Regression analysis of clustered failure time data with informative cluster size under the additive transformation models

Abstract

This paper discusses regression analysis of clustered failure time data, which occur when the failure times of interest are collected from clusters. In particular, we consider the situation where the correlated failure times of interest may be related to cluster sizes. For inference, we present two estimation procedures, the weighted estimating equation-based method and the within-cluster resampling-based method, when the correlated failure times of interest arise from a class of additive transformation models. The former makes use of the inverse of cluster sizes as weights in the estimating equations, while the latter can be easily implemented by using the existing software packages for right-censored failure time data. An extensive simulation study is conducted and indicates that the proposed approaches work well in both the situations with and without informative cluster size. They are applied to a dental study that motivated this study.

Appendix: Proofs of the asymptotic properties of the estimators

In this Appendix, we will first describe the regularity conditions needed for the asymptotic properties and then give more precise descriptions of the properties in the two theorems before sketching the proofs. First it can be seen from the definition of the matrix $A_{wee}$ that $A_{wee}$ converges in probability to a nonsingular deterministic matrix denoted by $\mathcal A_{wee}.$ In the following, assume that $n^{-1}\sum _{i=1}^n\sum _{j=1}^{n_i}n_i^{-1}Y_{ij}(t)\varvec{Z}_{ij}(t), n^{-1}\sum _{i=1}^n\sum _{j=1}^{n_i}n_i^{-1}Y_{ij}(t),\ n^{-1}\sum _{i=1}^nY_i^q(t)\varvec{Z}_i^q(t)$ and $n^{-1}\sum _{i=1}^nY_i^q(t)$ uniformly converge to $\kappa (t),\ \pi (t),$ $\tilde{\kappa }(t)$ and $\tilde{\pi }(t),$ respectively. For $i=1,\ldots ,n; \, j=1,\ldots ,n_i$ and some constant $\tau$, we assume the following regularity conditions: $P\{Y_{ij}(t)=1, 0\le t\le \tau \}>0;\ \int _0^\infty \lambda _0(t)dt<\infty ; \varvec{Z}_{ij}(t)$ is bounded; $\mathcal A_{wee}$ is positive definite and the cluster sizes are finite.

To give a precise description of the properties and the proof, define the marginal filtration by

$$\begin{aligned} \mathcal {F}_{ij}(t)=\sigma \{N_{ij}(u),Y_{ij}(u+),\varvec{Z}_{ij}(u+),\ 0\le u\le t\}, \end{aligned}$$

and

$$\begin{aligned} M_{ij}(t)=N_{ij}(t)-\int _0^tY_{ij}(s)\lambda _0(s)dt- \int _0^tY_{ij}(s)\varvec{\beta }_0^{\prime }\varvec{Z}_{ij}(s)ds. \end{aligned}$$

Obviously, $M_{ij}(t)$ is a local square-integrable martingale with respect to $\mathcal {F}_{ij}(t).$ Now we are ready to describe the properties and sketch the proofs.

1.1 Asymptotic properties of $\hat{\varvec{\beta }}_{wee}$ and the proof

Theorem 1

Under the regularity conditions given above, as $n\rightarrow \infty ,\ \sqrt{n}(\hat{\varvec{\beta }}_{wee}-\varvec{\beta }_0)$ converges in distribution to a zero-mean normal random vector, and the covariance matrix can be consistently estimated by $\hat{\varvec{\Sigma }}_{wee}=A_{wee}^{-1}B_{wee}A_{wee}^{-1}.$

