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.

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.

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

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.

Version History

Introduced in R2014a