2.3 Alternative formulation

A shared frailty model assuming a log-normal distribution for the frailty term has strong ties to random-effects models. A log-normal frailty model is formulated as \[ h_{ij}(t | \alpha_i) = \alpha_i h(t | X_{ij}) = \alpha_i h_0(t) \exp(X_{ij} \beta), \] with \(\alpha_i\) following a log-normal distribution. On the log-hazard scale: \[ h_{ij}(t | \alpha_i) = h_0(t) \exp(X_{ij} \beta + \eta_i), \] with \(\eta_i = \log \alpha_i\). \(\eta_i\) results being normally distributed with parameters \(\mu\) and \(\sigma ^ 2\) related to those of the log-normal distribution by the relationship \[ E(\alpha_i) = \exp(\mu + \sigma ^ 2 / 2) \] and \[ Var(\alpha_i) = \exp(2 \mu + \sigma ^ 2) (\exp(\sigma ^ 2) - 1) \]

By formulating the model on the log-hazard scale, the frailty term has a direct interpretation as a random intercept in the model. It is possible to further extend this model by allowing random covariates effects, potentially ranging over multiple levels of clustering. Using the usual mixed-effects model notation: \[ h_{ij}(t | b_i) = h_0(t) \exp(X_{ij} \beta + Z_{i} b_i), \] with \(X_{ij}\) representing the design matrix for the fixed effects \(\beta\) and \(Z_i\) representing the design matrix for the random effects \(b_i\). Any distribution or functional form can be assumed for \(h_0(t)\) (Crowther, Look, and Riley 2014), or it is possible to leave it unspecified altogether yielding a semi-parametric Cox model with random effects (Ripatti and Palmgren 2000; Therneau, Grambsch, and Pankratz 2003).

References

Crowther, Michael J, Maxime P Look, and Richard D Riley. 2014. “Multilevel Mixed Effects Parametric Survival Models Using Adaptive Gauss-Hermite Quadrature with Application to Recurrent Events and Individual Participant Data Meta-Analysis.” Statistics in Medicine 33 (22): 3844–58. doi:10.1002/sim.6191.

Ripatti, Samuli, and Juni Palmgren. 2000. “Estimation of Multivariate Frailty Models Using Penalized Partial Likelihood.” Biometrics 56 (4): 1016–22. doi:10.1111/j.0006-341x.2000.01016.x.

Therneau, Terry M, Patricia M Grambsch, and V SHare Pankratz. 2003. “Penalized Survival Models and Frailty.” Journal of Computational and Graphical Statistics 12 (1): 156–75. doi:10.1198/1061860031365.