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Quantifying violations of sphericity with epsilon

Quantifying violations of sphericity with epsilon

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Quantifying violations of sphericity with epsilon

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Deviations from sphericity in repeated measures ANOVA can be quantified by a value known as epsilon.  There are two methods for calculating it. Based on a recommendation from Maxwell and Delaney (p 545, reference below), Prism uses the method of Greenhouse and Geisser. While this method might be a bit conservative and underestimate deviations from the ideal, the alternative method by Huynh and Feldt tends to go too far in the other direction.

If you choose not to assume sphericity in repeated measures ANOVA, Prism  reports the value of epsilon. Its value can never be higher than 1.0, which denotes no violation of sphericity. The value of epsilon gets smaller with more violation of sphericity, but its value can never be lower than 1/(k - 1), where k is the number of treatment groups.

Number of treatments, k

Possible values of epsilon

3

0.5000 to 1.0000

4

0.3333 to 1.0000

5

0.2500 to 1.0000

6

0.2000 to 1.0000

7

0.1667 to 1.0000

8

0.1429 to 1.0000

9

0.1250 to 1.0000

10

0.1111 to 1.0000

11

0.1000 to 1.0000

12

0.0909 to 1.0000

13

0.0833 to 1.0000

14

0.0769 to 1.0000

15

0.0714 to 1.0000

20

0.0526 to 1.0000

25

0.0417 to 1.0000

50

0.0204 to 1.0000

k

1/(k-1) to 1.0000

Reference

Scott E. Maxwell, Harold D. Delaney, Designing Experiments and Analyzing Data: A Model Comparison Perspective, Second Edition. IBSN:0805837183.