Additive mixed effect model for recurrent gap time data

Abstract

Gap times between recurrent events are often of primary interest in medical and observational studies. The additive hazards model, focusing on risk differences rather than risk ratios, has been widely used in practice. However, the marginal additive hazards model does not take the dependence among gap times into account. In this paper, we propose an additive mixed effect model to analyze gap time data, and the proposed model includes a subject-specific random effect to account for the dependence among the gap times. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. In addition, some graphical and numerical procedures are presented for model checking. The finite sample behavior of the proposed methods is evaluated through simulation studies, and an application to a data set from a clinic study on chronic granulomatous disease is provided.

Appendices

Appendix A: Proofs of Asymptotic Properties

In order to study the asymptotic properties of the proposed estimators, we need the following regularity conditions:

• (C1) $P(X_{i1}\ge \tau )>0,$ and $E(\Vert Z_i\Vert ^2)<\infty .$

• (C2) $H_0(t)$ is a right-continuous increasing function and satisfies $H_0(\tau )<\infty$.

• (C3) The matrix A is nonsingular, where

  $$\begin{aligned} A=E\Big [\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau }Y_{ij}(t) \big \{Z_i-\bar{z}(t)\big \}^{\otimes 2}dt\Big ]. \end{aligned}$$

Proof of Theorem 1

Using the uniform strong law of large numbers (Pollard 1990, p. 41), we obtain

$$\begin{aligned} \frac{1}{n}\sum \limits _{i=1}^n\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*} Y_{ij}(t) \longrightarrow E\Big [\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*} I(X_{ij}\ge t)\Big ], \end{aligned}$$

and

$$\begin{aligned} \frac{1}{n}\sum \limits _{i=1}^n\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*} Y_{ij}(t)Z_i \longrightarrow E\Big [\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*} Z_iI(X_{ij}\ge t)\Big ] \end{aligned}$$

almost surely uniformly in $t\in [0,\tau ]$. Thus, $\bar{Z}(t)$ converges almost surely to $\bar{z}(t)$ uniformly in $t\in [0,\tau ]$. Similarly,

$$\begin{aligned} \hat{A}=\frac{1}{n}\sum \limits _{i=1}^n\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } Y_{ij}(t)\{Z_i-\bar{Z}(t)\}^{\otimes 2}dt \longrightarrow A \end{aligned}$$

(10)

almost surely. Define

$$\begin{aligned} dM^{(0)}_{ij}(t)=dN_{ij}(t)-Y_{ij}(t)\big \{dH_0(t)+ \beta _0' Z_i dt\big \}. \end{aligned}$$

Then we have

$$\begin{aligned}&\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \big \{Z_i-\bar{Z}(t)\big \}dN_{ij}(t)\nonumber \\&\qquad =\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \big \{Z_i-\bar{Z}(t)\big \} dM_{ij}^{(0)}(t) + \hat{A} \beta _0\nonumber \\&\qquad \longrightarrow A \beta _0 \end{aligned}$$

(11)

almost surely uniformly in $t\in [0,\tau ]$. Therefore, it follows from (10) and (11) that

$$\begin{aligned} \hat{\beta } = \hat{A}^{-1} \Big [\frac{1}{n} \sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \{Z_i-\bar{Z}(t)\} dN_{ij}(t)\Big ] \longrightarrow \beta _0 \end{aligned}$$

almost surely.

To prove the asymptotic normality of $\hat{\beta }$, write

$$\begin{aligned} \hat{\beta }-\beta _0= & {} \hat{A}^{-1}\Big [\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \big \{Z_i-\bar{Z}(t)\big \}dN_{ij}(t)\\&-\frac{1}{n}\sum \limits _{i=1}^n\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } Y_{ij}(t)\{Z_i-\bar{Z}(t)\}^{\otimes 2}\beta _0 dt\Big ]\\= & {} \hat{A}^{-1}\Big [\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \big \{Z_i-\bar{Z}(t)\big \}\big \{dN_{ij}(t)-Y_{ij}(t)\beta _0' Z_i dt\big \}\Big ]\\= & {} \hat{A}^{-1} \Big [\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*} \int _{0}^{\tau }\big \{Z_i-\bar{Z}(t)\big \} dM_{ij}^{(0)}(t)\Big ]. \end{aligned}$$

Since $\bar{Z}(t)$ converges almost surely to $\bar{z}(t)$ uniformly in $t\in [0,\tau ]$, using Lemma 1 in the Appendix of Lin et al. (2000) and (10), we have

$$\begin{aligned} n^{1/2}(\hat{\beta }-\beta _0) =A^{-1} \Big [ n^{-1/2} \sum \limits _{i=1}^{n} \frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau }\big \{Z_i-\bar{z}(t)\big \}dM_{ij}^{(0)}(t) \Big ] +o_p(1). \end{aligned}$$

