Frailty models are typically used when there are unobserved covariates. Here we explore their use in two situations where they provide important insights into two epidemiologic questions.
The first involves the question of type replacement after vaccination against the human papilloma virus (HPV). At least 13 types of HPV are known to cause cancer, especially cervix cancer. Recently vaccines have been developed against some of the more important types, notably types 16 and 18. These vaccines have been shown to prevent infection by the types used with almost 100% efficacy. However a concern has been raised that by eliminating these more common types, a niche will be created in which other types could now flourish and that the benefits of vaccination could be less than anticipated if this were to occur. It will be years before definitive data is available on this, but preliminary evidence could be obtained if it could be shown that there is a negative associated between the occurrence of multiple infections in the same individual. The virus is transmitted by sexual contact and testing for it has become part of cervical screening. As infection increases with greater sexual activity, a woman with one type is more likely to also harbour another type so the question can be phrase as to whether there is a negative association between specific pairs of types in the context of an overall positive association. A frailty model is used for this in which the total number of infections a women has is an unobserved covariate and the question can be rephrased to ask if specific types are negatively correlated conditional on the number of types present in a woman. This is modelled by assuming a multiplicative random variable τ having a log gamma distribution with unit mean and one additional parameter θ so that the occurrence of type j in individual i is modified to be τi pj where τi are iid copies of τ and the joint probability of being infected by types 1,..,k is Ѳkp1…pk with Ѳk = E(τk). A likelihood is obtained and moment based estimation procedures are developed and applied to a large data set.
A second example pertains to an extension of the widely used proportional hazards model for analysing time to event data with censoring. In practice hazards are often not proportional over time and converging hazards are observed, and the effect of a covariate is stronger in early follow up than it is subsequently. This can be modelled by assuming an unobserved multiplicative factor in the hazard function again having a log gamma distribution with unit mean and one additional parameter θ. Integrating out this term leads to a Pareto survival distribution. A (partial) likelihood is obtained and estimation procedures are developed and applied to a large data set.