A new joint model of recurrent event data with the additive hazards model for the terminal event time

Abstract

Recurrent event data are frequently encountered in clinical and observational studies related to biomedical science, econometrics, reliability and demography. In some situations, recurrent events serve as important indicators for evaluating disease progression, health deterioration, or insurance risk. In statistical literature, non informative censoring is typically assumed when statistical methods and theories are developed for analyzing recurrent event data. In many applications, however, there may exist a terminal event, such as death, that stops the follow-up, and it is the correlation of this terminal event with the recurrent event process that is of interest. This work considers joint modeling and analysis of recurrent event and terminal event data, with the focus primarily on determining how the terminal event process and the recurrent event process are correlated (i.e. does the frequency of the recurrent event influence the risk of the terminal event). We propose a joint model of the recurrent event process and the terminal event, linked through a common subject-specific latent variable, in which the proportional intensity model is used for modeling the recurrent event process and the additive hazards model is used for modeling the terminal event time.

Appendix: Proofs of asymptotic results

In this section, we will use the same notation defined above in Sect. 2.1, and all limits are taken as $n\rightarrow \infty$. Let $V_{i}=m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}$ and $\Omega _{i}=m_{i}(m_{i}-1)\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-2}$. Recall that $\left( N_{i}(\cdot ),Y_{i},\delta _{i},X_{i},D_{i},C_{i},v_{i}\right) ,i=1,2,\ldots$ are assumed to be independent and identically distributed copies of $\left( N(\cdot ),Y,\delta ,X,D,C,v\right)$, each i corresponding to a subject sampled at random.

In order to study the asymptotic distributions of $\widehat{\eta }$ and $\widehat{\theta },$ we need the following regularity conditions:

• (R1) $P(Y\ge \tau ,v>0)>0,$ $P(Y>\tau _{\varepsilon })=1,$ where $\tau _{\varepsilon }=\inf \{t:\Lambda _{0}(t)>\varepsilon \}$ for some $\varepsilon >0$, and $E\{N(\tau )^{2}\}<\infty .$

• (R2) $G(t)=E\{vI(Y\ge t)\exp (\gamma _{0}'X)\}$ is a continuous function for $t\in [0,\tau ].$

• (R3) The weight function Q(t) has bounded variation and converges to a deterministic function q(t) in probability uniformly in $t\in [0,\tau ];$

• (R4) A is non singular and $\Sigma =E(\xi _{i}\xi _{i}')$ exists.

$A=\left( \begin{array}{cc} A_{11} &{} \quad A_{12}\\ A_{12}' &{} \quad A_{22} \end{array}\right)$,

$$\begin{aligned}&A_{11}=E\Big \{\int _{0}^{\tau }q(t)\{X-\bar{x}(t)\}^{\otimes 2}\Delta (t)dN(t)\Big \}\;,\\&A_{22}=E\Big \{\int _{0}^{\tau }q(t)\{V-\bar{v}(t)\}^{2}\Delta (t)dN(t)\Big \},\\&A_{12}=E\Big \{\int _{0}^{\tau }q(t)\{X-\bar{x}(t)\}\{V-\bar{v}(t)\}\Delta (t)dN(t)\Big \}, \end{aligned}$$

where $\Delta (t)=I(t\le Y)$, $\xi _{i}$ is defined in Lemma 5.2 below, and $\bar{x}(t)$ and $\bar{v}(t)$ are the limits (in probability) respectively, of $\bar{X}(t)$ and $\bar{V}(t)$, which are well defined by regularity condition R1.

Define $R(t)=G(t)\Lambda _{0}(t)$,

$$\begin{aligned} H(t)= & {} \int _{0}^{t}G(u)d\Lambda _{0}(u),\\ D_{1}= & {} E\{\exp \{\alpha _{0}'X_{i}^{*}\}X_{i}^{*\otimes 2}\},\\ \kappa _{i}(t)= & {} \sum _{j=1}^{m_{i}}\Big \{\int _{t}^{\tau }\frac{I(T_{ij}\le u\le Y_{i})d{H}(u)}{{R}^{2}(u)}-\frac{I(t<T_{ij}\le \tau )}{{R}(T_{ij})}\Big \}, \end{aligned}$$

