manova

Multivariate analysis of variance

Syntax

manovatbl = manova(rm)

manovatbl = manova(rm,Name,Value)

[manovatbl,A,C,D]
= manova(___)

Description

example

manovatbl = manova(rm) returns the results of multivariate analysis of variance (manova) for the repeated measures model rm.

example

manovatbl = manova(rm,Name,Value) also returns manova results with additional options, specified by one or more Name,Value pair arguments.

example

[manovatbl,A,C,D] = manova(___) also returns arrays A, C, and D for the hypotheses tests of the form A*B*C = D, where D is zero.

Input Arguments

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`rm` — Repeated measures model
`RepeatedMeasuresModel` object

Repeated measures model, returned as a RepeatedMeasuresModel object.

For properties and methods of this object, see RepeatedMeasuresModel.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

`'WithinModel'` — Model specifying within-subjects hypothesis test
`'separatemeans'` (default) | model specification using formula | r-by-nc matrix

Model specifying the within-subjects hypothesis test, specified as one of the following:

'separatemeans' — Compute a separate mean for each group, and test for equality among the means.
Model specification — This is a model specification in the within-subject factors. Test each term in the model. In this case, tbl contains a separate manova for each term in the formula, with the multivariate response equal to the vector of coefficients of that term.
An r-by-nc matrix, C, specifying nc contrasts among the r repeated measures. If Y represents the matrix of repeated measures you use in the repeated measures model rm, then the output tbl contains a separate manova for each column of Y*C.

Example: 'WithinModel','separatemeans'

Data Types: single | double | char | string

`'By'` — Single between-subjects factor
character vector | string scalar

Single between-subjects factor, specified as the comma-separated pair consisting of 'By' and a character vector or string scalar. manova performs a separate test of the within-subjects model for each value of this factor.

For example, if you have a between-subjects factor, Drug, then you can specify that factor to perform manova as follows.

Example: 'By','Drug'

Data Types: char | string

Output Arguments

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`manovatbl` — Results of multivariate analysis of variance
table

Results of multivariate analysis of variance for the repeated measures model rm, returned as a table.

manova uses these methods to measure the contributions of the model terms to the overall covariance:

Wilks’ Lambda
Pillai’s trace
Hotelling-Lawley trace
Roy’s maximum root statistic

For details, see Multivariate Analysis of Variance for Repeated Measures.

manova returns the results for these tests for each group. manovatbl contains the following columns.

Column Name	Definition
`Within`	Within-subject terms
`Between`	Between-subject terms
`Statistic`	Name of the statistic computed
`Value`	Value of the corresponding statistic
`F`	F-statistic value
`RSquare`	Measure for variance explained
`df1`	Numerator degrees of freedom
`df2`	Denominator degrees of freedom
`pValue`	p-value for the corresponding F-statistic value

Data Types: table

`A` — Specification based on between-subjects model
matrix | cell array

Specification based on the between-subjects model, returned as a matrix or a cell array. It permits the hypothesis on the elements within given columns of B (within time hypothesis). If manovatbl contains multiple hypothesis tests, A might be a cell array.

Data Types: single | double | cell

`C` — Specification based on within-subjects model
matrix | cell array

Specification based on the within-subjects model, returned as a matrix or a cell array. It permits the hypotheses on the elements within given rows of B (between time hypotheses). If manovatbl contains multiple hypothesis tests, C might be a cell array.

Data Types: single | double | cell

`D` — Hypothesis value
0

Hypothesis value, returned as 0.

Examples

expand all

Perform Multivariate Analysis of Variance

Open Live Script

Load the sample data.

load fisheriris

The column vector species consists of iris flowers of three different species: setosa, versicolor, 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 = table([1 2 3 4]','VariableNames',{'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 multivariate analysis of variance.

manova(rm)

ans=8×9 table
     Within       Between      Statistic      Value        F       RSquare    df1    df2      pValue   
    ________    ___________    _________    _________    ______    _______    ___    ___    ___________

    Constant    (Intercept)    Pillai         0.99013    4847.5    0.99013     3     145    3.7881e-145
    Constant    (Intercept)    Wilks        0.0098724    4847.5    0.99013     3     145    3.7881e-145
    Constant    (Intercept)    Hotelling       100.29    4847.5    0.99013     3     145    3.7881e-145
    Constant    (Intercept)    Roy             100.29    4847.5    0.99013     3     145    3.7881e-145
    Constant    species        Pillai         0.96909    45.749    0.48455     6     292     2.4729e-39
    Constant    species        Wilks         0.041153    189.92    0.79714     6     290     2.3958e-97
    Constant    species        Hotelling       23.051    555.17    0.92016     6     288    4.6662e-155
    Constant    species        Roy              23.04    1121.3     0.9584     3     146    1.4771e-100

Perform multivariate anova separately for each species.

manova(rm,'By','species')

ans=12×9 table
     Within          Between          Statistic     Value        F       RSquare    df1    df2      pValue   
    ________    __________________    _________    ________    ______    _______    ___    ___    ___________

