8.2 Frailty models

Frailty models in survival analysis corresponds to hierarchical models in linear or generalized linear models. They are also called mixed effects models. A general theory, with emphasis on using R, of mixed effects models can be found in Pinheiro and Bates (2000).

8.2.1 The simple frailty model

Vaupel, Manton, and Stallard (1979) described an individual frailty model, \[\begin{equation*} h(t; \mathbf{x}, Z) = h_0(t) Z e^{\boldsymbol{\beta} \mathbf{x}}, \quad t > 0, \end{equation*}\] where \(Z\) is assumed to be drawn independently for each individual. Hazard rates for ``random survivors’’ are not proportional, but converging (to each other) if the frailty distribution has finite variance. Thus, the problem may be less pronounced in AFT than in PH regression. However, as indicated in the introductory example, with individual frailty the identification problems are large, and such models are best avoided.

References

Pinheiro, J., and D. Bates. 2000. Mixed-Effects Models in S and S-Plus. New York: Springer-Verlag.

Vaupel, J. G., K. G. Manton, and E. Stallard. 1979. “The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality.” Demography 16: 439–54.