The deviance for a model M0, based on a data set y, is defined as follows:
In the preceding equation:
denotes the fitted values of the parameters in the model M0.
denotes the fitted parameters for the full model (or saturated model).
Both sets of fitted values are implicitly functions of the observations y. In this case, the full model is a model with a parameter for every observation so that the data are fitted exactly.
The deviance is used to compare two models—in particular in the case of generalized linear models where it has a similar role to residual variance from ANOVA in linear models (RSS).
Suppose in the framework of the GLM that there are two nested models, M1 and M2. In particular, suppose that M1 contains the parameters in M2, and k additional parameters. Then, under the null hypothesis that M2 is the true model, the difference between the deviances for the two models follows an approximate chi-squared distribution with k-degrees of freedom. This provides us an alternative way for computing the log-likelihood ratio of two models.
Deviance is implemented in the GLM function. It computes residual deviance from model deviance and saturated deviance. The function does not compute null deviance.