    Constant    species=setosa        Pillai         0.9823    2682.7     0.9823     3     145    9.0223e-127
    Constant    species=setosa        Wilks        0.017698    2682.7     0.9823     3     145    9.0223e-127
    Constant    species=setosa        Hotelling      55.504    2682.7     0.9823     3     145    9.0223e-127
    Constant    species=setosa        Roy            55.504    2682.7     0.9823     3     145    9.0223e-127
    Constant    species=versicolor    Pillai           0.97    1562.8       0.97     3     145    3.7058e-110
    Constant    species=versicolor    Wilks        0.029999    1562.8       0.97     3     145    3.7058e-110
    Constant    species=versicolor    Hotelling      32.334    1562.8       0.97     3     145    3.7058e-110
    Constant    species=versicolor    Roy            32.334    1562.8       0.97     3     145    3.7058e-110
    Constant    species=virginica     Pillai        0.97261    1716.1    0.97261     3     145    5.1113e-113
    Constant    species=virginica     Wilks        0.027394    1716.1    0.97261     3     145    5.1113e-113
    Constant    species=virginica     Hotelling      35.505    1716.1    0.97261     3     145    5.1113e-113
    Constant    species=virginica     Roy            35.505    1716.1    0.97261     3     145    5.1113e-113

Return Arrays of the Hypothesis Test

Open Live Script

Load the sample data.

load fisheriris

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 multivariate analysis of variance. Also return the arrays for constructing the hypothesis test.

[manovatbl,A,C,D] = manova(rm)

manovatbl=8×9 table
     Within       Between      Statistic      Value        F       RSquare    df1    df2      pValue   
    ________    ___________    _________    _________    ______    _______    ___    ___    ___________

    Constant    (Intercept)    Pillai         0.99013    4847.5    0.99013     3     145    3.7881e-145
    Constant    (Intercept)    Wilks        0.0098724    4847.5    0.99013     3     145    3.7881e-145
    Constant    (Intercept)    Hotelling       100.29    4847.5    0.99013     3     145    3.7881e-145
    Constant    (Intercept)    Roy             100.29    4847.5    0.99013     3     145    3.7881e-145
    Constant    species        Pillai         0.96909    45.749    0.48455     6     292     2.4729e-39
    Constant    species        Wilks         0.041153    189.92    0.79714     6     290     2.3958e-97
    Constant    species        Hotelling       23.051    555.17    0.92016     6     288    4.6662e-155
    Constant    species        Roy              23.04    1121.3     0.9584     3     146    1.4771e-100

A=2×1 cell array
    {1x3 double}
    {2x3 double}

C = 4×3

     1     0     0
    -1     1     0
     0    -1     1
     0     0    -1

D = 0

Index into matrix A.

A{1}

ans = 1×3

     1     0     0

A{2}

ans = 2×3

     0     1     0
     0     0     1

Tips

The multivariate response for each observation (subject) is the vector of repeated measures.
To test a more general hypothesis A*B*C = D, use coeftest.

Documentation

manova

Syntax

Description

Input Arguments

`rm` — Repeated measures model
`RepeatedMeasuresModel` object

Name-Value Pair Arguments

`'WithinModel'` — Model specifying within-subjects hypothesis test
`'separatemeans'` (default) | model specification using formula | r-by-nc matrix

`'By'` — Single between-subjects factor
character vector | string scalar

Output Arguments

`manovatbl` — Results of multivariate analysis of variance
table

`A` — Specification based on between-subjects model
matrix | cell array

`C` — Specification based on within-subjects model
matrix | cell array

`D` — Hypothesis value
0

Examples

Perform Multivariate Analysis of Variance

Return Arrays of the Hypothesis Test

Tips

See Also

Topics

Statistics and Machine Learning Toolbox Documentation

Support

Documentation

manova

Syntax

Description

Input Arguments

rm — Repeated measures model RepeatedMeasuresModel object

Name-Value Pair Arguments

'WithinModel' — Model specifying within-subjects hypothesis test 'separatemeans' (default) | model specification using formula | r-by-nc matrix

'By' — Single between-subjects factor character vector | string scalar

Output Arguments

manovatbl — Results of multivariate analysis of variance table

A — Specification based on between-subjects model matrix | cell array

C — Specification based on within-subjects model matrix | cell array

D — Hypothesis value 0

Examples

Perform Multivariate Analysis of Variance

Return Arrays of the Hypothesis Test

Tips

See Also

Topics

Statistics and Machine Learning Toolbox Documentation

Support

`rm` — Repeated measures model
`RepeatedMeasuresModel` object

`'WithinModel'` — Model specifying within-subjects hypothesis test
`'separatemeans'` (default) | model specification using formula | r-by-nc matrix

`'By'` — Single between-subjects factor
character vector | string scalar

`manovatbl` — Results of multivariate analysis of variance
table

`A` — Specification based on between-subjects model
matrix | cell array

`C` — Specification based on within-subjects model
matrix | cell array

`D` — Hypothesis value
0