Conditional maximum likelihood estimation in semiparametric transformation model with LTRC data

Abstract

Left-truncated data often arise in epidemiology and individual follow-up studies due to a biased sampling plan since subjects with shorter survival times tend to be excluded from the sample. Moreover, the survival time of recruited subjects are often subject to right censoring. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of left-truncated and right-censored data. We propose a conditional likelihood approach and develop the conditional maximum likelihood estimators (cMLE) for the regression parameters and cumulative hazard function of these models. The derived score equations for regression parameter and infinite-dimensional function suggest an iterative algorithm for cMLE. The cMLE is shown to be consistent and asymptotically normal. The limiting variances for the estimators can be consistently estimated using the inverse of negative Hessian matrix. Intensive simulation studies are conducted to investigate the performance of the cMLE. An application to the Channing House data is given to illustrate the methodology.

Appendix: Proof of Theorem 1

The conditional likelihood function for the observations $X_i\le \tau _c~(i=1,\dots ,n)$ can be written as follows:

$$\begin{aligned} L_c(\beta ,R,\tau _c)=\prod _{i=1}^{n}\biggl (\prod _{t\le \tau _c} \bigl [dR(t) \bigr ]^{dN_i (t)}\biggr )\varPsi (O_i,\beta ,R), \end{aligned}$$

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where $O_i=(X_i,V_i,\delta _i,Z_i)$ denote the $i^{th}$ observation and

$$\begin{aligned} \varPsi (O_i,\beta ,R)=\biggl (\prod _{t\le \tau _c}[e^{\beta ^T Z_i} g(e^{\beta ^T Z_i}R(t))]^{dN_i(t)}\biggr )\exp \biggl \{-\int _{0}^{\tau _c}Y_i(u)d\varLambda (u|Z_i) \biggr \}. \end{aligned}$$

Similar to proof of Zeng and Lin (2010), we require the following conditions:

(C1) The true value $\beta _0$ lies in the interior of a compact set $\mathcal C$, and the true function $R_0 (\cdot )$ is strictly increasing and continuously differentiable in $[0,\tau _c]$.

(C2) With probability one, $P(\inf _{s\in [0,t]} Y_i (s)\ge 1|Z_i)> \delta _0 > 0$ for all $t\in [0,\tau _c]$.

(C3) There exists a constant $c_1 > 0$ and a random variable $r_1(O_i) > 0$ such that $E[\log r_1 (O_i)] < \infty$ and for any $\beta \in \mathcal{C}$ and any R,

$$\begin{aligned} \varPsi (O_i,\beta ,R)\le r_1(O_i)\prod _{t\le \tau _c} \bigg \{1+\int _{0}^{t}Y_i (t)dR(t)\bigg \}^{-dN_i (t)} \bigg \{1+\int _{0}^{\tau _c}Y_i (t)dR(t)\bigg \}^{-c_1} \end{aligned}$$

almost surely. In addition, for any constant $c_2$,

$$\begin{aligned} \inf \{\varPsi (O_i,\beta ,R):||R||_{V[0,\tau _c]}\le c_2,\beta \in \mathcal{C}\}> r_2(O_i) > 0, \end{aligned}$$

where $||w||_{V[0,\tau _c]}$ is the total variation of $w(\cdot )$ in $[0,\tau _c]$ and $r_2 (O_i)$, which may depend on $c_2$ is a finite random variable with $E[|\log r_2(O_i)|]<\infty$.

Furthermore, we require certain smoothness of $\varPsi$. Let ${\dot{\varPsi }}_{\beta }$ denote the derivative of $\varPsi$ with respect to $\beta$ and ${\dot{\varPsi }}(H)$ denote the derivative of $\varPsi$ along the path $R+\epsilon H$, where H belong the set of functions in which $R +\epsilon H$ is increasing with bounded total variation.

(C4) For any $(\beta _1,\beta _2)\in \mathcal{C}$ and $(R_1,R_2)$, $(H_1,H_2)$ with uniformly bounded total variations, there exists a random variable $R(O_i)\in L_4(P)$ and a stochastic process $\mu _i(t,O_i)\in L_6 (P)$ such that

