  In this work we deal with correlated failure time (age at onset) data arising
from population-based case-control studies, where case and control probands are
selected by population-based sampling and an array of risk factor measures is
collected for both cases and controls and their relatives. Parameters of
interest are effects of risk factors on the hazard function of failure times
and within-family dependencies of failure times after adjusting for the risk
factors. Due to the retrospective nature of sampling, a large sample theory for
existing methods has not been established. We develop a novel estimation
techniques for estimating these parameters under a general semiparametric
shared frailty model. We also present a simple, easily computed, and
non-iterative nonparametric estimator for the cumulative baseline hazard
function. A rigorous large sample theory for the proposed estimators of these
parameters is given along with simulations and a real data example illustrate
the utility of the proposed method.
