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			{{distinguish\|Deviation (statistics)}}

			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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			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 −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 – 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]]

en>LHcheM: Undid revision 508783890 by 120.56.179.196 (talk), possible vandalism

2012-08-23T16:55:44Z

Undid revision 508783890 by 120.56.179.196 (talk), possible vandalism

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en>Stesmo: Reverted 1 edit by 105.112.2.235 identified as test/vandalism using STiki

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en>Mean as custard: Undid revision 591160912 by Cajape96 (talk)

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