Semiparametric model for semi-competing risks data with application to breast cancer study

Abstract

For many forms of cancer, patients will receive the initial regimen of treatments, then experience cancer progression and eventually die of the disease. Understanding the disease process in patients with cancer is essential in clinical, epidemiological and translational research. One challenge in analyzing such data is that death dependently censors cancer progression (e.g., recurrence), whereas progression does not censor death. We deal with the informative censoring by first selecting a suitable copula model through an exploratory diagnostic approach and then developing an inference procedure to simultaneously estimate the marginal survival function of cancer relapse and an association parameter in the copula model. We show that the proposed estimators possess consistency and weak convergence. We use simulation studies to evaluate the finite sample performance of the proposed method, and illustrate it through an application to data from a study of early stage breast cancer.

Appendix

Proof of Theorem 2

Asymptotic results of $\hat{S}_1(t)$ are proved under the following regularity condition. Function $g(u, v, \alpha )$ is continuous and differentiable at u, v, and $\alpha$, respectively, and the parameter $\alpha$ lies in a compact set.

First, we show the consistency of $\hat{S}_1(t)$. We have that $\hat{S}_2(t)$ converges in probability to $S_2(t)$ uniformly for $t\in [0,\tau ]$, and $\hat{S}_Z(t)$ converges in probability to $S_Z(t)$ uniformly for $t\in [0,\tau ]$. By Theorem 1, $\hat{\alpha }$ converges in probability to $\alpha _0$. Since the function $g(u, v, \alpha )$ is a continuous function of u, v and $\alpha$, $g\{\hat{S}_2(t), \hat{S}_{Z}(t), \hat{\alpha }\}$ converges in probability to $g\{S_2(t), S_{Z}(t), \alpha _0\}$ uniformly for $t\in [0,\tau ]$. Therefore, $\hat{S}_1(t)$ is a consistent estimator of $S_1(t)$.

Next, we illustrate the asymptotic distribution of $\hat{S}_1(t)$. Applying the functional delta method to $g\{\hat{S}_2(t), \hat{S}_{Z}(t), \hat{\alpha }\}$ around $S_2(t)$, $S_Z(t)$ and $\alpha _0$, we have

$$\begin{aligned}&n^{1/2}\big [g\{\hat{S}_2(t), \hat{S}_{Z}(t), \hat{\alpha }\}-g\{S_2(t), S_Z(t),\alpha _0\}\big ]\nonumber \\&\quad \cong n^{1/2} \frac{\partial g\{S_2(t), S_Z(t),\alpha _0\}}{\partial u}(\hat{S}_2-S_2)(t)\nonumber \\&\qquad +\,n^{1/2} \frac{\partial g\{S_2(t), S_Z(t),\alpha _0\}}{\partial v} (\hat{S}_Z-S_Z)(t) \nonumber \\&\qquad +\,n^{1/2} \frac{\partial g\{S_2(t), S_Z(t),\alpha _0\}}{\partial \alpha } (\hat{\alpha }-\alpha _0). \end{aligned}$$

(8)

Using martingale representations for $\hat{S}_2$ and $\hat{S}_Z$ (Gill 1980), the sum of the first and second terms in (8) is asymptotically equivalent to

$$\begin{aligned} n^{-\frac{1}{2}}\left[ \sum _{i=1}^n \frac{\partial g\{S_2(t), S_Z(t),\alpha _0\}}{\partial u} I_1^0(Y_{i},\delta _{2i})+\frac{\partial g\{S_2(t), S_Z(t),\alpha _0\}}{\partial v} I_2^0(Z_{i},\delta _{zi})\right] , \end{aligned}$$

(9)

which is a sum of n independent and identically distributed random variables, with $I_1^0(Y_{i},\delta _{2i})$ and $I_2^0(Z_{i},\delta _{zi})$ defined as in the previous section. Also the expectation of each term in (9) is zero. By the central limit theorem, (9) converges weakly to a normal distribution with mean zero and variance $\omega ^2(t)$. By Theorem 1, $n^{1/2}(\hat{\alpha }-\alpha _0)$ converges weakly to a normal distribution with mean zero and variance $\rho ^2$. Therefore, the third term in (8) is asymptotically equivalent to

$$\begin{aligned} n^{-1/2} \frac{\partial g\{S_2(t), S_Z(t),\alpha \}}{\partial \alpha } \rho \sum _{i=1}^n \frac{\partial l_0^g\{\alpha , \hat{S}_2(Y_{i}), \hat{S}_{Z}(Z_i)\}}{\partial \alpha }, \end{aligned}$$

(10)

which is a sum of n independent and identically distributed random variables.

Moreover, we have

$$\begin{aligned}&E \left( \Bigg [\frac{\partial l_0^g\{\alpha , \hat{S}_2, \hat{S}_{Z}\}}{\partial \alpha }\Bigg ]\{I_1^0(Y_{i},\delta _{2i})+I_2^0(Z_{i},\delta _{zi})\}\right) \\&\quad =E\left( \{I_1^0(Y_{i},\delta _{2i})+I_2^0(Z_{i},\delta _{zi})\}E\Bigg [\frac{\partial l_0^g\{\alpha , \hat{S}_2, \hat{S}_{Z}\}}{\partial \alpha }\mid Y_{i}, Z_i,\delta _{2i},\delta _{zi} \Bigg ]\right) =0,\nonumber \end{aligned}$$

(11)

which means that (9) and (10) are asymptotically orthogonal. Therefore, (9), (10) and (11) imply that as $n\rightarrow \infty$, the process $n^{1/2}\{\hat{S}_1(t)-S_1(t)\}$ converges weakly to a zero-mean Gaussian process for $t\in [0,\tau ]$ with covariance function $\Big [\frac{\partial g\{S_2(t), S_{Z}(t), \alpha \}}{\partial \alpha }\Big ]^2\rho ^2+\omega ^2(t)$. $\square$