# stats

## Description

## Examples

### Return MANOVA Table for One-Way MANOVA

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 flower: the length and width of sepals and petals in centimeters.

Perform a one-way MANOVA with `species`

as the factor and the measurements in `meas`

as the response variables.

maov = manova(species,meas);

`maov`

is a `manova`

object that contains the results of the one-way MANOVA. Display the corresponding MANOVA table.

s = stats(maov)

`s=`*3×8 table*
Source DF TestStatistic Value F DFNumerator DFDenominator pValue
_______ ___ _____________ ______ ______ ___________ _____________ __________
Factor1 2 pillai 1.1919 53.466 8 290 9.7422e-53
Error 147
Total 149

The small *p*-value for `species`

indicates that the flower species has a statistically significant effect on at least one of the flower measurements.

### Return Hypothesis and Error Matrices for Two-Way MANOVA

Load the `carsmall`

data set.

`load carsmall`

The variable `Model_Year`

contains data for the year a car was manufactured, and the variable `Cylinders`

contains data for the number of engine cylinders in the car. The `Acceleration`

, `Displacement`

, and `Weight`

variables contain data for car acceleration, displacement, and weight.

Use the `table`

function to create a table from the data in `Model_Year`

, `Cylinders`

, `Acceleration`

, `Displacement`

, and `Weight`

.

tbl = table(Model_Year,Cylinders,Acceleration,Displacement,Weight,VariableNames=["Year" "Cylinders" "Acceleration" "Displacement" "Weight"]);

Perform a two-way MANOVA using the table variables `Year`

and `Cylinders`

as factors, and the `Acceleration`

, `Displacement`

, and `Weight`

variables as response variables.

`maov = manova(tbl,"Acceleration,Displacement,Weight ~ Cylinders + Year")`

maov = 2-way manova Acceleration,Displacement,Weight ~ 1 + Year + Cylinders Source DF TestStatistic Value F DFNumerator DFDenominator pValue _________ __ _____________ _______ ______ ___________ _____________ _________ Year 2 pillai 0.11134 1.8471 6 188 0.092099 Cylinders 2 pillai 0.96154 29.012 6 188 1.891e-24 Error 95 Total 99 Properties, Methods

`maov`

is a two-way `manova`

object that contains the results of the two-way MANOVA. The small *p*-value for `Cylinders`

indicates that enough evidence exists to conclude that `Cylinders`

has a statistically significant effect on the mean response vector.

Return the hypothesis and error matrices for the MANOVA model terms.

[~,H,E] = stats(maov)

`H=`*3×2 table*
Year Cylinders
___________________________________ __________________________________
33.703 -327.34 3443.7 278.01 -13017 -90619
-327.34 4835.3 -30382 -13017 7.1228e+05 4.9601e+06
3443.7 -30382 3.5753e+05 -90619 4.9601e+06 3.4541e+07

E =3×310^{7}× 0.0001 -0.0002 0.0021 -0.0002 0.0109 0.0451 0.0021 0.0451 1.3656

The variables in the table `H`

correspond to the MANOVA model terms of the same name. Each variable contains the hypothesis matrix for its corresponding MANOVA model term. The error matrix `E`

contains the irreducible error for the MANOVA model. You can use `H`

and `E`

to perform hypothesis tests that are not supported by MATLAB® or Statistics and Machine Learning Toolbox™.

## Input Arguments

`maov`

— MANOVA results

`manova`

object

MANOVA results, specified as a `manova`

object.
The properties of `maov`

contain the response data and factor values
used by `stats`

to calculate the statistics in the MANOVA
table.

`testStat`

— MANOVA test statistics

`maov.TestStatistic`

(default) | `"all"`

| `"pillai"`

| `"hotelling"`

| `"wilks"`

| `"roy"`

MANOVA test statistics, specified as `maov.TestStatistic`

,
`"all"`

, or one or more of the following values.

Value | Test Name | Equation |
---|---|---|

`"pillai"` (default) | Pillai's trace | $$V=trace\left({Q}_{h}{\left({Q}_{h}+{Q}_{e}\right)}^{-1}\right)={\displaystyle \sum {\theta}_{i},}$$ where Q –
_{h}θ(Q +
_{h}Q) = 0.
_{e}Q and
_{h}Q are, respectively, the
hypotheses and the residual sum of squares product matrices. _{e} |

`"hotelling"` | Hotelling-Lawley trace | $$U=trace\left({Q}_{h}{Q}_{e}^{-1}\right)={\displaystyle \sum {\lambda}_{i}},$$ where Q –
_{h}λQ| = 0._{e} |

`"wilks"` | Wilk's lambda |
$$\Lambda =\frac{\left|{Q}_{e}\right|}{\left|{Q}_{h}+{Q}_{e}\right|}={\displaystyle \prod \frac{1}{1+{\lambda}_{i}}}.$$ |

`"roy"` | Roy's maximum root statistic |
$$\Theta =\mathrm{max}\left(eig\left({Q}_{h}{Q}_{e}^{-1}\right)\right).$$ |

If you specify `testStat`

as `"all"`

,
`stats`

calculates all the test statistics in the table
above.

**Example: **`TestStatistic`

="hotelling"

**Data Types: **`char`

| `string`

| `cell`

## Output Arguments

`s`

— MANOVA table

table

MANOVA table, returned as a table. In addition to rows for the error and total,
`s`

contains *t* rows per model term, where
*t* is the number of test statistics in
`maov.TestStatistic`

. The table `s`

also has the
following columns:

`Source`

— MANOVA model term`DF`

— Degrees of freedom for the term in`Source`

`TestStatistic`

— Name of the test statistic used to calculate the*F*-statistic in the column`F`

and the*p*-value in the column`pValue`

`Value`

— Value of the test statistic named in`TestStatistic`

`F`

— Value of the*F*-statistic corresponding to the test statistic named in`TestStatistic`

`DFNumerator`

— Degrees of freedom for the numerator of the*F*-statistic`DFDenominator`

— Degrees of freedom for the denominator of the*F*-statistic`pValue`

—*p*-value for the*F*-statistic

**Data Types: **`table`

`H`

— Hypothesis matrices

table of matrices

Hypothesis matrices used to compute the *F*-statistics for the
MANOVA model terms, returned as a table of matrices. Each column of
`H`

corresponds to a MANOVA model term in
`maov.Formula`

. For more information about `H`

,
see *Q _{h}* in Multivariate Analysis of Variance for Repeated Measures.

**Data Types: **`table`

`E`

— MANOVA model error matrix

numeric matrix

MANOVA model error matrix used to compute the *F*-statistics for
the MANOVA model terms, returned as a numeric matrix. For more information about
`E`

, see *Q _{e}* in Multivariate Analysis of Variance for Repeated Measures.

**Data Types: **`single`

| `double`

## Version History

**Introduced in R2023b**

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