ABSTRACT
When a linear logistic model is used in the analysis of data in the form of proportions, the logistic transformation of the response probability is assumed to be linearly dependent on measured explanatory variables, and the observed numbers of successes are generally assumed to have a binomial distribution. If the fitted linear logistic model is to be satisfactory, the model must adequately fit the observed response probabilities, and, in addition, the random variation in the data must be modelled appropriately. When a linear logistic model fitted to n binomial proportions is satisfactory, the residual deviance has an approximate χ2-distribution on (n − p) degrees of freedom, where p is the number of unknown parameters in the fitted model. Since the expected value of a χ2 random variable on (n − p) degrees of freedom is (n − p), it follows that the residual deviance for a well-fitting model should be approximately equal to its number of degrees of freedom, or, equivalently, the mean deviance should be close to one. If the fitted model does not adequately describe the observed proportions, the residual mean deviance is likely to be greater than one. Similarly, if the variation in the data is greater than that under binomial sampling, the residual mean deviance is also likely to be greater than one.
