# mauchly

Mauchly’s test for sphericity

## Syntax

``tbl = mauchly(rm)``
``tbl = mauchly(rm,C)``

## Description

example

````tbl = mauchly(rm)` returns the result of the Mauchly’s test for sphericity for the repeated measures model `rm`. It tests the null hypothesis that the sphericity assumption is true for the response variables in `rm`.For more information, see Mauchly’s Test of Sphericity.```
````tbl = mauchly(rm,C)` returns the result of the Mauchly’s test based on the contrast matrix `C`.```

## Examples

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`load fisheriris`

The column vector `species` consists of iris flowers of three different species: setosa, versicolor, and virginica. The double matrix `meas` consists of four types of measurements on the flowers: the length and width of sepals and petals in centimeters, respectively.

Store the data in a table array.

```t = table(species,meas(:,1),meas(:,2),meas(:,3),meas(:,4),... 'VariableNames',{'species','meas1','meas2','meas3','meas4'}); Meas = dataset([1 2 3 4]','VarNames',{'Measurements'});```

Fit a repeated measures model, where the measurements are the responses and the species is the predictor variable.

`rm = fitrm(t,'meas1-meas4~species','WithinDesign',Meas);`

Perform Mauchly’s test to assess the sphericity assumption.

`mauchly(rm)`
```ans=1×4 table W ChiStat DF pValue _______ _______ __ __________ 0.55814 84.976 5 7.6149e-17 ```

The small $p$-value (in the `pValue` field) indicates that the sphericity, hence the compound symmetry assumption, does not hold. You should use epsilon corrections to compute the $p$-values for a repeated measures anova. You can compute the epsilon corrections using the `epsilon` method and perform the repeated measures anova with the corrected $p$-values using the `ranova` method.

## Input Arguments

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Repeated measures model, returned as a `RepeatedMeasuresModel` object.

For properties and methods of this object, see `RepeatedMeasuresModel`.

Contrasts, specified as a matrix. The default value of `C` is the Q factor in a QR decomposition of the matrix M, where M is defined so that Y*M is the difference between all successive pairs of columns of the repeated measures matrix Y.

Data Types: `single` | `double`

## Output Arguments

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Results of Mauchly’s test for sphericity for the repeated measures model `rm`, returned as a `table`.

`tbl` contains the following columns.

Column NameDefinition
`W`Value of Mauchly’s W statistic
`ChiStat`Chi-square statistic value
`DF`Degrees of freedom of the Chi-square statistic
`pValue`p-value corresponding to the Chi-square statistic

Data Types: `table`

## Version History

Introduced in R2014a