coefTest

Class: NonLinearModel

Linear hypothesis test on nonlinear regression model coefficients

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Syntax

p = coefTest(mdl) p = coefTest(mdl,H) p = coefTest(mdl,H,C) [p,F] = coefTest(mdl,...) [p,F,r] = coefTest(mdl,...)

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.

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

[p,F] = coefTest(mdl,...) returns the F test statistic.

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

Input Arguments

`mdl`	Nonlinear regression model, constructed by `fitnlm`.
`H`	Numeric matrix having one column for each coefficient in the model. When `H` is an input, the output `p` is the p-value for an F test that `H*B = 0`, where `B` represents the coefficient vector.
`C`	Numeric vector with the same number of rows as `H`. When `C` is an input, the output `p` is the p-value for an F test that `H*B = C`, where `B` represents the coefficient vector.

Output Arguments

`p`	p-value of the F test (see More About).
`F`	Value of the test statistic for the F test (see More About).
`r`	Numerator degrees of freedom for the F test (see More About). The F statistic has `r` degrees of freedom in the numerator and `mdl.DFE` degrees of freedom in the denominator.

Examples

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Test Nonlinear Regression Model Coefficients

Open Live Script

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.

More About

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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 matrix of size r-by-s, where 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:

$F = {(H \hat{β} - c)}^{'} {(H V H^{'})}^{- 1} (H \hat{β} - c) .$

Here $\hat{β}$ 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.

Documentation