Multivariate failure time distributions derived from shared frailty and copulas

Abstract

Copulas and frailty models have been two major tools for modeling dependence in multivariate failure time distributions. The objective of this paper is to investigate multivariate failure time models that include copula models and frailty models as special cases. To this end, we revisit a broad family of multivariate failure time models proposed by Marshall and Olkin (JASA 83:834–841, 1988). This family accommodates both frailty and copulas, unlike the models that accommodate only one of them. However, their work focused on very specific copulas and is limited to bivariate models. Instead, we focus more on popular members of copulas and some multivariate models. Another novel feature of our paper is to restrict our attention to the shared frailty model, and call our restricted class as the generic name “frailty-copula”. This name yields a taxonomic classification of all the members of distributions. We also consider somewhat complex frailty distributions (two-parameter gamma, lognormal, truncated-normal, and folded-normal), which were not considered in Marshall and Olkin (1988) and other papers of frailty models. To illustrate the usefulness of the proposed model, we briefly discuss maximum likelihood estimation methods with some numerical evaluations.

Appendices

Appendix A: The plots for the lognormal-Clayton and the gamma-Gumbel models

Figure 4 plots ${S}_{{T}_{1}{T}_{2}}({t}_{1},{t}_{2})$ under the lognormal-Clayton model with different values for $\theta$ and ${\sigma }^{2}$. The degree of dependence for the model is increased when $\theta$ is increased or ${\sigma }^{2}$ is increased. Hence, both $\theta$ and ${\sigma }^{2}$ contribute to the degree of dependence.

Fig. 4

The contour plot of the joint survival function for the lognormal-Clayton model

Figure 5 plots ${S}_{{T}_{1}{T}_{2}}({t}_{1},{t}_{2})$ under the gamma-Gumbel model with different values for $\theta$ and $\eta$. The degree of dependence is increased when $\theta$ is increased or $\eta$ is increased. Hence, both $\theta$ and $\eta$ contribute to the degree of dependence.

Fig. 5

The contour plot of the joint survival function for the gamma-Gumbel model

Appendix B: Mathematical derivations

2.1 Appendix B1: Derivations for the truncated-lognormal-Gumbel model:

By integrating out the frailty term $Z\sim TN(\mu ,{\sigma }^{2})$, the truncated-lognormal-Gumbel model is

$$ \begin{gathered} S_{{T_{1} ,\;...,\;T_{k} }} (\;t_{1} ,\;...,\;t_{k} \;) = \int\limits_{ - \infty }^{\infty } {S_{{T_{1} ,\;...,\;T_{k} |Z}} (\;t_{1} ,\;...,\;t_{k} |Z\;)f_{Z} (\;z\;)} {\text{d}}z \\ = \int\limits_{\mu }^{\infty } {\exp (\; - zA_{\theta } \;)\frac{1}{\sigma }\sqrt {\frac{2}{\pi }} \exp \left\{ {\; - \frac{{(\;z - \mu \;)^{2} }}{{2\sigma^{2} }}\;} \right\}} {\text{d}}z \\ = \int\limits_{\mu }^{\infty } {\frac{1}{\sigma }\sqrt {\frac{2}{\pi }} \exp \left[ {\; - \frac{1}{{2\sigma^{2} }}\{ \;2\sigma^{2} A_{\theta } z + (\;z - \mu \;)^{2} \;\} \;} \right]} {\text{d}}z \\ = 2\exp \left\{ {\;\frac{{(\;\mu - \sigma^{2} A_{\theta } \;)^{2} - \mu^{2} }}{{2\sigma^{2} }}\;} \right\}\int\limits_{\mu }^{\infty } {\frac{1}{{\sigma \sqrt {{2}\pi } }}\exp \left[ {\; - \frac{{\{ \;z - (\;\mu - \sigma^{2} A_{\theta } \;)\;\}^{2} }}{{2\sigma^{2} }}\;} \right]} {\text{d}}z, \\ \end{gathered} $$

