Semiparametric efficient estimation for additive hazards regression with case II interval-censored survival data

Abstract

Interval-censored data often arise naturally in medical, biological, and demographical studies. As a matter of routine, the Cox proportional hazards regression is employed to fit such censored data. The related work in the framework of additive hazards regression, which is always considered as a promising alternative, remains to be investigated. We propose a sieve maximum likelihood method for estimating regression parameters in the additive hazards regression with case II interval-censored data, which consists of right-, left- and interval-censored observations. We establish the consistency and the asymptotic normality of the proposed estimator and show that it attains the semiparametric efficiency bound. The finite-sample performance of the proposed method is assessed via comprehensive simulation studies, which is further illustrated by a real clinical example for patients with hemophilia.

Appendix: Proofs of Theorems

First we derive the integral equation for the least favorable direction. Denote

$$\begin{aligned} \xi _1(U,V,g)=&\frac{\exp \Big (-\int _0^{U}\exp \{g(s)\} \mathrm {d} s-{\varvec{\theta }}^\mathrm{T}(U\mathbf{Z})\Big )}{1-\exp \Big (-\int _0^{U}\exp \{g(s)\} \mathrm {d} s-{\varvec{\theta }}^\mathrm{T}(U\mathbf{Z})\Big )} \\ \xi _2(U,V,g)=&\frac{\exp \Big (-\int _0^{U}\exp \{g(s)\} \mathrm {d} s-{\varvec{\theta }}^\mathrm{T}(U\mathbf{Z})\Big )}{\exp \Big (-\int _0^{U}\exp \{g(s)\} \mathrm {d} s-{\varvec{\theta }}^\mathrm{T}(U\mathbf{Z})\Big )-\exp \Big (-\int _0^{V}\exp \{g(s)\}\mathrm {d} s- {\varvec{\theta }}^\mathrm{T}(V\mathbf{Z})\Big )} \\ \xi _3(U,V,g)=&\frac{\exp \Big (-\int _0^{V}\exp \{g(s)\}\mathrm {d} s- {\varvec{\theta }}^\mathrm{T}(V\mathbf{Z})\Big )}{\exp \Big (-\int _0^{U}\exp \{g(s)\} \mathrm {d} s-{\varvec{\theta }}^\mathrm{T}(U\mathbf{Z})\Big )-\exp \Big (-\int _0^{V}\exp \{g(s)\}\mathrm {d} s- {\varvec{\theta }}^\mathrm{T}(V\mathbf{Z})\Big )}\\ \varphi _1(U,V)=&E_\mathbf{z}\{ \xi _1(U,V,g)f(U,V|\mathbf{Z})\}, \varphi _2(U,V)=E_\mathbf{z}\{ \xi _2(U,V,g)f(U,V|\mathbf{Z})\}\\ \varphi _3(U,V)=&E_\mathbf{z}\{ \xi _3(U,V,g)f(U,V|\mathbf{Z})\}, \varphi _4(U,V)=E_\mathbf{z}\{ f(U,V|\mathbf{Z})\}\\ \psi _1(U,V)=&E_\mathbf{z}\{\mathbf{Z} \xi _1(U,V,g)f(U,V|\mathbf{Z})\}, \psi _2(U,V)=E_\mathbf{z}\{\mathbf{Z} \xi _2(U,V,g)f(U,V|\mathbf{Z})\}\\ \psi _2(U,V)=&E_\mathbf{z}\{\mathbf{Z} \xi _3(U,V,g)f(U,V|\mathbf{Z})\}, \psi _4(U,V)=E_\mathbf{z}\{\mathbf{Z}f(U,V|\mathbf{Z})\}, \end{aligned}$$

where $E_\mathbf{z}$ means taking expectation with respect $\mathbf{Z}$. Follow the similar steps of Huang et al. (2008), define function,

$$\begin{aligned} \varphi (t)=&\int _{t+c}^b\varphi _1(t,x)\mathrm {d} x+\int _{t+c}^b\varphi _2(t,x)\mathrm {d} x+\int _{a}^{t-c}\varphi _3(x,t)\mathrm {d} x +\int _{a}^{t-c}\varphi _4(x,t)\mathrm {d} x\\ \psi (t)=&\int _{t+c}^b\psi _1(t,x)\mathrm {d} x+\int _{t+c}^b\psi _1(t,x)\mathrm {d} x+\int _{a}^{t-c}\psi _3(x,t)\mathrm {d} x +\int _{a}^{t-c}\psi _4(x,t)\mathrm {d} x\\ r(t)=&-\psi (t)t+\int _{a}^{t-c}x\psi _2(x,t)\mathrm {d} x+\int _{t+c}^{b}x\psi _3(t,x)\mathrm {d} x\\ Q(t,x)=&\{\varphi _2(x,t)I_{a\le x\le t-c}+\varphi _3(t,x)I_{t+c\le x\le b}\}/\varphi (t). \end{aligned}$$

Then we can attain (4).

