Free-ranging dogs’ lifetime estimated by an approach for long-term survival data with dependent censoring

Abstract

Populations of free-ranging dogs are still a matter of concern in developing countries. The presence of stray dogs is associated with environmental and public health consequences such as the spread of zoonotic diseases. Therefore, public health managers base the promotion of public health on sanitary measures, including the control of the free-ranging dogs’ population. In this context, it is necessary to evaluate the free-ranging dogs’ life dynamics, taking into account all characteristics of the data, including long-term survival. In long-term studies, some causes of censoring are generally falsely assumed to be independent, leading to bias neglected. Therefore, we propose a likelihood-based approach for long-term clustered survival data, which is suitable to accommodate the dependent censoring. The association between lifetimes and dependent censoring is accommodated through the conditional approach of the frailty models. The marginal distributions can be adjusted assuming Weibull or piecewise exponential distributions, respectively. A Monte Carlo Expectation–Maximization algorithm is developed to estimate the proposed estimators. The simulation study results show a small relative bias and coverage probability near to the nominal level, indicating that the proposed approach works well. Moreover, the model identifiability is assured once data has a cluster structure. Finally, we analyze the survival times of free-ranging dogs from the West Bengal, India, collected between 2010 to 2015, and conclude that survival time (death due to natural cause) is negatively correlated to dependent censoring (missing cause).

Appendices

Appendix A: Approximate information matrix

Given the frailty terms $w_{i}, i=1,\cdots ,m$, the conditional likelihood function for the complete data $({\mathbf {d}}, {\mathbf {w}})$, where ${\mathbf {d}}$ denotes the observed data and ${\mathbf {w}}$ the unobserved frailty, is the following:

$$\begin{aligned} \begin{array}{llll} L_{ij}\left( \varvec{\Theta }^{(T)}, \varvec{\Theta }^{(C)}; {\mathbf {d}}_{ij}, w_{i}\right) = &{}\left[ h^{(Z)}\left( y_{ij}|\varvec{\psi }^{(Z)}\right) \exp \left( -H^{(Z)}\left( y_{ij}|\varvec{\psi }^{(Z)}\right) \right) \right. \\ &{}\left. \exp \left\{ {\mathbf {x}}_{ij}^{(T)}\varvec{\beta }^{(T)} + w_{i}\right\} \right] ^{\delta _{ij}^{(T)}}\\ &{}\times \exp \left\{ -\exp \left\{ {\mathbf {x}}_{ij}^{(T)}\varvec{\beta }^{(T)} + w_{i}\right\} \right. \\ &{}\left. \left[ 1-\exp \left\{ -H^{(Z)}\left( y_{ij}|\varvec{\psi }^{(Z)}\right) \right\} \right] \right\} \\ &{}\times \left[ h_{0}^{(C)}\left( y_{ij}|\varvec{\psi }^{(C)}\right) \exp \left\{ {\mathbf {x}}^{(C)}_{ij}\varvec{\beta }^{(C)}+\phi w_{i}\right\} \right] ^{\delta _{ij}^{(C)}}\\ &{}\times \exp \left\{ -H_{0}^{(C)}\left( y_{ij}|\varvec{\psi }^{(C)}\right) \exp \left\{ {\mathbf {x}}^{(C)}_{ij}\varvec{\beta }^{(C)} + \phi w_{i}\right\} \right\} , \end{array} \end{aligned}$$

