stats

Multivariate analysis of variance (MANOVA) table

Since R2023b

Syntax

s = stats(maov)

s = stats(maov,TestStatistic=testStat)

[s,H,E] = stats(___)

Description

s = stats(maov) returns a MANOVA table for the manova object maov. The MANOVA table contains statistics for the model terms, error, and total.

s = stats(maov,TestStatistic=testStat) specifies the test statistics to display in the MANOVA table.

[s,H,E] = stats(___) also returns a table H of the hypothesis matrices for the MANOVA model terms, and the error matrix E used to perform the MANOVA, using any of the input argument combinations in the previous syntaxes.

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 = 3×3
107 ×

    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 θi values are the solutions of the characteristic equation Qh – θ(Qh + Qe) = 0. Qh and Qe are, respectively, the hypotheses and the residual sum of squares product matrices.
"hotelling"	Hotelling-Lawley trace	$$U=trace\left(Q_h{Q}_e^{-1}\right)={\displaystyle \sum {\lambda }_i},$$ where λi are the solutions of the characteristic equation |Qh – λQe| = 0.
"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	$$\Theta =\max \left(eig\left(Q_h{Q}_e^{-1}\right)\right).$$

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