Next we present the proof for Theorem 1 and 2. Throughout the following proofs, for notation simplicity, we denote $P_nf= \frac{1}{n}\sum _{i=1}^n f(\mathbf{O}_i)$, $M(\tau )=P\ell (\tau ;\mathbf{O}) =P\ell ({\varvec{\theta }},g; \mathbf{O})$ and $M_n(\tau )=P_n\ell (\tau ; \mathbf{O})= P_n\ell ({\varvec{\theta }},g;\mathbf{O})$, let C represent a generic constant that may vary from place to place.

Proof of Theorem 1

To show the consistency and derive the convergence rate, we just need to verify the following conditions C1–C3 in Theorem 1 of Shen and Wong (1994), which are presented as follows:

• C1 $\inf _{\{d(\tau ,\tau _0)\ge \epsilon , \tau \in \varTheta \times \mathbb {G}_n\}} M(\tau _0)-M(\tau )\ge C \inf _{\{d(\tau ,\tau _0)\ge \epsilon , \tau \in \varTheta \times \mathbb {G}_n\}}d^2(\tau ,\tau _0)$ where $\tau _0=({\varvec{\theta }}_0,g_0)$, and C1 holds with $\alpha =1$.

• C2 $\sup _{\{d(\tau ,\tau _0)\le \epsilon , \tau \in \varTheta \times \mathbb {G}_n\}} \text{ var }( \ell (\tau _0;\mathbf{O})-\ell (\tau ;\mathbf{O}))\le \sup _{\{d(\tau ,\tau _0)\le \epsilon , \tau \in \varTheta \times \mathbb {G}_n\}}d^2(\tau ,\tau _0)$, and C2 holds with $\beta =1$.

• C3 Let $\mathcal {F}_n=\{\ell (\tau ;\cdot ): \tau \in \varTheta \times \mathbb {G}_n\}$, $H(\epsilon , \mathcal {F}_n)\le C n^{2\gamma _0}\log (1/\epsilon ),$ where $H(\epsilon , \mathcal {F}_n)$ is the $L_{\infty }$-metric entropy of the space $\mathcal {F}_n$ and C3 holds with $2\gamma _0=\nu$ and $\gamma =0^{+}$.

Condition C1 with $\alpha =1$ can be verified by similar contexts as in Zhang et al. (2010). Condition C2 can be easily obtained through a Taylor expansion combined with conditions A1–A5. By inequality $\log (x)\le x-1$, we have the following results, for $\tau \in \varTheta \times {\mathbb {G}_n}$,

