A New Approach for Regression Analysis of Multivariate Current Status Data with Informative Censoring

Abstract

Regression analysis of interval-censored failure time data has recently attracted a great deal of attention partly due to their increasing occurrences in many fields. In this paper, we discuss a type of such data, multivariate current status data, where in addition to the complex interval data structure, one also faces dependent or informative censoring. For inference, a sieve maximum likelihood estimation procedure is developed and the proposed estimators of regression parameters are shown to be asymptotically consistent and efficient. For the implementation of the method, an EM algorithm is provided, and the results from an extensive simulation study demonstrate the validity and good performance of the proposed inference procedure. For an illustration, the proposed approach is applied to a tumorigenicity experiment.

Appendix: Proofs of the Asymptotic Properties of \(\hat{\theta }_n\)

In this Appendix, we will sketch the proof of Theorems 4.1, 4.2 and 4.3. For this, we will mainly use some results about empirical processes given in van der Vaart and Wellner [26].

Proof of Theorem 4.1

To prove the consistency, we will verify the conditions of Theorem 5.7 of Van der Vaart [25]. First we will verify the condition $J_1=: \text{ lim}_{n}\text{ sup}_{\theta _n\in \Theta _n}|\text{ P}_n l(\theta , O)-\text{ P }l(\theta , O)|=o_p(1).$ Note that

$$\begin{aligned} J_1\leqslant \text{ lim}_n \text{ sup}_{\theta _n\in \Theta _n}|&\text{ P}_nl(\theta , O)-\text{ P }l(\theta _n,O)|+\text{ lim}_{n}\text{ sup}_{\theta _n\in \Theta _n}|\text{ P }l(\theta _n,O)-\text{ P }l(\theta ,O)|\\&=:J_{11}+J_{12}. \end{aligned}$$

Therefore, it is sufficient to prove that $J_{1k}=o_p(1), k=1,2$. To prove that $J_{11}=o_p(1)$, we just need verify that $\varepsilon =\{l(\theta _n, O), \theta _n\in \Theta _n\}$ is Euclidean class for its envelope function $\text{ max}_{\theta _n\in \Theta _n}l(\theta _n, O)$. According to (A1), (A2) and Lemma 2.14 in Pakes and Pollard [20], it is easy to see that class $\varepsilon$ is a Euclidean class. Hence, we have $J_{11}=o_p(1)$. For $J_{12}$, by Lemman A1 of Lu et al. [17] and contiguous property of log-likelihood function, we have $J_{12}=o(1)$. Thus, we could obtain that condition $J_1=: \text{ lim}_{n}\text{ sup}_{\theta _n\in \Theta _n}|\text{ P}_n l(\theta , O)-\text{ P }l(\theta , O)|=o_p(1)$ holds.

Now, we verify that another condition of Theorem 5.7 of Van der Vaart [25] holds. That is, for any $\epsilon$,

$$\begin{aligned} \text{ sup}_{d(\theta , \theta _0)>\epsilon }{\mathbb {P}}l(\theta , O)<{\mathbb {P}}l(\theta _0, O). \end{aligned}$$

Note that this condition is satisfied according to condition (A4). Now, by Theorem 5.7 of Van der Vaart [25], we have $d(\hat{\theta }_n, \theta _0)=o_p(1)$, which completes the proof of Theorem 4.1. $\square$

Proof of Theorem 4.2

To derive the convergence rate, for any $\omega >0$, define the class ${\mathcal {F}}_w=\{l(\theta _{n0}, O)-l(\theta , O): \theta \in \Theta _n, d(\theta , \theta _{n0})\leqslant w\}$ with $\theta _{n0}=(\beta _0,\gamma _0,\Sigma _0, \Lambda _{1n0}, \ldots , \Lambda _{Kn0}, \Lambda _{cn0})$. Following the calculation of Shen and Wong (1994, P.597), we can establish that $\text{ log }N_{[]}(\epsilon , {\mathcal {F}}_{\omega }, \parallel .\parallel _{2})\leqslant CN \text{ log }(\omega /\epsilon )$ with $N=2(s+k_n)$, where $N_{[]}(\epsilon , {\mathcal {F}}_{\omega }, d)$ denotes the bracketing number (see Definition 2.1.6 in [26]) with respect to the metric or semi-metric d of a function class ${\mathcal {F}}$. Moreover, some algebraic calculations lead to $\parallel l(\theta _{n0},O)-l(\theta , O)\parallel ^2\leqslant C\omega ^2$ for any $l(\theta _{n0},O)-l(\theta , O)\in {\mathcal {F}}_\omega$. Then Lemma 19.36 of van der Vaart [25] gives