(12)

Note that

$$\begin{aligned} E\Big [\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \big \{Z_i-\bar{z}(t)\big \} dM_{ij}^{(0)}(t)\Big ]=0. \end{aligned}$$

It then follows from the multivariate central limit theorem that $n^{1/2}(\hat{\beta }-\beta _0)$ is asymptotically normal with mean zero and covariance matrix $A^{-1}\Sigma A^{-1}$, where

$$\begin{aligned} \Sigma = E \Big [\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau } \big \{Z_i-\bar{z}(t)\big \}dM_{ij}^{(0)}(t)\Big ]^{\otimes 2}. \end{aligned}$$

$\square$

Proof of Theorem 2

Define

$$\begin{aligned} dQ^{(0)}_{ij}(t)=dN_{ij}(t)-Y_{ij}(t)\big \{ d\Lambda _0(t)+ \beta _0' Z_i dt+\xi _idt \big \}. \end{aligned}$$

In view of (6), we have

$$\begin{aligned} \hat{\Lambda }_0(t)-\Lambda _0(t) =\frac{1}{n}\sum \limits _{i=1}^{n}\int _0^t \frac{\sum \nolimits _{j=1}^{M_i^*} dQ^{(0)}_{ij}(u)}{ \sum \nolimits _{j=1}^{M_i^*} Y_{ij}(u)} -D(t)' (\hat{\beta }-\beta _0) + \Big (\frac{1}{n}\sum \limits _{i=1}^{n} \xi _i\Big )t, \end{aligned}$$

(13)

where $D(t)=n^{-1}\sum _{i=1}^n Z_i t.$ Note that

$$\begin{aligned} E\bigg [\int _0^t\frac{\sum \nolimits _{j=1}^{M_i^*} dQ^{(0)}_{ij}(u)}{\sum \nolimits _{j=1}^{M_i^*} Y_{ij}(u)}\bigg ]=0. \end{aligned}$$

Then it follows from the uniform strong law of large numbers that the first term on the right-hand side of (13) converges almost surely to 0 uniformly in $t\in [0,\tau ]$. Since $\hat{\beta }-\beta _0$ and $n^{-1}\sum _{i=1}^{n} \xi _i$ converge almost surely to 0, the second and third terms converge almost surely to 0 uniformly in $t\in [0,\tau ]$ as well. Thus, $\hat{\Lambda }_0(t)$ converges almost surely to $\Lambda _0(t)$ uniformly in $t\in [0,\tau ]$.

To show the weak convergence of $n^{1/2}\{\hat{\Lambda }_0(t)-\Lambda _0(t)\}$, it follows from (12) and (13) that

$$\begin{aligned} n^{1/2}\{\hat{\Lambda }_0(t)-\Lambda _0(t)\}= & {} n^{-1/2}\sum \limits _{i=1}^{n} \int _{0}^{t} \frac{\sum \nolimits _{j=1}^{M_i^*}\big [dQ^{(0)}_{ij}(u)+Y_{ij}(u)\xi _idu\big ]}{\sum \nolimits _{j=1}^{M_i^*}Y_{ij}(u)} \nonumber \\&-D(t)' n^{1/2} (\hat{\beta }-\beta _0)\nonumber \\= & {} n^{-1/2}\sum \limits _{i=1}^{n} \phi _i(t)+o_p(1), \end{aligned}$$

(14)

where

$$\begin{aligned} \phi _i(t)=&\int _{0}^{t} \frac{\sum \limits _{j=1}^{M_i^*}\big [dN_{ij}(u)-Y_{ij}(u)\big \{d\Lambda _0(u)+\beta _0'Z_i du\big \}\big ]}{\sum \nolimits _{j=1}^{M_i^*}Y_{ij}(u)}\\&- d(t)' A^{-1} \Big [ \frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{\tau }\big \{Z_i-\bar{z}(t)\big \}dM_{ij}^{(0)}(t)\Big ]. \end{aligned}$$

Because $\phi _i(t)$ $(i=1, \ldots , n)$ are independent zero-mean random variables for each t,  the multivariate central limit theorem implies that $n^{1/2}\{\hat{\Lambda }_0(t)-\Lambda _0(t)\}$ converges in finite-dimensional distribution to a zero-mean Gaussian process. Note that $Z_i = \max \{Z_i, 0 \}-\max \{-Z_i, 0 \}.$ Then the first term of $\phi _i(t)$ can be written as sums of monotone processes and are thus tight (Vaart and Wellner 1996). The second term is tight because d(t) is a deterministic function and $n^{1/2}(\hat{\beta }-\beta _0)$ converges in distribution. Thus, $n^{1/2}\{\hat{\Lambda }_0(t)-\Lambda _0(t)\}$ is tight and converges weakly to a zero-mean Gaussian process whose covariance function at (s, t) is given by $\Gamma (s,t)=E[\phi _i(s)\phi _i(t)],$ which can be consistently estimated by $\hat{\Gamma }(s,t)$ defined in Theorem 2. $\square$

