A new long-term survival model with dispersion induced by discrete frailty

Abstract

Frailty models are generally used to model heterogeneity between the individuals. The distribution of the frailty variable is often assumed to be continuous. However, there are situations where a discretely-distributed frailty may be appropriate. In this paper, we propose extending the proportional hazards frailty models to allow a discrete distribution for the frailty variable. Having zero frailty can be interpreted as being immune or cured (long-term survivors). Thus, we develop a new survival model induced by discrete frailty with zero-inflated power series distribution, which can account for overdispersion. A numerical study is carried out under the scenario that the baseline distribution follows an exponential distribution, however this assumption can be easily relaxed and some other distributions can be considered. Moreover, this proposal allows for a more realistic description of the non-risk individuals, since individuals cured due to intrinsic factors (immune) are modeled by a deterministic fraction of zero-risk while those cured due to an intervention are modeled by a random fraction. Inference is developed by the maximum likelihood method for the estimation of the model parameters. A simulation study is performed in order to evaluate the performance of the proposed inferential method. Finally, the proposed model is applied to a data set on malignant cutaneous melanoma to illustrate the methodology.

Appendix: Calculation details for the observed information matrix

The components of the observed information matrix, ${\varvec{J}}({\varvec{\vartheta }}),$ are derived in the form

$$\begin{aligned} {\varvec{J}}_{{\varvec{\beta }}{\varvec{\beta }}}= & {} \sum _{i=1}^{n} \left\{ -\frac{A(\theta _i)A^\prime (\theta _i) - (A^\prime (\theta _i))^2}{A(\theta _i)^2 } \right. \nonumber \\&\left. - \delta _i \left[ \frac{1}{\theta _i^2} - S^2_0(t_i;{\varvec{\gamma }})\frac{A^\prime (\theta _iS_0(t_i;{\varvec{\gamma }})) A^{\prime \prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))- (A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})))^2}{A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))^2} \right] \right. \nonumber \\&\left. + (1-\delta _i) \frac{c_1{\dot{c}}_2 - c_2{\dot{c}}_1}{c_1^2} \right\} \frac{\partial \theta _i}{\partial {\varvec{\beta }}} \frac{\partial \theta _i}{\partial {\varvec{\beta }}^\top } + {\varvec{D}} \frac{\partial ^2 \theta _i}{\partial {\varvec{\beta }}\partial {\varvec{\beta }}^\top }. \end{aligned}$$

(22)

$$\begin{aligned} {\varvec{J}}_{{\varvec{\beta }}{\varvec{\gamma }}}= & {} \sum _{i=1}^{n} \left\{ \delta _i \left[ \frac{A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))}{A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))} + \theta _i S_0(t_i;{\varvec{\gamma }}) \right. \right. \nonumber \\&\left. \left. \left( \frac{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})) - (A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }})))^2}{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))^2 }\right) \right] \right. \nonumber \\&\left. (1-\delta _i)\frac{c_1c_2^* - c_2 c_1^* }{c_1^2} \right\} \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}}\frac{\partial \theta _i}{\partial {\varvec{\beta }}}. \end{aligned}$$

(23)

$$\begin{aligned} {\varvec{J}}_{{\varvec{\beta }}\phi }= & {} \sum _{i=1}^{n} \left\{ (1-\delta _i)\frac{(\phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }}))) A^\prime (\theta _i) - (\phi A^\prime (\theta _i) + A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))) A(\theta _i)}{(\phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }})))^2 }\right\} \frac{\partial \theta _i}{\partial {\varvec{\beta }}}. \nonumber \\ \end{aligned}$$

(24)

$$\begin{aligned} {\varvec{J}}_{{\varvec{\gamma }}{\varvec{\gamma }}}= & {} \sum _{i=1}^{n} \left\{ \delta _i \left[ \frac{1}{f_0(t_i;{\varvec{\gamma }})} \frac{\partial ^2 f_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}^2}\right. \right. \nonumber \\&\left. \left. + \theta _i^2 \frac{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) A^{\prime \prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})) - (A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})))^2}{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))^2} \left( \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}}\right) ^2\right. \right. \nonumber \\&\left. \left. \theta _i\frac{A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))}{A^{\prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))} \frac{\partial ^2 S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}^2} \right] + (1-\delta _i)\theta _i\frac{c_1 d_1^\star - d_1 c_1^\star }{c_1^2} \left( \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}}\right) ^2 \right. \nonumber \\&\left. + \frac{d_1}{c_1}\frac{\partial ^2 S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}^2} \right\} . \end{aligned}$$

(25)

$$\begin{aligned} {\varvec{J}}_{{\varvec{\gamma }}\phi }= & {} \sum _{i=1}^{n} \left\{ - (1-\delta _i)\theta _i \frac{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))A(\theta _i)}{(\phi A(\theta _i)+A(\theta _i S_0(t_i;{\varvec{\gamma }})))^2} \frac{\partial S_0(t_i;{\varvec{\gamma }})}{\partial {\varvec{\gamma }}} \right\} , \end{aligned}$$

(26)

and

$$\begin{aligned} \begin{aligned} {\varvec{J}}_{\phi \phi } = \frac{n}{(1+\phi )^2} - \displaystyle \sum _{i=1}^{n} \left\{ \frac{(A(\theta _i))^2}{(\phi A(\theta _i)+ A(\theta _i S_0(t_i;{\varvec{\gamma }})))^2} \right\} , \end{aligned} \end{aligned}$$

(27)

where

$$\begin{aligned} c_1= & {} \phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }})),\\ c_2= & {} {\dot{c}}_1 =\phi A^\prime (\theta _i) + A^\prime (\theta _iS_0(t_i;{\varvec{\gamma }})) S_0(t_i;{\varvec{\gamma }}),\\ {\dot{c}}_2= & {} \phi A^{\prime \prime }(\theta _i) + A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))S_0(t_i;{\varvec{\gamma }})^2,\\ c_1^*= & {} A(\theta _i S_0(t_i;{\varvec{\gamma }})) \theta _i,\\ c_2^*= & {} A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) + \theta _i S_0(t_i;{\varvec{\gamma }})A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }})),\\ d_1= & {} A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})),\\ d_1^\star= & {} A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))\theta _i,\\ c_1^\star= & {} A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) \theta _i \end{aligned}$$

and

$$\begin{aligned} {\varvec{D}}= & {} \frac{-A^\prime (\theta _i)}{A(\theta _i)} + \delta _i \left[ \frac{1}{\theta _i} + \frac{A^{\prime \prime }(\theta _i S_0(t_i;{\varvec{\gamma }}))}{A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }}))} S_0(t_i;{\varvec{\gamma }}) \right] \\&+(1-\delta _i) \frac{\phi A^\prime (\theta _i) + A^\prime (\theta _i S_0(t_i;{\varvec{\gamma }})) S_0(t_i;{\varvec{\gamma }}) }{ \phi A(\theta _i) + A(\theta _i S_0(t_i;{\varvec{\gamma }}))}. \end{aligned}$$