ABSTRACT
This chapter focuses on the analysis of univariate data, for example, event times of unrelated individuals. Basic survival models deal with the simplest case of independent and identically distributed data. This is based on the assumption that the study population is homogeneous up to some observed covariates. Such kind of models were considered in the last chapter. However, it is a basic observation that individuals differ greatly, for example, with respect to the effects of a drug, a treatment, or the influence of various explanatory variables. This heterogeneity is often referred to as variability, and it is one of the important sources of variability in medical, epidemiological and biological applications. The issue of this chapter is unobserved heterogeneity in survival analysis. This heterogeneity may be difficult to assess, but it is nevertheless of great importance. In recent decades, a large amount of papers on frailty models have appeared. The key idea of these models is that individuals have different frailties, and that the most frail will die earlier than the less frail. Consequently, systematic selection of robust individuals takes place, which biases what is observed. When mortality rates are estimated, one may be interested in how they change over time or age. Quite often, they rise at the beginning of the observation period, reach a maximum, and then decline (unimodal hazard) or level off. This, for example, is typical for death rates of cancer patients, meaning that the longer the patient lives beyond diagnosis and treatment, the better her or his chances of survival are. But it is often an open question whether this reflects changes in the individual hazard. It is likely that unimodal hazards are often the result of selection and that they do not reflect an underlying development on the individual level. The population hazard starts to decline simply because high-risk individuals have already died, but the hazard of a given individual might well continue to increase.
