Statistical inference for shared frailty models

Abstract

This chapter considers the statistical inference for the shared frailty models described in the previous chapter. A main part of this is estimation procedures. Estimation difficulties have previously limited the applicability of the shared frailty models. There have, however, been a number of suggestions on how to estimate the parameters. Reasons for the many choices are, of course, that some formulas are complicated and that iteration can be time consuming. One basic direction to take is to integrate out the random frailties, but this is not the only possibility. Alternatively, one can use estimation routines where the frailties are included as unobserved random variables, similar to BLUP (best linear unbiased predictor) methods for normal distribution models. For non-parametric hazard functions, there is one parameter per time point with observed events. This can be specifically included in the model, or one can attempt to remove it from the likelihood, an approach that is inspired by the successful way of doing so in the Cox model. Also for the Nelson-Aalen estimate, it is easy to handle the hazard contributions, because there is a separate equation for each term allowing for an explicit solution. It is, unfortunately, not quite as easy in a frailty model; iteration is necessary as all expressions are non-linear and related to each other.