A logistic truncated normal mixture model for overdispersed binomial data

Abstract

A common model for overdispersion in binomial data is the logistic normal, in which the logit transform of the binomial parameter has a normal distribution. Motivated by an application in fission track analysis, we generalise this model by replacing the normal distribution with a truncated normal mixture, so that the binomial parameter has a logistic truncated normal distribution with an atom of probability at the truncation point. In fission track analysis, the truncation point corresponds to the minimum geological age of a set of mineral crystals extracted from a rock sample; the minimum age provides information about the genesis of the host rock and the geological history of a region.

Fitting the model by maximum likelihood requires the repeated numerical evaluation of an intractable integral. We propose an efficient method of doing this, and briefly discuss several variants. We fit the model to both real and simulated data sets, and conclude that it needs highly overdispersed data for the parameters to be well identified. We suggest a submodel for estimating the truncation point when the data are mildly overdispersed.

Keywords:

• Extra-binomial variation
• fission track analysis
• Half-Hermite quadrature
• logistic normal
• maximum likelihood estimation
• mixture model