where ${\mathbf {d}}_{ij}=( y_{ij},~ \delta ^{(T)}_{ij},~ \delta ^{(C)}_{ij}, ~ {\mathbf {x}}^{(T)}_{ij}, ~ {\mathbf {x}}^{(C)}_{ij})'$ corresponds to the observed data, for $i=1, \cdots , m$ and $j=1, \cdots , n_{i}$; $\varvec{\Theta }^{(T)}=\left( \varvec{\psi }^{(Z)}, \varvec{\beta }^{(T)}\right)$ and $\varvec{\Theta }^{(C)}=\left( \varvec{\psi }^{(C)}, \varvec{\beta }^{(C)},\phi \right)$, with $\varvec{\psi }^{(Z)}$ and $\varvec{\psi }^{(C)}$ being vectors of parameters associated with the baseline distributions of the promotion times Z and the dropout times, respectively; $\varvec{\beta }^{(T)}$ and $\varvec{\beta }^{(C)}$ are $p \times 1$ and $q \times 1$ vectors of regression coefficients related to ${\mathbf {x}}_{ij}^{(T)}$ and ${\mathbf {x}}_{ij}^{(C)}$; $w_{i}$ is the frailty term for the ith cluster; and $\phi$ is the association parameter.

1.1 A.1 Weibull cure model with dependent censoring

Consider the Weibull model with cure rate proposed in Section 2.2, the parameter vector is denoted by $\varvec{\Theta }= \left( \varvec{\Theta }^{(T)}, \varvec{\Theta }^{(C)}, \phi , \sigma ^{2}\right)$, where $\varvec{\Theta }^{(T)}=\left( \varvec{\psi }^{(Z)}, \varvec{\beta }^{(T)}\right)$ and $\varvec{\Theta }^{(C)}=\left( \varvec{\psi }^{(C)}, \varvec{\beta }^{(C)}\right)$, with $\varvec{\psi }^{(Z)} = (\alpha ^{(Z)}, \gamma ^{(Z)})$ and $\varvec{\psi }^{(C)} = (\alpha ^{(C)}, \gamma ^{(C)})$.

The first-order partial derivatives are given by

$$\begin{aligned} \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)}}&= \displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ x_{ik}^{(T)}\left[ \delta _{ik}^{(T)} - \exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k})\right. \right. \\&\left. \left. \left( 1-\exp (-y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)})\right) \right] \right\} , \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \alpha ^{(Z)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{\delta _{ik}^{(T)}\left[ \dfrac{1}{\alpha ^{(Z)}} + \log (y_{ik}) - y_{ik}^{\alpha ^{(Z)}}\log (y_{ik})\gamma ^{(Z)}\right] \\&+\exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+ w_{k})\exp (-y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)})\left( -y_{ik}^{\alpha ^{(Z)}}\log (y_{ik})\gamma ^{(Z)}\right) \Bigg \}, \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \gamma ^{(Z)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{\delta _{ik}^{(T)}\left[ \dfrac{1}{\gamma ^{(Z)}} - y_{ik}^{\alpha ^{(Z)}}\right] \\&+ \exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k})\exp (-y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)})\left( -y_{ik}^{\alpha ^{(Z)}}\right) \Bigg \}, \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)}}&= \displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ x_{ik}^{(C)}\left[ \delta _{ik}^{(C)} - y_{ik}^{\alpha ^{(C)}}\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)} + \phi w_{k})\right] \right\} , \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \phi }&= \displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ w_{k}\left[ \delta _{ik}^{(C)} - y_{ik}^{\alpha ^{(C)}}\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)} + \phi w_{k})\right] \right\} , \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \alpha ^{(C)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{\delta _{ik}^{(C)}\left[ \dfrac{1}{\alpha ^{(C)}} + \log (y_{ik})\right] \\&- y_{ik}^{\alpha ^{(C)}}\log (y_{ik})\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k})\Bigg \}, \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \gamma ^{(C)}}&= \displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ \delta _{ik}^{(C)}\left[ \dfrac{1}{\gamma ^{(C)}}\right] - y_{ik}^{\alpha ^{(C)}}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k})\right\} \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \sigma ^{2}} = \displaystyle \sum _{i=1}^{n_{k}}\left\{ \dfrac{w_{k}^{2}}{2\sigma ^{4}} - \dfrac{1}{2\sigma ^{2}} \right\} . \end{aligned}$$

