Does sphericity require repeated measurements to be independent?

No; correlated measurements can satisfy sphericity.

What happens with only two conditions?

There is only one contrast dimension, so the usual sphericity restriction imposes no additional constraint for that effect.

AI Glossary / Statistics and probability / Advanced

Mauchly's sphericity test

Definition

Mauchly's sphericity test assesses a covariance assumption commonly used in univariate repeated-measures analysis of variance. For a within-subject factor, sphericity requires the population variances of all pairwise differences between its levels to be equal. Equivalently, the covariance matrix of appropriately orthonormalized contrasts is proportional to an identity matrix. The test examines this covariance structure, not whether the condition means differ.

What it means

Repeated measurements on the same person or other unit are related. The relevant question is not simply whether each measurement occasion has the same variance, but whether contrasts between occasions have the covariance structure required by the chosen analysis.
If sphericity fails, an uncorrected repeated-measures F test can have an inappropriate reference distribution. Mauchly's test provides evidence about that assumption, but its result does not independently establish that the entire analysis is valid.

How it works

Identify the within-unit effect being tested and the corresponding contrast space. With three measurement conditions, two independent contrasts can describe their differences. More complicated within-subject designs can require different transformations for different effects.
Estimate the residual covariance structure after accounting for the relevant model. Under the classical assumptions, independent subjects supply multivariate normal observations, leading to the covariance distribution used by the test. Independence between subjects is distinct from dependence within a subject.
Transform the measurements to a set of orthonormal contrasts. Sphericity means that these contrast coordinates have equal variances and zero covariances in the population. Arbitrarily scaled, nonorthogonal differences should not simply be compared with an identity matrix without accounting for their transformation.
For a positive-definite covariance matrix S in a q-dimensional orthonormal contrast space, Mauchly's W is det(S) divided by [trace(S)/q]^q. In terms of eigenvalues, it compares their product with the qth power of their average. Equal eigenvalues give W = 1; unequal eigenvalues reduce it.
A significance calculation uses sample information and an appropriate reference approximation. W alone is not a p-value. Implementations can differ slightly in the approximation used, so record the method when reproducing results across software.
A small p-value provides evidence against sphericity under the test assumptions. A large p-value does not prove sphericity: a small study may have little power to detect an important departure. A very large study may detect departures whose practical impact needs separate consideration.
With only two levels of a within-subject factor, there is only one difference to consider. The usual sphericity restriction is automatically satisfied, so the standard multi-dimensional test is unnecessary for that effect.
If the assumption is unsuitable, options include an appropriate degrees-of-freedom correction, a multivariate analysis, or a model that represents within-unit covariance directly. Greenhouse-Geisser and Huynh-Feldt corrections address the repeated-measures F reference distribution; they do not alter the observed measurements to make them spherical.
Choose the inferential strategy with the study design and assumptions in mind. Treating an assumption test as an automatic switch between analyses can obscure uncertainty and encourage selective reporting. Explain covariance choices, missing-data handling, and sensitivity checks alongside the substantive results.

Three practical examples

A usability study measures the same participants under several interface conditions and checks the covariance assumption for its repeated-measures analysis.
An analyst evaluates repeated quality ratings from the same assessors while distinguishing mean differences from covariance structure.
A research team compares corrected and uncorrected within-subject tests as part of a planned sensitivity analysis.

Think of it this way

Imagine measuring variation along several perpendicular directions in a space of within-person differences. Sphericity says the population variation has the same scale in every such direction. Mauchly's test asks whether the estimated shape provides evidence against that condition.

Worked example

An original calculation considers three measurement conditions and two orthonormal contrast coordinates. Suppose their estimated covariance matrix is diagonal with entries 1 and 4.
Its determinant is 4 and its trace is 5. With q = 2, W = 4/(5/2)^2 = 4/6.25 = 0.64. If the two diagonal entries were both 2.5, the determinant would be 6.25 and W would equal 1.
As a related descriptive calculation, the Greenhouse-Geisser epsilon estimate from these two eigenvalues is (1+4)^2/[2*(1^2+4^2)] = 25/34, approximately 0.7353. This is a correction factor, not the Mauchly p-value.
The matrix and W do not determine a significance result without the sample's residual degrees of freedom and the chosen approximation. Therefore this example deliberately reports no rejection decision. It shows how unequal contrast variances affect W while keeping a diagnostic statistic distinct from its inferential interpretation.

Illustrative case study

In a fictional case, Larch Design asks the same participants to rate three prototype interfaces. Analysts initially focus on whether average ratings differ and overlook the dependence structure.
They review the within-person contrasts and report a prespecified corrected repeated-measures analysis together with the covariance diagnostics. They also investigate missing ratings before interpreting the comparison. The case illustrates how an assumption check belongs within a larger analysis; it does not imply that a nonsignificant Mauchly result guarantees reliable conclusions.

Common misconceptions and mistakes

Do not confuse sphericity with equality of the original measurement variances or with independence of repeated measurements. Compound symmetry is sufficient for sphericity but is a stronger requirement.
Do not interpret rejection as evidence that condition means differ. Do not interpret failure to reject as proof that the covariance assumption holds. W and epsilon are not interchangeable statistics.

Business use

This test can help teams understand repeated-measures analyses used in product experiments and evaluation studies. Its value is diagnostic: it directs attention to whether the uncertainty calculation matches the measurement design. Report it with the substantive effect estimates and analysis choices, rather than using it as an isolated quality stamp.

Check your understanding

Does W = 0.64 automatically mean rejection at 5%?

Answer: No. The significance calculation also needs sample information and a reference approximation.

Does a correction make the observed covariance spherical?

Answer: No; it adjusts the inferential procedure.

Related terms

Sphericity
Compound symmetry
Covariance matrix
Repeated-measures ANOVA
Orthonormal contrast
Greenhouse-Geisser correction
Huynh-Feldt correction

Learn next

Next, study covariance structures for repeated measurements. Compare univariate corrections, multivariate methods, and mixed models, including their assumptions about missing observations and within-unit dependence.

Sources

R documentation, Mauchly's Test of Sphericity: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/mauchly.test.html R Core implementation, multivariate linear-model and sphericity calculations: https://raw.githubusercontent.com/wch/r-source/trunk/src/library/stats/R/mlm.R IBM, GLM Repeated Measures documentation: https://www.ibm.com/docs/en/spss-statistics/31.0.0?topic=statistics-glm-repeated-measures

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