Regression analysis of informative current status data with the additive hazards model

Abstract

This paper discusses regression analysis of current status failure time data arising from the additive hazards model in the presence of informative censoring. Many methods have been developed for regression analysis of current status data under various regression models if the censoring is noninformative, and also there exists a large literature on parametric analysis of informative current status data in the context of tumorgenicity experiments. In this paper, a semiparametric maximum likelihood estimation procedure is presented and in the method, the copula model is employed to describe the relationship between the failure time of interest and the censoring time. Furthermore, I-splines are used to approximate the nonparametric functions involved and the asymptotic consistency and normality of the proposed estimators are established. A simulation study is conducted and indicates that the proposed approach works well for practical situations. An illustrative example is also provided.

Appendix: Proofs of asymptotic consistency and normality

In this Appendix, we will sketch the proofs for the asymptotic consistency and normality of the proposed estimators described in Sect. 3. First we will give the proofs for the consistency results given in (4) and (5) and then the proof for the asymptotic result described in (6). The following are the regularity conditions needed for these results.

• (A1) The covariates $Z_i$’s have a bounded support.

• (A2) The copula function $M ( \cdot ,\cdot )$ has bounded first order partial derivatives and both the partial derivatives are Lipschitz.

• (A3) Assume that $\displaystyle \inf _{{d(\theta ,\theta _0)<\epsilon }}Pl(\theta ,X)>Pl(\theta _0,X).$

• (A4) The $m$th derivative of $\varLambda _k(\cdot )$, denoted by $\varLambda _k^{(m)}(\cdot )$, is Holder continuous such that $|\varLambda _k^{(m)}(t_1)-\varLambda _k^{(m)}(t_2)| \le M|t_1-t_2|^\eta$ for some $\eta \in (0, 1]$ and all $t_1, t_2 \in (l,u)$, $k=1,2$, where $0 < l < u < \infty$ and $M$ are some constants.

• (A5) The matrix $E(S_\vartheta S_\vartheta ')$ is finite and positive definite with $\vartheta =(\beta ',\gamma ')'$, where $S_\vartheta$ is defined below.

First we will sketch the proof for the result described in (4). For this we suppose that the regularity conditions (A1)–(A4) given above hold and will employ the empirical process theory (van der Vaart and Wellner 1996). Define $\mathcal {L}_n = \{ l ( \theta , X) : \, \theta \in \varTheta _n \}$. Note that for any $\theta ^1=(\beta ^1, \gamma ^1, \varLambda _1^1,\varLambda _2^1)$ and $\theta ^2 = (\beta ^2, \gamma ^2, \varLambda _1^2,\varLambda _2^2)\in \varTheta _n$, it is easy to show that

$$\begin{aligned}&|l(\theta ^1,X)- l(\theta ^2,X)|\\&\quad \le K\big (\Vert \beta ^1-\beta ^2\Vert +\Vert \gamma ^1-\gamma ^2\Vert +\Vert \varLambda _1^1-\varLambda _1^2\Vert _\infty +\Vert \varLambda _2^1-\varLambda _2^2\Vert _\infty \big ) \end{aligned}$$

by the Taylor’s series expansion under conditions (A1) and (A2). Also note that according to the calculation of van der Vaart and Wellner (1996) (p. 94), we have

$$\begin{aligned}&N\Big (\epsilon ,{\mathcal {L}}_n,L_1(P_n)\Big )\\&\le N\Big (\frac{\epsilon }{3M},\mathcal {B},\Vert \cdot \Vert \Big )\cdot N\Big (\frac{\epsilon }{3M_n},\mathcal {M}_n^1,L_\infty \Big )\cdot N\Big (\frac{\epsilon }{3M_n},\mathcal {M}_n^2,L_\infty \Big )\\&\le \Big (\frac{9M}{\epsilon }\Big )^{2p}\cdot \Big (\frac{9M_n^2}{\epsilon }\Big )^{k+k_n}\cdot \Big (\frac{9M_n^2}{\epsilon }\Big )^{k +k_n}\\&\le KM^{2p}M_n^{4(k +k_n)}\epsilon ^{-p_m}, \end{aligned}$$

where $p_m = 2 p + 2 ( k + k_n)$. It follows from the inequality (31) of Pollard (1984) (p. 31) that

$$\begin{aligned} \displaystyle \sup _{\theta \in \varTheta _n} \big |P_nl(\theta , X)-Pl(\theta , X)\big |\rightarrow 0 \end{aligned}$$

(7)

in probability.

