An additive hazards cure model with informative interval censoring

Abstract

The existence of a cured subgroup happens quite often in survival studies and many authors considered this under various situations (Farewell in Biometrics 38:1041–1046, 1982; Kuk and Chen in Biometrika 79:531–541, 1992; Lam and Xue in Biometrika 92:573–586, 2005; Zhou et al. in J Comput Graph Stat 27:48–58, 2018). In this paper, we discuss the situation where only interval-censored data are available and furthermore, the censoring may be informative, for which there does not seem to exist an established estimation procedure. For the analysis, we present a three component model consisting of a logistic model for describing the cure rate, an additive hazards model for the failure time of interest and a nonhomogeneous Poisson model for the observation process. For estimation, we propose a sieve maximum likelihood estimation procedure and the asymptotic properties of the resulting estimators are established. Furthermore, an EM algorithm is developed for the implementation of the proposed estimation approach, and extensive simulation studies are conducted and suggest that the proposed method works well for practical situations. Also the approach is applied to a cardiac allograft vasculopathy study that motivated this investigation.

Appendix

In this appendix, we will sketch the proofs for Theorems 1–3 and for this, we will mainly use some results about the empirical processes given in Van der Vaart and Wellner (1996). Throughout the following proofs, we denote $Pf=\int f(x)dP(X)$ and $P_{n}f=n^{-1}\sum _{i=1}^{n}f(X_{i})$ for a function f and a random variable X with the distribution P and let J represent a generic constant that may vary from place to place. We first present the required regularity conditions.

• (C1) (i) $(\beta ^{\top }_{1}, \beta _{2}, \alpha ^{\top }, \eta ^{\top }, \sigma ^{2}) \in {\mathcal {B}}$, where ${\mathcal {B}}$ is a compact subset of ${\mathbb {R}}^{2p+q+2}$;(ii) The rth derivative of $\varLambda _{0}(\cdot )(\varLambda _{0h}(\cdot ))$ is bounded and continuous.

• (C2) (i) There exists a positive $\eta ^{*}$ such that $P(U_{i,j}-U_{i,j-1}\ge \eta ^{*})=1$ for subject i, where $j=2,\ldots ,K_{i}$; (ii) For subject i, the union of the support of $U_{ij}$ is contained in an interval [a, b], where $0<a<b<\infty$ and $j=1,\ldots ,K_{i}$.

• (C3) The covariates X and Z have bounded supports in ${\mathbb {R}}^{p}$ and ${\mathbb {R}}^{q}$, respectively, where p and q are the dimension of X and Z;

• (C4) Let $l(\theta ;O)=\log L_{O}(\theta )$ with $L_{O}(\theta )$ given in Sect. 2. For any $\theta ^{1},\theta ^{2}\in \varTheta$, if $l(\theta ^{1};O)=l(\theta ^{2};O)$, then $\theta ^{1}=\theta ^{2}$. In particular, O denotes the single observation data.

• (C5) For every $\theta$ in a neighbourhood of $\theta _{0}$, $P(l(\theta ;O)-l(\theta _{0};O))\preceq -d^{2}(\theta ,\theta _{0})$, where $\preceq$ means “smaller than, up to a constant”.

• (C6) (i) $0< E[(\dot{l}(\theta _{0};O)[\upsilon ])^{2}] < \infty$ for $\upsilon \ne 0, \upsilon \in V$, where $\dot{l}(\theta _{0};O)[\upsilon ]$ is defined in (E5) below; (ii) For $\theta \in \{\theta : d(\theta ,\theta _{0})=O(\delta _{n})\}$, $P\{\ddot{l}(\theta ;O)[\theta -\theta _{0},\theta -\theta _{0}]- \ddot{l}(\theta _{0};O)[\theta -\theta _{0},\theta -\theta _{0}]\}=O(\delta _{n}^{3})$ and $\delta _{n}^{3} = O(n^{-1})$, where $\ddot{l}(\theta ;O)[\upsilon ,{\tilde{\upsilon }}]$ is defined in (E8)

