devianceTest
Analysis of deviance for generalized linear regression model
Syntax
Description
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^2statistic vs. constant model: 2.95e+05, pvalue = 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 pvalue 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 ( 
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  Fstatistic or chisquared statistic, depending on whether the dispersion is estimated (Fstatistic) or not (chisquared statistic)

pValue  pvalue associated with the test: chisquared
statistic with p – 1 degrees of freedom, or Fstatistic with p – 1 numerator degrees of freedom and 
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 M_{1} is twice the difference between the loglikelihood of the model M_{1} 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}^{T}β, 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 M_{1} is
$$2\left(\mathrm{log}L\left({b}_{1},y\right)\mathrm{log}L\left({b}_{S},y\right)\right),$$
where b_{1} and b_{s} contain the estimated parameters for the model M_{1} and the saturated model, respectively. The deviance has a chisquared 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 M_{1}.
Assume you have two different generalized linear regression models M_{1} and M_{2}, and M_{1} has a subset of the terms in M_{2}. You can assess the fit of the models by comparing their deviances D_{1} and D_{2}. The difference of the deviances is
$$\begin{array}{l}D={D}_{2}{D}_{1}=2\left(\mathrm{log}L\left({b}_{2},y\right)\mathrm{log}L\left({b}_{S},y\right)\right)+2\left(\mathrm{log}L\left({b}_{1},y\right)\mathrm{log}L\left({b}_{S},y\right)\right)\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\left(\mathrm{log}L\left({b}_{2},y\right)\mathrm{log}L\left({b}_{1},y\right)\right).\end{array}$$
Asymptotically, the difference D has a chisquared distribution with
degrees of freedom v equal to the difference in the number of parameters
estimated in M_{1} and
M_{2}. You can obtain the
pvalue for this test by using 1 —
chi2cdf(D,v)
.
Typically, you examine D using a model M_{2} with a constant term and no predictors. Therefore, D has a chisquared 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
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