Greenhouse and Geisser epsilon
In this section we provide the motivation for the Greenhouse and Geisser epsilon correction factor.
Box proposed the measure of sphericity to be
where Σ = [cij] is the population covariance matrix. Greenhouse and Geisser epsilon is simply the sample approximation of this measure. If we now let C = [cij] be the sample covariance matrix and define S = [sij] where
then S approximates Σ and the Greenhouse and Geisser epsilon becomes
Example 1: Calculate the value of the GG epsilon for Example 1 of ANOVA with Repeated Measures with One Within Subjects Factor using the above approach.
Figure 1 – Calculation of GG epsilon for Example 1
Range A5:D8 in Figure 1 contains the sample covariance matrix (see Figure 3 of Sphericity). E5:E8 contains the row means and A9:D9 contain the column means. E9 contains the grand mean.
Range A14:D17 contains the estimated population matrix as described above. E.g. cell A15 contains the formula =A6-$E6-A$9+$E$9. The elements in row 18 are the values of the diagonal of this matrix formed in the usual way (e.g. cell B18 contains the formula =INDEX(B14:B17,B13).
The value of the GG epsilon is 0.493 (cell F17) is the same as that calculated previously (see Figure 3 of Sphericity).
Observation: An alternative way of calculating GG epsilon is using the eigenvalues λj of matrix S, as follows:
See Example 1 of Goal Seeking and Solver for the definition of an eigenvalue and a way of calculating it. The supplemental Excel function eVALUES contained in the Real Statistics Resource Pack can also be used to calculate the eigenvalues of a matrix as described in Real Multivariate Statistics using Excel [Za].
Example 2: Calculate the value of the GG epsilon for Example 1 using the eigenvalues of matrix .
Figure 2 – Calculation of GG epsilon for Example 1 using eigenvalues
The results are given in Figure 2. The eigenvalues of matrix S are produced by highlighting cells A22:C22 (k – 1 = 3 cells) and entering the Real Statistics array formula =eVALUES(A14:D17). The value of GG epsilon (cell G25) is once again .493.
Mauchly’s test for sphericity
In this section we present Mauchly’s test, a commonly used test to determine whether the sphericity assumption holds. Because of its lack of power, Mauchly’s test is not recommended and in fact it is simply better to apply either the GG or HF epsilon correction factor in all cases. We also present John, Nagao and Sugiura’s test for sphericity, a much more powerful and useful test.
Property 1: (Mauchly’s test) Define the following statistics related to the matrix S described above:
If W = 1 then the original data meets the sphericity assumption. If the original data meets the sphericity assumption (the null hypothesis) then
Example 3: Determine whether the data in Example 1 meets the Mauchly’s test for sphericity.
Figure 3 – Mauchly’s test for sphericity
From Figure 3, we reject the null hypothesis and conclude that the sphericity assumption has not been met.
John, Nagao and Sugiura’s Test
Property 2: (John, Nagao and Sugiura’s test): Define the following statistics related to the matrix S described above:
If the original data meets the sphericity assumption (the null hypothesis) then
Example 4: Determine whether the data in Example 1 meets the John, Nagao and Sugiura’s test for sphericity.
Figure 4 – John, Nagao and Sugiura’s test for sphericity
This test even more definitively shows (see Figure 4) that the data does not meet the sphericity assumption.