Special issue dedicated to Odd O. Aalen

Lifetime Data Analysis -     

Survival Models Based on the Ornstein-Uhlenbeck Process

When modelling survival data it may be of interest to imagine an underlying process leading up to the event in question. The Ornstein-Uhlenbeck process is a natural model to consider in a biological context because it stabilizes around some equilibrium point. This corresponds to the homeostasis often observed in biology, and also to some extent in the social sciences. First, we study the first-passage time distribution of an Ornstein-Uhlenbeck process, focussing especially on what is termed quasi-stationarity and the various shapes of the hazard rate. Next, we consider a model where the individual hazard rate is a squared function of an Ornstein-Uhlenbeck process. We extend known results on this model. The results on quasi-stationarity are relevant for recent discussions about mortality plateaus. In addition, we point out a connection to models for short-term interest rates in financial modeling.     

A hierarchical frailty model applied to two-generation melanoma data

We present a hierarchical frailty model based on distributions derived from non-negative Lévy processes. The model may be applied to data with several levels of dependence, such as family data or other general clusters, and is an alternative to additive frailty models. We present several parametric examples of the model, and properties such as expected values, variance and covariance. The model is applied to a case-cohort sample of age at onset for melanoma from the Swedish Multi-Generation Register, organized in nuclear families of parents and one or two children. We compare the genetic component of the total frailty variance to the common environmental term, and estimate the effect of birth cohort and gender.     

Recurrent events and the exploding Cox model

Counting process models have played an important role in survival and event history analysis for more than 30 years. Nevertheless, almost all models that are being used have a very simple structure. Analyzing recurrent events invites the application of more complex models with dynamic covariates. We discuss how to define valid models in such a setting. One has to check carefully that a suggested model is well defined as a stochastic process. We give conditions for this to hold. Some detailed discussion is presented in relation to a Cox type model, where the exponential structure combined with feedback lead to an exploding model. In general, counting process models with dynamic covariates can be formulated to avoid explosions. In particular, models with a linear feedback structure do not explode, making them useful tools in general modeling of recurrent events.