and

$$\begin{aligned} e_{i}=X_{i}^{*}\Big [\frac{m_{i}}{F(Y_{i})}-\exp \{\alpha _{0}'X_{i}^{*}\}\Big ]-\int \frac{x^{*}m\kappa _{i}(y)dP_{1}(x^{*},y,m)}{{F}(y)}, \end{aligned}$$

where $T_{ij}$ denotes the occurrence time of the jth event of the ith subject, and $P_{1}(x^{*},y,m)$ is the joint probability measure of $(X_{i}^{*},Y_{i},m_{i}).$ Note that $X_{i}$ and $X_{i}^{*}$ share the same random variables, hereafter we use $P_{1}(x,y,m)$ to denote the joint probability measure of $(X_{i},Y_{i},m_{i}).$ Let $\phi _{1i}$ denote the vector $D_{1}^{-1}e_{i}$ without the first entry and $\phi _{2i}$ denote the first entry of $D_{1}^{-1}e_{i}$. Set $\varphi _{i}(t)=\kappa _{i}(t)+\phi _{2i}$, and $b_{i}(y,x)=\varphi _{i}(y)+\phi _{1i}'x$.

Under conditions (R1) and (R2), it follows from Wang et al. (2001) that

$$\begin{aligned} n^{1/2}\{\widehat{\Lambda }_{0}(t)-\Lambda _{0}(t)\}=n^{-1/2}\Lambda _{0}(t)\sum _{i=1}^{n}\varphi _{i}(t)+o_{p}(1) \end{aligned}$$

(7)

and

$$\begin{aligned} n^{1/2}\{\widehat{\gamma }-\gamma _{0}\}=n^{-1/2}\sum _{i=1}^{n}\phi _{1i}+o_{p}(1) \end{aligned}$$

(8)

Lemma 5.1

(Lemma A.1 of Lin and Ying 2001) Let $H_{n}(t)$ and $M_{n}(t)$ be two sequences of bounded processes. Suppose that $H_{n}(t)$ is monotone and converges uniformly to H(t) in probability (i.e., $\sup _{t}|H_{n}(t)-H(t)|$ converges to 0 in probability) and that $M_{n}(t)$ converges weakly to a zero-mean process with continuous sample paths. Then $\int _{0}^{t}\{H_{n}(s)-H(s)\}dM_{n}(s)$ converges in probability to 0 uniformly in t.

Lemma 5.2

Under the joint models specified in Sect. 2 and condition (R4), $n^{-1/2}\sum _{i=1}^{n}\xi _{i}$ has an asymptotically normal distribution with mean zero and covariance matrix $\Sigma =E(\xi _{i}\xi _{i}')$, where $\xi _{i}=(\xi _{1i}',\xi _{2i}')'$ with

$$\begin{aligned} \xi _{1i}=&\int _{0}^{\tau }q(t)\{X_{i}-\bar{x}(t)\}dM_{i}^{D}(t)\\&+\theta _{0}\int _{0}^{\tau }q(t)\int \{x-\bar{x}(t)\}\frac{mI(y\ge t)}{\Lambda _{0}(y)e^{\gamma '_{0}x}}b_{i}(y,x)dP_{1}(x,y,m)dt, \end{aligned}$$

and

$$\begin{aligned} \xi _{2i}= & {} \int _{0}^{\tau }q(t)\Big [\{V_{i}-\bar{v}(t)\}\big \{ dN_{i}^{D}(t)-\big (\eta _{0}'X_{i}+\alpha _{0}(t)\big )\Delta _{i}(t)dt\big \}\\&-\,\theta _{0}\{\Omega _{i}(t)-V_{i}\bar{v}(t)\}\Delta _{i}(t)dt\Big ]-\int q(y)\frac{m\delta }{\Lambda _{0}(y)e^{\gamma '_{0}x}}b_{i}(y,x)dP_{2}(x,y,m,\delta )\\&+\,\int _{0}^{\tau }q(t)\int \{\eta _{0}'x+\alpha _{0}(t)-\theta _{0}\bar{v}(t)\}\frac{mI(y\ge t)}{\Lambda _{0}(y)e^{\gamma '_{0}x}}b_{i}(y,x)dP_{1}(x,y,m)dt\\&+\,\theta _{0}\int _{0}^{\tau }q(t)\int \frac{2m(m-1)I(y\ge t)}{\{\Lambda _{0}(y)e^{\gamma '_{0}x}\}^{2}}b_{i}(y,x)dP_{1}(x,y,m)dt, \end{aligned}$$