$$\begin{aligned} E^{*}\text{ sup}_{d(\theta , \theta _0)<\omega }\parallel \sqrt{n}({\mathbb {P}}_n-{\mathbb {P}})(l(\theta , O)-l(\theta _0, O))\parallel =O(1)\omega ^{1/2}(1+\frac{\omega ^{1/2}}{\epsilon ^2\sqrt{n}}M_1), \end{aligned}$$

where $E^{*}$ is the outer expectation and $M_1$ is a positive constant. Let $\phi _n(\omega )=\omega ^{1/2}(1+\frac{\omega ^{1/2}}{\epsilon ^2\sqrt{n}}M_1)$. Then $\phi _n(\omega )/\omega$ is a decreasing function, and $n^{\frac{2}{3}}\phi _n(n^{\frac{-1}{3}})=O(\sqrt{n})$ for large n. Furthermore, by Theorem 4.1, we know that $\hat{\theta }_n$ is consistent. According to theorem 3.4.1 of van der Vaart and Wellner [26], we can conclude that $d(\hat{\theta }_n, \theta _0)=\{ \parallel \hat{\zeta }_n-\zeta _0 \parallel ^{2}+\parallel \hat{\Lambda }_{cn}(c)-\Lambda _{c0}(c)\parallel ^{2}+\sum \nolimits _{k=1}^{K}\int [\hat{\Lambda }_{kn}(c)-\Lambda _{k0}(c)]^{2}f_k(c)\mathrm{d}c\}^{\frac{1}{2}}=O_p(n^{-1/3})$, which completes the proof of Theorem 4.2. $\square$

Proof of Theorem 4.3

The score functions for $\beta$ and $\gamma$ are denoted by $S_{\beta }(\theta )$ and $S_\gamma (\theta )$, respectively, where $S_\beta (\theta )=\frac{\partial l(\beta , \gamma , \Sigma , {\mathcal {A}})}{\partial \beta }$ and $S_\gamma (\theta )=\frac{\partial l(\beta , \gamma , \Sigma , {\mathcal {A}})}{\partial \gamma }$. For $k = 1, \ldots , K$, we let $h_k(t)$ be a nonnegative and nondecreasing function on $[\tau _1, \tau _2]$. Define $H =\{h = (h_1(t), \ldots , h_K(t))\}.$ Consider parametric submodels $\Lambda _\epsilon (t)=(\Lambda _{1, \epsilon }(t), \ldots , \Lambda _{K, \epsilon }(t))$, where $\Lambda _{k, \epsilon }(t)=\Lambda _k(t)+\epsilon h_k(t)$. For each k, the score function along the kth submodels is given by $S_{\Lambda _k(\theta )}[h(k)]=\frac{\partial l(\beta , \gamma , \Sigma , \Lambda _k, \epsilon )}{\partial \epsilon }|_{\epsilon =0}$. The efficient score for $\zeta$ at $(\zeta _0, {\mathcal {A}}_0)$ is ${\tilde{l}}(\zeta _0, {\mathcal {A}}_0)=S_{\zeta }(\zeta _0, {\mathcal {A}}_0)-\sum \nolimits _{k=1}^{K}S_{\Lambda _{k}}(\zeta _0, {\mathcal {A}}_0)[h_k^{*}]$, $h_k^{*}$is a $(d + 1)$-vector function satisfying

$$\begin{aligned} {\mathbb {P}}[(S_{\zeta }(\zeta _0, {\mathcal {A}}_0)-\sum \limits _{k=1}^{K}S_{\Lambda _{k}}(\zeta _0, {\mathcal {A}}_0)[h_k^{*}])'(\sum \limits _{k=1}^{K}S_{\Lambda _{k}}(\zeta _0, {\mathcal {A}}_0)[h_k^{*}])=0, \end{aligned}$$

for each $h_k$ in H. By following similar calculations in Section 3 of Chang et al. [1], we can establish the existence of $h_k$ in the above equation.

