Semiparametric methods for center effect measures based on the ratio of survival functions

Kevin He; Douglas E. Schaubel

doi:10.1007/s10985-014-9293-9

Lifetime Data Analysis

October 2014, Volume 20, Issue 4, pp 619–644 | Cite as

Semiparametric methods for center effect measures based on the ratio of survival functions

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Kevin HeEmail author
Douglas E. Schaubel

Article

First Online: 28 February 2014

Received: 24 October 2012

Accepted: 10 February 2014

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Abstract

The survival function is often of chief interest in epidemiologic studies of time to an event. We develop methods for evaluating center-specific survival outcomes through a ratio of survival functions. The proposed method assumes a center-stratified additive hazards model, which provides a convenient framework for our purposes. Under the proposed methods, the center effects measure is cast as the ratio of subject-specific survival functions under two scenarios: the scenario in which the subject is treated at center $j$; and that wherein the subject is treated at a hypothetical center with survival function equal to the population average. The proposed measure reduces to the ratio of baseline survival functions, but is invariant to the choice of baseline covariate level. We derive the asymptotic properties of the proposed estimators, and assess finite-sample characteristics through simulation. The proposed methods are applied to national kidney transplant data.

Keywords

Additive hazards model Failure time data Observational studies Stratification

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Notes

Acknowledgments

This work was supported in part by National Institutes of Health Grant 5R01-DK070869. The authors thank the Associate Editor and two Reviewers for constructive comments which strengthened the article. The data reported here have been supplied by the Minneapolis Medical Research Foundation (MMRF) as the contractor for the Scientific Registry of Transplant Recipients (SRTR). The interpretation and reporting of these data are the responsibility of the authors and in no way should be seen as an official policy of or interpretation by the SRTR or the U.S. Government.

Appendix 1: Large-sample distribution of $\hat{\varvec{\beta }}$:

With simple algebraic manipulation, formula (7) yields

$$\begin{aligned} U(\varvec{\beta }_0)=\sum _{j=1}^J \sum _{i=1}^n \int _0^{\tau } \left\{ \mathbf Z _i(t)-\overline{\mathbf{Z }}_j(t)\right\} dM_{ij}(t). \end{aligned}$$

In the argument below, we demonstrate that $\overline{\mathbf{Z }}_j(t)$ converges almost surely to $\overline{\mathbf{z }}_j(t)$ uniformly in $t\in [0, \tau ], $ where

$$\begin{aligned} \overline{\mathbf{z }}_j(t)=\frac{s_j^{(1)}(t)}{s_j^{(0)}(t)}, \end{aligned}$$

with $s_j^{(1)}(t)$ being the limiting value of $S_j^{(1)}(t)$, and $s_j^{(0)}(t)$ being the limiting value of $S_j^{(0)}(t)$. We then show below that $n^{-\frac{1}{2}}U(\varvec{\beta }_0)$ is essentially a sum of independent and identically distributed random variables:

$$\begin{aligned} n^{-\frac{1}{2}}U(\varvec{\beta }_0)=n^{-\frac{1}{2}}\sum _{i=1}^n u_i(\varvec{\beta }_0)+o_p(1), \end{aligned}$$

(11)

where

$$\begin{aligned} u_i(\varvec{\beta }_0)=\sum _{j=1}^J \int _0^{\tau } \left\{ \mathbf Z _i(t)-\overline{\mathbf{z }}_j(t)\right\} dM_{ij}(t). \end{aligned}$$

The Multivariate Central Limit Theorem can be used to demonstrate that the random vector $n^{-\frac{1}{2}} U(\varvec{\beta }_0)$ asymptotically follows a normal distribution with mean zero. Asymptotic properties of the regression coefficient estimator are then given in the following Lemma.

Lemma 1

The random vector, $n^{\frac{1}{2}}(\hat{\varvec{\beta }}- \varvec{\beta }_0)$, converges weakly to a zero-mean normal random vector with a variance-covariance matrix given by $\varvec{\sigma }_{\beta }=\mathbf A ^{-1} E[u_1(\varvec{\beta }_0)u_1(\varvec{\beta }_0)^T] \mathbf A ^{-1}$.

