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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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==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]]

Revision as of 21:22, 17 January 2014

Template:Distinguish

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.

Definition

The deviance for a model M0, based on a dataset y, is defined as:[1][2]

Here denotes the fitted values of the parameters in the model M0, while 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 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 (RSS).

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.[2]

Some usage of the term "deviance" can be confusing. According to Collett:[3]

"the quantity is sometimes referred to as a deviance. This is [...] inappropriate, since unlike the deviance used in the context of generalized linear modelling, 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

Notes

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References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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External links

Template:Statistics Template:Least squares and regression analysis

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  2. 2.0 2.1 McCullagh and Nelder (1989)Template:Page needed
  3. Collett (2003)Template:Page needed