A frailty model for interaction between multiple events

Most models for categorical data rely on linear models in which higher order interactions are limited, usually to second order terms. Here we explore a dataset where second order interactions lead unavoidably to high order interactions. However in certain cases these interactions have a simple form which can be explained by population heterogeneity or ‘frailty’ and the Rasch model which uses a logistic structure is one such model. However its parameters are difficult to interpret. When event probabilities are low, we develop a simple model based on a multiplicative structure and a one-parameter gamma frailty model is developed to accommodate this sort of data. Moment estimators estimates are also provided for more general cases and goodness of fit statistics are given. An example involving 35 different human papillomavirus types in 33 614 women with normal cervical cytology smears (stratified by age) is explored.