- Deviance (statistics)
The deviance for a model M0 is defined as
Here denotes the fitted values of the parameters in the model M0, while 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), "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."
- Pearson's chi-squared test, an alternative quality of fit statistic for generalized linear models.
- Akaike information criterion
- Deviance information criterion
- Peirce's criterion
- Discrepancy function
- McCullagh, Peter; Nelder, John (1989). Generalized Linear Models, Second Edition. Chapman & Hall/CRC. ISBN 0412317605.
- Collett, David (2003). Modelling Survival Data in Medical Research, Second Edition. Chapman & Hall/CRC. ISBN 1-58488-325-1.
Generalized Linear Models - Edward F. Connor
Statistics Descriptive statisticsSummary tables Data collectionDesigning studiesUncontrolled studies Statistical inferenceFrequentist inferenceSpecific tests Correlation and regression analysisNon-standard predictorsPartition of variance Categorical, multivariate, time-series, or survival analysis Applications Least squares and regression analysis Computational statistics Correlation and dependence Regression analysis Regression as a
statistical modelPredictor structureNon-standardNon-normal errors
Decomposition of variance Model exploration Background Design of experiments Numerical approximation Applications
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