Applying a change of variable by $w=\{z-(\mu -{\sigma }^{2}{A}_{\theta })\}/\sigma$, we obtain the desired result

$${S}_{{T}_{1}...{T}_{k}}\left({t}_{1},...,{t}_{k}\right)=2{\mathrm{exp}}\left(\frac{{\sigma }^{2}}{2}{A}_{\theta }^{2}-\mu {A}_{\theta }\right)\Phi \left(-\sigma {A}_{\theta }\right).$$

2.2 Appendix B2: Derivations for the folded-normal-Gumbel model:

Under $Z\sim FN(\mu ,{\sigma }^{2})$, what we need to compute is

$${S}_{{T}_{1}...{T}_{k}}\left({t}_{1},...,{t}_{k}\right)=\underset{0}{\overset{\infty }{\int }}{\mathrm{exp}}\left(-z{A}_{\theta }\right)\frac{1}{\sigma \sqrt{2\pi }}\left[{\mathrm{exp}}\left\{-\frac{{\left(z+\mu \right)}^{2}}{2{\sigma }^{2}}\right\}+{\mathrm{exp}}\left\{-\frac{{\left(z-\mu \right)}^{2}}{2{\sigma }^{2}}\right\}\right]{\text{d}}z.$$

By a similar calculation as Appendix B.1, one has

$$\underset{0}{\overset{\infty }{\int }}{\mathrm{exp}}\left(-z{A}_{\theta }\right)\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{exp}}\left\{-\frac{{\left(z+\mu \right)}^{2}}{2{\sigma }^{2}}\right\}{\text{d}}z={\mathrm{exp}}\left(\frac{{\sigma }^{2}}{2}{A}_{\theta }^{2}+\mu {A}_{\theta }\right)\Phi \left(-\frac{\mu }{\sigma }-\sigma {A}_{\theta }\right).$$

Replacing $\mu$ by $-\mu$, one also obtains another integral to get the desired result

$${S}_{{T}_{1}...{T}_{k}}\left({t}_{1},...,{t}_{k}\right)={\mathrm{exp}}\left(\frac{{\sigma }^{2}}{2}{A}_{\theta }^{2}+\mu {A}_{\theta }\right)\Phi \left(-\frac{\mu }{\sigma }-\sigma {A}_{\theta }\right)+{\mathrm{exp}}\left(\frac{{\sigma }^{2}}{2}{A}_{\theta }^{2}-\mu {A}_{\theta }\right)\Phi \left(\frac{\mu }{\sigma }-\sigma {A}_{\theta }\right).$$

2.3 Appendix B3: Derivations of the marginal distributios for the gamma-FGM model:

To derive the marginal distributions from the gamma-FGM model, we consider the Pascal triangle ${a}_{1,1}={1},{a}_{2,1}=1,{a}_{2,2}={1}, {a}_{j,1}=1,{a}_{j,j}={1}$ for$j=2,...,l$, and ${a}_{l,j}={a}_{l-1,j-1}+{a}_{l-1,j+1}$ for$j=2,...,l-1, l=3,4,...$. (Fig. 6).

Fig. 6

The representation of the coefficient $a_{i,\;j}$ based on the Pascal triangle

Then, setting ${t}_{i}=0$ for $\forall i\ne j$, the marginal survival function is

$$\begin{array}{c}{S}_{{T}_{j}}({t}_{j})={1+\beta {\Lambda }_{0j}({t}_{j})}^{-\alpha }+\theta \sum_{l=1}^{k}{(-1)}^{l+1}{a}_{k,l}{[1+\beta {\Lambda }_{0j}({t}_{j})]}^{-\alpha }\\ +\theta \sum_{l=1}^{k}{\left(-1\right)}^{l}{a}_{k,l}{{1+2\beta {\Lambda }_{0j}({t}_{j})}}^{-\alpha }, j=1,...,k.\end{array}$$

Since $\sum_{l=1}^{k}{(-1)}^{l+1}{a}_{k,l}=\sum_{l=1}^{k}{(-1)}^{l}{a}_{k,l}=0,\forall k=2,3,...$, the marginal survival functions can be written as ${S}_{{T}_{j}}({t}_{j})={(1+\beta {\Lambda }_{0j}({t}_{j}))}^{-\alpha }$. Note that the marginal can also be derived by integrating ${S}_{{T}_{k}|Z}({t}_{k}|Z)$ by the frailty distribution.