Deviance (statistics) - Definition

Definition

The deviance for a model M₀, based on a dataset y, is defined as

Here denotes the fitted values of the parameters in the model M₀, while denotes the fitted parameters for the "full model" (or "saturated model"): both sets of fitted values are implicitly functions of the observations y. Here the full model is a model with a parameter for every observation so that the data are fitted exactly. This expression is simply −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, we have 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.

Some usage of the term "deviance" can be confusing. According to Collett:

"the quantity is sometimes referred to as a deviance. This is inappropriate, since unlike the deviance used in the context of generalized linear modelling, does not measure deviation from a model that is a perfect fit to the data."