Proof

Some simple algebraic manipulation yields

$$\begin{aligned}&\sqrt{n}(\hat{\varvec{\beta }}_{wee}-\varvec{\beta }_0)\\&\quad =\sqrt{n}(nA_{wee})^{-1} \left( \sum _{i=1}^n\frac{1}{n_i}\sum _{j=1}^{n_i}\int _0^\infty ( \varvec{Z}_{ij}(t)-\bar{\varvec{Z}}(t))dN_{ij} (t)-nA_{wee}\varvec{\beta }_0\right) \\&\quad =\frac{1}{\sqrt{n}}A_{wee}^{-1}\sum _{i=1}^n \frac{1}{n_i}\sum _{j=1}^{n_i}\left( \int _0^\infty ( \varvec{Z}_{ij}(t)-\bar{\varvec{Z}}(t))dN_{ij} (t)\right. \\&\qquad \left. -\int _0^\infty Y_{ij}(t)(\varvec{Z}_{ij} (t)-\bar{\varvec{Z}}(t))^{\otimes 2}\varvec{\beta }_0dt\right) \\&\quad =\frac{1}{\sqrt{n}}A_{wee}^{-1}\sum _{i=1}^n\frac{1}{n_i} \sum _{j=1}^{n_i}\int _0^\infty (\varvec{Z}_{ij} (t)-\bar{\varvec{Z}}(t))\left( dN_{ij}(t)- Y_{ij}(t) \varvec{\beta }_0^{\prime }\varvec{Z}_{ij}(t)dt\right) \\&\quad =\frac{1}{\sqrt{n}}A_{wee}^{-1}\sum _{i=1}^n \frac{1}{n_i}\sum _{j=1}^{n_i}\int _0^\infty (\varvec{Z}_{ij} (t)-\bar{\varvec{Z}}(t))dM_{ij}(t). \end{aligned}$$

Applying the same arguments as those in Yin and Cai (2004), under the regularity conditions, the above quantity can be shown to be asymptotically equivalent to

$$\begin{aligned} \frac{1}{\sqrt{n}}A_{wee}^{-1}\sum _{i=1}^n\left( \frac{1}{n_i} \sum _{j=1}^{n_i}\int _0^\infty \left( \varvec{Z}_{ij}(t) -\frac{\kappa (t)}{\pi (t)}\right) dM_{ij}(t)\right) :=\frac{1}{\sqrt{n}}A_{wee}^{-1}\sum _{i=1}^n \Theta _i(\varvec{\beta }_0). \end{aligned}$$

Note that $A_{wee}$ converges in probability to $\mathcal A_{wee},$ and $\Theta _i(\varvec{\beta }_0)$ for $i=1,\ldots ,n$ are independent random vectors with zero-mean and bounded variance. By the multivariate central limit theorem and the Slutsky’s theorem, it yields that $\sqrt{n}(\hat{\varvec{\beta }}_{wee}-\varvec{\beta }_0)$ converges to a zero-mean normal random vector with zero-mean and covariance matrix that can be consistently estimated by

$$\begin{aligned} \hat{\varvec{\Sigma }}_{wee}=A_{wee}^{-1}\left( \frac{1}{n} \sum _{i=1}^n\hat{\Theta }_i(\hat{\varvec{\beta }}_{wee}) \hat{\Theta }_i^{\prime }(\hat{\varvec{\beta }}_{wee})\right) A_{wee}^{-1}=A_{wee}^{-1}B_{wee}A_{wee}^{-1}. \end{aligned}$$

So Theorem 1 holds. $\square$

1.2 Asymptotic properties of $\hat{\varvec{\beta }}_{wcr}$ and the proof

Theorem 2

Under regularity conditions, as $n\rightarrow \infty ,\ \sqrt{n}(\hat{\varvec{\beta }}_{wcr}-\varvec{\beta }_0)$ converges in distribution to a zero-mean normal random vector, and the covariance matrix can be consistently estimated by $\hat{\varvec{\Sigma }}_{wcr}.$

