manova
Multivariate analysis of variance
Description
also
returns manova results with additional options, specified by one or
more manovatbl
= manova(rm
,Name,Value
)Name,Value
pair arguments.
Examples
Perform Multivariate Analysis of Variance
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
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 = 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
{[ 1 0 0]}
{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
Input Arguments
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 Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: manovatbl =
manova(rm,'WithinModel','separatemeans')
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 outputtbl
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
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.
Tips
The multivariate response for each observation (subject) is the vector of repeated measures.
To test a more general hypothesis
A*B*C = D
, usecoeftest
.
Version History
Introduced in R2014a
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