Testing for center effects on survival and competing risks outcomes using pseudo-value regression

Abstract

In multi-center studies, the presence of a cluster effect leads to correlation among outcomes within a center and requires different techniques to handle such correlation. Testing for a cluster effect can serve as a pre-screening step to help guide the researcher towards the appropriate analysis. With time to event data, score tests have been proposed which test for the presence of a center effect on the hazard function. However, sometimes researchers are interested in directly modeling other quantities such as survival probabilities or cumulative incidence at a fixed time. We propose a test for the presence of a center effect acting directly on the quantity of interest using pseudo-value regression, and derive the asymptotic properties of our proposed test statistic. We examine the performance of our proposed test through simulation studies in both survival and competing risks settings. The proposed test may be more powerful than tests based on the hazard function in settings where the center effect is time-varying. We illustrate the test using a multicenter registry study of survival and competing risks outcomes after hematopoietic cell transplantation.

Appendix: Asymptotic distribution of the proposed test statistic

Here we assume that the cluster size $n_k$’s and the covariates vector ${\varvec{x}}_{kj}$ belong to a bounded set. We also assume standard regularity conditions on the risk set, namely that there exist functions $r^c(s)$ defined on [0, t] with $\inf _{s \in [0,t]} r^c(s) >0$ such that

$$\begin{aligned} \sup _{s \in [0,t]}\left| a_n^{-2}{R^c}^{(n)}(s)-r^c(s)\right| \rightarrow _p 0 \qquad as \quad n \rightarrow \infty , \end{aligned}$$

where $R^c(t)$ denotes the number at risk at time t, $\{a_n\}$ is a sequence of positive constants.

We derive the asymptotic distribution of the proposed test for the competing risks setting, and note that the survival setting can be obtained as a special case. Before we derive the mean, variance, and distribution of the pseudo-value test under $H_0$, we first present two Lemmas characterizing regularity conditions on the $\varphi _{kj}$.

Lemma 1

Under $H_0$, ${\text {var}}(\varphi _{kj})$ is bounded.

Proof of Lemma 1

$$\begin{aligned} {\text {var}}(\varphi _{kj})=&{\text {var}}\left\{ \frac{\Delta _{kj}N_{kj}(t)}{G(T_{kj})}\right\} \end{aligned}$$

(3)

$$\begin{aligned}&+{\text {var}}\left\{ \int _0^{T_{kj}}\frac{P(T\le t, \epsilon = 1 | T\ge u)}{G(u)} d M_{kj}^c (u) \right\} \end{aligned}$$

(4)

$$\begin{aligned}&+ 2{\text {cov}}\left\{ \frac{\Delta _{kj}N_{kj}(t)}{G(T_{kj})} , \int _0^{T_{kj}}\frac{P(T\le t, \epsilon = 1 | T\ge u)}{G(u)} d M_{kj}^c (u) \right\} . \end{aligned}$$

(5)

The term in (3) is bounded since $\Delta _{kj}(t)N_{kj}(t)$ is either 0 or 1 and $1/G(T_{kj})$ is bounded under the regularity conditions on the risk set. The term in (4) can be written as

$$\begin{aligned}&{\text {var}}\left\{ \int _0^{T_{kj}}\right. \left. \frac{P(T\le t, \epsilon = 1 | T\ge u)}{G(u)} d M_{kj}^c (u) \right\} \nonumber \\&\quad =\int _0^{T_{kj}}\frac{(P(T\le t, \epsilon = 1 | T\ge u))^2}{G^2(u)} I(T_{kj} \ge u) \lambda ^c(u)d u \nonumber \\&\quad \le \frac{1}{G^2(T_{kj})}\Lambda ^c(T_{kj}). \end{aligned}$$

which is bounded due to the regularity conditions on the risk set.

Finally the term in (5) can be rewritten as

$$\begin{aligned}&{\text {E}}\left\{ \frac{\Delta _{kj}N_{kj}(t)}{G(T_{kj})} \quad \right. \left. \int _0^{T_{kj}}\frac{P(T\le t, \epsilon = 1 | T\ge u)}{G(u)} d M_{kj}^c (u) \right\} \nonumber \\&\quad =-{\text {E}}\bigg \{\frac{I(T_{kj} \le t)I(\widetilde{T_{kj}} \le C_{kj})}{G(T_{kj})} \nonumber \\&\qquad \int _0^{T_{kj}}\frac{P(T\le t, \epsilon = 1 | T\ge u)}{G(u)}I(T_{kj}\ge u)\lambda ^c(u)du \bigg \}. \end{aligned}$$

(6)

which is bounded in absolute value by $\Lambda ^c(T_{kj})/G^2(T_{kj})$. Therefore ${\text {var}}(\varphi _{kj})$ is bounded.

The second lemma utilizes Lemma 1 to establish additional regularity conditions used to establish the asymptotic distribution of the proposed test statistic; because the results are straightforward given Lemma 1, it is stated without proof. $\square$

Lemma 2

Defining

$$\begin{aligned} W_k =\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}\left( \varphi _{kj}-\mu _{kj}^0\right) \left( \varphi _{kj'}-\mu _{kj'}^0\right) , \end{aligned}$$

the following conditions hold:

• (a) $\sum _{k=1}^{\infty }\left\{ {\text {var}}(\partial W_k/\partial \beta _l)/k^2\right\} < \infty \quad \forall \quad l=1,\dots ,p$.

