Deviance (statistics)

Short description: Measure of goodness of fit for a statistical model

In statistics, deviance is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing. It is a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood. It plays an important role in exponential dispersion models and generalized linear models.

Deviance can be related to Kullback-Leibler divergence.^[1]

Definition

The unit deviance^[2]^[3] [math]\displaystyle{ d(y,\mu) }[/math] is a bivariate function that satisfies the following conditions:

[math]\displaystyle{ d(y,y) = 0 }[/math]
[math]\displaystyle{ d(y,\mu) \gt 0 \quad\forall y \neq \mu }[/math]

The total deviance [math]\displaystyle{ D(\mathbf{y},\hat{\boldsymbol{\mu}}) }[/math] of a model with predictions [math]\displaystyle{ \hat{\boldsymbol{\mu}} }[/math] of the observation [math]\displaystyle{ \mathbf{y} }[/math] is the sum of its unit deviances: [math]\displaystyle{ D(\mathbf{y},\hat{\boldsymbol{\mu}}) = \sum_i d(y_i, \hat{\mu}_i) }[/math].

The (total) deviance for a model M₀ with estimates [math]\displaystyle{ \hat{\mu} = E[Y|\hat{\theta}_0] }[/math], based on a dataset y, may be constructed by its likelihood as:^[4]^[5] [math]\displaystyle{ D(y,\hat{\mu}) = 2 \left(\log \left[p(y\mid\hat \theta_s)\right] - \log \left[ p(y\mid\hat \theta_0)\right]\right). }[/math]

Here [math]\displaystyle{ \hat \theta_0 }[/math] denotes the fitted values of the parameters in the model M₀, while [math]\displaystyle{ \hat \theta_s }[/math] denotes the fitted parameters for the saturated model: both sets of fitted values are implicitly functions of the observations y. Here, the saturated 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 full model compared to the reduced model. The deviance is used to compare two models – in particular in the case of generalized linear models (GLM) where it has a similar role to residual sum of squares 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, based on Wilks' theorem, an approximate chi-squared distribution with k-degrees of freedom.^[5] This can be used for hypothesis testing on the deviance.

Some usage of the term "deviance" can be confusing. According to Collett:^[6]

"the quantity [math]\displaystyle{ -2 \log \big[ p(y\mid\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]\displaystyle{ -2 \log \big[ p(y\mid\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.

Examples

The unit deviance for the Poisson distribution is [math]\displaystyle{ d(y,\mu) = 2\left(y\log\frac{y}{\mu}-y+\mu\right) }[/math], the unit deviance for the Normal distribution is given by [math]\displaystyle{ d(y,\mu) = \left(y-\mu\right)^2 }[/math].

Notes

↑ Hastie, Trevor. "A closer look at the deviance." The American Statistician 41.1 (1987): 16-20.
↑ Jørgensen, B. (1997). The Theory of Dispersion Models. Chapman & Hall.
↑ Song, Peter X. -K. (2007). Correlated Data Analysis: Modeling, Analytics, and Applications. Springer Series in Statistics. Springer Series in Statistics. doi:10.1007/978-0-387-71393-9. ISBN 978-0-387-71392-2.
↑ Nelder, J.A.; Wedderburn, R.W.M. (1972). "Generalized Linear Models". Journal of the Royal Statistical Society. Series A (General) 135 (3): 370–384. doi:10.2307/2344614.
↑ ^5.0 ^5.1 McCullagh and Nelder (1989): page 17
↑ Collett (2003): page 76