Mauchly's sphericity test

Mauchly's sphericity test is a statistical test used to validate repeated measures factor ANOVAs. The test was introduced by ENIAC co-inventor John Mauchly in 1940.^[1]

What is sphericity?

Sphericity relates to the equality of the variances of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. Sphericity is an assumption of an ANOVA with a repeated measures factor (RMF). Thus, results from ANOVAs violating this assumption can not be trusted.

Formulation of the test

Mauchly's sphericity test is a special case of a test that the covariance matrix of a multivariate normal distribution is proportional to a given matrix, in this case the identity matrix. The test is based on the likelihood ratio criterion^[2] and involves a scaled comparison between the determinant and the trace of the sample covariance matrix.

Interpreting Mauchly's sphericity test

When the significance level of the Mauchly’s test is < 0.05 then sphericity cannot be assumed.

Violations of sphericity

Departures from the assumption of sphericity affect the validity of various statistical tests used in the analysis of variance. Corrections for violations of sphericity include the Greenhouse–Geisser, the Huynh–Feldt and the lower-bound corrections. To correct for sphericity, these corrections alter the degrees of freedom, thereby altering the significance value of the F-ratio. There are different opinions about the best correction to apply. A good rule of thumb is to use the Greenhouse-Geisser estimate unless it leads to a different conclusion from the other two.^{[citation needed]}

See also

• Sphericity#Sphericity in statistics

References

1. ^ Mauchly, John W. (June 1940). "Significance Test for Sphericity of a Normal n-Variate Distribution". The Annals of Mathematical Statistics 11 (2): 204–209. doi:10.1214/aoms/1177731915. JSTOR 2235878. 
2. ^ Anderson, T.W. (1958) An Introduction to Multivariate Statistical Analysis. Wiley. ISBN 0471026409. p.262