The second-order partial derivatives are given by

$$\begin{aligned} \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)} \partial \varvec{\beta }^{(T)'}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -\exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+w_{k}){\varvec{x}}_{ik}^{(T)}{\varvec{x}}_{ik}^{(T)'}\left[ 1-\exp (-y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)})\right] \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)} \partial \alpha ^{(Z)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -\exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+w_{k})x_{ik}^{(T)}\gamma ^{(Z)}y_{ik}^{\alpha ^{(Z)}}\log (y_{ik}) \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)} \partial \gamma ^{(Z)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -\exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+w_{k}){\varvec{x}}_{ik}^{(T)}y_{ik}^{\alpha ^{(Z)}} \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\alpha ^{(Z)})^{2}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\Bigg \{ \delta _{ik}^{(T)}\left[ \dfrac{1}{\alpha ^{2}} - y_{ik}^{\alpha ^{(Z)}}\log (y_{ik})\log (y_{ik})\gamma ^{(Z)}\right] \\&-\exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+w_{k})\exp (-y_{ik}^{\alpha ^{(Z)}} \gamma ^{(Z)})\gamma ^{(Z)}\log (y_{ik})y_{ik}^{\alpha ^{(Z)}}\log (y_{ik})\\&\times \left[ 1- y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)}\right] \Bigg \},\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \alpha ^{(Z)} \partial \gamma ^{(Z)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\Bigg \{ \delta _{ik}^{(T)} \left[ - y_{ik}^{\alpha ^{(Z)}}\log (y_{ik})\right] \\&\quad - \exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+w_{k}) \exp (-y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)})\\&\times y_{ik}^{\alpha ^{(Z)}}\log (y_{ik})\left[ 1- y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)}\right] \Bigg \}, \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\gamma ^{(Z)})^{2}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\Bigg \{ \delta _{ik}^{(T)}\left[ -\dfrac{1}{(\gamma ^{(Z)})^{2}}\right] + \exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)}+w_{k}) \\&\times \exp (-y_{ik}^{\alpha ^{(Z)}}\gamma ^{(Z)})y_{ik}^{2\alpha ^{(Z)}} \Bigg \} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \varvec{\beta }^{(C)'}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k}){\varvec{x}}_{ik}^{(C)}{\varvec{x}}_{ik}^{(C)'} \right\} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \phi } =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k}){\varvec{x}}_{ik}^{(C)}w_{k} \right\} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\alpha ^{(C)})^{2}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}} \Bigg \{ \delta _{ik}^{(C)}\left[ -\dfrac{1}{(\alpha ^{(C)})^{2}}\right] -y_{ik}^{\alpha ^{(C)}}\log (y_{ik})\log (y_{ik})\gamma ^{(C)}\\&\times \exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k}) \Bigg \} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \alpha ^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\log (y_{ik})\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k}){\varvec{x}}_{ik}^{(C)} \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \phi \partial \alpha ^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\log (y_{ik})\gamma ^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k})w_{k} \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\gamma ^{(C)})^{2}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ \delta _{ik}^{(C)}\left[ -\dfrac{1}{(\gamma ^{(C)})^{2}}\right] \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \gamma ^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k}){\varvec{x}}_{ik}^{(C)} \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \phi \partial \gamma ^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\exp (x_{ik}^{(C)}\beta ^{(C)}+\phi w_{k})w_{k} \right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \alpha ^{(C)} \partial \gamma ^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -y_{ik}^{\alpha ^{(C)}}\log (y_{ik})\exp (x_{ik}^{(C)}\beta ^{(C)}+\phi w_{k}) \right\} \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\sigma ^{2})^{2}} =&\displaystyle \sum _{k=1}^{m} \left\{ -\dfrac{w_{k}^{2}}{(\sigma ^{2})^{3}} + \dfrac{1}{2(\sigma ^{2})^{2}}\right\} . \end{aligned}$$

The other elements of the matrix are null.

