\theta}_{k}$

  From (8),

(18)

With the uniform consistency of $\widehat{\boldsymbol{\beta}}$ and $\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}},t)$ as n→∞,

In addition,

by the SLLN as n→∞. Hence the numerator in (18) converges almost surely to 0 as n→∞. With the boundedness condition (c) and (e), the denominator in (18) is also bounded. Hence,

$$\widehat{\theta}_k-\theta_k\stackrel{\mathrm{a.s.}}{ \rightarrow} 0. $$

1.4 Weak convergence of $\widehat{\theta}_{k}$

  Let $\zeta_{k}=\theta_{k}^{-1}$ and $\widehat{\zeta}_{k}=\widehat{\theta}_{k}^{-1}$ for k=1,…,K. In the following, we will first show the asymptotic approximation of $n^{1/2}(\widehat{\zeta}_{k}-\eta_{k})$ and then use the delta method to obtain the asymptotic properties for $n^{1/2}(\widehat{\theta}_{k}-\theta_{k})$. Since

$n^{1/2}(\widehat{\eta}_{k}-\eta_{k})$ can be decomposed into three terms as

(19)

A Taylor series expansion of the first term of the right-hand side of equation (19) equals

$$n^{1/2}\overline{N}_k(\tau)^{-1} \biggl\{\int _0^{\tau} \mathbf{S}^{(1)}_k\bigl( \boldsymbol{\beta}^{\dagger}, t\bigr) \,d\widehat{\mu}_0\bigl( \boldsymbol{\beta}^{\dagger}, t\bigr)+\int_0^{\tau} S^{(0)}_k\bigl(\boldsymbol{\beta}^{\dagger}, t\bigr)\, dH\bigl( \boldsymbol{\beta}^{\dagger}, t\bigr) \biggr\}^T\bigl(\widehat{\boldsymbol{\beta}}- \boldsymbol{\beta}_0\bigr), $$

where β ^† is on the line segment between $\widehat{\boldsymbol{\beta}}$ and β ₀. With the results in (7), condition (e) and the uniform consistency of $\widehat{\boldsymbol{\beta}}$ and $\widehat{\mu}_{0}(\widehat{\boldsymbol{\beta}}, t)$, the first term of the right-hand side of equation (19) equals

(20)

With the results in (7) and condition (e), the second term of the right-hand side of equation (19) equals

(21)

The third term of the right-hand side of equation (19) equals

(22)

Combining (19), (20), (21), and (22), we see that

$$n^{1/2}(\widehat{\eta}_k-\eta_k)=n^{-1/2} \sum_{i=1}^n-\zeta_k^2 \varGamma_{ki}(\boldsymbol{\beta}_0)+o_p(1). $$

Therefore, $n^{1/2}(\widehat{\eta}_{k}-\eta_{k})$ converges in distribution to a normal variable with mean 0 and variance is asymptotically E{ζ ⁴ Γ _ki (β ₀)²}. Since $\widehat{\theta}_{k}=1/\widehat{\zeta}_{k}$, applying the delta method, we see that $n^{1/2}(\widehat{\theta}_{k}-\theta_{k})$ is asymptotically normally distributed with mean 0 and covariance Σ _k =E{Γ _ki (β ₀)²}.