The deviance for a model M 0, based on a data set y, is defined as follows:
In the preceding equation:
denotes the fitted values of the parameters in the model M 0.
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. This expression is -2 times the log-likelihood ratio of the reduced model compared to the full model.
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, M 1 and M 2. In particular, suppose that M 1 contains the parameters in M 2, and k additional parameters. Then, under the null hypothesis that M 2 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.