1.2 A.2 Piecewise exponential cure model with dependent censoring

Consider the piecewise exponential model with cure rate proposed in Sect. 2.3, the parameter vector is denoted by $\varvec{\Theta }= \left( \varvec{\Theta }^{(T)}, \varvec{\Theta }^{(C)}, \phi , \sigma ^{2}\right)$, where $\varvec{\Theta }^{(T)}\!\!=\left( \varvec{\psi }^{(Z)}, \varvec{\beta }^{(T)}\right) \!\!$ and $\varvec{\Theta }^{(C)}=\left( \varvec{\psi }^{(C)}, \varvec{\beta }^{(C)}\right)$, with $\varvec{\psi }^{(Z)}\!\!=\!\!\left( \lambda _{1}^{(Z)}, \cdots , \lambda _{b_{Z}}^{(Z)}\right)$ and $\varvec{\psi }^{(C)}=\left( \lambda _{1}^{(C)}, \cdots , \lambda _{b_{C}}^{(C)}\right)$ are vectors of failure rates, with $\lambda ^{(Z)}_{k}>0$ and $\lambda ^{(C)}_{l}>0$, for $k = 1, \cdots , b_{(Z)}$ and $l = 1, \cdots , b_{(C)}$.

The first-order partial derivatives are given by

$$\begin{aligned} \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{x_{ik}^{(T)}\Bigg [ \delta _{ik}^{(T)} - \exp \left( \varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k}\right) \\&\times \left( 1-\exp \left( -\displaystyle \sum _{j=1}^{b}\lambda ^{(Z)}_{j}(y_{ij}-s_{j-1})\right) \right) \Bigg ] \Bigg \},\\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \lambda _{j}^{(Z)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{\delta _{ikj}^{(T)}\left[ \dfrac{1}{\lambda _{j}^{(Z)}} + (y_{ij}-s_{j-1})\right] \\&+ \exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k})\exp \left( -\displaystyle \sum _{j=1}^{b}\lambda ^{(Z)}_{j}(y_{ij}-s_{j-1})\right) (-(y_{ij}-s_{j-1}))\Bigg \}, \\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ x_{ik}^{(C)}\left[ \delta _{ik}^{(C)} - \displaystyle \sum _{l=1}^{d}\lambda ^{(C)}_{l}(y_{il}-s_{l-1})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k})\right] \right\} ,\\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \phi } =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ w_{k}\left[ \delta _{ik}^{(C)} - \displaystyle \sum _{l=1}^{d}\lambda ^{(C)}_{l}(y_{il}-s_{l-1})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k})\right] \right\} ,\\ \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \lambda _{l}^{(C)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\left\{ \delta _{ikl}^{(C)}\left[ \dfrac{1}{\lambda _{l}^{(C)}} \right] - (y_{il}-s_{l-1})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k}) \right\} \end{aligned}$$

and

$$\begin{aligned} \dfrac{\partial l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \sigma ^{2}} = \displaystyle \sum _{i=1}^{n_{k}}\left\{ \dfrac{1}{2\sigma ^{4}}(w_{k}^{2}) - \dfrac{1}{2\sigma ^{2}} \right\} . \end{aligned}$$