A consistent estimator of the covariance matrix is given by

$$\begin{aligned} \hat{\varvec{\sigma }}_\beta =\hat{\mathbf{A }} ^{-1} \left\{ n^{-1}\sum _{i=1}^{n} \hat{u}_i(\hat{\varvec{\beta }}) \hat{u}_i(\hat{\varvec{\beta }}) ^T \right\} \hat{\mathbf{A }}^{-1}, \end{aligned}$$

with $ \hat{u}_i(\hat{\varvec{\beta }})$ obtained by replacing limiting values in $ u_i(\varvec{\beta }_0)$ by their empirical counterparts.

To show that (11) holds, i.e.,

$$\begin{aligned} n^{-\frac{1}{2}} \sum _{j=1}^J \sum _{i=1}^n \int _0^{\tau } \left\{ \overline{\mathbf{Z }}_j(t)-\overline{\mathbf{z }}_j(t)\right\} dM_{ij}(t) \xrightarrow {P} 0, \end{aligned}$$

we first prove that $\overline{\mathbf{Z }}_j(t)$ converges almost surely to $\overline{\mathbf{z }}_j(t)$ uniformly in $t\in [0, \tau ] $. Note that the collection of all cells $[t, \infty )$ in the real line is a VC class of index 2 (van der Vaart 1998) and, hence, satisfies the entropy conditions for the Glivenko-Cantelli theorem (Vaart and Wellner 1996). Therefore, $\{ I(X \ge t)I(G=j) , t \in [0, \tau ]\}$ belong to some Glivenko-Cantelli class, and $\mathbb {P}_n [I(X \ge t)I(G=j)] \xrightarrow {a.s.} \mathbb {P} [I(X \ge t)I(G=j)]$ uniformly in $t\in [0, \tau ] $, where $\mathbb {P}$ denotes expectations and $\mathbb {P}_n$ is the empirical measure (i.e. $\mathbb {P}_n [I(X\ge t) I(G=j)]=n^{-1} \sum _{i=1}^n Y_{ij}(t)$). Next, note that $\mathbf Z (t)$ is a stochastic process with bounded variation, and every function with bounded variation is the difference of two bounded monotonic functions. We then have that $\mathbf Z (t)$ belongs to some Glivenko-Cantelli class indexed by $t$. Moreover, both $I(X \ge t)$ and $\mathbf Z (t)$ are uniformly bounded. Hence, the product of these two classes form a Glivenko-Cantelli class. Therefore, $\mathbb {P}_n [I(X \ge t)I(G=j)\mathbf Z (t)] \xrightarrow {a.s.} \mathbb {P} [I(X \ge t)I(G=j)\mathbf Z (t)]$ uniformly in $t\in [0, \tau ] $. Finally, arguments analogous to the proof of Glivenko-Cantelli Theorem for the empirical distribution function (e.g., Theorem 19.1 of van der Vaart 1998), the bounded and monotonic conditions lead to $ [{\mathbb {P}_n \{Y_{j}(t)\}}]^{-1} \xrightarrow {a.s.} [{\mathbb {P} \{Y_{j}(t)\}}]^{-1}$ uniformly in $t\in [0, \tau ] $. Hence, we have $\overline{\mathbf{Z }}_j(t) \xrightarrow {a.s.} \overline{\mathbf{z }}_j(t)$ uniformly in $t\in [0, \tau ] $.

Next, we prove that $n^{-\frac{1}{2}} \sum _{j=1}^J \sum _{i=1}^n M_{ij}(t)$ converges weakly to a zero-mean Gaussian process.