Proof of Theorem 1

We first define the covering number of the class ${\mathcal {L}}_{n}= \{l(\theta ;O):\theta \in \varTheta _{n}\}$. In particular, O denotes the single observation data. For any $\epsilon >0$, define the covering number $N(\epsilon , {\mathcal {L}}_{n},L_{1}(P_{n}) )$ as the smallest value of $\kappa$ for which there exists $\{ \theta ^{(1)}, \ldots , \theta ^{(\kappa )}\}$ such that

$$\begin{aligned} \min _{j\in \{1, \ldots , \kappa \}} \frac{1}{n}\sum _{i=1}^{n}| l(\theta ;O_{i})-l(\theta ^{(j)};O_{i})| < \epsilon \end{aligned}$$

for all $\theta \in \varTheta _{n}$, where $\theta ^{(j)}=(\beta _{1}^{(j)},\beta _{2}^{(j)}, \alpha ^{(j)},\eta ^{(j)},\sigma ^{2(j)}, \varLambda _{n}^{(j)}, \varLambda _{nh}^{(j)})^{\top }\in \varTheta _{n}$, $j=1,\ldots ,\kappa$. We will define $N(\epsilon , {\mathcal {L}}_{n}, L_{1}(P_{n}))= \infty$ if no such $\kappa$ exists.

Let $l(\theta ;O)$ denote the log likelihood function based on the single observation as defined in Sect. 2. Then the covering number(definition 2.1.5 in Van der Vaart and Wellner 1996) of the class ${\mathcal {L}}_{n}=\{l(\theta ;O):\theta \in \varTheta _{n}\}$ satisfies

$$\begin{aligned} N(\epsilon , {\mathcal {L}}_{n}, L_{1}(P_{n})) \le JM^{2p+q+2}M_{n}^{2(m+1)}\epsilon ^{-p_{m}}, \end{aligned}$$

where $p_{m}=2p+q+2+2(m+1)$, $M_{n}=O(n^{a})$ with $0< a < \frac{1}{2}$. Adopting similar proofs of inequality (31) in Pollard (1984, p. 34), we have

$$\begin{aligned} \sup _{\theta \in \varTheta _{n}}|P_{n}l(\theta ;O)-Pl(\theta ;O)|\rightarrow 0. \hbox {a.s.} \end{aligned}$$

(E1)

Let $M(\theta ;O)=-l(\theta ;O)$,

$$\begin{aligned} \zeta _{1n}= & {} \sup _{\theta \in \varTheta _{n}}|P_{n}M(\theta ;O)-PM(\theta ;O)|, \\ \zeta _{2n}= & {} P_{n}M(\theta _{0};O)-PM(\theta _{0};O). \end{aligned}$$

Denote $C_{\epsilon }=\{\theta ;d(\theta ,\theta _{0})\ge \epsilon ,\theta \in \varTheta _{n}\}$.

$$\begin{aligned}&\inf _{C_{\epsilon }}PM(\theta ;O)= \inf _{C_{\epsilon }}\{PM(\theta ;O)- P_{n}M(\theta ;O)+P_{n}M(\theta ;O)\} \\&\quad \le \zeta _{1n}+\inf _{C_{\epsilon }}P_{n}M(\theta ;O) \end{aligned}$$

(E2)

If ${\hat{\theta }}_{n}\in C_{\epsilon }$, we have

$$\begin{aligned} \inf _{C_{\epsilon }}P_{n}M(\theta ;O)&=P_{n}M({\hat{\theta }}_{n};O) \le P_{n}M(\theta _{0};O)=P_{n}M(\theta _{0};O)-PM(\theta _{0};O)\\&\quad +PM(\theta _{0};O) \\&=\zeta _{2n}+PM(\theta _{0};O) \end{aligned}$$

(E3)

By identification condition(C4), we obtain that

$$\begin{aligned} \inf _{C_{\epsilon }}PM(\theta ;O)-PM(\theta _{0};O)=\delta _{\epsilon }>0. \end{aligned}$$

(E4)

By (E2) and (E3), we have

$$\begin{aligned} \inf _{C_{\epsilon }}PM(\theta ;O)\le \zeta _{1n}+\zeta _{2n}+ PM(\theta _{0};O)=\zeta _{n}+PM(\theta _{0};O) \end{aligned}$$

