epsilon

Epsilon adjustment for repeated measures anova

Syntax

tbl = epsilon(rm)

tbl = epsilon(rm,C)

Description

tbl = epsilon(rm) returns the epsilon adjustment factors for repeated measures model rm.

tbl = epsilon(rm,C) returns the epsilon adjustment factors for the test based on the contrast matrix C.

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.

C — Contrasts

matrix

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

tbl — Epsilon adjustment factors

table

Epsilon adjustment factors for the repeated measures model rm, returned as a table. tbl contains four different adjustments for epsilon.

Correction	Definition
Uncorrected	No adjustments, epsilon = 1
Greenhouse-Geisser	Greenhouse-Geisser approximation
Huynh-Feldt	Huynh-Feldt approximation
Lower bound	Lower bound on the p-value

For details, see Compound Symmetry Assumption and Epsilon Corrections.

Data Types: table

Examples

Epsilon Corrections for Repeated Measures ANOVA

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 repeated measures analysis of variance.

ranovatbl = ranova(rm)

ranovatbl=3×8 table
                                SumSq     DF      MeanSq       F         pValue        pValueGG       pValueHF       pValueLB  
                                ______    ___    ________    ______    ___________    ___________    ___________    ___________

    (Intercept):Measurements    1656.3      3      552.09    6873.3              0    9.4491e-279    2.9213e-283    2.5871e-125
    species:Measurements        282.47      6      47.078     586.1    1.4271e-206    4.9313e-156    1.5406e-158     9.0151e-71
    Error(Measurements)         35.423    441    0.080324                                                                      

ranova computes the last three $$p$$-values using Greenhouse-Geisser, Huynh-Feldt, and lower bound corrections, respectively.

Display the epsilon correction values.

epsilon(rm)

ans=1×4 table
    Uncorrected    GreenhouseGeisser    HuynhFeldt    LowerBound
    ___________    _________________    __________    __________

         1              0.75179          0.76409       0.33333  

You can check the compound symmetry (sphericity) assumption using the mauchly method.

Tips

• The mauchly method tests for sphericity.

• The ranova method contains p-values based on each epsilon value.

Algorithms

ranova computes the regular p-value (in the pValue column of the rmanova table) using the F-statistic cumulative distribution function:

p-value = 1 – fcdf(F,v₁,v₂).

When the compound symmetry assumption is not satisfied, ranova uses a correction factor epsilon, ε, to compute the corrected p-values as follows:

p-value_corrected = 1 – fcdf(F,ε*v₁,ε*v₂).

The epsilon method returns the epsilon adjustment values.