A new method for regression analysis of interval-censored data with the additive hazards model

Abstract

The additive hazards model is one of the most popular regression models for analyzing failure time data, especially when one is interested in the excess risk or risk difference. Although a couple of methods have been developed in the literature for regression analysis of interval-censored data, a general type of failure time data, they may be complicated or inefficient. Corresponding to this, we present a new maximum likelihood estimation procedure based on the sieve approach and in particular, develop an EM algorithm that involves a two-stage data augmentation with the use of Poisson latent variables. The method can be easily implemented and the asymptotic properties of the proposed estimators are established. A simulation study is conducted to assess the performance of the proposed method and indicates that it works well for practical situations. Also the method is applied to a set of interval-censored data from an AIDS cohort study.

Appendix: Proofs of the Theorems

Proof of Theorem 1

Let $l(\theta ; O)$ denote the log-likelihood function based on a single observation O and consider the class of functions ${\mathcal {L}}=\{l(\theta ; O):\theta \in \Theta _n\}$. Then, following the similar calculation by Zhang et al. (2010), the bracketing number of ${\mathcal {L}}$ is bounded by $K(1/\varepsilon )^{K(m+1)+d}$. Hence ${\mathcal {L}}$ is Glivenlo–Cantelli by Theorem 2.4.1 of van der Vaart and Wellner (1996). Thus

$$\begin{aligned} \sup _{\theta \in \Theta _n}|P_nl(\theta ;O)-Pl(\theta ;O)|\rightarrow 0 \quad a.s. \end{aligned}$$

Let $M(\theta ;O)=-l(\theta ;O)$, and define $K_\epsilon =\{\theta :d(\theta ,\theta _0)\ge \epsilon ,\theta \in \Theta _n\}$ for $\epsilon >0$, $\zeta _{1n}=\sup _{\theta \in \Theta _n}|P_nM(\theta ;O)-PM(\theta ;O)|$, and $\zeta _{2n}=P_nM(\theta _0;O)-PM(\theta _0;O)$. Then

$$\begin{aligned}&\inf _{K_{\epsilon }}PM(\theta ;O) =\inf _{K_{\epsilon }}\{PM(\theta ;O)-P_nM(\theta ;O)+P_nM(\theta ;O)\} \le \zeta _{1n}\nonumber \\&\quad +\inf _{K_{\epsilon }}P_nM(\theta ;O). \end{aligned}$$

(3)

If ${\widehat{\theta }}_n\in K_{\epsilon }$, then

$$\begin{aligned} \inf _{K_{\epsilon }}P_nM(\theta ;O)=P_nM({\widehat{\theta }}_n;O) \le P_nM(\theta _0;O)=\zeta _{2n}+PM(\theta _0;O). \end{aligned}$$

(4)

It follows from (3) and (4) that

$$\begin{aligned} \inf _{K_{\epsilon }}PM(\theta ;O)\le \zeta _{1n}+\zeta _{2n}+PM(\theta _0;O). \end{aligned}$$

and hence $\zeta _{1n}+\zeta _{2n}\ge \delta _{\epsilon }>0$, where $\delta _{\epsilon }=\inf _{K_{\epsilon }}PM(\theta ;O)-PM(\theta _0;O)$. This gives $\{{\widehat{\theta }}_n\in K_{\epsilon }\}\subseteq \{\zeta _{1n}+\zeta _{2n}\ge \delta _{\epsilon }\}$. Together with strong law of large numbers, we have $\zeta _{1n}+\zeta _{2n}\rightarrow 0$ almost surely. Therefore, $d({\widehat{\theta }}_n,\theta _0)\rightarrow 0$ almost surely. $\square$

Proof of Theorem 2

We verify the conditions of Theorem 3.4.1 of van der Vaart and Wellner (1996) in order to derive the convergence rate. First note from Theorem 1.6.2 of Lorentz (1986) that there exists a Bernstein polynomial $\Lambda _{n0}$ such that $||\Lambda _{n0}-\Lambda ||_{\infty }=O(m^{-r/2})$. Define $\theta _{n0}=(\beta _0,\Lambda _{n0})$. Then we have $d(\theta _{n0},\theta _0)=O(n^{-r\nu /2})$.

