Asymptotic justification of maximum likelihood estimation for the proportional excess hazard model in analysis of cancer registry data

Abstract

Population-based cancer registry studies are conducted to investigate the various cancer question and have important impacts on cancer control. In order to investigate cancer prognosis from cancer registry data, it is necessary to adjust the effect of deaths from other causes, since cancer registry data include deaths from causes other than cancer. To correct for the effect of deaths from other causes, excess hazard models are often used. The concept of the excess hazard model is that the hazard function for any death in a cancer registry population is the sum of the hazard for cancer deaths, refer to the excess hazard, and the hazard for deaths from other causes. The Cox proportional hazard model for the excess hazard has been developed, and for this model, Perme et al. (Biostatistics 10:136–146, 2009) proposed the inference procedure of the regression coefficients using the techniques of the EM algorithm to compute the maximum likelihood estimator. In this article, we present the large sample properties for the maximum likelihood estimator. We introduce a consistent estimator of the variance for the regression coefficients based on the technique of the semiparametric theory and the consistency and the asymptotic normality of the estimator. The empirical property of variance estimator is investigated by the finite sample simulation studies. We also apply the variance estimator to cancer registry data for stomach, lung, and liver cancer patients from the Surveillance, Epidemiology, and End Results (SEER) database in U.S.

Appendices

Appendix

Appendix A: Existence of the maximum likelihood estimator and identifiability of $\beta _0$ and $\Lambda _0$

The existence of the pair of the parameters $(\beta ,\Lambda )$ maximizing the observed likelihood (3) is proved by using the similar arguments to the proof of theorem 1 in Fang et al. (2005). In this Appendix, along this line, we prove the identifiability of $(\beta _0,\Lambda _0)$ in the sense that $L(\Lambda ,\beta ;t,\delta ,z) = L(\Lambda _0, \beta _0;t,\delta ,z)$ implies $\beta = \beta _0$ and $\Lambda (t)=\Lambda _0$ for all $t \in [0,\tau ]$.

Suppose the parameter space $\mathscr {B} \in R^p$ of $\beta$ is compact, where p is the dimension of $\beta$. Since the vector of covariates Z is bounded, $e^{\beta ^TZ}$ is also bounded, and its lower and upper bounds are denoted by $K_l$ and $K_u$, respectively. Let $t_1<t_2<\cdots <t_k$ be the distinct failure times. Then, for any right-continuous and non-decreasing function $\Lambda (t)$, it holds that

$$\begin{aligned} 0 \le L(\beta , \Lambda )&\le \prod _{i=1}^n{ \left\{ K_u \Lambda (T_i) + \Lambda _P(T_i|Z_i) \right\} ^{\Delta _i} e^{ - \Lambda (T_i)K_l } } \nonumber \\&\le \prod _{i:T_i < t_k}{ \left\{ \frac{ K_u }{ K_l } + \Lambda _P(T_i|Z_i) \right\} ^{\Delta _i} }\nonumber \\&\times \prod _{i:T_i = t_k}{ \left\{ K_u \Lambda (t_k)e^{ - \Lambda (t_k)K_l } + \Lambda _P(T_i|Z_i) \right\} ^{\Delta _i} }. \end{aligned}$$

(12)

Because forcing $\Lambda (T_i)=\Lambda (t_k)$ for all $T_i \ge t_k$ will increase the likelihood if $t_k$ is sufficiently large value satisfying $\Lambda (t_k) \ge 1$, it suffices to restrict the space of $\Lambda (t)$ to the space $\Omega _0$, where

$$\begin{aligned} \Omega _0 =&\left\{ \Lambda : \Lambda (t) \ \text{ is } \text{ the } \text{ right } \text{ continuous } \text{ and } \text{ non-decreasing } \text{ function } \right. \\&\qquad \left. \text{ with } \ \Lambda (t)=\Lambda (t_k) \ \mathrm{for \ all} \ t \ge t_k \right\} . \end{aligned}$$

Let $A_M = \left\{ \Lambda \in \Omega _0: \Lambda (t_k) \le M \right\}$ for any $0<M<\infty$. Because $L(\beta ,\Lambda )$ is continuous in $\beta$ and $\Lambda$, it has a maximum in the compact subspace $\mathscr {B} \times A_M$ for any given M. Let $L^{(M)}$ be the maximum value of $L(\beta ,\Lambda )$ in $\mathscr {B} \times A_M$. By $e^{-MK_l}M \rightarrow 0$ as $M \rightarrow \infty$, there exists an $M_0 \ge 1$ such that the right-hand side of (12) is less than $L^{(M_0)}$ for all $\Lambda$ out of $A_{M_0}$. Therefore, the likelihood evaluated at any sequence $\Lambda _m$ of $\Lambda$ with $\Lambda _m(t_k)$ diverging to infinity as $m \rightarrow \infty$ will not approach the maximum value of $L(\beta ,\Lambda )$. As a consequence, when maximizing the observed likelihood (3), we can restrict the compact subspace $\mathscr {B} \times A_{M_0}$. The existence of the maximum likelihood estimator can be proved by the continuity of the likelihood.

We prove that both of $\beta _0$ and $\Lambda _0$ are identifiable. By considering $L(\Lambda ,\beta ;t,0,z) = L(\Lambda _0, \beta _0;t,0,z)$, we have that $\Lambda (t)/\Lambda _0(t)=e^{-\left( \beta - \beta _0 \right) ^TZ}$ for all $t \le \tau$ and Z such that $\Pr (T> \tau |Z)>0$. Therefore, since $\left( \beta - \beta _0 \right) ^TZ$ is constant for all Z, it hold that $\beta =\beta _0$ if the covariance of Z is nondegenerate, and also we have $\Lambda (t)=\Lambda _0(t)$ for all $t \le \tau$. By considering $L(\Lambda ,\beta ;t,1,z) = L(\Lambda _0, \beta _0;t,1,z)$, we also have $\mathrm{{d}}\Lambda (t)=\mathrm{{d}}\Lambda _0(t)$ for all $t \le \tau$.