(E5)

with $\zeta _{n}=\zeta _{1n}+\zeta _{2n}$. By (E4) and (E5), we get $\zeta _{n}\ge \delta _{\epsilon }$. Here, we have $\{{\hat{\theta }}_{n}\in C_{\epsilon }\}\subseteq \{\zeta _{n}\ge \delta _{\epsilon }\}$. By (E1) and Strong Law of Large Numbers, we have $\zeta _{1n}=o(1),\zeta _{2n}=o(1)$, a.s. Therefore, by $\cup _{k=1}^{\infty }\cap _{n=k}^{\infty }\{{\hat{\theta }}_{n}\in C_{\epsilon }\} \subseteq \cup _{k=1}^{\infty }\cap _{n=k}^{\infty } \{\zeta _{n}\ge \delta _{\epsilon }\}$, we complete the proof. $\square$

Proof of Theorem 2

To establish the convergence rate, note that by Theorem 1.6.2 of Lorentz (1986), there exists Bernstein polynomials $\varLambda _{n0}$ and $\varLambda _{nh0}$ such that $\parallel \varLambda _{0}- \varLambda _{n0} \parallel _{\infty }=O(n^{-\frac{r\nu }{2}})$ and $\parallel \varLambda _{0h}- \varLambda _{nh0} \parallel _{\infty }=O(n^{-\frac{r\nu }{2}})$, respectively. For any $\chi >0$, define the class ${\mathcal {F}}_{\chi }= \{l(\theta ;O)-l(\theta _{n0};O):\theta \in \varTheta _{n},d(\theta ,\theta _{n0})\le \chi \}$ with $\theta _{n0}=(\beta _{1n0}^{\top },\beta _{2n0},\alpha _{n0}^{\top },\eta _{n0}^{\top }, \sigma ^{2}_{n0},\varLambda _{1n0},\varLambda _{2n0})^{\top }$. Following the calculation of Shen and Wong (1994, p. 597), we can establish that $\log N_{[~]}(\epsilon ,{\mathcal {F}}_{\chi }, \parallel \cdot \parallel )\le JN\log (\chi /\epsilon )$ with $N=2(m+1)$, where $N_{[~]}(\epsilon ,{\mathcal {F}},d)$ denotes the bracketing number(definition 2.1.6 in Van der Vaart and Wellner 1996) with respect to the metric or semi-metric d of a function class ${\mathcal {F}}$. Condition (C5) lead to $\parallel l(\theta ;O)-l(\theta _{n0};O)\parallel ^{2}_{2} \le J\chi ^{2}$ for any $l(\theta ;O)-l(\theta _{n0};O) \in {\mathcal {F}}_{\chi }$. Therefore, by Lemma 3.4.2 of Van der Vaart and Wellner (1996), we obtain

$$\begin{aligned}&E_{P}\parallel n^{\frac{1}{2}}(P_{n}-P)\parallel _{{\mathcal {F}}_{\chi }} \le CH_{\chi }(\epsilon ,{\mathcal {F}}_{\chi },\parallel \cdot \parallel _{2}) \left\{ 1+\frac{H_{\chi }(\epsilon ,{\mathcal {F}}_{\chi },\parallel \cdot \parallel _{2})}{\chi ^{2}n^{\frac{1}{2}}}\right\} ,\\&\quad \le C\cdot CN^{\frac{1}{2}}\chi \left\{ 1+\frac{CN^{\frac{1}{2}\chi }}{\chi ^{2}N^{\frac{1}{2}}}\right\} =C(N^{\frac{1}{2}}+N/n^{\frac{1}{2}})\doteq \phi _{n}(\chi ). \end{aligned}$$

(E6)

where $H_{\chi }(\epsilon ,{\mathcal {F}}_{\chi },\parallel \cdot \parallel _{2})= \int _{0}^{\chi }\{1+\log N_{[~]}(\epsilon ,{\mathcal {F}}_{\chi }, \parallel \cdot \parallel _{2})\}^{1/2}d\epsilon$. It is easy to see that $\phi _{n}(\chi )/\chi$ decreasing in $\chi$, and $r^{2}_{n}\phi _{n}(1/r_{n})=r_{n}N^{\frac{1}{2}}+r^{2}_{n}N/n^{\frac{1}{2}}<2n^{\frac{1}{2}}$, where $r_{n}=N^{-\frac{1}{2}}n^{\frac{1}{2}}=n^{(-\nu +1)/2},0<\nu <\frac{1}{2}$. Here, $n^{(1-\nu )/2}\cdot$ $d({\hat{\theta }},\theta _{n0})=O_{P}(1)$ by Theorem 3.2.5 of var der Van der Vaart and Wellner (1996). This, together with $d(\theta _{n0},\theta _{0})=O_{p}(n^{-\frac{r\nu }{2}})$ (Theorem 1.6.2 of Lorentz 1986),yields that $d({\hat{\theta }},\theta _{0})=O_{P}(n^{-(1-\nu )/2}+n^{-\frac{r\nu }{2}})$. This completes the proof. $\square$

