Likelihood inference for the destructive exponentially weighted Poisson cure rate model with Weibull lifetime and an application to melanoma data

Abstract

In this paper, we develop the steps of the expectation maximization algorithm (EM algorithm) for the determination of the maximum likelihood estimates (MLEs) of the parameters of the destructive exponentially weighted Poisson cure rate model in which the lifetimes are assumed to be Weibull. This model is more flexible than the promotion time cure rate model as it provides an interesting and realistic interpretation of the biological mechanism of the occurrence of an event of interest by including a destructive process of the initial number of causes in a competitive scenario. The standard errors of the MLEs are obtained by inverting the observed information matrix. An extensive Monte Carlo simulation study is carried out to evaluate the performance of the developed method of estimation. Finally, a known melanoma data are analyzed to illustrate the method of inference developed here. With these data, a comparison is also made with the scenario when the destructive mechanism is not included in the analysis.

Appendix: Expressions useful in M-step

$$\begin{aligned} \frac{\partial Q}{\partial \beta _{1l}}\!= & {} \!\sum _{J_1}x_{il}\left( 1\!-\!p_i\right) \left\{ 1\!-\!\eta _ip_ie^{\phi }F\left( t_i\right) \right\} -\sum _{J_0}x_{il}\left( 1-\pi _i^{\left( k\right) }\right) \eta _ip_i\left( 1\!-\!p_i\right) e^{\phi }\\&+\,\sum _{J_0}x_{il}\pi _i^{\left( k\right) }\eta _ip_i\left( 1-p_i\right) e^{\phi }\frac{\left\{ q_{0i}-S_{ pop }\left( t_i\right) F\left( t_i\right) \right\} }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} },\\ \frac{\partial Q}{\partial \beta _{2k}}= & {} \sum _{J_1}z_{ik}\left\{ 1-\eta _ip_ie^{\phi }F\left( t_i\right) \right\} -\sum _{J_0}z_{ik}\left( 1-\pi _i^{\left( k\right) }\right) \eta _ip_ie^{\phi }\\&+\,\sum _{J_0}z_{ik}\pi _i^{\left( k\right) }\eta _ip_ie^{\phi }\frac{\left\{ q_{0i}-S_{ pop }\left( t_i\right) F\left( t_i\right) \right\} }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} },\\ \frac{\partial Q}{\partial \gamma _1}= & {} \frac{1}{\gamma _1}\sum _{J_1}\left\{ \frac{\left( \gamma _2t_i\right) ^{\frac{1}{\gamma _1}}}{\gamma _1}\log \left( \gamma _2t_i\right) -\frac{\log \left( \gamma _2t_i\right) }{\gamma _1}-1\right\} -\sum _{J_1}\eta _ip_ie^{\phi }F_1\left( t_i\right) \\&-\,\sum _{J_0}\pi _i^{\left( k\right) }\frac{\eta _ip_ie^{\phi }S_{ pop }\left( t_i\right) F_1\left( t_i\right) }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} },\\ \frac{\partial Q}{\partial \gamma _2}= & {} \frac{1}{\gamma _1\gamma _2}\sum _{J_1}\left\{ 1-\left( \gamma _2 t_i\right) ^{\frac{1}{\gamma _1}}\right\} \\&-\,\sum _{J_1}\eta _ip_ie^{\phi }F_2\left( t_i\right) -\sum _{J_0}\pi _i^{\left( k\right) }\frac{\eta _ip_ie^{\phi }S_{ pop }\left( t_i\right) F_2\left( t_i\right) }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} },\\ \frac{\partial ^2Q}{\partial \beta _{1l}\partial \beta _{1l^{\prime }}}= & {} -\sum _{J_1}x_{il}x_{il^\prime }p_i\left( 1-p_i\right) \left\{ 1+\eta _i\left( 1-2p_i\right) e^{\phi }F\left( t_i\right) \right\} \\ \end{aligned}$$