$$\begin{aligned} n^{-\frac{1}{2}} \sum _{j=1}^J \sum _{i=1}^n M_{ij}(t)&= \mathbb {G}_n \left[ \sum _{j=1}^J I(G=j) \left\{ I( X \le t) \Delta \right. \right. \nonumber \\&\left. \left. -\int _0^{t} \ I(X \ge u)\{d \Lambda _{0j}(u)+\varvec{\beta }_0^T \mathbf Z (u)du\} \right\} \right] , \end{aligned}$$

where $\mathbb {G}_n$ is the empirical process defined by $\mathbb {G}_n f=\sqrt{n}(\mathbb {P}_n-\mathbb {P}) f$ for a measurable function $f$. Since the VC class with finite index also satisfies the entropy conditions for the Donsker Theorem (Vaart and Wellner 1996), $\{I(X \ge t), t\in [0, \tau ]\}$ and $\{I(X \le t), t\in [0, \tau ]\}$ belong to some Donsker classes. The same holds for the stochastic process with bounded variation, $\{\mathbf{Z }(t), t\in [0, \tau ]\}$, and the bounded monotonic stochastic process $\{\Lambda _{0j}(t), t\in [0, \tau ]\}$. The set of functions $\{\varvec{\beta }^T\mathbf Z (t), t\in [0, \tau ], \varvec{\beta } \in \textit{B} \}$ is also well-behaved and belongs to some Donsker class when the parameter space, $\textit{B}$, is compact. Finally, the class of functions of Lipschitz transformations of Donsker classes is a Donsker class. Therefore, with the various bounded conditions, we have that the class of functions $\sum _{j=1}^J I(G=j) \left[ I( X \le t) \Delta -\int _0^{t} \ I(X \ge u)\{d \Lambda _{0j}(u)+\varvec{\beta }^T \mathbf Z (u)du\} \right] $ indexed by $t$ and $\varvec{\beta }$ are Donsker classes and, hence, $n^{-\frac{1}{2}} \sum _{j=1}^J \sum _{i=1}^n M_{ij}(t)$ converges weakly to a zero-mean Gaussian process. It then follows from the Strong Embedding Theorem (Shorack and Wellner 1986) and Lemma 1 of Lin et al. (2000) that (11) holds.

Note that (11) can be also derived through Lemma 19.24 of van der Vaart (1998). For instance, define the following random map:

$$\begin{aligned} \Psi (\varvec{\beta }, \eta )=\sum _{j=1}^J \int _0^{\tau } I(G=j) \eta (t) \Big [ dN_{ij}(t) -\ I(X \ge t)\{d \Lambda _{0j}(t)+\varvec{\beta }^T \mathbf Z (t)dt\} \Big ], \end{aligned}$$

with $\hat{\eta }(t)=\overline{\mathbf{Z }}_j(t)$ and $\eta _0(t)=\overline{\mathbf{z }}_j(t)$. Let $\mathcal {F}$ be the functional space $\{\Psi (\varvec{\beta }, \eta ): \varvec{\beta } \in \textit{B}, \eta \in \varvec{\eta }\}$. The regularity conditions, (b), from Sect. 3 ensures that $\mathbb {P}_n [ I(\ge t)I(G=j)]$ is bounded away from zero, and hence, $\overline{\mathbf{Z }}_j(t)$ has bounded variation. Therefore, the class of functions $\mathcal {F}$ is a Donsker class when $\textit{B}$ is compact. With the bounded condition and the conclusion that $\overline{\mathbf{Z }}_j(t) \xrightarrow {a.s.} \overline{\mathbf{z }}_j(t)$ uniformly in $t\in [0, \tau ] $, it follows from the Dominant Convergence Theorem that $\Psi (\varvec{\beta }, \hat{\eta })$ converges to $\Psi (\varvec{\beta }, \eta _0)$ in probability with a norm defined by $d(\Psi (\cdot , \eta ), \Psi (\cdot , \eta '))=\sup _{\varvec{\beta } \in \textsc {B}} \{\mathbb {P}(\Psi (\varvec{\beta }, \eta )-\Psi (\varvec{\beta }, \eta '))^2\}^{\frac{1}{2}}$. We then have all the conditions to apply Lemma 19.24 of van der Vaart (1998). Thus, $\mathbb {G}_n \{\Psi (\varvec{\beta }, \hat{\eta })-\Psi (\varvec{\beta }, \eta _0)\} \xrightarrow {P} 0$ and (11) holds.