$$\begin{aligned}&E\{\ell (\tau _0)-\ell (\tau )\}^2\\&\quad =E\Big (\varDelta _{1}\log \frac{1- \exp \{-\phi _0(\mathbf{Z},U)\}}{1-\exp \{-\phi (\mathbf{Z},U)\}} +\varDelta _{2}\log \frac{\exp \{-\phi _0(\mathbf{Z},U)\}- \exp \{-\phi _0(\mathbf{Z},V)\}}{\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi (\mathbf{Z},V)\}}\\&\qquad -(1-\varDelta _{1}-\varDelta _{2})\{\phi _0(\mathbf{Z},V)-\phi (\mathbf{Z},V)\}\Big )^2\\&\quad \le CE\Big (\varDelta _1 \frac{\exp \{-\phi (\mathbf{Z},U)\} -\exp \{-\phi _0(\mathbf{Z},U)\}}{1-\exp \{-\phi (\mathbf{Z},U)\}}\\&\qquad + \varDelta _2 \frac{\exp \{-\phi _0(\mathbf{Z},U)\}- \exp \{-\phi _0(\mathbf{Z},V)\}-(\exp \{-\phi (\mathbf{Z},U)\} -\exp \{-\phi (\mathbf{Z},V)\})}{\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi (\mathbf{Z},V)\}}\\&\qquad (1-\varDelta _1-\varDelta _2)\{\phi _0(\mathbf{Z},V)-\phi (\mathbf{Z},V)\}\Big )^2\\&\quad \le C E\Bigg [\left( \varDelta _1 \frac{\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi _0(\mathbf{Z},U)\}}{1-\exp \{-\phi (\mathbf{Z},U)\}}\right) ^2\\&\qquad +\left( \varDelta _2 \frac{\exp \{-\phi _0(\mathbf{Z},U)\}- \exp \{-\phi _0(\mathbf{Z},V)\}-(\exp \{-\phi (\mathbf{Z},U)\} -\exp \{-\phi (\mathbf{Z},V)\})}{\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi (\mathbf{Z},V)\}}\right) ^2\\&\qquad +\left[ (1-\varDelta _1-\varDelta _2)\{\phi _0(\mathbf{Z},V) -\phi (\mathbf{Z},V)\}\right] ^2\bigg ]\\&\quad \le CE\big [(\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi _0(\mathbf{Z},U)\})^2\\&\qquad +(\exp \{-\phi _0(\mathbf{Z},U)\}-\exp \{-\phi _0(\mathbf{Z},V)\} -(\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi (\mathbf{Z},V)\}))^2\\&\qquad +(\{\phi _0(\mathbf{Z},V)-\phi (\mathbf{Z},V)\})^2\big ]\\&\quad \le CE\big [(\exp \{-\phi (\mathbf{Z},U)\}-\exp \{-\phi _0(\mathbf{Z},U)\})^2\\&\qquad + (\exp \{-\phi _0(\mathbf{Z},V)\}-\exp \{-\phi (\mathbf{Z},V)\})^2 +(\{\phi _0(\mathbf{Z},V)-\phi (\mathbf{Z},V)\})^2\big ]\\&\quad \le C E\Bigg [\Vert {\varvec{\theta }}_0-{\varvec{\theta }}\Vert ^2+ \left\{ \int _0^U[\exp \{g_0(s)\}-\exp \{g(s)\}]\mathrm {d}s\right\} ^2\\&\qquad +\left\{ \int _0^V[\exp \{g_0(s)\}-\exp \{g(s)\}]\mathrm {d}s\right\} ^2\Bigg ]\\&\quad \le C E\Big [\Vert {\varvec{\theta }}_0-{\varvec{\theta }}\Vert ^2+ \int _0^U[\exp \{g_0(s)\}-\exp \{g(s)\}]^2\mathrm {d}s\\&\qquad +\int _0^V[\exp \{g_0(s)\}-\exp \{g(s)\}]^2\mathrm {d}s\Big ] \\&\quad \le C E\Big [\Vert {\varvec{\theta }}_0-{\varvec{\theta }}\Vert ^2+ \int _0^U[\exp \{g^\star (s)\}]^2\{g_0(s)-g(s)\}^2\mathrm {d}s\\&\qquad +\int _0^V[\exp \{g^\star (s)\}]^2\{g_0(s)-g(s)\}^2\mathrm {d}s\Big ]\\&\quad \le Cd^2(\tau _0,\tau ), \end{aligned}$$

where the second and the fourth inequality follow from the inequality $(a+b)^2\le C(a^2+b^2)$, the sixth inequality is obtained by Cauchy–Schwartz inequality and $g^\star (s)$ is a value between $g_0(s)$ and g(s). With condition C1 which we have already shown, we can verify condition C2 with $\beta =1$.

Next we verify the condition C3. Let $L_1=\{\ell (\tau ;\mathbf{O}): \tau \in \varTheta \times \mathbb {G}_n\}$. We can easily construct a set of brackets $\{[\ell _{s,i}^{L}(\mathbf{O}),\ell _{s,i}^{U}(\mathbf{O})]: s=1,2, \ldots , [C(1/\epsilon )^d]; i=1, 2, \ldots , [C(1/\epsilon )^{Cq_n}]\}$ for any $\ell (\tau ;\mathbf{O}) \in L_1$, Specifically,

$$\begin{aligned} \ell _{si}^{L}(\mathbf{O})= & {} \varDelta _{1}\log \Big \{1-\exp \Big (-\int _0^{U}\exp \{g_i(t)^L \}\mathrm {d}t-((U\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s-UC\epsilon )\Big )\Big \}\\&+\varDelta _{2}\log \Big \{\exp \Big (-\int _0^{U}\exp \{g_i(t)^U\} \mathrm {d}t-((U\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s+UC\epsilon )\Big )\\&-\exp \Big (-\int _0^{V}\exp \{g_i(t)^L\}\mathrm {d}t -((V\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s-VC\epsilon )\Big )\Big \}\\&- \Big (1-\varDelta _{1}-\varDelta _{2}\Big )\Big \{\int _0^{V}\exp \{g_i(t)^U\} \mathrm {d}t+((V\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s+VC\epsilon )\Big \} \end{aligned}$$