where $dM_{i}^{D}(t)=dN_{i}^{D}(t)-\Delta _{i}(t)\{\eta _{0}'X_{i}+\theta _{0}V_{i}+\alpha _{0}(t)\}dt$, and $P_{1}(x,y,m)$ and $P_{2}(x,y,m,\delta )$ denote the joint probability measures of $(X_{i},Y_{i},m_{i})$ and $(X_{i},Y_{i},m_{i},\delta _{i})$, respectively.

Proof

Note that under the joint models specified in Sect. 2.1, $\xi _{i},i=1,\ldots ,n$ are independent and identically distributed. Recalling the Central Limit Theorem and condition (R4), it suffices to verify that $E(\xi _{i})=0$. By the definition of $dM_{i}^{D}(t)$, under model (2) and the Poisson model (1), it has mean zero given the covariates. Hence, the first term of $\xi _{1i}$ has mean zero. In addition, for fixed x and y, under model (1), we have

$$\begin{aligned} E(b_{i}(y,x))=E(\kappa _{i}(y))+E(\phi _{2i})+x'E(\phi _{1i})=0. \end{aligned}$$

By the Fubini theorem, we exchange the order of expectation, and can verify that the second term of $\xi _{1i}$ has mean zero. So, we have verified that $E(\xi _{1i})=0$. Similarly, we can verify $E(\xi _{2i})=0$, which completes the proof.

Lemma 5.3

Under the regularity conditions (R1)-(R3), we have

$$\begin{aligned} \widehat{V}_{i}-V_{i}=-m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}n^{-1}\left\{ \sum _{j=1}^{n}\varphi _{j}(t)+\sum _{j=1}^{n}\phi _{1j}'X_{i}\right\} +o_{p}(n^{-1/2}) \end{aligned}$$

(9)

and

$$\begin{aligned} \widehat{\Omega }_{i}-\Omega _{i}=-2m_{i}(m_{i}-1)\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-2}n^{-1}\big \{\sum _{j=1}^{n}\varphi _{j}(t)+\sum _{j=1}^{n}\phi _{1j}'X_{i}\big \}+o_{p}(n^{-1/2}) \end{aligned}$$

(10)

Proof

By using the Taylor expansion theorem and (7), (8), we have for (9)

$$\begin{aligned} \widehat{V}_{i}-V_{i}&=m_{i}\{\widehat{\Lambda }_{0}(Y_{i})e^{\widehat{\gamma }'X_{i}}\}^{-1}-m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}\\&=\,-\,m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-2}\big \{ e^{\gamma _{0}'X_{i}}\{\widehat{\Lambda }_{0}(Y_{i}){-}\Lambda _{0}(Y_{i})\}{+}\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}(\widehat{\gamma }{-}\gamma _{0})'X_{i}\big \}\\&~~~~~+\,o(||\widehat{\gamma }-\gamma _{0}||+|\widehat{\Lambda }_{0}(Y_{i})-\Lambda _{0}(Y_{i})|)\\&=\,-\,m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}n^{-1}\big \{\sum _{j=1}^{n}\varphi _{j}(t)+\sum _{j=1}^{n}\phi _{1j}'X_{i}\big \}+o_{p}(n^{-1/2}), \end{aligned}$$

where $o_{p}(.)$ is independent of i since (7) holds uniformly in t (Huang and Wang 2004). Similarly, we can show that (10) holds.

