Problem 19
Two individuals with cumulative hazard functions \(u H_{1}\left(y_{1}\right)\) and \(u H_{2}\left(y_{2}\right)\) are independent conditional on the value \(u\) of a frailty \(U\) whose density is \(f(u)\) (a) For this shared frailty model, show that $$ \mathcal{F}\left(y_{1}, y_{2}\right)=\operatorname{Pr}\left(Y_{1}>y_{1}, Y_{2}>y_{2}\right)=\int_{0}^{\infty} \exp \left\\{-u H_{1}\left(y_{1}\right)-u H_{2}\left(y_{2}\right)\right\\} f(u) d u $$ If \(f(u)=\lambda^{\alpha} u^{\alpha-1} \exp (-\lambda u) / \Gamma(\alpha)\), for \(u>0\) is a gamma density, then show that $$ \mathcal{F}\left(y_{1}, y_{2}\right)=\frac{\lambda^{\alpha}}{\left\\{\lambda+H_{1}\left(y_{1}\right)+H_{2}\left(y_{2}\right)\right\\}^{\alpha}}, \quad y_{1}, y_{2}>0 $$ and deduce that in terms of the marginal survivor functions \(\mathcal{F}_{1}\left(y_{1}\right)\) and \(\mathcal{F}_{2}\left(y_{2}\right)\) of \(Y_{1}\) and \(Y_{2}\) $$ \mathcal{F}\left(y_{1}, y_{2}\right)=\left\\{\mathcal{F}_{1}\left(y_{1}\right)^{-1 / \alpha}+\mathcal{F}_{2}\left(y_{2}\right)^{-1 / \alpha}-1\right\\}^{-\alpha}, \quad y_{1}, y_{2}>0 $$ What happens to this joint survivor function as \(\alpha \rightarrow \infty\) ? (b) Find the likelihood contributions when both individuals are observed to fail, when one is censored, and when both are censored. (c) Extend this to \(k\) individuals with parametric regression models for survival.
Problem 20
A positive stable random variable \(U\) has \(\mathrm{E}\left(e^{-s U}\right)=\exp \left(-\delta s^{\alpha} / \alpha\right), 0<\alpha \leq 1\) (a) Show that if \(Y\) follows a proportional hazards model with cumulative hazard function \(u \exp \left(x^{\mathrm{T}} \beta\right) H_{0}(y)\), conditional on \(U=u\), then \(Y\) also follows a proportional hazards model unconditionally. Are \(\beta, \alpha\), and \(\delta\) estimable from data with single individuals only? (b) Consider a shared frailty model, as in the previous question, with positive stable \(U\). Show that the joint survivor function may be written as $$ \mathcal{F}\left(y_{1}, y_{2}\right)=\exp \left(-\left[\left\\{-\log \mathcal{F}_{1}\left(y_{1}\right)\right\\}^{1 / \alpha}+\left\\{-\log \mathcal{F}_{2}\left(y_{2}\right)\right\\}^{1 / \alpha}\right]^{\alpha}\right), \quad y_{1}, y_{2}>0 $$ in terms of the marginal survivor functions \(\mathcal{F}_{1}\) and \(\mathcal{F}_{2}\). Show that if the conditional cumulative hazard functions are Weibull, \(u H_{r}(y)=u \xi_{r} y^{\gamma}, \gamma>0, r=1,2\), then the marginal survivor functions are also Weibull. Show also that the time to the first event has a Weibull distribution.
Problem 22
Let \(Y_{1}, \ldots, Y_{n}\) be independent exponential variables with hazards \(\lambda_{j}=\exp \left(\beta^{\mathrm{T}} x_{j}\right)\). (a) Show that the expected information for \(\beta\) is \(X^{\mathrm{T}} X\), in the usual notation. (b) Now suppose that \(Y_{j}\) is subject to uninformative right censoring at time \(c_{j}\), so that \(y_{j}\) is a censoring time or a failure time as the case may be. Show that the log likelihood is $$ \ell_{U}(\beta)=\sum_{f} \beta^{\mathrm{T}} x_{j}-\sum_{j=1}^{n} \exp \left(\beta^{\mathrm{T}} x_{j}\right) y_{j} $$ where \(\sum_{f}\) denotes a sum over observations seen to fail. If the \(j\) th censoring-time is exponentially distributed with rate \(\kappa_{j}\), show that the expected information for \(\beta\) is \(X^{\mathrm{T}} X-\) \(X^{\mathrm{T}} C X\), where \(C=\operatorname{diag}\left\\{c_{1}, \ldots, c_{n}\right\\}\), and \(c_{j}=\kappa_{j} /\left(\kappa_{j}+\lambda_{j}\right)\) is the probability that the \(j\) th observation is censored. What is the implication for estimation of \(\beta\) if the \(c_{j}\) are constant? (c) Sometimes a variable \(W_{j}\) has been measured which can act as a surrogate response variable for censored individuals. We formulate this as \(W_{j}=Z_{j} / U_{j}\), where \(Z_{j}\) is the unobserved remaining life-time of the \(j\) th individual from the moment of censoring, and \(U_{j}\) is a noise component which has a fixed distribution independent of the censoring time and of \(x_{j}\). Owing to the exponential assumption, the excess life \(Z_{j}\) is independent of \(Y_{j}\) if censoring occurred. If \(U_{j}\) has gamma density $$ \alpha^{K} u^{\kappa-1} \exp (-\alpha u) / \Gamma(\kappa), \quad \alpha, \kappa>0, u>0 $$ show that \(W_{j}\) has density $$ \lambda_{j} \kappa \alpha^{\kappa} /\left(\alpha+\lambda_{j} w\right)^{\kappa+1}, \quad w>0 $$ Show that the log likelihood for the data, including the additional information in the \(W_{j}\), is $$ \ell(\beta)=L_{U}(\beta)+\sum_{c}\left\\{\beta^{\mathrm{T}} x_{j}+\log \kappa+\kappa \log \alpha-(\kappa+1) \log \left(\alpha+e^{\beta^{\mathrm{T}}} x_{j} w_{j}\right)\right\\} $$ where \(\sum_{c}\) denotes a sum over censored individuals, and we have assumed that \(\alpha\) and \(\kappa\) are known. Show that the expected information for \(\beta\) is $$ X^{\mathrm{T}} X-2 /(\kappa+2) X^{\mathrm{T}} C X $$ and compare this with (b). Explain qualitatively in terms of the variability of the distribution of \(U\) why the loss of information decreases as \(\kappa\) increases. \((\operatorname{Cox}, 1983)\)