$$\begin{aligned}&|\varPsi (O_i,\beta _1,R_1)-\varPsi (O_i,\beta _2,R_2)|+ |{\dot{\varPsi }}_{\beta }(O_i,\beta _1,R_1)-{\dot{\varPsi }}_{\beta }(O_i,\beta _2,R_2)|\\&\quad +\;|{\dot{\varPsi }}(O_i,\beta _1,R_1)(H_1)-{\dot{\varPsi }}(O_i,\beta _2,R_2)(H_2)|\\&\quad +\;\biggl |{{{\dot{\varPsi }}(O_i,\beta _1,R_1)(H_1)}\over {\varPsi (O_i,\beta _1,R_1)}} \quad -{{{\dot{\varPsi }}(O_i,\beta _2,R_2)(H_2)}\over {\varPsi (O_i,\beta _2,R_2)}}\biggr |\\&\le R(O_i)\biggl [|\beta _1-\beta _2|+\int _{0}^{\tau _c}|R_1(t)-R_2(t)| d\mu _i(t,O_i)\\&\quad +\;\int _{0}^{\tau _c}|H_1(t)-H_2(t)|d\mu _i(t,O_i)\biggr ]. \end{aligned}$$

In addition, $\mu _i (t,O_i)$ is non-decreasing and $E[R(O_i)\mu _i(t,O_i)]$ is left-continuous with uniformly bounded left- and right-derivatives for any $t\in [0,\tau _c]$.

(C5) If

$$\begin{aligned} \prod _{t\le \tau _c} \bigl [dR^{*}(t) \bigr ]^{dN_i (t)}\varPsi (O_i,\beta ^{*},R^{*})= \prod _{t\le \tau _c} \bigl [dR_0(t) \bigr ]^{dN_i (t)}\varPsi (O_i,\beta _0,R_0) \end{aligned}$$

almost surely, then $\beta ^{*}=\beta _0$ and $R^{*}(t)=R_0 (t)$ for $t\in [0,\tau _c]$.

(C6) Let $BV[0,\tau _c]$ denote the space of functions with bounded total variations in $[0,\tau _c]$. There exists a bounded function $\zeta (t,\beta _0,R_0)\in BV[0,\tau _c]$ and a matrix $M(\beta _0,R_0)$ such that

$$\begin{aligned}&\biggl |E\biggl [{{{\dot{\varPsi }}_{\beta }(O_i,\beta ,R)}\over {\varPsi (O_i,\beta ,R)}} -{{{\dot{\varPsi }}_{\beta }(O_i,\beta _0,R_0)}\over {\varPsi (O_i,\beta _0,R_0)}}\biggr ] -M(\beta _0,R_0)(\beta -\beta _0)\\&\qquad -\int _{0}^{\tau _c}\zeta (t,\beta _0,R_0)d(R(t)-R_0 (t))\biggr |\\&\quad =o(|\beta -\beta _0|+||R-R_0||_{V[0,\tau _c]}). \end{aligned}$$

In addition,

$$\begin{aligned}&\sup _{s\in [0,\tau _c]}\biggl |[\eta (s,\beta ,R)-\eta (s,\beta _0),R_0)] -\eta _{1}(s,\beta _0,R_0)(\beta -\beta _0)\\&\qquad -\int _{0}^{\tau _c}\eta _{2}(s,t,\beta _0,R_0)d(R-R_0)(t)\biggr |\\&\quad =o(|\beta -\beta _0|+||R-R_0||_{V[0,\tau _c]}), \end{aligned}$$

where $\eta _1(s,\beta ,R)$ is a p-dimensional bounded function and $\eta _2(s,t,\beta ,R)$ is a bounded bivariate function. Furthermore, there exists a constant $c_3$ such that $|\eta _2(s,t_1\beta _0,R_0)-\eta _2(s,t_2,\beta _0,R_0)|\le c3|t_1 -t_2|$ for any $s,t_1,t_2\in [0,\tau _c]$.

(C7)

$$\begin{aligned} \int _{0}^{\tau _c} w(t)Y_i (t)dN_i (t)+ {{{\dot{\varPsi }}_{\beta }(O_i,\beta _0,R_0)^{T}v+{\dot{\varPsi }}(O_i,\beta _0,R_0) (\int _{0}^{\tau _c} w(t)dR_0 (t))}\over {\varPsi (O_i,\beta _0,R_0)}}=0 \end{aligned}$$

for some vector $v\in \mathcal{R}^{p}$ and $w\in BV[0,\tau _c]$, then $v=0$ and $w=0$.

Define $\mathcal{V}=\{v\in \mathcal{R}^{p},|v|\le 1\}$ and $\mathcal{D}=\{w(t):||w(t)||_{V[0,\tau _c]}\le 1\}$.

