Mauchly's sphericity test or Mauchly's W is a statistical test used to validate a repeated measures analysis of variance (ANOVA). It was developed in 1940 by John Mauchly.
Sphericity Sphericity is an important assumption of a repeated-measures ANOVA. It is the condition of equal variances among the differences between all possible pairs of within-subject conditions (i.e., levels of the independent variable). If sphericity is violated (i.e., if the variances of the differences between all combinations of the conditions are not equal), then the variance calculations may be distorted, which would result in an inflated F-ratio. Sphericity can be evaluated when there are three or more levels of a repeated measure factor and, with each additional repeated measures factor, the risk for violating sphericity increases. If sphericity is violated, a decision must be made as to whether a univariate or multivariate analysis is selected. If a univariate method is selected, the repeated-measures ANOVA must be appropriately corrected depending on the degree to which sphericity has been violated.
Measurement of sphericity
To further illustrate the concept of sphericity, consider a matrix representing data from patients who receive three different types of drug treatments in Figure 1. Their outcomes are represented on the left-hand side of the matrix, while differences between the outcomes for each treatment are represented on the right-hand side. After obtaining the difference scores for all possible pairs of groups, the variances of each group difference can be contrasted. From the example in Figure 1, the variance of the differences between Treatment A and B (17) appear to be much greater than the variance of the differences between Treatment A and C (10.3) and between Treatment B and C (10.3). This suggests that the data may violate the assumption of sphericity. To determine whether statistically significant differences exist between the variances of the differences, Mauchly's test of sphericity can be performed.
Interpretation Developed in 1940 by John W. Mauchly, Mauchly's test of sphericity is a popular test to evaluate whether the sphericity assumption has been violated. The null hypothesis of sphericity and alternative hypothesis of non-sphericity in the above example can be mathematically written in terms of difference scores.
H 0 : σ Tx A − Tx B 2 = σ Tx A − Tx C 2 = σ Tx B − Tx C 2 {\displaystyle H_{0}:\sigma _{{\text{Tx A}}-{\text{Tx B}}}^{2}=\sigma _{{\text{Tx A}}-{\text{Tx C}}}^{2}=\sigma _{{\text{Tx B}}-{\text{Tx C}}}^{2}}
H 1 : The variances are not all equal . {\displaystyle H_{1}:{\text{The variances are not all equal}}.}
Interpreting Mauchly's test is fairly straightforward. When the probability of Mauchly's test statistic is greater than or equal to α {\displaystyle \alpha } (i.e., p > α {\displaystyle \alpha } , with α {\displaystyle \alpha } commonly being set to .05), we fail to reject the null hypothesis that the variances are equal. Therefore, we could conclude that the assumption has not been violated. However, when the probability of Mauchly's test statistic is less than or equal to α {\displaystyle \alpha } (i.e., p < α {\displaystyle \alpha } ), sphericity cannot be assumed and we would therefore conclude that there are significant differences between the variances of the differences. Sphericity is always met for two levels of a repeated measure factor and is, therefore, unnecessary to evaluate. Statistical software should not provide output for a test of sphericity for two levels of a repeated measure factor; however, some versions of SPSS produce an output table with degrees of freedom equal to 0, and a period in place of a numeric p value.
Violations of sphericity
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