# coefTest

Class: LinearModel

Linear hypothesis test on linear 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, except for the intercept term.

`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` Linear model, as constructed by `fitlm` or `stepwiselm`. `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 model represented by the formula `mdl``.Formula`.

• The observations are independent conditional on the predictor values.

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

collapse all

### Test Linear Regression Model

Make a linear model of mileage as a function of the weight, weight squared, and model year from the `carsmall` data set. Test the coefficients to see if all should be zero.

Load the data and make a table, where the model year is an ordinal variable.

```load carsmall tbl = table(MPG,Weight); tbl.Year = ordinal(Model_Year); mdl = fitlm(tbl,'MPG ~ Year + Weight + Weight^2');```

Test the model for significant differences from a constant model.

`p = coefTest(mdl)`
```p = 5.5208e-41```

There is no doubt that the model contains more than the intercept term.

### Test a Particular Coefficient

Test the `Weight^2` coefficient in a linear model of mileage as a function of the weight, weight squared, and model year.

Load the data and make a table, where the model year is an ordinal variable.

```load carsmall tbl = table(MPG,Weight); tbl.Year = ordinal(Model_Year); mdl = fitlm(tbl,'MPG ~ Year + Weight + Weight^2');```

Test the significance of the `Weight^2` coefficient. To do so, find the coefficient corresponding to `Weight^2`.

`mdl.CoefficientNames`
```ans = '(Intercept)' 'Weight' 'Year_76' 'Year_82' 'Weight^2'```

`Weight^2` is the fifth (final) coefficient.

Test the significance of the `Weight^2` coefficient.

`p = coefTest(mdl,[0 0 0 0 1])`
```p = 0.0022```

## Alternatives

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

`anova` provides a test for each model term.