By (C3), the conditional likelihood function is bounded by

$$\begin{aligned} \prod _{i=1}^{n}\varPsi (O_i,\beta ,R)\bigg [\prod _{t\le \tau _c} dR(t)Y_i (t) \bigg \{1+\int _{0}^{t}Y_i (s)dR(s)\bigg \}^{-1}\bigg ]^{dN_i (t)} \bigg \{1+\int _{0}^{\tau _c}Y_i (t)dR(t)\bigg \}^{-c_1}. \end{aligned}$$

Furthermore, (C2) implies that the maximum of (A.1) can be attained only for ${\hat{R}}(\tau _c) <\infty$. By differentiating (A.1) with respect to $dR(X_i)$ for which $dN_i (X_i)=1$ and $Y_i (X_i)=1$, it follows that ${\hat{R}}$ satisfied

$$\begin{aligned} {\hat{R}}(t)=-\sum _{i=1}^{n}\int _{0}^{t}\bigg \{\sum _{j=1}^{n} {{{\dot{\varPsi }}(O_j,{\hat{\beta }},{\hat{R}})(I_{[X_j\ge s]})} \over {\varPsi (O_j,{\hat{\beta }},{\hat{R}})}}\bigg \}^{-1} Y_i (s)dN_i (s). \end{aligned}$$

Let ${\tilde{R}}$ be a step function with jumps only at the $X_i$ for which $dN_i (X_i)=1$ and $Y_i(X_i)=1$, i.e. ${\tilde{R}}$ satisfies

$$\begin{aligned} {\tilde{R}}(t)=-\sum _{i=1}^{n}\int _{0}^{t}\bigg \{\sum _{j=1}^{n} {{{\dot{\varPsi }}(O_j,\beta _0,R_0)(I_{[X_j\ge s]})} \over {\varPsi (O_j,\beta _0,R_0)}}\bigg \}^{-1} Y_i (s)dN_i (s). \end{aligned}$$

Under condition (C4), by Lemma 1 of Zeng and Lin (2010) and the Glivenko-Cantelli Theorem, ${\tilde{R}}$ converges uniformly to $R_0$ in $[0,\tau _c]$. Let $l_n(\beta ,R,\tau _c)$ denote the log-likelihood function of $L_c(\beta ,R,\tau _c)$. Similar to the arguments of Step 2 of Zeng and Lin (2010), the difference between $l_n({\hat{\beta }},{\hat{R}},\tau _c)$ and $l_n(\beta _0,{\tilde{R}},\tau _c)$, is negative eventually if ${\hat{R}}(\tau _c)$ diverges, which will induce a contradiction. Hence, $\lim \sup _{n}{\hat{R}}(\tau _c) <\infty$ almost surely. Since ${\hat{R}}$ is bounded and monotone, Helly’s Theorem implies that for any subsequence, we can always choose a further subsequence such that ${\hat{R}}$ converges point-wisely to some monotone function $R_{*}$. Without loss of generality, assume that ${\hat{\beta }}$ converges to $\beta _{*}$. Note that

$$\begin{aligned} {\hat{R}} (t)=\int _{0}^{t}{{|n^{-1}\sum _{j=1}^{n} {\dot{\varPsi }}(O_j,\beta _0,R_0)(I_{[X_j\ge s]})/\varPsi (O_j,\beta _0,R_0)|}\over {|n^{-1}\sum _{j=1}^{n} {\dot{\varPsi }}(O_j,{\hat{\beta }}_n,{\hat{R}}_n)(I_{[X_j\ge s]})/\varPsi (O_j,{\hat{\beta }},{\hat{R}})|}} d{\tilde{R}}(s). \end{aligned}$$

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Under condition (C3), by Lemma 1 of Zeng and Lin (2010) and the Glivenko-Cantelli Theorem, the numerator and denominator in the integrand of (A.2) converges uniformly to deterministic functions. Under condition (C5), it follows by arguments of Step 3 of Zeng and Lin (2010) $\beta _{*}=\beta _0$ and $R_{*}=R_0$. The proof is complete.

Remark 5

Notice that by the arguments of section 10.1 of Zeng and Lin (2010), a sufficient condition for (C3), (C4) and (C6) is the (S1) given as follows.

(S1) G(t) is four-times differentiable such that $G(0)=0$, $G^{'}(0) > 0$ for any integer $m\ge 0$ and any sequence $0<t_1< \dots < t_m\le y$,

$$\begin{aligned} \prod _{i=1}^{m}\bigg \{(1+t_i)G^{'}(t_i)\exp \{-G(y)\}\le \mu _0^{m}(1+y)^{-k_0} \end{aligned}$$

for some constants $\mu _0$ and $k_0$. In addition, there exits a constant $\rho _0$ such that

$$\begin{aligned} \sup _{t}\biggl \{{{|G^{''}(t)|+|G^{(3)}(t)|+|G^{(4)}(t)|}\over {G^{'}(t)(1+t)^{\rho _0}}}\biggr \} < \infty . \end{aligned}$$