Multivariate survival analysis using piecewise gamma frailty.

In this note we propose a frailty model called piecewise gamma frailty for correlated survival data with random effects having a nested structure. In frailty models, a dependence function defined as a hazard ratio of one member given the failure time of another member in a unit is determined by the distributional assumptions on frailty. In the piecewise gamma frailty model, the nested structure of random effects or frailty allows the dependence function to vary over the time periods. This model includes existing models such as the piecewise exponential model (Breslow, 1974, Biometrics 30, 89-100) and the gamma frailty model (Clayton, 1978, Biometrika 65, 141-151; Oakes, 1982, Journal of the Royal Statistical Society, Series B 44, 414-428) as special cases. A study of familial aggregation of epilepsy is used to illustrate the proposed method.