The second-order partial derivatives are given by

$$\begin{aligned} \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)}\partial \varvec{\beta }^{(T)'}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{-x_{ik}^{(T)}x_{ik}^{(T)'}\exp \left( \varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k}\right) \\&\times \left( 1-\exp \left( -\displaystyle \sum _{j=1}^{b}\lambda ^{(Z)}_{j}(y_{ij}-s_{j-1})\right) \right) \Bigg \},\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(T)}\partial \lambda _{j}^{(T)}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{x_{ik}^{(T)}\exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k})\exp \left( -\displaystyle \sum _{j=1}^{b}\lambda ^{(Z)}_{j}(y_{ij}-s_{j-1})\right) \\&\times (-(y_{ij}-s_{j-1})) \Bigg \},\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\lambda _{j}^{(Z)})^{2}} =&\displaystyle \sum _{k=1}^{m}\sum _{i=1}^{n_{k}}\Bigg \{\delta _{ikj}^{(T)}\left[ \dfrac{-1}{(\lambda _{j}^{(Z)})^{2}} \right] \\&+ \exp (\varvec{\beta }^{(T)'}{\varvec{x}}_{ik}^{(T)} + w_{k})\exp \left( -\displaystyle \sum _{j=1}^{b}\lambda ^{(Z)}_{j}(y_{ij}-s_{j-1})\right) (-(y_{ij}-s_{j-1}))^{2}\Bigg \}, \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \varvec{\beta }^{(C)'}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -\displaystyle \sum _{l=1}^{d}\lambda _{l}^{(C)}(y_{ikl} - c_{l})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)} + \phi w_{k}){\varvec{x}}_{ik}^{(C)}{\varvec{x}}_{ik}^{(C)'}\right\} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \phi } =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -\displaystyle \sum _{l=1}^{d}\lambda _{l}^{(C)}(y_{ikl} - c_{l})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)} + \phi w_{k})x_{ik}^{(C)}w_{k} \right\} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \varvec{\beta }^{(C)} \partial \lambda _{j}^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -(y_{ikl} - c_{l}){\varvec{x}}_{jk}^{(C)}\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{jk}^{(C)} + \phi w_{k})\right\} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \phi ^{2}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -\displaystyle \sum _{l=1}^{d}\lambda _{l}^{(C)}(y_{ikl} - c_{l})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{ik}^{(C)}+\phi w_{k})w_{k}^{2}\right\} , \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \phi \partial \lambda _{j}^{(C)}} =&\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\left\{ -(y_{ikl} - c_{l})\exp (\varvec{\beta }^{(C)'}{\varvec{x}}_{jk}^{(C)} + \phi w_{k})w_{k}\right\} ,\\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial \left\{ \lambda _{l}^{(C)}\right\} ^{2}} =&- \dfrac{\displaystyle \sum _{k=1}^{m}\displaystyle \sum _{i=1}^{n_{k}}\delta _{ikl}^{(C)}}{\left\{ \lambda _{l}^{(C)}\right\} ^{2} }, \\ \dfrac{\partial ^{2} l(\varvec{\Theta }; {\mathbf {d}}, {\mathbf {w}})}{\partial (\sigma ^{2})^{2}} =&\displaystyle \sum _{k=1}^{m} \left\{ -\dfrac{w_{k}^{2}}{(\sigma ^{2})^{3}} + \dfrac{1}{2(\sigma ^{2})^{2}}\right\} . \end{aligned}$$

Table 7 Results of the MC study for survival data with dependent censoring

Table 8 Simulations study when the frailty distribution is misspecified

Table 9 Results of the MC study for survival data with number of individuals per cluster varying

The other elements of the matrix are null.

Appendix B: Simulation studies

In this Section we present the results for the adjustments with 3 and 7 intervals in the exponential piecewise model (Table 7). In addition, we evaluated the impact of the frailty distribution, Table 8, and the variation in the number of individuals per cluster (Table 9).Figure 3 shows the 500 means of the ratios $RB^{indep} /RB^{dep}$, for the scenario with positive correlation and Weibull adjustment.

Fig. 3

Distribution of the 500 means of the ratio $\text {RB}^{indep}/\text {RB}^{dep}$ calculated as in the Eq. (14)

To assess the impact of the frailty distribution misspecification, we performed a simulation study considering 500 datasets for each scenario (positive correlation ($\phi =2$), negative correlation ($\phi =-2$), and independence ($\phi =0$)). The frailty terms were generated from a log-gamma distribution wich parameters location = 0, scale = 1, and shape = 1.4 (resulting in mean$\approx$ 0 and variance $\approx$ 1) from the VGAM R package (Yee 2010). The adjustment was performed using the proposed Weibull model and the standard Weibull model (assuming independence between T and C), and the results can be found in Table 8 of the Appendix. Even if the frailty distribution is incorrectly specified, we can observe that the proposed approach performs well. In general, all regression coefficients have a relative bias less than 5%, standard deviation close to standard errors, coverage probability close to nominal value, except for the parameter $\sigma ^2$, which is expected since the frailty distribution is misspecified.