For any $\eta >0$, define the class of functions ${\mathcal {F}}_{\eta }=\{l(\theta ;O)-l(\theta _{n0};O):\theta \in \Theta _n, \eta /2<d(\theta ,\theta _{n0})\le \eta \}$. One can easily show that $P(l(\theta _0;O)-l(\theta _{n0};O))\le K d^2(\theta _0,\theta _{n0})\le Kn^{-r\nu }$. Hence, we have for large n,

$$\begin{aligned} P(l(\theta ;O)-l(\theta _{n0};O))&= {} P(l(\theta ;O)-l(\theta _{0};O))+P(l(\theta _0;O)-l(\theta _{n0};O))\\ &\le -K\eta ^2+Kn^{-r\nu }=-K\eta ^2, \end{aligned}$$

for any $l(\theta ;O)-l(\theta _{n0};O)\in {\mathcal {F}}_{\eta }$.

Following the calculations in Shen and Wong (1994), we can derive that for $0<\varepsilon <\eta$, $\log N_{[]}(\varepsilon ,{\mathcal {F}}_{\eta },L_2(P))\le K(m+1)\log (\eta /\varepsilon )$. Under Conditions (3), (4) and (6), ${\mathcal {F}}_{\eta }$ is uniformly bounded. Together with $P(l(\theta ;O)-l(\theta _{n0};O))^2\le K\eta ^2$ for $l(\theta ;O)-l(\theta _{n0};O)\in {\mathcal {F}}_{\eta }$, by Lemma 3.4.2 of van der Vaart and Wellner (1996), one can obtain

$$\begin{aligned} E^*\Vert n^{1/2}(P_n-P)\Vert _{{\mathcal {F}}_{\eta }}\le K J_{[]}(\eta ,{\mathcal {F}}_{\eta },L_2(P)) \left\{ 1+\frac{J_{[]}(\eta ,{\mathcal {F}}_{\eta },L_2(P))}{\eta ^2n^{1/2}}\right\} , \end{aligned}$$

where $J_{[]}(\eta ,{\mathcal {F}}_{\eta },L_2(P))= \int _0^\eta (1+\log N_{[]}(\varepsilon ,{\mathcal {F}}_{\eta },L_2(P)))^{1/2}d\varepsilon \le K(m+1)^{1/2}\eta$. This yields $\phi _n(\eta )=(m+1)^{1/2}\eta +(m+1)/n^{1/2}$. It is easy to see that if we choose $r_n=n^{(1-\nu )/2}$, $r_n^2\phi _n(1/r_n)\le K n^{1/2}$.

Together with the facts that $P_n(l({\widehat{\theta }}_n;O)-l(\theta _{n0};O))\ge 0$ and $d({\widehat{\theta }}_n,\theta _{n0})\le d({\widehat{\theta }}_n,\theta _{0})+d(\theta _0,\theta _{n0})\rightarrow 0$ in probability, by Theorem 3.4.1 of van der Vaart and Wellner (1996), one can get $d({\widehat{\theta }}_n,\theta _{n0})=O_P(n^{-(1-\nu )/2})$. This together with $d(\theta _{n0},\theta _0)=O(n^{-r\nu /2})$ yields that $d({\widehat{\theta }}_n,\theta _{0})=O_P(n^{-(1-\nu )/2}+n^{-r\nu /2})$. $\square$

Proof of Theorem 3

We first calculate the information matrix based on the general semiparametric information theory described by Bickel et al. (1993). Define functions $A_i, i=1,2,3$, by

$$\begin{aligned} A_1(u,v,z)=\frac{ \exp ( - \Lambda (u) - \beta ^T z u ) }{1 - \exp ( - \Lambda (u) - \beta ^T z u ) }, \end{aligned}$$

$$\begin{aligned} A_2(u,v,z)=\frac{ \exp ( - \Lambda (u) - \beta ^T z u ) }{\exp (- \Lambda (u) - \beta ^T z u ) - \exp ( - \Lambda (v) - \beta ^T z v ) }, \end{aligned}$$

and

$$\begin{aligned} A_3(u,v,z)=\frac{ \exp ( - \Lambda (v) - \beta ^T z v ) }{\exp (- \Lambda (u) - \beta ^T z u ) - \exp ( - \Lambda (v) - \beta ^T z v ) }. \end{aligned}$$

By conditions (3) and (4), $A_1$, $A_2$ and $A_3$ are positive functions of (u, v, z). Then the score function for $\beta$ is

$$\begin{aligned} {\dot{l}}_{\beta }(O)=\frac{\partial }{\partial \beta }l(\beta , \Lambda ;O)=z \{ \delta _1 u A_1- \delta _2 (u A_2 - v A_3) - \delta _3 v \}. \end{aligned}$$

The score operator for $\Lambda$ is

$$\begin{aligned} {\dot{l}}_{\Lambda }\phi (O)\,=\frac{\partial }{\partial s}l(\beta , \Lambda + s \phi ;O) \mid _{s=0} \,=\delta _1 \phi (u) A_1 - \delta _2 (\phi (u) A_2 - \phi (v) A_3) - \delta _3 \phi (v) . \end{aligned}$$

