# coefTest

## Syntax

## Description

## Examples

### Test Significance of Multinomial Regression Model

Load the `fisheriris`

data set.

`load fisheriris`

The column vector `species`

contains iris flowers of three different species: setosa, versicolor, and virginica. The matrix `meas`

contains four types of measurements for the flowers: the length and width of sepals and petals in centimeters.

Create a table from the iris measurements and species data by using the `array2table`

function.

tbl = array2table(meas,... VariableNames=["SepalLength","SepalWidth","PetalLength","PetalWidth"]); tbl.Species = species;

Fit a multinomial regression model using the petal measurements as the predictor data and the species as the response data.

`mdl = fitmnr(tbl,"Species ~ PetalLength + PetalWidth^2")`

mdl = Multinomial regression with nominal responses Value SE tStat pValue _______ ______ _______ __________ (Intercept_setosa) 136.9 12.587 10.876 1.4933e-27 PetalLength_setosa -17.351 7.0021 -2.478 0.013211 PetalWidth_setosa -77.383 24.06 -3.2163 0.0012987 PetalWidth^2_setosa -24.719 8.3324 -2.9666 0.0030111 (Intercept_versicolor) 8.2731 14.489 0.571 0.568 PetalLength_versicolor -5.7089 2.0638 -2.7662 0.0056709 PetalWidth_versicolor 35.208 21.97 1.6026 0.10903 PetalWidth^2_versicolor -14.041 7.1653 -1.9596 0.050037 150 observations, 292 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 309.3988, p-value = 7.9151e-64

`mdl`

is a multinomial regression model object that contains the results of the fitting a nominal multinomial regression model to the data. The chi-squared statistic and *p*-value correspond to the null hypothesis that the fitted model does not outperform a degenerate model consisting of only an intercept term. The large *p*-value indicates that not enough evidence exists to reject the null hypothesis.

Perform an *F*-test to test the null hypothesis that all coefficients, except the intercept term, are zero. Use the default 95% significance level.

p = coefTest(mdl)

p = 3.5512e-133

The small *p*-value in the output indicates that enough evidence exists to reject the null hypothesis that all coefficients are zero. Enough evidence exists to conclude that at least one of the fitted model coefficients is statistically significant at the 95% significance level.

### Test Significance of Multinomial Model Coefficient

Load the `carsmall`

data set.

`load carsmall`

The variables `Acceleration`

, `Weight`

, and `Model_Year`

contain data for car acceleration, weight, and model year, respectively. The variable `MPG`

contains car mileage data in miles per gallon (MPG).

Sort the data in `MPG`

into four response categories by using the `discretize`

function.

MPG = discretize(MPG,[9 19 29 39 48]); tbl = table(MPG,Acceleration,Weight,Model_Year);

Fit a multinomial regression model of the car mileage as a function of the acceleration, weight, and model year.

mdl = fitmnr(tbl,"MPG ~ Acceleration + Model_Year + Weight",CategoricalPredictors="Model_Year")

mdl = Multinomial regression with nominal responses Value SE tStat pValue ________ _________ _______ ___________ (Intercept_1) 154.38 15.697 9.835 7.9576e-23 Acceleration_1 -11.31 0.53323 -21.21 7.7405e-100 Weight_1 0.098347 0.0034745 28.306 2.9244e-176 Model_Year_76_1 182.33 4.5868 39.75 0 Model_Year_82_1 -1690.4 4.6231 -365.64 0 (Intercept_2) 177.87 14.211 12.516 6.0891e-36 Acceleration_2 -11.28 0.48884 -23.076 8.1522e-118 Weight_2 0.090009 0.0030349 29.658 2.6661e-193 Model_Year_76_2 187.19 4.2373 44.176 0 Model_Year_82_2 -136.5 3.4781 -39.244 0 (Intercept_3) 103.66 14.991 6.9146 4.6928e-12 Acceleration_3 -11.359 0.48805 -23.274 8.2157e-120 Weight_3 0.080071 0.0033652 23.794 3.8879e-125 Model_Year_76_3 283.31 4.7309 59.885 0 Model_Year_82_3 -34.727 4.0878 -8.4953 1.9743e-17 94 observations, 267 error degrees of freedom Dispersion: 1 Chi^2-statistic vs. constant model: 169.6193, p-value = 5.7114e-30

`mdl`

is a multinomial regression model object that contains the results of fitting a nominal multinomial regression model to the data. By default, the fourth response category is the reference category. Each row of the table output corresponds to the coefficient of the model term in the first column. The `tStat`

and `pValue`

columns contain the *t*-statistics and *p*-values, respectively, for the null hypothesis that the corresponding coefficient is zero. The small *p*-values for the `Model_Year`

terms indicate that the model year has a statistically significant effect on `mdl`

. For example, the *p*-value for the term `Model_Year_76_2`

indicates that a car being manufactured in 1976 has a statistically significant effect on $$\mathrm{ln}\left(\frac{{\pi}_{2}}{{\pi}_{4}}\right)$$, where $${\pi}_{i}$$ is the *i*th category probability.

You can use a numeric index matrix to investigate whether a group of coefficients contains a coefficient that is statistically significant. Use a numeric index matrix to test the null hypothesis that all coefficients corresponding to the `Model_Year`

terms are zero.

idx_Model_Year = [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0;... 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0;... 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;... 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;... 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0;... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1;... ]; [p_Model_Year,F_Model_Year,r_Model_Year] = coefTest(mdl,idx_Model_Year)

p_Model_Year = 0

F_Model_Year = 4.8985e+04

r_Model_Year = 6

The returned *p*-value indicates that at least one of the category coefficients corresponding to `Model_Year`

is statistically different from zero. This result is consistent with the small *p*-value for each of the `Model_Term`

coefficients.

## Input Arguments

`mdl`

— Multinomial regression model object

`MultinomialRegression`

model object

Multinomial regression model object, specified as a `MultinomialRegression`

model object created with the `fitmnr`

function.

`H`

— Hypothesis matrix

numeric index matrix | logical 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.

**Example: **`[1 0 0 0 0]`

tests the first coefficient among five
coefficients.

**Data Types: **`single`

| `double`

| `logical`

`C`

— Hypothesized value

numeric vector

## 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.

## Version History

**Introduced in R2023a**

## See Also

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