$$\begin{aligned}&-\,\sum _{J_0}x_{il}x_{il^\prime }\left( 1-\pi _i^{\left( k\right) }\right) \eta _ip_i\left( 1-p_i\right) \left( 1-2p_i\right) e^{\phi }\\&+\,\sum _{J_0}x_{il}x_{il^\prime }\pi _i^{\left( k\right) }\frac{\eta _ip_i\left( 1-p_i\right) e^{\phi }}{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} }\left[ \left( 1-2p_i\right) \left( q_{0i}-S_{ pop }\left( t_i\right) F\left( t_i\right) \right) \right. \\&\left. -\frac{\eta _ip_i\left( 1-p_i\right) e^{\phi }q_{0i}S_{ pop }\left( t_i\right) \left( 1-F\left( t_i\right) \right) ^2}{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} }\right] ,\\ \frac{\partial ^2Q}{\partial \beta _{1l}\partial \beta _{2k}}\!= & {} \!-\sum _{J_1}x_{il}z_{ik}\eta _ip_i\left( 1\!-\!p_i\right) e^{\phi }F\left( t_i\right) \!-\!\sum _{J_0}x_{il}z_{ik}\left( 1\!-\!\pi _i^{\left( k\right) }\right) \eta _ip_i\left( 1-p_i\right) e^{\phi }\\&+\,\sum _{J_0}x_{il}z_{ik}\pi _i^{\left( k\right) }\frac{\eta _ip_i\left( 1-p_i\right) e^{\phi }}{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} }\\&\times \left[ q_{0i}-S_{ pop }\left( t_i\right) F\left( t_i\right) -\frac{\eta _ip_ie^{\phi }q_{0i}S_{ pop }\left( t_i\right) \left( 1-F\left( t_i\right) \right) ^2}{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} }\right] ,\\ \frac{\partial ^2Q}{\partial \beta _{2k}\partial \beta _{2k^{\prime }}}= & {} -\sum _{J_1}z_{ik}z_{ik^\prime }\eta _ip_ie^{\phi }F\left( t_i\right) -\sum _{J_0}z_{ik}z_{ik^\prime }\left( 1\!-\!\pi _i^{\left( k\right) }\right) \eta _ip_ie^{\phi }\\&+\,\sum _{J_0}z_{ik}z_{ik^\prime }\pi _i^{\left( k\right) }\frac{\eta _ip_ie^{\phi }}{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} }\\&\times \left[ q_{0i}-S_{ pop }\left( t_i\right) F\left( t_i\right) -\frac{\eta _ip_ie^{\phi }q_{0i}S_{ pop }\left( t_i\right) \left( 1-F\left( t_i\right) \right) ^2}{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} }\right] , \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2Q}{\partial \beta _{1l}\partial \gamma _1}= & {} -\sum _{J_1}x_{il}\eta _ip_i\left( 1-p_i\right) e^{\phi }F_1\left( t_i\right) +\sum _{J_0}x_{il}\pi _i^{\left( k\right) }\eta _ip_i\left( 1-p_i\right) e^{\phi }\\&\times \,\left[ \frac{S_{ pop }\left( t_i\right) F_1\left( t_i\right) \left\{ q_{0i}-S_{ pop }\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}\left( 1-F\left( t_i\right) \right) \right\} }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\right] ,\\ \frac{\partial ^2Q}{\partial \beta _{1l}\partial \gamma _2}= & {} -\sum _{J_1}x_{il}\eta _ip_i\left( 1-p_i\right) e^{\phi }F_2\left( t_i\right) +\sum _{J_0}x_{il}\pi _i^{\left( k\right) }\eta _ip_i\left( 1-p_i\right) e^{\phi }\\&\times \,\left[ \frac{S_{ pop }\left( t_i\right) F_2\left( t_i\right) \left\{ q_{0i}-S_{ pop }\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}\left( 1-F\left( t_i\right) \right) \right\} }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\right] ,\\ \frac{\partial ^2Q}{\partial \beta _{2k}\partial \gamma _1}= & {} -\sum _{J_1}z_{ik}\eta _ip_ie^{\phi }F_1\left( t_i\right) +\sum _{J_0}z_{ik}\pi _i^{\left( k\right) }\eta _ip_ie^{\phi }\\&\times \,\left[ \frac{S_{ pop }\left( t_i\right) F_1\left( t_i\right) \left\{ q_{0i}-S_{ pop }\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}\left( 1-F\left( t_i\right) \right) \right\} }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\right] ,\\ \frac{\partial ^2Q}{\partial \beta _{2k}\partial \gamma _2}= & {} -\sum _{J_1}z_{ik}\eta _ip_ie^{\phi }F_2\left( t_i\right) +\sum _{J_0}z_{ik}\pi _i^{\left( k\right) }\eta _ip_ie^{\phi }\\&\times \,\left[ \frac{S_{ pop }\left( t_i\right) F_2\left( t_i\right) \left\{ q_{0i}-S_{ pop }\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}\left( 1-F\left( t_i\right) \right) \right\} }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\right] ,\\ \frac{\partial ^2 Q}{\partial {\gamma _1^2}}= & {} \frac{1}{\gamma _1^2}\sum _{J_1}\left[ 1+\frac{2\log \left( \gamma _2t_i\right) }{\gamma _1}-\frac{\log \left( \gamma _2t_i\right) \left( \gamma _2t_i\right) ^{\frac{1}{\gamma _1}}}{\gamma _1}\left\{ 2+\frac{\log \left( \gamma _2t_i\right) }{\gamma _1}\right\} \right] \\ \end{aligned}$$