Proof of Theorem 3

To complete the proof of Theorem 3, we need following notations. Define V as the linear span of $\varTheta -\theta _{0}$, where $\theta _{0}=(\beta _{10}^{\top },\beta _{20},\alpha _{0}^{\top }, \eta _{0}^{\top }, \sigma ^{2}_{0},\varLambda _{0},\varLambda _{0h})^{\top }$ denotes the true value of $\theta$ and $\varTheta _{0}$ the true parameter space. Let $l(\theta ;O)$ be the log-likelihood function and $\delta _{n}=n^{-r\upsilon /2}+n^{-(1-\upsilon )/2}$. For any $\theta \in \{\theta \in \varTheta _{0}:d(\theta ,\theta _{0})=O(\delta _{n})\}$, define the first order directional derivative of $l(\theta ;O)$ at the direction $\upsilon \in V$ as

$$\begin{aligned} \dot{l}(\theta ;O)[\upsilon ] = \frac{dl(\theta +s\upsilon ;O)}{ds}| _{s=0}, \end{aligned}$$

(E7)

and the second order directional derivative as

$$\begin{aligned} \ddot{l}(\theta ;O)[\upsilon ,{\tilde{\upsilon }}]= \frac{d^{2}l(\theta +s\upsilon +{\tilde{s}}{\tilde{\upsilon }};O)}{d{\tilde{s}}}\mid _{s=0,{\tilde{s}}=0} =\frac{d\dot{l}(\theta +{\tilde{s}}{\tilde{\upsilon }};O)}{d{\tilde{s}}}\mid _{{\tilde{s}}=0}. \end{aligned}$$

(E8)

Define the Fisher inner product on the space V as $\langle \upsilon , {\tilde{\upsilon }}\rangle = P\{\dot{l}(\theta ;O)[\upsilon ] \dot{l}(\theta ;O)[{\tilde{\upsilon }}]\}$ and the Fisher norm for $\upsilon \in V$ as $\parallel \upsilon \parallel ^{2}= \langle \upsilon ,\upsilon \rangle$. Let ${\bar{V}}$ be the closed linear span of V under the Fisher norm. Then $({\bar{V}},\parallel \cdot \parallel )$ is a Hilbert space.

Furthermore, define the smooth functional of $\theta$ as $\gamma (\theta )=h_{1}^{\top }\beta _{1}+h_{2}\beta _{2}+h_{3}^{\top }\alpha + h_{4}^{\top }\eta + h_{5}\sigma ^{2}$, where $h=(h_{1}^{\top },h_{2},h_{3}^{\top },h_{4}^{\top }, h_{5})^{\top }$ is any vector of $2p+q+2$ dimension with $\parallel h\parallel \le 1$. Let $\vartheta = (\beta _{1}^{\top },\beta _{2},\alpha ^{\top },\eta ^{\top },\sigma ^{2})^{\top }$. Also for any $\upsilon =(\upsilon _{\vartheta },b_{1},b_{2}) \in V$, we denote

$$\begin{aligned} {\dot{\gamma }}(\theta _{0})[\upsilon ]=\frac{d\gamma (\theta _{0}+s\upsilon )}{ds}\mid _{s=0}= h^{\top }\upsilon _{\vartheta }. \end{aligned}$$

Note that $\gamma (\theta )-\gamma (\theta _{0})={\dot{\gamma }}(\theta _{0})[\theta -\theta _{0}]$. By the Riesz representation theorem, there exists $\upsilon ^{*} \in {\bar{V}}$ such that ${\dot{\gamma }}(\theta _{0})[\upsilon ]=\langle \upsilon ^{*},\upsilon \rangle$ for all $\upsilon \in {\bar{V}}$ and $\parallel \upsilon ^{*}\parallel ^{2}=\parallel {\dot{\gamma }}(\theta _{0})\parallel$. Thus it follows from the proof method of Theorem 3 of Ma et al. (2015), we can show that

$$\begin{aligned} \sqrt{n}\langle {\hat{\theta }}-\theta _{0},\upsilon ^{*}\rangle + o_{p}(1) \rightarrow N(0,\parallel \upsilon ^{*}\parallel ^{2}) \end{aligned}$$

in distribution.