Formula (11) entails that $n^{-\frac{1}{2}}U(\varvec{\beta }_0)$ is essentially a sum of independent and identically distributed random variables:

$$\begin{aligned} n^{-\frac{1}{2}} U({\varvec{\beta }}_0) \!&= \! \mathbb {G}_n \Bigg [ \sum _{j=1}^J \Big \{ \{\mathbf{Z}(X)-{\overline{\mathbf{z}}}_j(X)\} I(G=j)\Delta \\&\quad -\,\int _0^{\tau }\{\mathbf{Z}(t) \!-\! {\overline{\mathbf{z}}}_j(t)\} I(G \!=\! j) I(X \!\ge \! t) \{d\varvec{\Lambda }_0(t) \!+\! \varvec{\beta }_0^T\mathbf Z (t)dt\} \Big \} \Bigg ] \!+\! o_p(1). \end{aligned}$$

Using the Multivariate Central Limit Theorem, the random vector $n^{-\frac{1}{2}} U(\varvec{\beta }_0)$ asymptotically follows a $p$-variate normal distribution with mean zero. Similarly, we have that

$$\begin{aligned} n^{\frac{1}{2}} ({\hat{\varvec{\beta }}}-{\varvec{\beta }}_0) =&n^{-\frac{1}{2}} {\hat{\mathbf{A}}}^{-1} \sum _{j=1}^J \sum _{i=1}^n \int _0^{\tau }\{\mathbf{Z}_i(t)-{\overline{\mathbf{z}}}_j(t)\}dM_{ij}(t) +o_p(1)\\ =&\mathbf{A}^{-1} \mathbb {G}_n \Bigg [\sum _{j=1}^J \Big \{ \{\mathbf{Z}(X)-\overline{\mathbf{z}}_j(X)\} I(G=j)\Delta \\&\,-\int _0^{\tau }\{\mathbf{Z}(t) \!-\! \overline{\mathbf{z}}_j(t)\} I(G \!=\! j) I(X \!\ge \! t) \{d\varvec{\Lambda }_0(t) \!+\! \varvec{\beta }_0^T\mathbf Z (t)dt\} \Big \} \Bigg ] \!+\! o_p(1). \end{aligned}$$

By the Multivariate Central Limit Theorem and Slutsky’s Theorem, Theorem 1 follows.

Appendix 2: Weak convergence of $\varvec{\Lambda }_0(t)$ and $\varvec{\psi }(t)$:

Applying the Functional Delta Method (van der Vaart 1998), we now consider the asymptotic properties of the random processes $n^{\frac{1}{2}}(\hat{\varvec{\Lambda }}_0- \varvec{\Lambda }_0)$, where $\varvec{\Lambda }_0=[\Lambda _{01}, \ldots , \Lambda _{0J}]^T$, with $\Lambda _{0j} =\{\Lambda _{0j}(t); \; t\in (0,\tau ) \}$.

Lemma 2

$n^{\frac{1}{2}}(\hat{\varvec{\Lambda }}_0- \varvec{\Lambda }_0)$ converges weakly to a zero-mean Gaussian process with a variance-covariance matrix between $n^{\frac{1}{2}}\{\hat{\Lambda }_{0j}(s)- \Lambda _{0j}(s)\}$ and $n^{\frac{1}{2}}\{\hat{\Lambda }_{0k}(t)- \Lambda _{0k}(t)\}$ given by $\sigma _{ jk}(s,t)=E[\xi _{1j}(s)\xi _{1k}(t)]$, where

$$\begin{aligned} \xi _{ij}(s)= \int _0^s \frac{dM_{ij}(u)}{s_j^{(0)}(u)} -\left\{ \int _0^s \overline{\mathbf{z }}_j(u) du \right\} ~ \mathbf A ^{-1} \sum _{\ell =1}^J \int _{0}^{\tau } \left\{ \mathbf Z _{i}(u)-\overline{\mathbf{z }}_\ell (u)\right\} dM_{i\ell }(u).\nonumber \\ \end{aligned}$$

(12)

The covariance function can be consistently estimated by $\hat{\sigma }_{ jk}(s,t)=n^{-1}\sum _{i=1}^{n} \hat{\xi }_{ij}(s) \hat{\xi }_{ik}(t)$, with $\hat{\xi }_{ij}(s)$ obtained by replacing limiting values in $\xi _{ij}(s)$ with their empirical counterparts.

