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{{distinguish|Deviation (statistics)}}
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In [[statistics]], '''deviance''' is a quality of fit statistic for a model that is often used for [[statistical hypothesis testing]]. It is a generalization of the idea of using the sum of squared residuals in [[ordinary least squares]] to cases where model-fitting is achieved by [[maximum likelihood]].
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==Definition==
The deviance for a model ''M''<sub>0</sub>, based on a dataset ''y'', is defined as:<ref>{{cite journal | last= Nelder | first = J.A. |authorlink = John Nelder | first2 = R.W.M. |last2 = Wedderburn |authorlink2 = Robert Wedderburn (statistician) | title = Generalized Linear Models | year=1972 | journal = [[Journal of the Royal Statistical Society]]. Series A (General) | volume= 135 |issue=3 | pages=370–384 | doi= 10.2307/2344614 | jstor= 2344614 }}</ref><ref name=McN>McCullagh and Nelder (1989){{Page needed|date=November 2011}}</ref>
 
:<math>D(y) = -2 \Big( \log \big( p(y|\hat \theta_0)\big)-\log \big(p(y|\hat \theta_s)\big)\Big).\,</math>
 
Here <math>\hat \theta_0</math> denotes the fitted values of the parameters in the model ''M''<sub>0</sub>, while <math>\hat \theta_s</math> 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  &minus;2 times the [[likelihood function|log-likelihood ratio]] of the reduced model compared to the full model. The deviance is used to compare two models &ndash; in particular in the case of [[generalized linear model]]s where it has a similar role to residual variance from [[ANOVA]] in linear models ([[Residual sum of squares|RSS]]).
 
Suppose in the framework of the GLM, we have two nested models, ''M<sub>1</sub>'' and ''M<sub>2</sub>''. In particular, suppose that ''M<sub>1</sub>'' contains the parameters in ''M<sub>2</sub>'', and ''k'' additional parameters. Then, under the null hypothesis that ''M<sub>2</sub>'' is the true model, the difference between the deviances for the two models follows an approximate [[chi-squared distribution]] with ''k''-degrees of freedom.<ref name=McN/>
 
Some usage of the term "deviance" can be confusing. According to Collett:<ref>Collett (2003){{Page needed|date=November 2011}}</ref>
: "the quantity <math>-2 \log \big( p(y|\hat \theta_0)\big) </math> is sometimes referred to as a ''deviance''. This is [...] inappropriate, since unlike the deviance used in the context of generalized linear modelling, <math>-2 \log \big( p(y|\hat \theta_0)\big)</math> does not measure deviation from a model that is a perfect fit to the data." However, since the principal use is in the form of the difference of the deviances of two models, this confusion in definition is unimportant.
 
==See also==
 
*[[Pearson's chi-squared test]], an alternative quality of fit statistic for [[generalized linear models]] for count data.
*[[Hosmer-Lemeshow test]], a quality of fit statistic that can be used for binary data.
*[[Akaike information criterion]]
*[[Deviance information criterion]]
*[[Peirce's criterion]]
*[[Discrepancy function]]
 
==Notes==
 
{{reflist}}
 
==References==
*{{cite book|last1=McCullagh|first1=Peter|last2=Nelder|first2=John|authorlink1 = Peter McCullagh|authorlink2 = John Nelder|title = Generalized Linear Models, Second Edition|publisher = Chapman & Hall/CRC|year = 1989|isbn=0-412-31760-5
}}
 
*{{cite book|last1=Collett|first1=David|authorlink1 =|title = Modelling Survival Data in Medical Research, Second Edition|publisher = Chapman & Hall/CRC|year = 2003|isbn=1-58488-325-1
}}
 
==External links==
*[http://userwww.sfsu.edu/~efc/classes/biol710/Glz/Generalized%20Linear%20Models.htm Generalized Linear Models] - Edward F. Connor
*[http://www.unc.edu/courses/2006spring/ecol/145/001/docs/lectures/lecture22.htm Lectures notes on Deviance]
 
{{Statistics|correlation}}
{{Least squares and regression analysis}}
 
[[Category:Hypothesis testing]]
[[Category:Statistical deviation and dispersion]]

Latest revision as of 00:26, 7 January 2015

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