Frailty models for survival data

A frailty model is a random effects model for time variables, where the random effect (the frailty) has a multiplicative effect on the hazard. It can be used for univariate (independent) failure times, i.e. to describe the influence of unobserved covariates in a proportional hazards model. More interesting, however, is to consider multivariate (dependent) failure times generated as conditionally independent times given the frailty. This approach can be used both for survival times for individuals, like twins or family members, and for repeated events for the same individual. The standard assumption is to use a gamma distribution for the frailty, but this is a restriction that implies that the dependence is most important for late events. More generally, the distribution can be stable, inverse Gaussian, or follow a power variance function exponential family. Theoretically, large differences are seen between the choices. In practice, using the largest model makes it possible to allow for more general dependence structures, without making the formulas too complicated.This paper is a revised version of a review, which together with ten papers by the author made up a thesis for a Doctor of Science degree at the University of Copenhagen.     

Rationing in the presence of baselines

We analyze a general model of rationing in which agents have baselines, in addition to claims against the (insufficient) endowment of the good to be allocated. Many real-life problems fit this general model (e.g., bankruptcy with prioritized claims, resource allocation in the public health care sector, water distribution in drought periods). We introduce (and characterize) a natural class of allocation methods for this model. Any method within the class is associated with a rule in the standard rationing model, and we show that if the latter obeys some focal properties, the former obeys them too.     

Fisher Information for Two Gamma Frailty Bivariate Weibull Models

The asymptotic properties of frailty models for multivariate survival data are not well understood. To study this aspect, the Fisher information is derived in the standard bivariate gamma frailty model, where the survival distribution is of Weibull form conditional on the frailty. For comparison, the Fisher information is also derived in the bivariate gamma frailty model, where the marginal distribution is of Weibull form.     

Rationalising inefficiency: Staff utilisation in branches of a large Canadian bank

In this paper we consider staffing decisions in branches of a large Canadian bank. The bank has well-developed staffing models and the branches work in a highly competitive environment. One would therefore expect limited ‘inefficiency’ in the sense of wasted resources and over-staffing. Using Data Envelopment Analysis (DEA) we nevertheless find considerable ‘inefficiency’ which raises the question whether this is best interpreted as waste or if the apparent inefficiency may serve other purposes. To investigate this, we invoke the theoretical framework of rational inefficiency (Bogetoft and Hougaard [8]).     

Multi-state Models: A Review

Multi-state models are models for a process, for example describing a life history of an individual, which at any time occupies one of a few possible states. This can describe several possible events for a single individual, or the dependence between several individuals. The events are the transitions between the states. This class of models allows for an extremely flexible approach that can model almost any kind of longitudinal failure time data. This is particularly relevant for modeling different events, which have an event-related dependence, like occurrence of disease changing the risk of death. It can also model paired data. It is useful for recurrent events, but has limitations. The Markov models stand out as much simpler than other models from a probability point of view, and this simplifies the likelihood evaluation. However, in many cases, the Markov models do not fit satisfactorily, and happily, it is reasonably simple to study non-Markov models, in particular the Markov extension models. This also makes it possible to consider, whether the dependence is of short-term or long-term nature. Applications include the effect of heart transplantation on the mortality and the mortality among Danish twins.