coefTest

Linear hypothesis test on generalized linear regression model coefficients

Syntax

p = coefTest(mdl)

p = coefTest(mdl,H)

p = coefTest(mdl,H,C)

[p,F] = coefTest(___)

[p,F,r] = coefTest(___)

Description

p = coefTest(mdl) computes the p-value for an F test that all coefficient estimates in mdl, except the intercept term, are zero.

p = coefTest(mdl,H) performs an F-test that H × B = 0, where B represents the coefficient vector. Use H to specify the coefficients to include in the F-test.

p = coefTest(mdl,H,C) performs an F-test that H × B = C.

[p,F] = coefTest(___) also returns the F-test statistic F using any of the input argument combinations in previous syntaxes.

[p,F,r] = coefTest(___) also returns the numerator degrees of freedom r for the test.

Examples

Test Significance of Generalized Linear Regression Model

Fit a generalized linear regression model, and test the coefficients of the fitted model to see if they differ from zero.

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 fitted model has coefficients that differ significantly from zero.

p = coefTest(mdl)

p = 
4.1131e-153

The small p-value indicates that the model fits significantly better than a degenerate model consisting of only an intercept term.

Test Significance of Generalized Linear Regression Model Coefficient

Fit a generalized linear regression model, and test the significance of a specified coefficient in the fitted 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 the significance of the x1 coefficient. According to the model display, x1 is the second predictor. Specify the coefficient by using a numeric index vector.

p = coefTest(mdl,[0 1 0])

p = 
2.8681e-145

The returned p-value indicates that x1 is statistically significant in the fitted model.

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.

H — Hypothesis matrix

numeric index matrix

Hypothesis matrix, specified as a full-rank numeric index matrix of size r-by-s, where r is the number of linear combinations of coefficients being tested, and s is the total number of coefficients.

• If you specify H, then the output p is the p-value for an F-test that H × B = 0, where B represents the coefficient vector.

• If you specify H and C, then the output p is the p-value for an F-test that H × B = C.

Example: [1 0 0 0 0] tests the first coefficient among five coefficients.

Data Types: single | double

C — Hypothesized value

numeric vector

Hypothesized value for testing the null hypothesis, specified as a numeric vector with the same number of rows as H.

If you specify H and C, then the output p is the p-value for an F-test that H × B = C, where B represents the coefficient vector.

Data Types: single | double

Output Arguments

p — p-value for F-test

numeric value in the range [0,1]

p-value for the F-test, returned as a numeric value in the range [0,1].

F — Value of test statistic for F-test

numeric value

Value of the test statistic for the F-test, returned as a numeric value.

r — Numerator degrees of freedom for F-test

positive integer

Numerator degrees of freedom for the F-test, returned as a positive integer. The F-statistic has r degrees of freedom in the numerator and mdl.DFE degrees of freedom in the denominator.

Algorithms

The p-value, F-statistic, and numerator degrees of freedom are valid under these assumptions:

• The data comes from a model represented by the formula in the Formula property of the fitted model.

• The observations are independent, conditional on the predictor values.

Under these assumptions, let β represent the (unknown) coefficient vector of the linear regression. Suppose H is a full-rank numeric index matrix of size r-by-s, where r is the number of linear combinations of coefficients being tested, and s is the total number of coefficients. Let c be a column vector with r rows. The following is a test statistic for the hypothesis that Hβ = c:

$$F={\left(H\widehat{\beta }-c\right)}^{\prime }{\left(HV{H}^{\prime }\right)}^{-1}\left(H\widehat{\beta }-c\right)/r.$$

Here $$\widehat{\beta }$$ is the estimate of the coefficient vector β, stored in the Coefficients property, and V is the estimated covariance of the coefficient estimates, stored in the CoefficientCovariance property. When the hypothesis is true, the test statistic F has an F Distribution with r and u degrees of freedom, where u is the degrees of freedom for error, stored in the DFE property.

Alternative Functionality

The values of commonly used test statistics are available in the Coefficients property of a fitted model.

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