devianceTest

Analysis of deviance for generalized linear regression model

Syntax

tbl = devianceTest(mdl)

Description

tbl = devianceTest(mdl) returns an analysis of deviance table for the generalized linear regression model mdl. The table tbl gives the result of a test that determines whether the model mdl fits significantly better than a constant model.

Examples

Perform Deviance Test

Perform a deviance test on a generalized linear regression model.

Generate sample data using Poisson random numbers with two underlying predictors X(:,1) and X(:,2).

rng('default') % For reproducibility
rndvars = randn(100,2);
X = [2 + rndvars(:,1),rndvars(:,2)];
mu = exp(1 + X*[1;2]);
y = poissrnd(mu);

Create a generalized linear regression model of Poisson data.

mdl = fitglm(X,y,'y ~ x1 + x2','Distribution','poisson')

mdl = 
Generalized linear regression model:
    log(y) ~ 1 + x1 + x2
    Distribution = Poisson

Estimated Coefficients:
                   Estimate       SE        tStat     pValue
                   ________    _________    ______    ______

    (Intercept)     1.0405      0.022122    47.034      0   
    x1              0.9968      0.003362    296.49      0   
    x2               1.987     0.0063433    313.24      0   


100 observations, 97 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 2.95e+05, p-value = 0

Test whether the model differs from a constant in a statistically significant way.

tbl = devianceTest(mdl)

tbl=2×4 table
                             Deviance     DFE     chi2Stat     pValue
                            __________    ___    __________    ______

    log(y) ~ 1              2.9544e+05    99                         
    log(y) ~ 1 + x1 + x2         107.4    97     2.9533e+05       0  

The small p-value indicates that the model significantly differs from a constant. Note that the model display of mdl includes the statistics shown in the second row of the table.

Input Arguments

mdl — Generalized linear regression model

GeneralizedLinearModel object | CompactGeneralizedLinearModel object

Generalized linear regression model, specified as a GeneralizedLinearModel object created using fitglm or stepwiseglm, or a CompactGeneralizedLinearModel object created using compact.

Output Arguments

tbl — Analysis of deviance summary statistics

table

Analysis of deviance summary statistics, returned as a table.

tbl contains analysis of deviance statistics for both a constant model and the model mdl. The table includes these columns for each model.

Column	Description
Deviance	Deviance is twice the difference between the loglikelihoods of the corresponding model (mdl or constant) and the saturated model. For more information, see Deviance.
DFE	Degrees of freedom for the error (residuals), equal to n – p, where n is the number of observations, and p is the number of estimated coefficients
chi2Stat	F-statistic or chi-squared statistic, depending on whether the dispersion is estimated (F-statistic) or not (chi-squared statistic) F-statistic is the difference between the deviance of the constant model and the deviance of the full model, divided by the estimated dispersion. Chi-squared statistic is the difference between the deviance of the constant model and the deviance of the full model.
pValue	p-value associated with the test: chi-squared statistic with p – 1 degrees of freedom, or F-statistic with p – 1 numerator degrees of freedom and DFE denominator degrees of freedom, where p is the number of estimated coefficients

More About

Deviance

Deviance is a generalization of the residual sum of squares. It measures the goodness of fit compared to a saturated model.

The deviance of a model M₁ is twice the difference between the loglikelihood of the model M₁ and the saturated model M_s. A saturated model is a model with the maximum number of parameters that you can estimate.

For example, if you have n observations (y_i, i = 1, 2, ..., n) with potentially different values for X_i^Tβ, then you can define a saturated model with n parameters. Let L(b,y) denote the maximum value of the likelihood function for a model with the parameters b. Then the deviance of the model M₁ is

$$-2\left(\log L\left(b_1,y\right)-\log L\left(b_S,y\right)\right),$$

where b₁ and b_s contain the estimated parameters for the model M₁ and the saturated model, respectively. The deviance has a chi-squared distribution with n – p degrees of freedom, where n is the number of parameters in the saturated model and p is the number of parameters in the model M₁.

Assume you have two different generalized linear regression models M₁ and M₂, and M₁ has a subset of the terms in M₂. You can assess the fit of the models by comparing their deviances D₁ and D₂. The difference of the deviances is

$$\begin{array}{l}D=D_2-D_1=-2\left(\log L\left(b_2,y\right)-\log L\left(b_S,y\right)\right)+2\left(\log L\left(b_1,y\right)-\log L\left(b_S,y\right)\right)\\ =-2\left(\log L\left(b_2,y\right)-\log L\left(b_1,y\right)\right).\end{array}$$

Asymptotically, the difference D has a chi-squared distribution with degrees of freedom v equal to the difference in the number of parameters estimated in M₁ and M₂. You can obtain the p-value for this test by using 1  —  chi2cdf(D,v).

Typically, you examine D using a model M₂ with a constant term and no predictors. Therefore, D has a chi-squared distribution with p – 1 degrees of freedom. If the dispersion is estimated, the difference divided by the estimated dispersion has an F distribution with p – 1 numerator degrees of freedom and n – p denominator degrees of freedom.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2012a