$$\begin{aligned}&\quad -\,\sum _{J_1}\eta _ip_ie^{\phi }F_{11}\left( t_i\right) \\&-\,\sum _{J_0}\pi _i^{\left( k\right) }\frac{\eta _ip_ie^{\phi }S_{ pop }\left( t_i\right) }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\\&\times \left[ \left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} F_{11}\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}F_1^2\left( t_i\right) \right] ,\\ \frac{\partial ^2 Q}{\partial {\gamma _2^2}}= & {} -\frac{1}{\gamma _1\gamma _2^2}\sum _{J_1}\left\{ 1+\left( \frac{1}{\gamma _1}-1\right) \left( \gamma _2t_i\right) ^{\frac{1}{\gamma _1}}\right\} -\sum _{J_1}\eta _ip_ie^{\phi }F_{22}\left( t_i\right) \\&-\,\sum _{J_0}\pi _i^{\left( k\right) }\frac{\eta _ip_ie^{\phi }S_{ pop }\left( t_i\right) }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\\&\times \left[ \left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} F_{22}\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}F_2^2\left( t_i\right) \right] ,\\ \frac{\partial ^2 Q}{\partial {\gamma _2}\partial {\gamma _1}}= & {} \frac{1}{\gamma _1^2\gamma _2}\sum _{J_1}\left[ \left( \gamma _2t_i\right) ^{\frac{1}{\gamma _1}}\left\{ 1+\frac{\log \left( \gamma _2t_i\right) }{\gamma _1}\right\} -1\right] -\sum _{J_1}\eta _ip_ie^{\phi }F_{12}\left( t_i\right) \\&-\,\sum _{J_0}\pi _i^{\left( k\right) }\frac{\eta _ip_ie^{\phi }S_{ pop }\left( t_i\right) }{\left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} ^2}\\&\times \left[ \left\{ S_{ pop }\left( t_i\right) -q_{0i}\right\} F_{12}\left( t_i\right) +\eta _ip_ie^{\phi }q_{0i}F_1\left( t_i\right) F_2\left( t_i\right) \right] , \end{aligned}$$

where

$$\begin{aligned} F_1\left( t_i\right)= & {} \frac{\partial F\left( t_i\right) }{\partial \gamma _1}=-\frac{1}{\gamma _1^2}S\left( t_i\right) \left( \gamma _2 t_i\right) ^{\frac{1}{\gamma _1}}\log \left( \gamma _2 t_i\right) , F_2\left( t_i\right) =\frac{\partial F\left( t_i\right) }{\partial \gamma _2}\\= & {} \frac{1}{\gamma _1\gamma _2}S\left( t_i\right) \left( \gamma _2 t_i\right) ^{\frac{1}{\gamma _1}},\\ F_{11}\left( t_i\right)= & {} \frac{\partial ^2 F\left( t_i\right) }{\partial \gamma _1^2}=-F_1\left( t_i\right) \left\{ \frac{1}{\gamma _1^2}\left\{ \log \left( \gamma _2t_i\right) +2\gamma _1\right\} +\frac{F_1\left( t_i\right) }{S\left( t_i\right) }\right\} ,\\ F_{22}\left( t_i\right)= & {} \frac{\partial ^2 F\left( t_i\right) }{\partial \gamma _2^2}=F_2\left( t_i\right) \left\{ \frac{1}{\gamma _2}\left( \frac{1}{\gamma _1}-1\right) -\frac{F_2\left( t_i\right) }{S\left( t_i\right) }\right\} ,\\ F_{12}\left( t_i\right)= & {} \frac{\partial ^2 F\left( t_i\right) }{\partial \gamma _2\partial \gamma _1}=-F_2\left( t_i\right) \left\{ \frac{1}{\gamma _1}\left( 1+\frac{\log \left( \gamma _2t_i\right) }{\gamma _1}\right) +\frac{F_1\left( t_i\right) }{S\left( t_i\right) }\right\} . \end{aligned}$$

The above quantities are all defined for $l,l^{\prime }=0,1,\ldots ,s,\,x_{i0}=1 \forall i=1,2,\ldots ,n,$ and $k,k^{\prime }=1,2,\ldots ,s^\prime .$