Applying the Functional Delta method again, we have

$$\begin{aligned} n^{\frac{1}{2}}\{\hat{\psi }_{0j}(t)- \psi _{0j}(t)\} = \mathbf H _j^T(t) n^{\frac{1}{2}}\{\hat{\varvec{\Lambda }}_{0}(t)- {\varvec{\Lambda }}_{0}(t)\}+o_p(1), \end{aligned}$$

where $\mathbf H _j^T(t)=[H_{j1}(t), \ldots , H_{jJ}(t)]$, with $H_{jk}(t)= \psi _{0j}(t) \psi _{0k}(t) p_k- I (k=j)\psi _{0j}(t)$ for $k=1, \ldots , J$.

We begin the proof by considering the large-sample behavior of the estimator of $\Lambda _{0j}(t)$. In particular,

$$\begin{aligned}&n^{\frac{1}{2}} \{{\hat{\Lambda }}_{0j}(t)-\Lambda _{0j}(t)\} \\&\quad = n^{\frac{1}{2}} \{{\hat{\Lambda }}_{0j}(t;{\hat{\varvec{\beta }}})- {\hat{\Lambda }}_{0j}(t; {\varvec{\beta }})\}+ n^{\frac{1}{2}} \{{\hat{\Lambda }}_{0j}(t; {\varvec{\beta }})- \Lambda _{0j}(t) \} \\&\quad = n^{\frac{1}{2}} \int _0^t \frac{\sum _{i=1}^n dM_{ij}(u)}{\sum _{i=1}^n Y_{ij}(u)}- n^{\frac{1}{2}} \int _0^t \frac{\sum _{i=1}^n Y_{ij}(u) ({\hat{\varvec{\beta }}}- {\varvec{\beta }_0})^T \mathbf{Z }_{i}(u)du}{\sum _{i=1}^n Y_{ij}(u)} \\&\quad = n^{\frac{1}{2}} \int _0^t \frac{\sum _{i=1}^n dM_{ij}(u)}{ \mathbb {P} [ I(\ge t)I(G=j)]}- \left\{ \int _0^t {\overline{\mathbf{z}}}_j(u) du \right\} ~ \mathbf{A }^{-1} \sum _{j=1}^J \int _{0}^{\tau } \left\{ \mathbf{Z }(u)-{\overline{\mathbf{z }}}_j(u)\right\} \nonumber \\&\qquad \times dM_{j}(u)+o_p(1)\\&\quad ={} \mathbb {G}_n \Bigg [ \frac{I( X \le t)(G=j) \Delta - \int _0^t I(X \ge u)I(G=j)\{\varvec{\beta }^T \mathbf{Z }_{i}(u)du+d\Lambda _{0j}(u)\} }{ \mathbb {P} [ I(X\ge t)I(G=j)]} \\ {}&\qquad - \{\int _0^t {\overline{\mathbf{z }}}_j(u) du \} ~ \mathbf{A }^{-1} \sum _{\ell =1}^J I(G=\ell ) \Big [ \{\mathbf{Z }(X)-{\overline{\mathbf{z }}}_\ell (X) \} \Delta \\ {}&\qquad -\int _{0}^{\tau } \left\{ \mathbf{Z }(u)-{\overline{\mathbf{z }}}_\ell (u)\right\} I(X \ge u)\{\varvec{\beta }^T \mathbf{Z }_{i}(u)du+d\Lambda _{0\ell }(u)\} \Big ] \Bigg ]+o_p(1) , \end{aligned}$$

where the third equality can be obtained by either Lemma 1 of Lin et al. (2000) or Lemma 19.24 of van der Vaart (1998). Applying the Donsker Theorem once again, Lemma 2 follows.