Define $K_\epsilon =\{\theta : d(\theta , \theta _0)\ge \epsilon , \theta \in \varTheta _n\}$ and let $M(\theta , X) = -l (\theta , X)$ and

$$\begin{aligned} \zeta _{1n}=\displaystyle \sup _{\theta \in \varTheta _n} |P_nM(\theta , X)-PM(\theta ,X)|,\,\,\zeta _{2n}=P_nM(\theta _0,X)-PM(\theta _0,X), \end{aligned}$$

where $\theta _0$ denotes the true value of $\theta$. Then we have

$$\begin{aligned} \inf _{K_\epsilon }PM(\theta ,X)&= \inf _{K_\epsilon }\Big \{P M(\theta ,X)-P_nM(\theta ,X)+P_nM(\theta ,X)\Big \}\nonumber \\&\le \zeta _{1n}+\displaystyle \inf _{K_\epsilon }P_nM(\theta ,X). \end{aligned}$$

(8)

Furthermore, if $\hat{\theta }_n\in K_\epsilon$, one can show that

$$\begin{aligned} \displaystyle \inf _{K_\epsilon } P_nM(\theta ,X)=P_nM(\hat{\theta }_n,X)\le P_nM(\theta _0,X) = \zeta _{2n}+PM(\theta _0,X). \end{aligned}$$

(9)

It thus follows from the condition (A3) that

$$\begin{aligned} \displaystyle \inf _{K_\epsilon }P M(\theta ,X)-PM(\theta _0,X)=\delta _\epsilon >0 \end{aligned}$$

and (8) and (9) give

$$\begin{aligned} \displaystyle \inf _{K_\epsilon }PM(\theta ,X)\le \zeta _{1n}+\zeta _{2n} + PM(\theta _0,X)=\zeta _n+ PM(\theta _0,X), \end{aligned}$$

where $\zeta _n=\zeta _{1n}+\zeta _{2n}$. Hence we have that $\zeta _n \ge \delta _\epsilon$ and furthermore, $\{\hat{\theta }_n \in K_{\epsilon } \}\subseteq \{\zeta _n \ge \delta _{\epsilon }\}$. Then by using the condition (A1) and the strong law of large numbers, one can show that $\zeta _{1n} = o(1)$ and $\zeta _{2n}=o(1)$ almost surely. The consistency result thus follows from $\cup _{k=1}^{\infty }\cap _{n=k}^{\infty }\{\hat{\theta }_n \in K_{\epsilon } \} \subseteq \cup _{k=1}^{\infty }\cap _{n=k}^{\infty }\{\zeta _n \ge \delta _{\epsilon }\}$.

Now we show the convergence rate result given in (5) and assume that the regularity conditions (A1)–(A4) given above hold. For any $\eta >0,$ define the class ${\mathcal {F}}_{\eta }=\{l(\theta _{n0},X)-l(\theta ,X): \theta \in \varTheta _n, d(\theta ,\theta _{n0})\le \eta \}$ with $\theta _{n0}=(\beta _0,\gamma _0, \varLambda _{1n0},\varLambda _{2n0} ).$ Following the calculation of Shen and Wong (1994) (p. 597), we can establish that $\log N_{[]}(\varepsilon ,{\mathcal {F}}_{\eta },\Vert \cdot \Vert _{2})\le C N\log (\eta /\varepsilon )$ with $N= 2 ( k +k_n).$ Moreover, some algebraic calculations lead to $\Vert l(\theta _{n0},X)-l(\theta ,X)\Vert _{2}^2\le C \eta ^2$ for any $l(\theta _{n0},X)-l(\theta ,X)\in {\mathcal {F}}_{\eta }.$ Therefore it follows from Lemma 3.4.2 of van der Vaart and Wellner (1996) that

$$\begin{aligned} E_P\Vert n^{1/2}(P_n-P)\Vert _{{\mathcal {F}}_{\eta }}\le CJ_{\eta }(\varepsilon ,{\mathcal {F}}_{\eta },\Vert \cdot \Vert _{2}) \biggl \{1+\frac{J_{\eta }(\varepsilon ,{\mathcal {F}}_{\eta }, \Vert \cdot \Vert _{2})}{\eta ^2n^{1/2}}\biggl \}, \end{aligned}$$

(10)

where $J_{\eta }(\varepsilon ,{\mathcal {F}}_{\eta },\Vert \cdot \Vert _{2})=\int _0^\eta \{1+\log N_{[]}(\varepsilon ,{\mathcal {F}}_{\eta },\Vert \cdot \Vert _{2})\}^{1/2}d\varepsilon \le C N^{1/2}\eta .$