The efficient Fisher information matrix $I_0$ for $\zeta$ at $(\zeta _0, {\mathcal {A}}_0)$ is defined as ${\mathbb {P}}({\tilde{l}}(\zeta _0, {\mathcal {A}}_0){\tilde{l}}'(\zeta _0, {\mathcal {A}}_0)).$ By Taylor expansion, we can obtain

$$\begin{aligned}&{\mathbb {P}}{\tilde{l}}(\zeta _0, {\mathcal {A}})={\mathbb {P}}{\tilde{l}}(\zeta _0, {\mathcal {A}}_0)\\&\quad +{\mathbb {P}}\{\sum \limits _{k=1}^{k}S_{\zeta , k}(\theta )[\Lambda _k-\Lambda _{k0}]-\sum \limits _{k=1}^{K}\sum \limits _{j=1}^{K}S_{\zeta , j}(\theta )[h_k^{*}, \Lambda _k-\Lambda _{k0}]\}+O_p(\sum \limits _{k=1}^{K}\parallel \Lambda _k-\Lambda _{k0}\parallel ^{2}). \end{aligned}$$

Note that ${\mathbb {P}}{\tilde{l}}(\zeta _0, {\mathcal {A}}_0)=0$, ${\mathbb {P}}(S_{\zeta }(\theta )S_{\Lambda _k}(\theta )[h_k])=-{\mathbb {P}}(S_{\zeta , k}(\theta )[h_k])$, ${\mathbb {P}}(S_{\Lambda _k}(\theta )[{\tilde{h}}_k]S_{\Lambda _j}(\theta )[h_j])=-{\mathbb {P}}(S_{k, j}(\zeta )[{\tilde{h}}_k, h_j]),$ by the consistency and the convergence rate of ${\hat{\Lambda }}_n$, we can conclude that ${\mathbb {P}}{\tilde{l}}(\zeta _0, {\hat{\mathcal {A}}}_n)=O_p(n^{-2/3})$. Therefore, $\sqrt{n}({\mathbb {P}}_n-{\mathbb {P}})({\tilde{l}}({\hat{\zeta }}_n, {\hat{\mathcal {A}}}_n)- {\tilde{l}}({\hat{\zeta }}_0, {\hat{\mathcal {A}}}_0))=o_p(1)$. Due to the fact that ${\mathbb {P}}_n{\tilde{l}}({\hat{\theta }}_n)={\mathbb {P}}{\tilde{l}}(\theta _0)=0$ and ${\mathbb {P}}{\tilde{l}}(\zeta _0, {\hat{\mathcal {A}}})=o_p(1)$, we have

$$\begin{aligned} -\sqrt{n}{\mathbb {P}}({\tilde{l}}({\hat{\theta }}_n)-{\tilde{l}}(\zeta _0, {\hat{\mathcal {A}}})=\sqrt{n}{\mathbb {P}}_n({\tilde{l}}(\theta _0))+o_p(1). \end{aligned}$$

By the mean value theorem, we have

$$\begin{aligned} -\sqrt{n}{\mathbb {P}}\frac{\partial {\tilde{l}}(\zeta ', {\hat{\mathcal {A}}}_n)}{\partial \zeta }(\hat{\zeta }_n-\zeta _0)=\sqrt{n}{\mathbb {P}}_n({\tilde{l}}(\theta _0))+o_p(1), \end{aligned}$$

where $\zeta '$ is a point between $\hat{\zeta }_n$ and $\zeta _0$. Since $\hat{\theta }_n$ is consistency and ${\mathbb {P}}(-\frac{\partial {\tilde{l}}(\theta _0)}{\partial \zeta }={\mathbb {P}}({\tilde{l}}(\theta _0){\tilde{l}}'(\theta _0))=I_0$, we can conclude that

$$\begin{aligned} \sqrt{n}(\hat{\zeta }_n-\zeta _0)=I_0^{-1}\sqrt{n}{\mathbb {P}}({\tilde{l}}(\theta _0)+o_p(1){\mathop {\rightarrow }\limits ^{d}}N(0, I_0^{-1}). \end{aligned}$$

This completes the proof of Theorem 4.3. $\square$