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RM-ANOVA

Mauchly's Test of Sphericity — Repeated-Measures ANOVA Worked Example

Step-by-step Mauchly's sphericity test on a 4-condition repeated-measures design. Orthonormal contrasts, W = 0.136, chi-square approximation, and when to apply Greenhouse-Geisser correction.

📊 W = 0.136, χ²(5) = 11.41, p = .044 → violated
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🌀 26 · Mauchly's Test of Sphericity — Repeated-Measures ANOVA

Testing the Sphericity Assumption Before a Repeated-Measures ANOVA
RM-ANOVA
Research Question: 8 participants each tasted 4 foods and rated their disgust. Before running a repeated-measures ANOVA, we must check the sphericity assumption: are the variances of all the pairwise differences between conditions equal? Mauchly's test answers this.
Participant Food 1 Food 2 Food 3 Food 4
18716
29525
36238
45319
58458
67567
710272
812681
Mauchly's W (via orthonormal contrasts)
W = det ( S ) [ tr ( S ) k − 1 ] k − 1
where S is the covariance matrix of the (k−1) ORTHONORMAL contrast scores.
χ 2 = − [ n − 1 − 2 ( k − 1 ) 2 + ( k − 1 ) + 2 6 ( k − 1 ) ] ln ⁡ W , d f = ( k − 1 ) k 2 − 1
  1. 1
    Build orthonormal contrasts: Transform the 4 conditions into k−1 = 3 orthonormal (Helmert) contrast scores per participant. This step is essential — using raw successive differences gives a WRONG W.
  2. 2
    Covariance matrix S (3×3) of the contrast scores:
    tr ( S ) = 21.91 , det ( S ) = 53.1
  3. 3
    W:
    W = 53.1 ( 21.91 / 3 ) 3 = 53.1 389.6 = 0.1362
  4. 4
    Chi-square approximation:
    χ 2 = − [ 8 − 1 − 2 ( 9 ) + 3 + 2 18 ] ln ⁡ ( 0.1362 ) = − 6.72 × ( − 1.994 ) = 11.41 , d f = 5
  5. 5
    Decision rule: p = .044 < .05 → sphericity is violated → apply a Greenhouse-Geisser or Huynh-Feldt correction to the RM-ANOVA degrees of freedom. If p had been > .05, no correction would be needed.
0.1362
W
11.41
χ²
5
df
.044
p
🔴 Sphericity violated — W = 0.136, χ²(5) = 11.41, p = .044. Report the Greenhouse-Geisser corrected F for the RM-ANOVA.
APA-7
Mauchly's test indicated that the assumption of sphericity had been violated, χ²(5) = 11.41, p = .044; therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = .62).

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