Note that Lemma 2 can also be established by Functional Delta Method. The proof is essentially the same as those used by Kosorok (2008) for the Nelson-Aalen estimator. Adapting the argument of Kosorok (2008), the center-specific cumulative hazard function can be formulated as:

$$\begin{aligned} \hat{\Lambda }_{0j}(t)=\int _0^t \frac{dB_{nj}(u)}{ Y_{nj}(u)}, \end{aligned}$$

where

$$\begin{aligned} B_{nj}(t)=\mathbb {P}_n \left[ \Delta I(G=j) I(X \le t) - \int _0^t I(X \ge u) I(G=j) \hat{\varvec{\beta }}^T\mathbf Z (u)du \right] \end{aligned}$$

and

$$\begin{aligned} Y_{nj}(t)= \mathbb {P}_n \left[ I(X \ge t) I(G=j)\right] . \end{aligned}$$

After defining

$$\begin{aligned} B_{0j}&= \mathbb {P} \left[ \Delta I(G=j) I(X \le t) - \int _0^t I(X \ge u) I(G=j) {\varvec{\beta }}^T\mathbf Z (u)du\right] \\&= \mathbb {P} \left[ \int _0^t I(X \ge t) I(G=j) d\Lambda _{0j}(u)\right] , \end{aligned}$$

we have

$$\begin{aligned} n^{\frac{1}{2}}\{B_{nj}-B_{0j}\} ={}&\mathbb {G}_n \Bigg [\Delta I(G=j) I(X \le t) - \int _0^t I(X \ge u) I(G=j) {\varvec{\beta }}^T\mathbf Z (u)du \\ {}&- \mathbb {P}\Big [\int _0^t I(X \!\ge \! u) I(G \!=\! j) \mathbf Z (u)du\Big ] ~ \mathbf A ^{-1} \sum _{\ell =1}^J I(G \!=\! \ell ) \Big [\left\{ \mathbf Z (X) \!-\! \overline{\mathbf{z }}_\ell (X)\right\} \Delta \\ {}&- \int _{0}^{\tau } \left\{ \mathbf Z (u)-\overline{\mathbf{z }}_\ell (u)\right\} I(X \ge u)\{\varvec{\beta }^T \mathbf Z _{i}(u)du+d\Lambda _{0\ell }(u)\}\Big ]\Bigg ]+o_p(1) \end{aligned}$$

and

$$\begin{aligned} n^{\frac{1}{2}}\{Y_{nj}-s_{0j}\}= \mathbb {G}_n \left[ I(X \ge t) I(G=j)\right] . \end{aligned}$$

Using the Donsker Theorem, we have that the pair $(B_{nj}, Y_{nj})$ is asymptotically normal:

$$\begin{aligned} n^{\frac{1}{2}} \genfrac(){0.0pt}{}{B_{nj}-B_{0j}}{Y_{nj}-s_{0j}} \Longrightarrow \genfrac(){0.0pt}{}{\mathbb {G}_{1j}}{\mathbb {G}_{2j}}, \end{aligned}$$

where $\mathbb {G}_{1j}$ and $\mathbb {G}_{2j}$ are tight Gaussian processes.

We are ready to show the asymptotic properties of $\varvec{\Lambda }_0(t)$. Note that the center-specific cumulative hazard function is constructed through the map $ (A, B) \mapsto (A, \frac{1}{B}) \mapsto \int _0^t \frac{dA}{B}. $ This composition map is Hadamard-differentiable (van der Vaart 1998) on appropriate domains; e.g., the regularity conditions in our setting guarantee that $B$ is bounded away from zero and $A$ has uniformly bounded variation. By the Functional Delta Method (van der Vaart 1998), we conclude that

$$\begin{aligned} n^{\frac{1}{2}} (\hat{\Lambda }_{0j} - \Lambda _{0j}) \Longrightarrow \int _0^t \frac{d \mathbb {G}_{1j}}{s_{0j}}-\int _0^t \frac{\mathbb {G}_{2j} dB_{0j}}{s_{0j}^2}, \end{aligned}$$

which leads to Lemma 2. Applying Functional Delta Method once again, Theorem 1 follows.

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Kevin He
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Douglas E. Schaubel
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1.Department of BiostatisticsUniversity of MichiganAnn ArborUSA

Semiparametric methods for center effect measures based on the ratio of survival functions

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