Lemma 5.4

Under the regularity conditions (R1)-(R3),we have

$$\begin{aligned} n^{-1/2}U_{1}(\eta _{0},\theta _{0})&=n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\{X_{i}{-}\bar{x}(t)\}dM_{i}^{D}(t)\nonumber \\&\quad -\theta _{0}n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\{X_{i}-\bar{x}(t)\}\big \{\widehat{V}_{i}-V_{i}\big \}\Delta _{i}(t)dt+o_{p}(1), \end{aligned}$$

(11)

and

$$\begin{aligned} n^{-1/2}U_{2}(\eta _{0},\theta _{0})&=n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\Big [\{V_{i}{-}\bar{v}(t)\}\big \{ dN_{i}^{D}(t){-}\big (\eta _{0}'X_{i}{+}\alpha _{0}(t)\big )\Delta _{i}(t)dt\big \}\nonumber \\&\quad -\,\theta _{0}\{\Omega _{i}(t)-V_{i}\bar{v}(t)\}\Delta _{i}(t)dt\Big ]\nonumber \\&\quad +\,n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\big \{\widehat{V}_{i}{-}{V}_{i}\big \}\big \{ dN_{i}^{D}(t){-}\big (\eta _{0}'X_{i}{+}\alpha _{0}(t)\big )\Delta _{i}(t)dt\big \}\nonumber \\&\quad -\,\theta _{0}n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\big \{\widehat{\Omega }_{i}{-}\Omega _{i}{-}\{\widehat{V}_{i}{-}{V}_{i}\}\bar{v}(t)\big \}\Delta _{i}(t)dt{+}o_{p}(1). \end{aligned}$$

(12)

Proof

Straightforward calculations give

$$\begin{aligned} n^{-1/2}U_{1}(\eta _{0},\theta _{0})&=n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\{X_{i}-\bar{x}(t)\}dM_{i}^{D}(t)\\&-\theta _{0}n^{-1/2}\sum _{i=1}^{n}\int _{0}^{\tau }Q(t)\{X_{i}-\bar{x}(t)\}\big \{\widehat{V}_{i}-V_{i}\big \}\Delta _{i}(t)dt\\&-\!\int _{0}^{\tau }\!\!Q(t)\{\bar{X}(t)-\bar{x}(t)\}n^{-1/2} \sum _{i=1}^{n}\!\big \{ dM_{i}^{D}(t)-\theta _{0}\{\widehat{V}_{i}-V_{i}\}\Delta _{i}(t)dt\big \}. \end{aligned}$$

To obtain (11), it suffices to show that the third term is $o_{p}(1)$. We first show that

$$\begin{aligned} \int _{0}^{\tau }Q(t)\{\bar{X}(t)-\bar{x}(t)\}n^{-1/2}\sum _{i=1}^{n}dM_{i}^{D}(t)=o_{p}(1) \end{aligned}$$

(13)

To see this, let $M^{n}(t)=\int _{0}^{t}{-1/2}\sum _{i=1}^{n}dM_{i}^{D}(t)$. Then, by the functional Central Limit Theorem(Vaart and Wellner 1996), under conditions(R1)-(R3), $M^{n}(t)$ converges weakly to a Gaussian process G with continuous paths. By the Skorohod embedding theorem, there exists a sequence $\tilde{M}^{n}(t)$ which is equal to $M^{n}(t)$ in law and converges uniformly to a Gaussian process $\tilde{G}$ with continuous paths. In addition, by the uniform law of large numbers, $Q(t)\{\bar{X}(t)-\bar{x}(t)\}$ converges uniformly to 0 almost surely. Then, integrating by parts and applying Lemma 5.1, we have that

$$\begin{aligned} \int _{0}^{\tau }Q(t)\{\bar{X}(t)-\bar{x}(t)\}d\tilde{M}^{n}(t)=o_{p}(1), \end{aligned}$$

which implies that

$$\begin{aligned} \int _{0}^{\tau }Q(t)\{\bar{X}(t)-\bar{x}(t)\}dM^{n}(t)=o_{p}(1). \end{aligned}$$

Hence, (13) is obtained. Applying the expansion (9), we have

$$\begin{aligned}&\int _{0}^{\tau } Q(t)\{\bar{X}(t)-\bar{x}(t)\}n^{-1/2}\sum _{i=1}^{n}\{\widehat{V}_{i}-V_{i}\}\Delta _{i}(t)dt\nonumber \\&\quad =-\int _{0}^{\tau }Q(t)\{\bar{X}(t)-\bar{x}(t)\}\Big \{\{n^{-1}\sum _{i=1}^{n}m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}\Delta _{i}(t)\}\{n^{-1/2}\sum _{j=1}^{n}\varphi _{j}(t)\}\nonumber \\&\,\,\,\quad +\{n^{-1}\sum _{i=1}^{n}m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}X_{i}\Delta _{i}(t)\}'\{n^{-1/2}\sum _{j=1}^{n}\phi _{1j}\}\Big \} dt+o_{p}(1). \end{aligned}$$