and

$$\begin{aligned} \ell _{si}^{U}(\mathbf{O})= & {} \varDelta _{1}\log \Big \{1-\exp \Big (-\int _0^{U}\exp \{g_i(t)^U\}\mathrm {d}t- ((U\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s+UC\epsilon )\Big )\Big \}\\&+\varDelta _{2}\log \Big \{\exp \Big (-\int _0^{U}\exp \{g_i(t)^L \}\mathrm {d}t-((U\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s-UC\epsilon )\Big )\\&-\exp \Big (-\int _0^{V}\exp \{g_i(t)^U\}\mathrm {d}t -((V\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s+VC\epsilon )\Big )\Big \}\\&-\Big (1-\varDelta _{1}-\varDelta _{2}\Big )\Big \{\int _0^{V}\exp \{g_i(t)^L\}\mathrm {d}t+ ((V\mathbf{Z}^\mathrm{T}){\varvec{\theta }}_s-VC\epsilon )\Big \}. \end{aligned}$$

where $\{[g_i^L,g_i^U]: i=1,\ldots ,[(1/\epsilon )]^{Cq_n}\}$ is the brackets set for any $g \in S_n$. Then, using a Taylor expansion along with conditions A1–A3, we can conclude that the $\epsilon$-bracketing number for $L_1$ with $L_1(P)$-norm is bounded by $C(1/\epsilon )^{Cq_n+d}$ and $H(\epsilon , L_1)\le C n^{-\nu }\log (1/\epsilon )$. Hence, condition C3 in Theorem 1 of Shen and Wong (1994) holds with $2\gamma _0=\nu$ and $r=0^{+}$.

With condition A4, for $g_0 \in \mathbb {G}$, employing Corollary 6.21 in Schumaker (1981), there exists a function $g_{0n}\in S_n$ of order $m\ge p+2$ such that $\Vert g_{0n}-g_0\Vert _\infty =O(n^{-p \nu })$, where $\Vert \cdot \Vert _{\infty }$ is the sup-norm, which also means $\Vert g_{0n}-g_0\Vert _{\mathbb {G}}=O(n^{-p \nu })$. Now denote $\tau _{0,n}=({\varvec{\theta }}_0,g_{0,n})$. Then we have

$$\begin{aligned} M_n(\widehat{\tau }_n)-M_n(\tau _0)= & {} M_n(\widehat{\tau }_n)-M_n(\tau _{0,n})+M_n({\tau }_{0,n})-M_n(\tau _0) \\\ge & {} P_n \ell (\tau _{0,n};\mathbf{O})-P_n \ell (\tau _{0};\mathbf{O})\\= & {} (P_n-P)\{\ell (\tau _{0,n};\mathbf{O})-\ell (\tau _{0};\mathbf{O})\}+ M(\widehat{\tau }_{0,n})-M(\tau _0). \end{aligned}$$

Similar as (Zhang et al. 2010), we can conclude that

$$\begin{aligned} M_n(\widehat{\tau }_{n})-M_n(\tau _0)\ge o_p(n^{-1/2})-o(1)=-o_p(1), \end{aligned}$$

and then $\widehat{\tau }_n$ satisfies inequality (1.1) in Shen and Wong (1994).

Next, we derive the convergence rate. We have obtained that condition C3 in Theorem 1 of Shen and Wong (1994) holds with constants $2\gamma _0=\nu$ and $r=0^{+}$ in their notation. Furthermore, the constant $\tau$ in Theorem 1 of Shen and Wong (1994) is $(1-\nu )/2-(\log \log n)/( 2\log n)$. On the other hand, we can pick a ${\bar{\nu }}$ slightly greater than $\nu$ such that $(1-{\bar{\nu }})/2 \le (1-\nu )/2-(\log \log n)/(2\log n)$ for large n. We still denote ${\bar{\nu }}$ by $\nu$ and then $\tau =(1-\nu )/2$. The Kullback-Leibler distance between $\tau _0=({\varvec{\theta }}_0,g_0)$ and $\tau _{0,n}=({\varvec{\theta }}_0,g_{0n})$ is given by

