# Documentation

### This is machine translation

Translated by
Mouse over text to see original. Click the button below to return to the English verison of the page.

# coefTest

Class: NonLinearModel

Linear hypothesis test on nonlinear regression model coefficients

## 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 Definitions). `F` Value of the test statistic for the F test (see Definitions). `r` Numerator degrees of freedom for the F test (see Definitions). The F statistic has `r` degrees of freedom in the numerator and `mdl.DFE` degrees of freedom in the denominator.

## Definitions

### 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 v be a vector the same size as β. The following is a test statistic for the hypothesis that  = v:

`$F={\left(H\stackrel{^}{\beta }-v\right)}^{\prime }{\left(HC{H}^{\prime }\right)}^{-1}\left(H\stackrel{^}{\beta }-v\right).$`

Here $\stackrel{^}{\beta }$ is the estimate of the coefficient vector β in `mdl.Coefs`, and C 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.

## Examples

expand all

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.

## Alternatives

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