(14)

By the uniform law of large numbers, we have that $\bar{X}(t)-\bar{x}(t)=o_{p}(1)$, $n^{-1}\sum _{i=1}^{n}m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}\Delta _{i}(t)=O_{p}(1)$ and $n^{-1}\sum _{i=1}^{n}m_{i}\{\Lambda _{0}(Y_{i})e^{\gamma _{0}'X_{i}}\}^{-1}X_{i}\Delta _{i}(t)=O_{p}(1)$ uniformly in t. By the functional form of the Central Limit Theorem, we have $n^{-1/2}\sum _{j=1}^{n}\varphi _{j}(t)=O_{p}(1)$ uniformly in t. $n^{-1/2}\sum _{j=1}^{n}\phi _{1j}=O_{p}(1)$ follows by the Central Limit Theorem. Note that Q(t) converges uniformly to q(t) by assumption (R3), so applying Lemma 5.1 again we obtain

$$\begin{aligned} \int _{0}^{\tau }Q(t)\{\bar{X}(t)-\bar{x}(t)\}n^{-1/2}\sum _{i=1}^{n}\{\widehat{V}_{i}-V_{i}\}\Delta _{i}(t)dt=o_{p}(1), \end{aligned}$$

which completes the proof of (11). In a similar way, we can show that (12) also holds.

Lemma 5.5

Under the regularity conditions (R1)-(R3), $n^{-1/2}U(\eta _{0},\theta _{0})$ has asymptotically a normal distribution with mean zero and covariance matrix $\Sigma =E(\xi _{i}\xi _{i}')$, where $U(\eta ,\theta )=(U_{1}(\eta ,\theta )',U_{2}(\eta ,\theta )')'$ and $\xi _{i}$ is defined in Lemma 5.2.

Proof

Combing the results of Lemmas 5.3 and 5.4, with justifications similar to those in the proof of (11), we obtain that

$$\begin{aligned} n^{-1/2}U(\eta _{0},\theta _{0})=n^{-1/2}\sum _{i=1}^{n}\xi _{i}+o_{p}(1). \end{aligned}$$

Hence, the proof can be completed by applying Lemma 5.2.

Lemma 5.6

Under the regularity conditions (R1)–(R4),

$$\begin{aligned} n^{1/2}\left( \begin{array}{c} \widehat{\eta }-\eta _{0}\\ \widehat{\theta }-\theta _{0} \end{array}\right) =A^{-1}n^{-1/2}{U}(\eta _{0},\theta _{0})+o_{p}(1). \end{aligned}$$

Proof

Note that $-n^{-1}\partial {U}(\eta _{0},\theta _{0})/\partial (\eta ',\theta ')=\widehat{A}$ is independent of $\eta _{0}$ and $\theta _{0}$ and converges in probability to A,  which is defined in condition (R4). Straightforward calculation gives

$$\begin{aligned} n^{-1}{U}(\widehat{\eta },\widehat{\theta })-n^{-1}{U}(\eta _{0},\theta _{0})=-\widehat{A}\left( \begin{array}{c} \widehat{\eta }-\eta _{0}\\ \widehat{\theta }-\theta _{0} \end{array}\right) . \end{aligned}$$

Therefore, based on the consistency of $\widehat{\eta }$ and $\widehat{\theta }$ proved in Sect. 2.1, we complete the proof.

Proof of Theorem 2.1

Proof

Combining Lemmas 5.5 and 5.6, we have, $\widehat{\eta }$ and $\widehat{\theta }$ converge in probability to $\eta _{0}$ and ${\theta }_{0}$, respectively. In addition $n^{1/2}(\widehat{\eta }-\eta _{0})$ and $n^{1/2}(\widehat{\theta }-\theta _{0})$ have asymptotically a joint normal distribution with mean zero and covariance matrix $A^{-1}\Sigma A^{-1}.$