Using the attached SPSS data file...

Sphericity Violated Data.savDownload Sphericity Violated Data.sav

1^stRun the Repeated Measure ANOVA in SPSS and look over the results for Mauchly'sWtest (Table 1 below)

1a. If the Sig. value for the Approx . Chi-Square test is greater than (>) .05, then the assumption of Sphericity has been met. See 2a.

1b. If the Sig. value for the Approx . Chi-Square test is less than (<) .05, then the assumption of Sphericity has been violated. See 2b, 3, 3a, 3b.

2a If the assumption of Sphericity had been met, you can interpret the F-ratio for the line Sphericity Assumed (see table 2)

2b If the assumption of Sphericity had been violated then interpret the F-ratio for the Greenhouse-Geisser adjustment or the Huynh-Feldt Adjustment. (See table 2). Here's how to decide which.

1. Assuming that Sphericity had been violated (the Chi-square test for Mauchly's W is significant) examine Epsilon values in Table 1.

3a. IfW is significant and the Huynh-Feldt .75, then interpret the F-ratio for the Huynh-Feldt adjustment in Table 2. (Note Huynh is pronounced Win).

3b. IfWis significant and the Greenhouse-Geisser < .75, then interpret the F-ratio for the Greenhouse-Geisser adjustment in Table 2.

1. Therefore, in the example below:

A preliminary analysis employing Mauchly's test indicated that the assumption of sphericity had violated (W= .312, ²(2) = 11.65,p= .003,Greenhouse-Geisser < .622).Therefore, as recommended by Field (2018), theGreenhouse-Geisser adjusted F-test was employed. The results for the test indicated that the three sample means do not differ significantly (F(1.185, 13.033) = 3.75,p= .070, ² = .254). The Null Hypothesis (M1 = M2 = M3) is accepted).

(Note that if I were to interpret the F-test for the Sphericity Assumed findings, I would have concluded that the sample means do differ significantly (F)2, 22) = 3.75, p = .04). The Null Hypothesis is rejected. Rather the Alternative Hypotheses (M1 M2 M3) is supported by the analysis.

Table 1

Mauchly's Test of Sphericitya
Measure: MEASURE_1
Within	Mauchly's	Approx	df	Sig.	Epsilonb
Subjects	W	Chi-Sq.			Greenhouse	Huynh-	Lower-
Effect					-Geisser	Feldt	bound
Time	0.312	11.649	2	0.003	0.592	0.622	0.5
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
a. Design: Intercept
Within Subjects Design: Time
b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.

Table 2.

Tests of Within-Subjects Effects
Source	Type III SS	df	MS	F	p	p2
Time	Sphericity Assumed	956.22	2	478.11	3.75	0.04	0.254
Greenhouse-Geisser	956.22	1.185	807.07	3.75	0.07	0.254
Huynh-Feldt	956.22	1.245	768.19	3.75	0.067	0.254
Lower-bound	956.22	1	956.22	3.75	0.079	0.254
Error(Time)	Sphericity Assumed	2806.44	22	127.57
Greenhouse-Geisser	2806.44	13.033	215.34
Huynh-Feldt	2806.44	13.693	204.96
Lower-bound	2806.44	11	255.13

Reference.

Field, A. (2018). Discovering Statistics Using SPSS (5^thedition). London: Sage