Appendix B: Nuisance tangent space and its orthogonal complement

Let $\mathcal {H}$ be a Hilbert space consisted of all p-dimensional measurable functions of $(T,\Delta , Z)$ with mean-zero and finite variance equipped with inner product $\left<h_1,h_2\right>=E\left[ h_1^T(T,\Delta ,Z)h_2(T,\Delta ,Z)\right]$. To derive the nuisance tangent space for the nuisance parameter $\eta =\left\{ \Lambda , \Lambda _C, F_Z \right\}$, we consider parametric submodels $\Lambda _{h_1}(t;\gamma _1)$, $\Lambda _{C,h_2}(t|Z;\gamma _2)$, and $F_{Z,h_3}(z;\gamma _3)$ for $\Lambda$, $\Lambda _C$, and $F_Z$, respectively, which were defined in Sect. 3, where $\gamma _1$, $\gamma _2$, and $\gamma _3$ are the finite-dimensional nuisance parameters. Then, the nuisance tangent spaces for each nuisance parameter will be derived as the mean-square closure of all parametric submodel nuisance tangent spaces. Since the derivations of the nuisance tangent spaces $\Gamma _2$ and $\Gamma _3$ in Sect. 4, which are for the nuisance parameters $\Lambda _C$ and $F_Z$, respectively, are the same as those of Section 5.2 in Tsiatis (2006), we only derive here the nuisance tangent space $\Gamma _1$, which is for the nuisance parameter $\Lambda$, in Theorem 1.

Again, we consider a parametric submodel $\Lambda _{h_1}(t;\gamma _1) = \int _0^t{ \left\{ 1 + \gamma _1 h_1(u) \right\} \mathrm{{d}}\Lambda _0(u) } = \int _0^t{ \left\{ 1 + \gamma _1 h_1(u) \right\} \lambda _0(u) \mathrm{{d}}u }$, where $h_1(u)$ is an arbitrary p-dimensional bounded function. The contribution to the log-likelihood function under the parametric submodel is

$$\begin{aligned} \ell _n(\beta , \gamma _1; h_1)&= \sum _{i=1}^n{ \Delta _i \log { \left[ \left\{ 1 + \gamma _1 h_1(T_i) \right\} \mathrm{{d}}\Lambda _0(T_i)e^{\beta ^TZ_i} + \mathrm{{d}}\Lambda _P(T_i|Z_i) \right] } } \\&\quad - \sum _{i=1}^n{ \int _{0}^{T_i}{ \left\{ 1 + \gamma _1 h_1(t) \right\} \mathrm{{d}}\Lambda _0(t) }e^{\beta ^TZ_i } }. \end{aligned}$$

Taking derivatives of $\ell _n(\beta , \gamma _1; h_1)$ with respect to $\gamma _1$, and evaluating $\beta =\beta _0$ and $\gamma _1=0$, we obtain the score function

$$\begin{aligned} U_{n,\gamma _1}(\beta _0;h)&= \sum _{i=1}^n{ \int _{0}^{\tau }{ h_1(t) W(t|Z_i;\beta _0, \Lambda _0) \mathrm{{d}}M_i(t)} }. \end{aligned}$$

Then, the score function for this parametric submodel is in the nuisance tangent space $\Gamma _1$. Since any element of $\mathcal {H}$ can be approximated by a sequence of bounded function (Tsiatis 2006, Section 4), the score function with parametric submodel without the boundedness of $h_1(t)$ is also in $\Gamma _1$.

For any parametric submodel $\Lambda (t;\gamma _1) = \int _0^t{ \lambda (u;\gamma _1) \mathrm{{d}}u }$, the score function with respect to $\gamma _1$, setting $\gamma _1=0$ and $\beta =\beta _0$, is expressed as

$$\begin{aligned} U_{1,\gamma _1}(\beta _0)&= \int _{0}^{\tau }{ \left\{ \left. \frac{\partial }{\partial \gamma _1 }\log {\lambda (t;\gamma _1)} \right| _{\gamma _1=0} \right\} W(t|Z;\beta _0, \Lambda _0) \mathrm{{d}}M(t)}. \nonumber \end{aligned}$$

Then, this score function is in the nuisance tangent space $\Gamma _1$. On the other hand, we can demonstrate that the score function for the some parametric submodel included in $\Gamma _1$, such as $\Lambda _{h_1}(t;\gamma _1) = \int _0^t{ \left\{ 1 + \gamma _1 h_1(u) \right\} \mathrm{{d}}\Lambda _0(u) }$, is an element of a parametric submodel nuisance tangent space. Therefore, it holds that the nuisance tangent space for $\Lambda (t)$ is equal to $\Gamma _1$.

$\Gamma _1 \perp \Gamma _2$ can be easily proved under the assumption (A2) and $\Gamma _i \perp \Gamma _3 \ (i=1,2)$ can be also proved by $E[\alpha _i^Th_3(Z)]=0$, where $\alpha _i \in \Gamma _i\ (i=1,2)$ and $h_3(Z) \in \Gamma _3$. Then the nuisance tangent space for the nuisance parameter $\eta =\left\{ \Lambda , \Lambda _C, F_Z \right\}$ is given by the direct sum of three orthogonal spaces, $\Gamma = \Gamma _1 \oplus \Gamma _2 \oplus \Gamma _3$. The orthogonal complement $\Gamma ^{\perp }$ is obtained by applying the almost same procedures as the proof of Theorem 5.5 in Tsiatis (2006).