Mauchly's sphericity

When to use this test:

The Mauchly test (1940[120]) checks the sphericity assumption – a condition similar to, but stronger than, equality of variances, required before performing the ANOVA for dependent groups. This assumption is met if the variances of the differences between all pairs of repeated measurements are close to each other. A significant test result (\(p\le\alpha\)) means the sphericity assumption is violated and an appropriate correction must be used (see ANOVA for dependent groups with Epsilon correction and MANOVA).

Assumptions

measurement on an interval scale,
multivariate normal distribution, or normality of distribution of each analysed variable,
a dependent model.

Instead of the sphericity assumption, a simpler but more stringent condition is sometimes considered – compound symmetry – which assumes symmetry of the covariance matrix: equal variances of the variables and equal covariances between them. Meeting the compound symmetry condition entails meeting the sphericity assumption.

Performing and interpreting the analysis

In practice, this test is performed as part of a broader analysis – an example of its use and interpretation can be found in the ANOVA for dependent groups with Epsilon correction and MANOVA chapter (Example 1).

FOR THOSE INTERESTED

Hypotheses:

\[\begin{array}{cl} \mathcal{H}_0: & \sigma_{diff(1)}=\sigma_{diff(2)}=...=\sigma_{diff(s)},\\ \mathcal{H}_1: & \text{not all }\sigma_{diff(i)}\text{ are equal }(i=1,2,...,s), \end{array}\]

where \(\sigma_{diff(i)}\) – population variance of the differences between the \(i\)-th pair of repeated measurements, \(s\) – number of pairs.

Formulas: Mauchly's \(W\) value is defined as follows:

\[ W=\frac{\prod_{j=1}^{k-1}\lambda_j}{\left(\frac{1}{k-1}\sum_{j=1}^{k-1}\lambda_j\right)^{k-1}}. \]

The test statistic is defined by:

\[ \chi^2=(f-1)(n-1)\ln W, \]

where:
\(f=\frac{2(k-1)^2+(k-1)+2}{6(k-1)(n-1)}\),
\(\lambda_j\) – eigenvalue of the expected covariance matrix,
\(k\) – number of variables analyzed.

Interpretation: this statistic asymptotically (for large sample sizes) has the \(\chi^2\) distribution with \(df=\frac{k(k-1)}{2}-1\) degrees of freedom.

The \(p\) value, determined on the basis of the test statistic, is compared with the significance level \(\alpha\): if \(p \le \alpha\), we reject \(\mathcal{H}_0\) in favor of \(\mathcal{H}_1\); if \(p > \alpha\), there is no reason to reject \(\mathcal{H}_0\).

A value of \(W\approx1\) indicates that the sphericity assumption is met. When interpreting the results of this test, however, it is important to remember that it is sensitive to violations of the normality assumption.