Appendix B: Proof of (7) in Sect. 5

Define $dM_i^{(0)}(t)=\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}dM_{ij}^{(0)}(t).$ Due to (12), we have that uniformly in t and z,

$$\begin{aligned} n^{1/2}\{\hat{H}(t)-H_0(t)\}= & {} n^{-1/2}\sum \limits _{i=1}^{n}\int _{0}^{t} \frac{dM_{i}^{(0)}(u)}{n^{-1}\sum \limits _{i=1}^{n}M_i^{*-1}\sum \limits _{j=1}^{M_i^*}Y_{ij}(u)}\nonumber \\&-n^{-1/2}\sum \limits _{i=1}^{n}\int _{0}^{t}\bar{Z}(u)'du A^{-1} \Big [\int _{0}^{\tau }\{Z_i-\bar{z}(t)\}dM_i^{(0)}(t)\Big ] +o_p(1).\nonumber \\ \end{aligned}$$

(15)

Thus, it follows from (12) and (15) that uniformly in t and z,

$$\begin{aligned} \mathcal {F}(t,z)= & {} n^{-1/2}\sum \limits _{i=1}^{n}\int _{0}^{t}I(Z_i\le z)\ d M_{i}^{(0)}(u)\\&-\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{t}I(Z_i\le z)Y_{ij}(u)n^{1/2} d \big \{\hat{H}(u)-H_0(u)\big \}\\&-\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*} \int _{0}^{t}I(Z_i\le z)Y_{ij}(u)Z_i'du\big \{n^{1/2}(\hat{\beta }-\beta _0)\big \}\\= & {} n^{-1/2}\sum \limits _{i=1}^{n}\int _{0}^{t} \Bigg [I(Z_i\le z)-\frac{\sum \nolimits _{i=1}^{n}M_i^{*-1}\sum \nolimits _{j=1}^{M_i^*}I(Z_i\le z)Y_{ij}(u)}{\sum \nolimits _{i=1}^{n}M_i^{*-1}\sum \nolimits _{j=1}^{M_i^*}Y_{ij}(u)}\Bigg ]dM_{i}^{(0)}(u)\\&-n^{-1/2}\sum \nolimits _{i=1}^{n} \Big [\frac{1}{n}\sum \limits _{i=1}^{n}\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{t}I(Z_i\le z)Y_{ij}(u)\big \{Z_i-\bar{Z}(u)\big \}'du\Big ]\\&\times A^{-1} \int _{0}^{\tau }\big \{Z_i-\bar{z}(t)\big \}dM_i^{(0)}(t) +o_p(1). \end{aligned}$$

Therefore, we obtain that uniformly in t and z,

$$\begin{aligned} \mathcal {F}(t,z)= & {} n^{-1/2}\sum \limits _{i=1}^{n} \Big [ \int _{0}^{t}\big \{I(Z_i\le z)-B(u,z)\big \}dM_{i}^{(0)}(u) \nonumber \\&\qquad \qquad \qquad - C(t,z)'A^{-1}\int _{0}^{\tau }\{Z_i-\bar{z}(t)\}dM_i^{(0)}(t) \Big ]\nonumber \\&+o_p(1), \end{aligned}$$

(16)

where

$$\begin{aligned} B(t,z)=\frac{E\Big [M_i^{*-1}\sum \nolimits _{j=1}^{M_i^*}I(Z_i\le z)Y_{ij}(t)\Big ]}{E\Big [M_i^{*-1}\sum \nolimits _{j=1}^{M_i^*}Y_{ij}(t)\Big ]}, \end{aligned}$$

and

$$\begin{aligned} C(t,z)=E\Big [\frac{1}{M_i^*}\sum \limits _{j=1}^{M_i^*}\int _{0}^{t}I(Z_i\le z)Y_{ij}(u)\{Z_i-\bar{z}(u)\} du \Big ]. \end{aligned}$$

The multivariate central limit theorem implies that $\mathcal {F}(t,z)$ converges in finite-dimensional distribution to a zero-mean Gaussian process. $\mathcal {F}(t,z)$ is tight by using the same arguments as the tightness of $n^{1/2}\{\hat{\Lambda }_0(t)-\Lambda _0(t)\}$. Hence $\mathcal {F}(t,z)$ converges weakly to a zero-mean Gaussian process which can be approximated by $\tilde{\mathcal {F}}(t,z)$ given in (7).