Then we will proof that $\parallel \upsilon ^{*} \parallel ^{2}=h^{\top }\varSigma h$. For each component $\vartheta _{k}, k=1,2,\ldots ,2p+q+2$, we denote by $\varphi _{k}^{*}=(b_{1k}^{*},b_{2k}^{*})$ the solution to

$$\begin{aligned} \inf _{\varphi _{k}^{*}}E\{l_{\vartheta }\cdot e_{k}-l_{b_{1k}^{*}}[b_{1k}^{*}]-l_{b_{2k}^{*}}[b_{2k}^{*}]\}^{2}, \end{aligned}$$

(E9)

where $l_{\vartheta }=(l_{\beta _{1}}^{\top },l_{\beta _{2}},l_{\alpha }^{\top }, l_{\eta }^{\top },l_{\sigma ^{2}})^{\top }$, $l_{\vartheta }$ is the derivatives of $l(\theta ;O)$ with respect to $\vartheta$, $l_{b_{1k}^{*}}[b_{1k}^{*}]$ and $l_{b_{2k}^{*}}[b_{2k}^{*}]$ are the direction derivatives of $l(\theta ;O)$ with respect to $\varLambda _{0},\varLambda _{0h}$ at the direction $b_{1k}^{*},b_{2k}^{*}$, respectively. $e_{k}$ is a $(2p+q+2)$-dimensional vector of zeros expect the k-th element being equal to 1. Define the k-th element of $S_{\vartheta }$ as $l_{\vartheta }\cdot e_{k}-l_{b_{1k}^{*}}[b_{1k}^{*}]-l_{b_{2k}^{*}}[b_{2k}^{*}]$, $k=1,2,\ldots ,2p+q+2$.

Thus, we can show that

$$\begin{aligned} \sup _{\upsilon \in {\bar{V}}:\parallel \upsilon \parallel>0}\frac{\mid {\dot{\gamma }}(\theta _{0})[\upsilon ]\mid ^{2}}{\parallel \upsilon \parallel ^{2}} =\sup _{\upsilon \in {\bar{V}}:\parallel \upsilon \parallel>0} \frac{\mid h^{\top }\upsilon _{\vartheta } \mid ^{2}}{\parallel \upsilon \parallel ^{2}} =\sup _{\upsilon \in {\bar{V}}:\parallel \upsilon \parallel >0} \frac{\mid h^{\top }\upsilon _{\vartheta } \mid ^{2}}{P\{\dot{l}(\theta ;O)[\upsilon ]\}^{2}}. \end{aligned}$$

From the minimization procedure defined in (E9), using similar arguments to Sect. 3.2 in Chen et al. (2006), we obtain

$$\begin{aligned} \sup _{\upsilon \in {\bar{V}}:\parallel \upsilon \parallel >0}\frac{\mid {\dot{\gamma }}(\theta _{0})[\upsilon ]\mid ^{2}}{\parallel \upsilon \parallel ^{2}} =\sup _{(b_{1},b_{2})\ne 0} \frac{\mid h^{\top }\upsilon _{\vartheta } \mid ^{2}}{E(l_{\vartheta }[\upsilon _{\vartheta }]+l_{b_{1}}[b_{1k}]+ l_{b_{2}}[b_{2k}])^{2}}\\ =h^{\top }[E(S_{\vartheta }S_{\vartheta }^{\top })]^{-1}h^{\top }=h^{\top }\varSigma h. \end{aligned}$$

Thus the conclusion of the theorem follows by $h^{\top }\left( (\hat{\beta _{1}}-\beta _{10})^{\top },(\hat{\beta _{2}}-\beta _{20}),\right. \left. ({\hat{\alpha }}-\alpha _{0})^{\top }, ({\hat{\eta }}-\eta _{0})^{\top }, (\hat{\sigma ^{2}}-\sigma _{0}^{2}) \right) =\gamma ({\hat{\theta }})-\gamma (\theta _{0}) ={\dot{\gamma }}(\theta _{0}) [{\hat{\theta }}-\theta _{0}]=\langle {\hat{\theta }}-\theta _{0},\upsilon ^{*}\rangle$ and the Cramér–Wold device. The semiparametric efficiency can be established by applying the result of Theorem 4 in Shen (1997). $\square$