Let F is the distribution corresponding to $\Lambda$ and P is the joint probability measure of $(\delta _1, \delta _2 , \delta _3, U , V , Z)$, then the score operator ${\dot{l}}_{\Lambda }$ maps $L_2^0(F)$ to $L_2^0(P)$, where $L_2^0(F)\equiv \{a:\int a dF=0 \text{ and } \int a^2 dF<\infty \}$, and $L_2^0(P)$ is defined similarly as $L_2^0(F)$. Let ${\dot{l}}_{\Lambda }^T$ : $L_2^0(P)\rightarrow L_2^0(F)$ be the adjoint operator of ${\dot{l}}_{\Lambda }$, i.e., for any $a \in L_2^0(F)$ and $b \in L_2^0(P)$,

$$\begin{aligned} \langle b , {\dot{l}}_{\Lambda }a \rangle _P \, = \, \langle {\dot{l}}_{\Lambda }^Tb , a \rangle _F \, , \, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle _P$ and $\langle \cdot ,\cdot \rangle _F$ are the inner products in $L_2^0(P)$ and $L_2^0(F)$, respectively. We need to find $\phi ^*$ such that ${\dot{l}}_\beta -{\dot{l}}_{\Lambda }\phi ^*$ is orthogonal to ${\dot{l}}_{\Lambda }\phi$ in $L_2^0(P)$. This amounts to solving the following normal equation:

$$\begin{aligned} {\dot{l}}_{\Lambda }^T {\dot{l}}_{\Lambda }\phi ^* ={\dot{l}}_{\Lambda }^T {\dot{l}}_\beta . \end{aligned}$$

(5)

First note that we have

$$\begin{aligned} {\dot{l}}_{\Lambda }^T {\dot{l}}_{\Lambda }\phi (t) =E[{\dot{l}}_{\Lambda }\phi (O)|T = t]=E_ZE[{\dot{l}}_{\Lambda }\phi (O)|T = t, Z = z] \end{aligned}$$

by Groeneboom and Wellner (1992), pages 8–9, or Bickel et al. (1993), pages 271–272.

Let $B_1(u,v) = E_Z[A_1(u,v,Z)g(u,v|Z)]$, $B_2(u,v)= E_Z[A_2(u,v,Z)g(u,v|Z)]$, $B_3(u,v) = E_Z[A_3(u,v,Z)g(u,v|Z)]$ and $B_4(u,v) = E_Z[g(u,v|Z)]$. By the definition of A’s, B’s are all positive functions, and $B_2(u,v) = B_3(u,v) + B_4(u,v)$. We calculate

$$\begin{aligned} L(t)&\equiv {\dot{l}}_{\Lambda }^T {\dot{l}}_{\Lambda }\phi (t) = \int _{u=t}^{\tau _1} \int _{v=u+\eta }^{\tau _1} \phi (u) B_1(u,v) dvdu \\&\quad-\, \int _{u=\tau _0}^{t} \int _{v=t}^{\tau _1}[\phi (u)B_2(u,v) - \phi (v)B_3(u,v)] 1_{[v-u\ge \eta ]} dvdu \\&\quad-\, \int _{u=\tau _0}^{t} \int _{v=u+\eta }^{t} \phi (v) B_4(u,v) dvdu \end{aligned}$$

Let $C_1(u,v)=E_Z[ZA_1(u,v,Z)g(u,v|Z)]$, $C_2(u,v)=E_Z[ZA_2(u,v,Z)g(u,v|Z)]$, $C_3(u,v)=E_Z[ZA_3(u,v,Z)g(u,v|Z)]$ and $C_4(u,v)=E_Z[Zg(u,v|Z)]$. Then further calculation yields

$$\begin{aligned} R(t)&\equiv {\dot{l}}_{\Lambda }^T {\dot{l}}_\beta (t) = \int _{u=t}^{\tau _1} \int _{v=u+\eta }^{\tau _1} uC_1(u,v) dvdu \\&\quad- \int _{u=\tau _0}^{t} \int _{v=t}^{\tau _1} [uC_2(u,v)+vC_3(u,v)] 1_{[v-u\ge \eta ]} dvdu \\&\quad- \int _{u=\tau _0}^{t} \int _{v=u+\eta }^{t} vC_4(u,v) dvdu \end{aligned}$$