Note that the right-hand side of (10) gives $\phi _n(\eta )=C(N^{1/2}\eta +N/n^{1/2})$. Also it is easy to see that $\phi _n(\eta )/\eta$ decreases in $\eta$ and $r_n^2\phi _n(1/r_n)=r_nN^{1/2}+r_n^2N/n^{1/2} <2n^{1/2},$ where $r_n=N^{-1/2}n^{1/2}=n^{(1-\nu )/2}$ with $0<\nu <0.5$. Hence we have $n^{(1-\nu )/2}d(\hat{\theta },\theta _{n0})=O_P(1)$ by Theorem 3.2.5 of van der Vaart and Wellner (1996). This together with $d(\theta _{n0},\theta _0)=O_p(n^{-r\nu })$ (Lemma A1 in Lu et al. 2007) yields that $d(\hat{\theta },\theta _0)=O_p(n^{-(1-\nu )/2}+n^{-r\nu })$. The choice of $\nu =1/(1+2r)$ gives the rate of convergence as $d(\hat{{\theta }}_n, {\theta }_0) = O_p ( n^{ - r / (1+2r ) } )$ and completes the proof.

Finally we will provide the sketch for the proof of the asymptotic distribution result given in (6). For this, suppose that the regularity conditions (A1)–(A5) given above hold and $r > 2$. For this, denote $V$ as the linear span of $\varTheta _0-\theta _0$ where $\varTheta _0$ denotes the true parameter space. Let $l(\theta ,W)$ be the log-likelihood for a sample of size one and $\delta _n=(n^{-(1-\nu )/2}+n^{-r\nu }).$ For any $\theta \in \{\theta \in \varTheta _0: \Vert \theta -\theta _0\Vert =O(\delta _n)\},$ define the first order directional derivative of $l(\theta ,X)$ at the direction $v\in V$ as

$$\begin{aligned} \dot{l}(\theta ,X)[v]=\frac{dl(\theta +sv,X)}{ds}\Big |_{s=0}, \end{aligned}$$

(11)

and the second order directional derivative as

$$\begin{aligned} \ddot{l}(\theta ,X)[v,\tilde{v}]=\frac{d^2l(\theta +sv+\tilde{s} \tilde{v},X)}{d\tilde{s}ds}\Big |_{s=0}\Big |_{\tilde{s}=0} =\frac{d\dot{l}(\theta +\tilde{s}\tilde{v},X)}{d\tilde{s}}\Big |_{\tilde{s}=0}. \end{aligned}$$

Also define the Fisher inner product on the space $V$ as

$$\begin{aligned} <v,\tilde{v}>\,\,=P\Big \{\dot{l}(\theta ,X)[v]\dot{l}(\theta ,X)[\tilde{v}]\Big \} \end{aligned}$$

and the Fisher norm for $v\in V$ as $\Vert v\Vert ^{1/2}=<v,v>.$ Let $\bar{V}$ be the closed linear span of $V$ under the Fisher norm. Then $(\bar{V}, \Vert \cdot \Vert )$ is a Hilbert space.

In addition, define the smooth functional of $\theta$ as

$$\begin{aligned} \gamma (\theta )=b_1'\beta +b_2'\gamma , \end{aligned}$$

where $b=(b_1',b_2')'$ is any vector of $2p$ dimension with $\Vert b\Vert \le 1.$ For any $v\in V,$ we denote

$$\begin{aligned} \dot{\gamma }(\theta _0)[v]=\frac{d \gamma (\theta _0+sv)}{ds}\Big |_{s=0}=r(v) \end{aligned}$$

whenever the right hand-side limit is well defined. Note that $\gamma (\theta )-\gamma (\theta _0)=\dot{\gamma }(\theta _0) [\theta -\theta _0].$ It follows by the Riesz representation theorem that there exists $v^*\in \bar{V}$ such that $\dot{\gamma }(\theta _0)[v]=<v^*,v>$ for all $v \in \bar{V}$ and $\Vert v^*\Vert ^2=\Vert \dot{\gamma }(\theta _0)\Vert .$

