coefTest

Linear hypothesis test on nonlinear 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 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 Nonlinear Regression Model Coefficients

Make a nonlinear model of mileage as a function of the weight from the carsmall data set. Test the coefficients to see if all should be zero.

Create an exponential model of car mileage as a function of weight from the carsmall data. Scale the weight by a factor of 1000 so all the variables are roughly equal in size.

load carsmall
X = Weight;
y = MPG;
modelfun = 'y ~ b1 + b2*exp(-b3*x/1000)';
beta0 = [1 1 1];
mdl = fitnlm(X,y,modelfun,beta0);

Test the model for significant differences from a constant model.

p = coefTest(mdl)

p = 
1.3708e-36

There is no doubt that the model contains nonzero terms.

Input Arguments

mdl — Nonlinear regression model

NonLinearModel object

Nonlinear regression model, specified as a NonLinearModel object created using fitnlm.

H — Hypothesis matrix

numeric index matrix

Hypothesis matrix, specified as a numeric index matrix with one column for each coefficient in the model.

• 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.

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.

More About

Test Statistics

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

• The data comes from a normal distribution.

• The entries are independent.

Suppose these assumptions hold. 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 number of terms in β. Let c be a vector the same size as β. 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 in mdl.Coefs, and V is the estimated covariance of the coefficient estimates in mdl.CoefCov. When the hypothesis is true, the test statistic F has an F Distribution with r and u degrees of freedom.

Alternatives

The values of commonly used test statistics are available in the mdl.Coefficients table.

Version History

Introduced in R2012a