Deviance (statistics)

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In statistics, deviance is a quantity whose expected values can be used for statistical hypothesis testing.

It is defined as

$D(y,\theta) = -2 \log[p(y|\theta)].\,$

As a function of θ with y treated as fixed, it is −2 times the log-likelihood. The deviance of a model is usually not interpreted directly, but rather 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.

Suppose in the framework of the GLM, we have two nested models, M₁ and M₂. Under the null hypothesis that M₁ is the true model, then in the limit of a large sample, the difference between the deviances for the two models follows an approximate chi-squared distribution with degrees of freedom equal to the change in the number of estimated parameters. We therefore reject more complex models when they fail to give improvements in deviance much greater than the change in numbers of parameters.