Let $\varepsilon _n$ be any positive sequence satisfying $\varepsilon _n=o(n^{-1/2}).$ For any $v^*\in \varTheta _0,$ by (A4) and Corollary 6.21 of Schumaker (1981) (p. 227), there exists $\varPi _nv^*\in \varTheta _n$ such that $\Vert \varPi _nv^*-v^*\Vert =o(1)$ and $\delta _n\Vert \varPi _nv^*-v^*\Vert =o(n^{-1/2}).$ Also define $g[\theta -\theta _0,X]=l(\theta ,X)-l(\theta _0,X)-\dot{l}(\theta ,X)[\theta -\theta _0].$ Then by the definition of $\hat{\theta },$ we have

$$\begin{aligned} 0&\le P_n[l(\hat{\theta },W)-l(\hat{\theta }\pm \varepsilon _n \varPi _nv^*,W)]\\&= (P_n-P)[l(\hat{\theta },W)-l(\hat{\theta }\pm \varepsilon _n \varPi _nv^*,W)]+P[l(\hat{\theta },W)-l(\hat{\theta }\pm \varepsilon _n \varPi _nv^*,W)]\\&= \pm \varepsilon _n P_n \dot{l}(\theta ,W)[\varPi _nv^*]+(P_n-P)\Big \{g[\hat{\theta }-\theta _0,W] -g[\hat{\theta }\pm \varepsilon _n\varPi _nv^*-\theta _0,W]\Big \}\\&+\, P\Big \{g[\hat{\theta }-\theta _0,W]-g[\hat{\theta }\pm \varepsilon _n\varPi _nv^* -\theta _0,W]\Big \}\\&= \mp \varepsilon _n P_n \dot{l}(\theta ;W)[v^*]\pm \varepsilon _n P_n \dot{l}(\theta ,W)[\varPi _nv^*-v^*]+(P_n-P)\Big \{g[\hat{\theta }-\theta _0,W]\\&-\,g[\hat{\theta }\pm \varepsilon _n\varPi _nv^*-\theta _0,W]\Big \}+ P\Big \{g[\hat{\theta }-\theta _0,W]-g[\hat{\theta }\pm \varepsilon _n\varPi _nv^* -\theta _0,W]\Big \}\\&:= \mp \varepsilon _n P_n \dot{l}(\theta ,W)[v^*]+I_1+I_2+I_3. \end{aligned}$$

Note that for $I_1$, it follows from Conditions (A1)–(A2), Chebyshev inequality and $\Vert \varPi _nv^*-v^*\Vert =o(1)$ that $I_1=\varepsilon _n\times o_p(n^{-1/2}).$ For $I_2,$ we have

$$\begin{aligned} I_2&= (P_n-P)\Big \{l(\hat{\theta },W)-l(\hat{\theta }\pm \varepsilon _n\varPi _nv^*,W)\pm \varepsilon _n\dot{l}(\theta _0,W)[\varPi _nv^*]\Big \}\\&= \mp \varepsilon _n(P_n-P)\Big \{\dot{l}(\tilde{{\theta }},W) -\dot{l}(\theta _0,W)[\varPi _nv^*]\Big \}, \end{aligned}$$

where $\tilde{\theta }$ lies between $\hat{\theta }$ and $\hat{\theta }\pm \varepsilon _n\varPi _n v^*.$ By Theorem 2.8.3 in of van der Vaart and Wellner (1996), we know that $\{\dot{l}(\theta ;W)[\varPi _n v^*]: \Vert \theta -\theta _0\Vert =O(\delta _n)\}$ is Donsker class. Therefore, by Theorem 2.11.23 of van der Vaart and Wellner (1996), we have $I_2=\varepsilon _n\times o_p(n^{-1/2}).$

For $I_3$, note that

$$\begin{aligned} P(g[\theta -\theta _0,W])&= P\{l(\theta ,W)-l(\theta _0,W)-\dot{l}(\theta _0,W[\theta -\theta _0])\}\\&= 2^{-1}P\{\ddot{l}(\tilde{\theta },W)[\theta -\theta _0,\theta -\theta _0] -\ddot{l}(\theta _0,W)[\theta -\theta _0,\theta -\theta _0]\}\\&+\,2^{-1} P\{\ddot{l}(\theta _0,W)[\theta -\theta _0,\theta -\theta _0]\}\\&= 2^{-1}P\{\ddot{l}(\theta _0,W)[\theta -\theta _0,\theta -\theta _0]\}+\varepsilon _n\times o_p(n^{-1/2}), \end{aligned}$$

where $\tilde{\theta }$ lies between $\theta _0$ and $\theta$ and the last equation is due to Taylor expansion, (A1)–(A2) and $r>2.$ Therefore,