Proof

Since $\hat{\varvec{\beta }}^q$ is the solution of the estimating equation $U^q(\varvec{\beta })=0$, and by the Taylor’s expansion, we have

$$\begin{aligned} -U^q(\varvec{\beta }_0)=U^q(\hat{\varvec{\beta }}^q) -U^q(\varvec{\beta }_0)=\frac{\partial U^q(\varvec{\beta }^*)}{\partial {\varvec{\beta }}^*} (\hat{\varvec{\beta }}^q-\varvec{\beta }_0), \end{aligned}$$

(6)

where $\hat{\varvec{\beta }}^*$ is on the line segment between $\hat{\varvec{\beta }}^q$ and $\varvec{\beta }_0.$ Rewriting (6) yields that

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}^q-\varvec{\beta }_0)= \left( -\frac{1}{n}\frac{\partial U^q(\varvec{\beta }^*)}{\partial {\varvec{\beta }}^*}\right) ^{-1} \left( \frac{1}{\sqrt{n}}U^q(\varvec{\beta }_0)\right) . \end{aligned}$$

Note that

$$\begin{aligned} \frac{1}{n}\frac{\partial U^q(\varvec{\beta })}{\partial {\varvec{\beta }}} =-\frac{1}{n}\sum _{i=1}^n\int _0^\infty Y_i^q(t)\left( \varvec{Z}_i^q(t)-\bar{\varvec{Z}}^q(t)\right) ^{\otimes 2}dt=-A_q, \end{aligned}$$

which is negative definite, and so $A_q$ is positive definite. Since the Q resamples are identically distributed, it can be seen that $A_q$ converges in probability to a deterministic and positive definite matrix denoted by $\mathcal A_{wcr}.$

Averaging over $q=1,\ldots ,Q$ resamples, it yields

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}_{wcr}-\varvec{\beta }_0)= & {} \frac{1}{Q}\sum _{q=1}^Q\sqrt{n}(\hat{\varvec{\beta }}^q-\varvec{\beta }_0) =\frac{1}{Q}\sum _{q=1}^QA_q^{-1}\frac{1}{\sqrt{n}}U^q(\varvec{\beta }_0)\\= & {} {\mathcal A_{wcr}}^{-1}\frac{1}{\sqrt{n}Q} \sum _{q=1}^QU^q(\varvec{\beta }_0)+o_p(1). \end{aligned}$$

It is sufficient to show that $n^{-1/2}Q^{-1}\sum _{q=1}^QU^q(\varvec{\beta }_0)$ converges to a normal distribution as $n\rightarrow \infty ,$ changing the order of summation yields that

$$\begin{aligned} \frac{1}{\sqrt{n}Q}\sum _{q=1}^QU^q(\varvec{\beta }_0)= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{1}{Q}\sum _{q=1}^Q \int _0^\infty \left( \varvec{Z}_i^q(t)-\bar{\varvec{Z}}^q(t)\right) dM_i^q(t)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{1}{Q}\sum _{q=1}^Q \int _0^\infty \left( \varvec{Z}_i^q(t)-\frac{\tilde{\kappa }(t)}{\tilde{\pi }(t)}\right) dM_i^q(t)+o_p(1)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n{\mathcal U_i(\varvec{\beta }_0)}+o_p(1), \end{aligned}$$

where ${\mathcal U_i(\varvec{\beta }_0)},\ i=1,\ldots ,n$ are independent with zero mean and finite variance. By the multivariate central limit theorem, $n^{-1/2}Q^{-1}\sum _{q=1}^QU^q(\varvec{\beta }_0)$ is asymptotically normal with zero mean and some positive definite covariance matrix. Combining with Slutsky’s theorem, $\sqrt{n}(\hat{\varvec{\beta }}_{wcr}-\varvec{\beta }_0)$ converges in distribution to a normal random vector with zero mean and covariance matrix can be consistently estimated by $n\hat{\varvec{\Sigma }}_{wcr}$.

