Deviance (statistics)

In statistics, deviance is a quality of fit statistic for a model that is often used for statistical hypothesis testing.

Definition

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

$D(y) = -2 \Big( \log \big( p(y|\hat \theta_0)\big)-\log \big(p(y|\hat \theta_s)\big)\Big).\,$

Here $\hat \theta_0$ denotes the fitted values of the parameters in the model M₀, while $\hat \theta_s$ denotes the fitted parameters for the "full 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.

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 $-2 \log \big( p(y|\hat \theta_0)\big)$ is sometimes referred to as a deviance. This is [...] inappropriate, since unlike the deviance used in the context of generalized linear modelling, $-2 \log \big( p(y|\hat \theta_0)\big)$ does not measure deviation from a model that is a perfect fit to the data."

See also

• Pearson's chi-squared test, an alternative quality of fit statistic for generalized linear models.
• Akaike information criterion
• Deviance information criterion
• Peirce's criterion
• Discrepancy function

References

• McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Chapman & Hall/CRC. ISBN 0412317605. 

• Collett, David (2003). Modelling Survival Data in Medical Research, Second Edition. Chapman & Hall/CRC. ISBN 1-58488-325-1. 

External links

• Generalized Linear Models - Edward F. Connor

• Lectures notes on Deviance