The new Neyman type A beta Weibull model with long-term survivors

Abstract

For the first time, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the Neyman type A distribution and the time to this event has the beta Weibull distribution. This new model can be used to analyze survival data when the hazard rate function is increasing, decreasing, bathtub or unimodal-shaped. It includes some commonly used lifetime distributions and some well-known cure rate models as special cases. Maximum likelihood and non-parametric bootstrap are used to estimate the regression parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the new model is illustrated by means of an application in the medical area.

Appendix A: Matrix of second derivatives \(\ddot{{\mathbf L}}(\varvec{\theta })\)

Here, we derive the formulae to obtain the second-order partial derivatives of the log-likelihood function. After some algebraic manipulations, we obtain

$$\begin{aligned} {\mathbf L}_{\phi \phi }&= -\frac{r}{\phi ^2}+\sum _{i=1}^n [F(t_i)]^2\exp [{\mathbf x}_i^{\top } \varvec{\beta }-\phi F(t_i)],\\ {\mathbf L}_{\phi \beta _j}&= -\sum _{i=1}^n x_{ij}F(t_i)\exp [{\mathbf x}_i^{\top } \varvec{\beta }-\phi F(t_i)],\\ {\mathbf L}_{\phi \gamma _q}&= \sum _{i=1}^n[\dot{F}(t_i)]_{\gamma _q}\exp [{\mathbf x}_i^{\top } \varvec{\beta }-\phi F(t_i)][\phi F(t_i)-1]-\sum _{i=1^n}\delta _i[\dot{F}(t_i)]_{\gamma _q},\\ {\mathbf L}_{\beta _j\beta _s}&= -\sum _{i=1}^n x_{ij}x_{is}\exp ({\mathbf x}_i^{\top } \varvec{\beta })+\sum _{i=1}^n x_{ij} x_{is}\exp [{\mathbf x}_i^{\top } \varvec{\beta }-\phi F(t_i)],\\ {\mathbf L}_{\beta _j\gamma _q}&= -\phi \sum _{i=1}^n x_{ij}[\dot{F}(t_i)]_{\gamma _q}\exp [{\mathbf x}_i^{\top } \varvec{\beta }-\phi F(t_i)],\\ {\mathbf L}_{\gamma _q \gamma _r}&= -\phi \sum _{i=1}^n\exp [{\mathbf x}_i^{\top } \varvec{\beta }-\phi F(t_i)]\left\{ [\ddot{F}(t_i)]_{\gamma _q\gamma _r}-[\dot{F} (t_i)]_{\gamma _q}[\dot{F}(t_i)]_{\gamma _r}\right\} \\&\quad +\sum _{i=1}^n\frac{\delta _i}{f(t_i)} \left\{ [\ddot{f}(t_i)]_{\gamma _q\gamma _r}-\frac{[\dot{f}(t_i)]_{\gamma _q} [\dot{f}(t_i)]_{\gamma _r}}{f(t_i)}\right\} - \phi \sum _{i=1}^n \delta _i[\ddot{F}(t_i)]_{\gamma _q\gamma _r}, \end{aligned}$$

where $f(t_i)$ and $F(t_i)$ are defined in (2) and (3), respectively,

$$\begin{aligned} \left[\dot{f}(t_i)\right]_{\gamma _q}&= \frac{\partial f(t_i)}{\partial \gamma _q},\quad \left[\dot{f}(t_i)\right]_{\gamma _r}=\frac{\partial f(t_i)}{\partial \gamma _r},\quad \left[\dot{F}(t_i)\right]_{\gamma _q}=\frac{\partial F(t_i)}{\partial \gamma _q},\\ \left[\dot{F}(t_i)\right]_{\gamma _r}&= \frac{\partial F(t_i)}{\partial \gamma _r}, \left[\ddot{f}(t_i)\right]_{\gamma _q \gamma _r}=\frac{\partial ^{2} f(t_i)}{\partial \gamma _q\partial \gamma _r},\quad \left[\ddot{f}(t_i)\right]_{\gamma _q \gamma _r}=\frac{\partial ^{2} f(t_i)}{\partial \gamma _q\partial \gamma _r}, \end{aligned}$$

for $i=1,\ldots ,n$; $q,r=1,\ldots ,4\,$ and $\,j,s=1,\ldots ,k$.