Fitting mixtures of von Mises distributions: a case study involving sudden infant death syndrome

Abstract

Sudden infant death syndrome (SIDS) exhibits a seasonal pattern with a winter peak. This pattern is not symmetric. It rises rapidly to a winter peak before falling more slowly to a dip in the summer. It has been suggested that the relatively flat peak may be due to the presence of more than one population, where each population corresponds to a different cause of SIDS. Various models based on the von Mises distribution are fitted to monthly data for England, Wales, Scotland and Northern Ireland for the years 1983–1998, including a single von Mises distribution, a mixture of a von Mises and a uniform distribution and a mixture of two von Mises distributions. There are a number of ways of fitting such models ( Fisher, 1993; Spurr and Koutbeiy, 1991). Various computational problems arise with the fitting procedures. Attempts to tackle these problems for the SIDS data are discussed. A bootstrap likelihood ratio approach (Polymenis and Titterington, 1998) is used to assess how many components are required in the model. Its properties are investigated by simulation. The improvement in fit of two components compared to one is not significant in most years, and hence there is little evidence of two populations in the seasonality of SIDS. In most years, it was also impossible to fit a mixture of von Mises and uniform components, with a single von Mises distribution being sufficient.