For the consistent estimator of the covariance matrix, similar to Hoffman et al. (2001), we first write

$$\begin{aligned} \text {var}(\hat{\varvec{\beta }}_q)=E\left( \text {var}(\hat{\varvec{\beta }}_q|\text {data})\right) +\text {var}\left( {E}(\hat{\varvec{\beta }}_q|\text {data})\right) , \end{aligned}$$

where the expectations on the right-hand side are over the resampling distribution for $\hat{\varvec{\beta }}_q$ given the data. By the fact of $E(\hat{\varvec{\beta }}_q|\text {data})=\hat{\varvec{\beta }}_{wcr},$ it yields that

$$\begin{aligned} \text {var}(\hat{\varvec{\beta }}_{wcr})=\text {var}( \hat{\varvec{\beta }}_q)-E(\text {var} (\hat{\varvec{\beta }}_q|\text {data})). \end{aligned}$$

(7)

For each resampled data, $\text {var}(\hat{\varvec{\beta }}_q)$ can be consistently estimated by $\tilde{\varvec{\Sigma }}_q = \hat{\varvec{\Sigma }}_q/n$. By averaging over the Q resamples, the resulting estimator denoted by $Q^{-1}\sum _{q=1}^Q\tilde{\varvec{\Sigma }}_q$ is also consistent. For the second term on the right-hand side of (7), since

$$\begin{aligned} E(\text {var}(\hat{\varvec{\beta }}_q|\text {data}))= E\left( \frac{1}{Q}\sum _{q=1}^Q(\hat{\varvec{\beta }}_q- \hat{\varvec{\beta }}_{wcr}) (\hat{\varvec{\beta }}_q-\hat{\varvec{\beta }}_{wcr})^{\prime }\right) , \end{aligned}$$

it can be estimated as the covariance matrix based on the Q resampling estimators $\hat{\varvec{\beta }}_q$, $q = 1,\ldots ,Q$, that is

$$\begin{aligned} \tilde{\Omega }=\frac{1}{Q}\sum _{q=1}^Q(\hat{\varvec{\beta }}_q -\hat{\varvec{\beta }}_{wcr})(\hat{\varvec{\beta }}_q -\hat{\varvec{\beta }}_{wcr})^{\prime }\ \ \ \text {or}\ \ \ \Omega =\frac{1}{Q-1}\sum _{q=1}^Q(\hat{\varvec{\beta }}_q -\hat{\varvec{\beta }}_{wcr})(\hat{\varvec{\beta }}_q -\hat{\varvec{\beta }}_{wcr})^{\prime }\,. \end{aligned}$$

Here similar to Cong et al. (2007), let $\Omega$ denote the estimator of $\text {var}(\hat{\varvec{\beta }}_q|\text {data}).$ Thus the estimated variance-covariance matrix of $\hat{\varvec{\beta }}_{wcr}$ is

$$\begin{aligned} \hat{\varvec{\Sigma }}_{wcr}=\frac{1}{Q}\sum _{q=1}^Q \tilde{\varvec{\Sigma }}_q-\frac{1}{Q-1}\sum _{q=1}^Q (\hat{\varvec{\beta }}_q-\hat{\varvec{\beta }}_{wcr}) (\hat{\varvec{\beta }}_q-\hat{\varvec{\beta }}_{wcr})^{\prime }. \end{aligned}$$

To show the consistency of $\hat{\varvec{\Sigma }}_{wcr},$ it suffices to show that $\Omega -E(\tilde{\Omega })=(\Omega -\tilde{\Omega })+ (\tilde{\Omega }-E(\tilde{\Omega }))\rightarrow 0$ in probability as $n\rightarrow \infty .$ It is easy to see that $\Omega -\tilde{\Omega }\rightarrow 0$ in probability as $n\rightarrow \infty .$ Additionally, by applying the same arguments as those in the proof of Cong et al. (2007), it can be shown that $\tilde{\Omega }-E(\tilde{\Omega })\rightarrow 0$ in probability as $n\rightarrow \infty .$ This completes the proof of Theorem 2. $\square$