An aspect of discrete data analysis: fitting a beta-binomial distribution to the hospitals' data

Statistical analysis for discrete data, particularly for probability models such as the binomial, Poisson and multinomial, is by now very well understood, with a wealth of suitable software. It can happen that the standard generalized linear modelling (glm) software is not completely appropriate, since over-dispersion is present, relative to the standard distributions such as the Poisson or the binomial. Failure to take account of this over-dispersion, for example in fitting a model such as log(p/(1 - p)) = alpha + beta x (where the covariate x is the dose) will mean that our estimates of beta will be less precise than the binomial-based formula gives us. Thus for example we will be quoting confidence intervals for beta that are too narrow. One way of coping with this problem is to use a probability model which is more general than the binomial, and one such model is the beta-binomial. This paper discusses beta-binomial modelling (in S-Plus) in relation to the interesting data set given in the 1998 BMJ paper by Spiegelhalter and Marshall on success rates of 52 in vitro fertilisation clinics in the UK.