After some straightforward calculations, the derivative of L(t) is

$$\begin{aligned} L'(t) = -b(t)\phi (t)+ \int _{\tau _0}^{t-\eta } \phi (x)B_2(x,t) dx+ \int _{t+\eta }^{\tau _1} \phi (x)B_3(t,x) dx , \end{aligned}$$

where $b(t) = \int _{t+\eta }^{\tau _1} [B_1(t,x)+B_2(t,x)] dx + \int _{\tau _0}^{t-\eta } [B_3(x,t)+B_4(x,t)] dx$. Similarly, the derivative of R(t) is

$$\begin{aligned} r(t) \equiv R'(t) = -c(t)t+\int _{\tau _0}^{t-\eta } x C_2(x,t) dx+ \int _{t+\eta }^{\tau _1} x C_3(t,x) dx , \end{aligned}$$

where $c(t) = \int _{t+\eta }^{\tau _1} [C_1(t,x)+C_2(t,x)] dx + \int _{\tau _0}^{t-\eta } [C_3(x,t)+C_4(x,t)] dx$.

By conditions (3)–(7), r has a bounded derivative $r'$ on $[\tau _0,\tau _1]$. So equation (5) reduces to

$$\begin{aligned} -b(t)\phi (t)+ \int _{\tau _0}^{t-\eta } \phi (x)B_2(x,t) dx+ \int _{t+\eta }^{\tau _1} \phi (x)B_3(t,x) dx=r(t). \end{aligned}$$

(6)

By conditions (3) and (4), we have $\inf _{\tau _0\le t\le \tau _1}b(t)>0$. Let $d(t)=-r(t)/b(t)$ and

$$\begin{aligned} K(t,x)= [B_2(x,t)1_{[\tau _0\ge x \le t-\eta ]}+B_3(t,x)1_{[t+\eta \ge x \tau _1]}]/b(t). \end{aligned}$$

Then $\phi ^*(t)$ is the solution of a Fredholm integral equation of the second kind,

$$\begin{aligned} \phi ^*(t)-\int K(t,x)\phi ^*(x)dx \, = \, d(t). \end{aligned}$$

Therefore, the efficient score is $l_\beta ^*={\dot{l}}_\beta -{\dot{l}}_{\Lambda }\phi ^*$ and the information is $I(\beta )=E[l_\beta ^*]^{\otimes 2}$.

In the following, we will establish the asymptotic normality for ${\widehat{\beta }}_n$. Since ${\widehat{\theta }}_{n}$ maximizes the likelihood function, we have $P_n {\dot{l}}_{\beta } ({\widehat{\theta }}_n)=0$. Since $\phi ^*$ is obtained from the Fredholm integral equation given above, it has a bounded derivative and it is a function with bounded variation. Then there exists a $\phi ^*_n\in {\mathcal {M}}$ such that $P_n {\dot{l}}_{\Lambda }\phi ^*_n ({\widehat{\theta }}_n)=0$ and $||\phi ^*_n-\phi ^*||=O(n^{-\nu })$. Then

$$\begin{aligned}&P_n {\dot{l}}_{\Lambda }\phi ^* ({\widehat{\theta }}_n) =(P_n-P) {\dot{l}}_{\Lambda }[\phi ^*-\phi ^*_n] ({\widehat{\theta }}_n) +P ({\dot{l}}_{\Lambda }[\phi ^*-\phi ^*_n] ({\widehat{\theta }}_n)-{\dot{l}}_{\Lambda }[\phi ^*-\phi ^*_n] (\theta _0))\nonumber \\&\quad =I+II. \end{aligned}$$

(7)

Hence the first term $I=o_p(n^{-1/2})$ is followed by uniform asymptotic equicontinuity of empirical processes indexed by a Donsker class of functions. By Theorem 2 and the Cauchy-Schwartz inequality, the second term II is $o_{p}(n^{-1/2})$. Then $P_n {\dot{l}}_{\Lambda }\phi ^* ({\widehat{\theta }}_n)=o_{p}(n^{-1/2})$.

By the calculation above, the Fisher information matrix for $\beta$ is positive definite. By Theorem 2, the rate of convergence is proved. And there are two more facts. One is the uniform asymptotic equicontinuity of $(P_n-P) {\dot{l}}_{\beta } (\theta )$ and $(P_n-P) {\dot{l}}_{\Lambda }\phi ^* (\theta )$ in a small neighborhood of $\theta _0$, and this follows from uniform asymptotic equicontinuity of empirical processes indexed by a Donsker class of functions. The other is the smoothness of $P {\dot{l}}_{\beta } (\theta )$ and $P {\dot{l}}_{\Lambda }\phi ^* (\theta )$ in a small neighborhood of $\theta _0$, which follows from the Taylor expansion. Thus all conditions in Theorem 6.1 of Huang (1996) have been confirmed and the proof is complete. $\square$