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

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Generalized Linear Models - Edward F. Connor

The deviance for a model M_{0} 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 M_{0}, 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, M_{1} and M_{2}. In particular, suppose that M_{1} contains a set of the parameters in M_{2}, and k additional parameters. Then, under the null hypothesis that M_{2} is the true model, the difference between the deviances for the two models follows an approximate chi-squared distribution with k-degrees of freedom. Note that here both models M_{1} and M_{2} would be subsets of the full model used to define the zero of the deviance criterion.

- Pearson X-squared statistic, an alternative quality of fit statistic for generalized linear models.
- Akaike information criterion
- Deviance information criterion
- Peirce's criterion
- Bayesian deviance

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

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Last updated on Tuesday September 16, 2008 at 01:33:11 PDT (GMT -0700)

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Last updated on Tuesday September 16, 2008 at 01:33:11 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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