$$\begin{aligned}&K(\tau _0,\tau _{0,n})\\&\quad =P(l(\tau _0;X)-l(\tau _{0n};X)) \\&\quad =E\Big (\Big [1-\exp \{-\phi _{0n}(\mathbf{Z},U)\}\Big ] m\Big [\frac{1-\exp \{-\phi _0(\mathbf{Z},U)\}}{1-\exp \{-\phi _{0n}(\mathbf{Z},U)\}}\Big ]\\&\qquad +\Big [\exp \{-\phi _{0n}(\mathbf{Z},U)\}- \exp \{-\phi _{0n}(\mathbf{Z},V)\}\Big ]\\&\qquad \times m \Big [\frac{\exp \{-\phi _0(\mathbf{Z},U)\}- \exp \{-\phi _0(\mathbf{Z},V)\}}{\exp \{-\phi _{0n}(\mathbf{Z},U)\} -\exp \{-\phi _{0n}(\mathbf{Z},V)\}}\Big ]\\&\qquad +\exp \Big \{-\phi _{0n}(\mathbf{Z},V)\Big \} m\Big [\frac{\exp \{-\phi _0(\mathbf{Z},V)\}}{\exp \{-\phi _{0n}(\mathbf{Z},V)\}}\Big ]\Big )\\&\quad \le CE([\exp \{-\phi _{0}(\mathbf{Z},U)\}- \exp \{-\phi _{0n}(\mathbf{Z},U)\}]^2\\&\qquad + [\exp \{-\phi _{0}(\mathbf{Z},V)\}-\exp \{-\phi _{0n}(\mathbf{Z},V)\}]^2\\&\qquad +[\phi _{0}(\mathbf{Z},V)-\phi _{0n}(\mathbf{Z},V)]^2)\\&\quad \le C\Vert g_0-g_{0n}\Vert ^2_2\le C\Vert g_0-g_{0n}\Vert ^2_\infty =O(n^{-2p \nu }), \end{aligned}$$

where $m(x)=x \log x-x+1\le x(x-1)-x+1\le (x-1)^2$. Then, we can obtain $K^{\frac{1}{2}}(\tau _0,\tau _{0n})=O(n^{-p \nu })$. Following Theorem 1 of Shen and Wong (1994), we have $d(\widehat{\tau }_n,\tau _0)=O_p\{n^{-\min (p \nu ,(1-\nu )/2)}\}$, which completes the proof of Theorem 1. $\square$

Proof of Theorem 2

By Zhang et al. (2010), it is sufficient to derive the asymptotic normality for $\widehat{{\varvec{\theta }}}_n$ by verifying the following conditions.

• B1 $P_n \dot{\ell }_1(\widehat{\tau }_n; \mathbf{O})= o_p(n^{-1/2})$ and $P_n\dot{\ell }_2(\widehat{\tau }_n; \mathbf{O})[h_0]=o_p(n^{-1/2})$.

• B2 $(P_n-P)\{{\ell }^*(\widehat{\tau }_n; \mathbf{O})-{\ell }^*(\tau _0; \mathbf{O})\}=o_p(n^{-1/2})$.

• B3 $P\{{\ell }^*(\widehat{\tau }_n; \mathbf{O}) -{\ell }^*(\tau _0; \mathbf{O})\}=-I({\varvec{\theta }}_0)(\widehat{\varvec{\theta }}_n-{\varvec{\theta }}_0)+ O_p(\Vert \widehat{\varvec{\theta }}_n-{\varvec{\theta }}_0\Vert )+o_p(n^{-1/2})$.

Conditions B1 and B2 can be verified by similar arguments as Zhang et al. (2010). As for condition B3, using (a1) minus (a2) of conditions A7 and A8, we have

$$\begin{aligned}&P\{\ell ^*(\widehat{{\varvec{\theta }}}_n,\widehat{g}_n;\mathbf{O})-\ell ^*({\varvec{\theta }}_0,g_0;\mathbf{O})\}\\&\quad =-I({\varvec{\theta }}_0)(\widehat{{\varvec{\theta }}}_n-{\varvec{\theta }}_0)+o_p(\Vert \widehat{{\varvec{\theta }}}_n -{\varvec{\theta }}_0\Vert )+O(\Vert \widehat{g}_n-g_0\Vert ^\alpha ). \end{aligned}$$

By Theorem 1 and the fact $\alpha p \nu > \frac{1}{2}$, we have, $O(\Vert \widehat{g}_n-g_0\Vert ^\alpha )=o_p(n^{-1/2})$. So B3 holds. Then Theorem 2 can be established follow the general procedure which has stated in Zhang et al. (2010). $\square$