• (b) ${\text {E}}\{\partial ^2W_k/\partial \beta _l \partial \beta _m\}$ is bounded $\forall \quad l,m=1,\dots ,p \quad$ and $\forall \quad k$.

• (c) $\sum _{k=1}^{\infty }\left\{ {\text {var}}(\partial ^2W_k/\partial \beta _l\beta _m)/k^2\right\} < \infty \quad \forall \quad l,m=1,\dots ,p$.

• (d) $\forall \quad \epsilon >0 \sum _{k=1}^K\int _{\vert z\vert \ge \epsilon }z^2 dF_k \rightarrow 0$ where $z_k=W_k/I^{1/2}$ with distribution function $F_k$ (Lindeberg’s condition).

Based on the conditions established in Lemmas 1 and 2, we can prove the following theorem on the asymptotic distribution of the score test statistic under $H_0$.

Theorem

Under $H_0$ and the regularity conditions described above,

$$\begin{aligned} \frac{T_{PC}({{\widehat{\varvec{\beta }}}})}{\sqrt{I}} \overset{D}{\rightarrow } N(0,1), \end{aligned}$$

where ${\widehat{\varvec{\beta }}}$ is a consistent estimator of $\varvec{\beta }$ under $H_0$.

Proof

First we show that $T_{PC}$ is asymptotically equivalent to $K^{-1/2}W$, where

$$\begin{aligned} W&=\sum _{k=1}^K\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}W_k. \nonumber \\ T_{PC}&= K^{-1/2}\sum _{k=1}^K\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}\left\{ \varphi _{kj}-\mu _{kj}^0+O_p(K^{-1/2})\right\} \left\{ \varphi _{kj'}-\mu _{kj'}^0+O_p(K^{-1/2})\right\} \nonumber \\&= K^{-1/2}W + O_p(1)2K^{-1} \sum _{k=1}^K(n_k-1)\sum _{j=1}^{n_k}\left( \varphi _{kj}-\mu _{kj}^0\right) +O_p\left( K^{-1/2}\right) . \end{aligned}$$

Note that ${\text {E}}\{\sum _{j=1}^{n_k}(\varphi _{kj}-\mu _{kj}^0)\}=0$. Then by the law of large numbers, since $n_k$ is bounded and from Lemma 1${\text {var}}(\varphi _{kj}-\mu _{kj}^0)$ is bounded,

$$\begin{aligned} K^{-1} \sum _{k=1}^K\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}\left( \varphi _{kj'}-\mu _{kj'}^0\right) =o_p(1). \end{aligned}$$

Therefore we have

$$\begin{aligned} T_{PC}=K^{-1/2}W+o_p(1), \end{aligned}$$

and the two statistics will have the same limiting distribution. The mean of the limiting distribution of $K^{-1/2}W$ is 0 because under $H_0$, $\varphi _{kj}$ is independent of $\varphi _{kj'}$, so that

$$\begin{aligned} E(K^{-1/2}W)=K^{-1/2}\sum _{k=1}^K\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}E\left( \varphi _{kj}-\mu _{kj}^0\right) E\left( \varphi _{kj'}-\mu _{kj'}^0\right) =0. \end{aligned}$$

To derive the variance of the limiting distribution first note that since $\varphi _{kj}$ and $\varphi _{kj'}$ are independent under $H_0$ for $j \ne j'$, ${\text {E}}[(\varphi _{kj}-\mu _{kj}^0)(\varphi _{kj'}-\mu _{kj'}^0)]=0$ and

$$\begin{aligned} {\text {cov}}\left[ \left( \varphi _{kj}-\mu _{kj}^0\right) \left( \varphi _{kj'}-\mu _{kj'}^0\right) , \left( \varphi _{kl}-\mu _{kl}^0\right) \left( \varphi _{kl'}-\mu _{kl'}^0\right) \right] =0, \end{aligned}$$

for $(l,l') \ne (j,j')$. Then

$$\begin{aligned} {\text {var}}(K^{-1/2}W)&=K^{-1}\sum _{k=1}^K\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}{\text {var}}\left[ \left( \varphi _{kj}- \mu _{kj}^0\right) \left( \varphi _{kj'}-\mu _{kj'}^0\right) \right] \nonumber \\&=K^{-1}\sum _{k=1}^K\sum _{j=1}^{n_k}\sum _{j'\ne j}^{n_k}{\text {E}}\left[ \left( \varphi _{kj}-\mu _{kj}\right) ^2\left( \varphi _{kj'}-\mu _{kj'}^0\right) ^2\right] \nonumber \\&=I. \end{aligned}$$

Proof of asymptotic normality of the test statistic follows closely the derivation in Jacqmin-Gadda and Commenges applied to W, using the regularity conditions in Lemma 2, and so the details are omitted. $\square$