Models and Distributions for Frequency Data
In this chapter, the authors consider various models from very simple models implying that alternative outcomes are equally likely, through more structured models to formal distributions of counts, the binomial and Poisson distributions. In each case, testing whether the data are compatible with the model provides knowledge about the assumptions underpinning the model. The X2 goodness of fit statistic can be used to test the deviations of observed frequencies from theoretical frequencies for any model proposed for frequency data. The authors look at particular forms of generalized linear models for data from an experimental design structure, complex contingency tables, and regression relationships when the observed response is a proportion based on counts. As will be plain from the discussion of the hawk data, the models are based on a log link function, corresponding to proportional changes in frequencies. The authors investigate possible hypotheses about the dependencies that they have found in the hawk data.