A class of semiparametric transformation models for survival data with a cured proportion

Abstract

We propose a new class of semiparametric regression models based on a multiplicative frailty assumption with a discrete frailty, which may account for cured subgroup in population. The cure model framework is then recast as a problem with a transformation model. The proposed models can explain a broad range of nonproportional hazards structures along with a cured proportion. An efficient and simple algorithm based on the martingale process is developed to locate the nonparametric maximum likelihood estimator. Unlike existing expectation-maximization based methods, our approach directly maximizes a nonparametric likelihood function, and the calculation of consistent variance estimates is immediate. The proposed method is useful for resolving identifiability features embedded in semiparametric cure models. Simulation studies are presented to demonstrate the finite sample properties of the proposed method. A case study of stage III soft-tissue sarcoma is given as an illustration.

Appendix: information matrix

Let $\{t_j\}$ denote the sets of the observed failure times and $d\Lambda _j=\Delta \Lambda (t_j)$. Define $h_i(t;\varvec{\theta })=H_{\varvec{\gamma }}^{\prime }\big (\xi _i(t;\varvec{\beta },\Lambda )\big )$, $\psi _i(t;\varvec{\theta })=\psi _{\varvec{\gamma }}\big (\xi _i(t;\varvec{\beta },\Lambda )\big )$, $\phi _i(t;\varvec{\theta })=\phi _{\varvec{\gamma }}\big (\xi _i(t;\varvec{\beta },\Lambda )\big )$, and

$$\begin{aligned} \mathbf{Z }_i(t;\varvec{\theta })= \mathbf{Z }_i(t)+\psi _i(t-;\varvec{\theta }) \int _0^{t-} Y_i(u) \mathbf{Z }_i(u)e^{\varvec{\beta }^T \mathbf{Z }_i(u)} d\Lambda (u). \end{aligned}$$

The observed information matrix $\mathcal I$ for the NPMLE can be approximated by (3.8). The closed-form expressions for submatrices for $\mathcal I$, obtained from the idea of Chen (2009), are presented as follows;

$$\begin{aligned} \mathcal I _{d\Lambda _j d\Lambda _j}&= -\partial U_{d\Lambda _j}/\partial (d\Lambda _j) \\&= \sum _{i=1}^n dN_i(t_j)+\left\{ \sum _{i=1}^n Y_i(t_j)e^{2\varvec{\beta }^T \mathbf{Z }_j(t_j)} \mu _i(t_j;\varvec{\theta })\right\} (d\Lambda _j)^2, \end{aligned}$$

where

$$\begin{aligned} \mu _i(t_j;\varvec{\theta })=\int _{t_j+}^\tau Y_i(u) e^{\varvec{\beta }^T \mathbf{Z }_i(u)} h_i(u-;\varvec{\theta })\psi _i^2(u-;\varvec{\theta })d\Lambda (u). \end{aligned}$$

For $j<k$,

$$\begin{aligned} \mathcal{I }_{d\Lambda _{j}d\Lambda _{k}}&= -\partial U_{d\Lambda _j}/\partial (d\Lambda _k)\\&= \left[ \sum _{i=1}^n Y_i(t_j) e^{\varvec{\beta }^T\{\mathbf{Z }_i(t_k)+ \mathbf{Z }_i(t_j)\}} \{h_i(t_j-;\varvec{\theta })\psi _i(t_j-;\varvec{\theta })+\mu _i(t_j; \varvec{\theta })\}\right] (d\Lambda _{j}d\Lambda _{k}). \end{aligned}$$

By symmetry, $\mathcal I _{d\Lambda _kd\Lambda _j}$ can also be obtained from the above expression.

$$\begin{aligned} \mathcal{I }_{{\varvec{\beta }} d{\Lambda _{j}}}&= -\partial U_{\varvec{\beta }}/\partial (d{\Lambda _{j}}) =-\partial U_{d\Lambda _j}/\partial \varvec{\beta }=\mathcal{I }_{d{\Lambda _{j}}\varvec{\beta }} \\&= \left[ \sum _{i=1}^n Y_i(t_j)e^{\varvec{\beta }^T \mathbf{Z }_i(t_j)}\{\mathbf{Z }_i(t_j;\varvec{\theta }) h_i(t_j-;\varvec{\theta })+v_i(t_j;\varvec{\theta })\}\right] (d{\Lambda _{j}}), \end{aligned}$$

where

$$\begin{aligned} v_i(t_j;\varvec{\theta })&= \int _{t_j+}^\tau Y_i(u) \mathbf{Z }_i(u;\varvec{\theta }) e^{\varvec{\beta }^T \mathbf{Z }_i(u)} h_i(u-;\varvec{\theta })\psi _i(u-;\varvec{\theta })d\Lambda (u).\\ \mathcal I _{\varvec{\gamma }d\Lambda _j}&= -\partial U_{\varvec{\gamma }}/\partial (d{\Lambda _{j}}) =-\partial U_{d{\Lambda _{j}}}/\partial \varvec{\gamma }= \mathcal I _{d{\Lambda _{j}}\varvec{\gamma }} \\&= \left[ \sum _{i=1}^n Y_i(t_j) e^{\varvec{\beta }^T \mathbf{Z }_i(t_j)} \{ \phi _i(t_j;\varvec{\theta })h_i(t_j-;\varvec{\theta }) +w_i(t_j;\varvec{\theta })\}\right] (d{\Lambda _{j}}), \end{aligned}$$

where

$$\begin{aligned} w_i(t_j;\varvec{\theta })&= \int _{t_j+}^\tau Y_i(u) \phi _i(u;\varvec{\theta }) e^{\varvec{\beta }^T \mathbf{Z }_i(u)} h_i(u-;\varvec{\theta })\psi _i(u-;\varvec{\theta })d\Lambda (u).\\ \mathcal I _{\varvec{\beta }\varvec{\beta }}&= -\partial U_{\varvec{\beta }}/\partial \varvec{\beta }=\sum _{i=1}^n \int _0^\tau Y_i(t) \mathbf{Z }_i(t;\varvec{\theta })^{\otimes 2} e^{\varvec{\beta }^T \mathbf{Z }_i(t)} h_i(t-;\varvec{\theta })d\Lambda (t),\\ \mathcal I _{\varvec{\gamma }\varvec{\gamma }}&= -\partial U_{\varvec{\gamma }}/\partial \varvec{\gamma }= \sum _{i=1}^n \int _0^\tau Y_i(t) \phi _i(t;\varvec{\theta })^{\otimes 2} e^{\varvec{\beta }^T \mathbf{Z }_i(t)}g_i(t-;\varvec{\theta })d\Lambda (t), \end{aligned}$$

and

$$\begin{aligned} \mathcal I _{\varvec{\beta }\varvec{\gamma }}&= -\partial U_{\varvec{\beta }}/\partial \varvec{\gamma }= -\partial U_{\varvec{\gamma }} /\partial \varvec{\beta }= -\mathcal I _{\varvec{\gamma }\varvec{\beta }}\\&= \sum _{i=1}^n \int _0^\tau Y_i(t) \mathbf{Z }_i(t;\varvec{\theta }) \phi _i(t;\varvec{\theta }) e^{\varvec{\beta }^T \mathbf{Z }_i(t)}h_i(t-;\varvec{\theta })d\Lambda (t). \end{aligned}$$