Mauchly's sphericity test is a statistical test

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 used to validate repeated measures factor ANOVAs. The test was introduced by ENIAC

ENIAC

ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

 co-inventor John Mauchly

John Mauchly

John William Mauchly was an American physicist who, along with J. Presper Eckert, designed ENIAC, the first general purpose electronic digital computer, as well as EDVAC, BINAC and UNIVAC I, the first commercial computer made in the United States.Together they started the first computer company,...

 in 1940.

What is sphericity?

Sphericity relates to the equality of the variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

s 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

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

. The test is based on the likelihood ratio criterion and involves a scaled comparison between the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 and the trace

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

 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.