# Deviance (statistics)

﻿
Deviance (statistics)

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

The deviance for a model M0 is defined as

$D(y) = -2 [\log \lbrace p(y|\hat \theta_0)\rbrace -\log \lbrace p(y|\hat \theta_s)\rbrace ].\,$

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 is fit 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 (see McCullagh and Nelder).

Some usage of the term "deviance" can be confusing. According to Collett (2003)[citation needed], "the quantity $-2 \log \lbrace p(y|\hat \theta_0)\rbrace$ is sometimes referred to as a deviance. This is [...] inappropriate, since unlike the deviance used in the context of generalized linear modelling, $-2 \log \lbrace p(y|\hat \theta_0)\rbrace$ does not measure deviation from a model that is a perfect fit to the data."

## References

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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Deviance — can refer to: Deviance (sociology) Deviance (statistics) (statistical deviance) This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the …   Wikipedia

• Deviance information criterion — The deviance information criterion (DIC) is a hierarchical modeling generalization of the AIC (Akaike information criterion) and BIC (Bayesian information criterion, also known as the Schwarz criterion). It is particularly useful in Bayesian… …   Wikipedia

• List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

• List of psychology topics — This page aims to list all topics related to psychology. This is so that those interested in the subject can monitor changes to the pages by clicking on Related changes in the sidebar. It is also to see the gaps in Wikipedia s coverage of the… …   Wikipedia

• List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia

• Pearson's chi-squared test — (χ2) is the best known of several chi squared tests – statistical procedures whose results are evaluated by reference to the chi squared distribution. Its properties were first investigated by Karl Pearson in 1900.[1] In contexts where it is… …   Wikipedia

• Sociology — For the journal, see Sociology (journal). Sociology …   Wikipedia

• Counterproductive work behavior — (CWB) is employee behavior that goes against the goals of an organization.[1] These behaviors can be intentional or unintentional and result from a wide range of underlying causes and motivations. It has been proposed that a person by environment …   Wikipedia

• Criminology — and penology Theories Causes and correlates of crime Anomie Differential association theory …   Wikipedia

• Gender and crime — For crimes involving sexuality, see Sex and law or Gender crime. Attempts in various fields have tried to explore a possible relation between gender and crime. Such studies may belong to criminology, sociobiology (which attempts to demonstrate a… …   Wikipedia