Unobserved individual heterogeneity, also called frailty, is a major concern in the application of survival analysis. Hazard rates do not give direct information on the change over time in the individual risk, but are strongly influenced by selection effects operating in the population. The individuals surviving up to a certain time will on average be less frail than the original population. Models are reviewed that account for this phenomenon, and some medical examples are discussed. It is emphasized that the frailty phenomenon may be modelled in many different ways, and a stochastic process approach is discussed as an alternative to the common proportional frailty model.

Directed acyclic graphs (DAGs) play a large role in the modern approach to causal inference. DAGs describe the relationship between measurements taken at various discrete times including the effect of interventions. The causal mechanisms, on the other hand, would naturally be assumed to be a continuous process operating over time in a cause–effect fashion. How does such immediate causation, that is causation occurring over very short time intervals, relate to DAGs constructed from discrete observations? We introduce a time-continuous model and simulate discrete observations in order to judge the relationship between the DAG and the immediate causal model. We find that there is no clear relationship; indeed the Bayesian network described by the DAG may not relate to the causal model. Typically, discrete observations of a process will obscure the conditional dependencies that are represented in the underlying mechanistic model of the process. It is therefore doubtful whether DAGs are always suited to describe causal relationships unless time is explicitly considered in the model. We relate the issues to mechanistic modeling by using the concept of local (in)dependence. An example using data from the Swiss HIV Cohort Study is presented.