2.2 Shared frailty models

Further generalising the model presented in Section 2.1, it is possible for the frailty effect \(\alpha\) to be shared between clusters of study subjects. Specifically, for the \(j\)-th observation in the \(i\)-th cluster: \[ h_{ij}(t | \alpha_i) = \alpha_i h(t|X_{ij}). \] The conditional survival function is: \[ S_{ij}(t | \alpha_i) = S_{ij}(t) ^ {\alpha_i}. \]

In this setting, the cluster-specific contribution to the likelihood is obtained by calculating the cluster-specific likelihood conditional on the frailty, consequently integrating out the frailty itself: \[ L_i = \int_A L_i(\alpha_i) f(\alpha_i) \, d\alpha, \] with \(f(\alpha)\) the distribution of the frailty, \(A\) its domain, and \(L_i(\alpha_i)\) the cluster-specific contribution to the likelihood, conditional on the frailty. The cluster-specific contribution to the likelihood is \[ L_i(\alpha_i) = \alpha_i ^ {D_i} \prod_{j = 1} ^ {n_i} S_{ij}(t_{ij}) ^ {\alpha_i} h_{ij}(t_{ij}) ^ {d_{ij}}, \] with \(D_i = \sum_{j = 1} ^ {n_i} d_{ij}\). Analogously as before, analytical formulae can be obtained when \(\alpha_i\) follows a Gamma distribution: \[ L_i = \left[ \prod_{j = 1} ^ {n_i} h_{ij}(t_{ij}) ^ {d_{ij}} \right] \frac{\Gamma (1 / \theta + D_i)}{\Gamma (1 / \theta)} \left[ 1 - \theta \sum_{j = 1} ^ {n_i} \log S_{ij}(t_{ij}) \right] ^ {-1 / \theta - D_i}; \] further details in Gutierrez (2002). As in the univariate setting, assuming a log-normal distribution requires some numerical approximation to be performed, being the resulting model analytically intractable.

References

Gutierrez, Roberto G. 2002. “Parametric Frailty and Shared Frailty Survival Models.” The Stata Journal 2 (1): 22–44.