$$\begin{aligned} I_3&= -2^{-1}\{\Vert \hat{\theta }-\theta _0\Vert ^2-\Vert \hat{\theta }\pm \varepsilon _n\varPi _nv^*-\theta _0\Vert ^2\}+\varepsilon _n\times o_p(n^{-1/2})\\&= \pm \varepsilon _n <\hat{\theta }-\theta _0,\varPi _nv^*>+2^{-1}\Vert \varepsilon _n\varPi _nv^*\Vert ^2+\varepsilon _n\times o_p(n^{-1/2})\\&= \pm \varepsilon _n <\hat{\theta }-\theta _0,v^*>+2^{-1}\Vert \varepsilon _n\varPi _nv^*\Vert ^2+\varepsilon _n\times o_p(n^{-1/2})\\&= \pm \varepsilon _n <\hat{\theta }-\theta _0,v^*>+\varepsilon _n\times o_p(n^{-1/2}). \end{aligned}$$

In the above, the last equality holds since $\delta _n\Vert \varPi _nv^*-v^*\Vert =o(n^{-1/2}),$ Cauchy-Schwartz inequality, and $\Vert \varPi _nv^*\Vert ^2\rightarrow \Vert v^*\Vert ^2.$

By combing the above facts together with $P\dot{l}(\theta _0,W[v^*])=0$, one can show that

$$\begin{aligned} 0&\le P_n\{l(\hat{\theta },W)-l(\hat{\theta }\pm \varepsilon _n \varPi _nv^*,W)\}\\&= \mp \varepsilon _nP_n\dot{l}(\theta _0,W)[v^*]\pm \varepsilon _n <\hat{\theta }-\theta _0,v^*>+\varepsilon _n\times o_p(n^{-1/2})\\&= \mp \varepsilon _n(P_n-P)\{\dot{l}(\theta _0,W)[v^*]\}\pm \varepsilon _n <\hat{\theta }-\theta _0,v^*>+\varepsilon _n\times o_p(n^{-1/2}). \end{aligned}$$

Therefore, we have $\sqrt{n}<\hat{\theta }-\theta _0,v^*>=\sqrt{n}(P_n-P)\{\dot{l}(\theta _0,W)[v^*]\}+o_p(1)\rightarrow N(0,\Vert v^*\Vert ^2)$ by the central limits theorem with the the asymptotic variance being equal to $\Vert v^*\Vert ^2=\Vert \dot{l}(\theta _0,W)[v^*]\Vert ^2.$ This implies that $n^{1/2}(\gamma (\hat{\theta })-\gamma (\theta _0))=n^{1/2}<\hat{\theta } -\theta _0,v^*>+o_p(1)\rightarrow N(0,\Vert v^*\Vert ^2)$ in distribution. Furthermore, the semiparametric efficiency can be established by applying the result of Bickel and Kwon (2001) or Theorem 4 in Shen (1997).

For each component $\vartheta _q,$ $q=1,2,\ldots ,2p,$ denote by $\psi ^*_q=(b_{1q}^*,b_{2q}^*)$ the solution to

$$\begin{aligned} \displaystyle \inf _{\psi _q^*}E\Big \{l_\vartheta \cdot e_q-l_{b_1^*}[b_{1q}^*]-l_{b_2^*} [ b_{2q}^*]\Big \}^2, \end{aligned}$$

where $l_\vartheta =(l_\beta ', l_\gamma ')',$ $l_{b_1^*}[b_{1q}^*]$ and $l_{b_2^*}[b_{1q}^*]$ are defined similar to (11). Now let $\psi ^*=(\psi _1^*,\cdots ,\psi _q^*).$ By the calculations of Chen et al. (2006), we have $\Vert v^*\Vert ^2=\Vert \dot{\gamma }(\theta _0)\Vert =\sup _{v\in \bar{V}:\Vert v\Vert >0}\frac{|\dot{\gamma }(\theta _0)[v]|}{\Vert v\Vert }=b'\varSigma b,$ where $\varSigma =E(S_\vartheta S_\vartheta '),$ $S_\vartheta =\{l_\vartheta -l_{b_1^*}b_1^*-l_{b_2^*}b_2^*\}.$ Now, since $b'((\hat{\beta }-\beta _0)',(\hat{\gamma }-\gamma _0)')=<\hat{\theta }-\theta _0,v^*>,$ the final results follows from the Cram$\acute{e}$r-Wold device (Theorem 3.2 in Chap. 13 of Shorack 2000).