The R Book

Overdispersion and Hypothesis Testing

All the different statistical procedures that we have met in earlier chapters can also be used with data on proportions. Factorial analysis of variance, multiple regression, and a variety of models in which different regression lines are fitted in each of several levels of one or more factors, can be carried out. The only difference is that we assess the significance of terms on the basis of chi-squared – the increase in scaled deviance that results from removal of the term from the current model.

The important point to bear in mind is that hypothesis testing with binomial errors is less clear-cut than with normal errors. While the chi-squared approximation for changes in scaled deviance is reasonable for large samples (i.e. larger than about 30), it is poorer with small samples. Most worrisome is the fact that the degree to which the approximation is satisfactory is itself unknown. This means that considerable care must be exercised in the interpretation of tests of hypotheses on parameters, especially when the parameters are marginally significant or when they explain a very small fraction of the total deviance. With binomial or Poisson errors we cannot hope to provide exact p-values for our tests of hypotheses.

As with Poisson errors, we need to address the question of overdispersion (see p. 522). When we have obtained the minimal adequate model, the residual scaled deviance should be roughly equal to the residual degrees offreedom. When ...