Diagnosing and Reporting Sphericity Violations

Why Diagnose Sphericity?

For a within-subject effect with more than two levels, the ordinary univariate repeated-measures $F$ test relies on a covariance condition called sphericity. When that condition fails, the usual reference distribution can produce p-values that are too small. A degrees-of-freedom correction reduces this risk, but diagnosing the issue is not as simple as asking whether a single test is significant.

A useful analysis combines three pieces of information: Mauchly’s test, the estimated epsilon values, and the inferential result after a stated correction. Mauchly’s test addresses evidence against sphericity; epsilon describes the estimated degree of departure relevant to the univariate test; and the correction determines the degrees of freedom used to obtain the adjusted p-value. None of these pieces should be treated as a substitute for the others.

Mauchly’s Test in Contrast Coordinates

Suppose each of $n$ independent subjects is measured at $k$ occasions, and let $q=k-1$. Choose a $k \times q$ matrix $C$ whose columns are orthonormal contrasts, so that $C^\mathsf{T}C=I_q$ and $C^\mathsf{T}\mathbf{1}=0$. For a subject’s measurement vector $Y$, the contrast vector is $Z=C^\mathsf{T}Y$. Write $\Sigma_Z=\operatorname{Cov}(Z)$. Sphericity means that the covariance in this contrast space is a scalar multiple of the identity matrix.

Definition: Sphericity holds when $\Sigma_Z=\sigma^2 I_q$ for some $\sigma^2>0$. In the standard one-group setup, let $S$ be the sample covariance matrix of the $n$ subject-level contrast vectors. Mauchly’s statistic is $$ W=\frac{|S|}{\left\{\operatorname{tr}(S)/q\right\}^{q}}. $$ Here $|S|$ is the determinant and $\operatorname{tr}(S)$ is the trace. Mauchly’s test evaluates the null hypothesis of sphericity.

The statistic compares the determinant of $S$ with the determinant of a spherical covariance matrix having the same average variance. If the contrast variances are very unequal, or the contrasts are correlated, the determinant tends to be smaller relative to that spherical benchmark, and $W$ tends to be small. The test is conventionally reported using an approximate chi-square statistic. In the standard one-group setting with multivariate normal contrast vectors and positive-definite covariance, a common corrected form is

\chi^2_M=-\left[(n-1)-\frac{2q^2+q+2}{6q}\right]\log W, \qquad \text{df}=\frac{q(q+1)}{2}-1.

This is an approximation, not a universal exact finite-sample test. Its stated form assumes the standard one-group covariance setup; software may use model-specific calculations in designs with additional between-subject terms. In such cases, report the test and degrees of freedom as produced for the fitted model rather than applying this formula mechanically.

Result: For positive-definite $S$, $0<W\leq 1$, with $W=1$ if and only if $S$ is spherical. To see this, let $\lambda_1,\ldots,\lambda_q$ be the positive eigenvalues of $S$. Then $$ W=\frac{\prod_{j=1}^q\lambda_j}{\left\{(\sum_{j=1}^q\lambda_j)/q\right\}^q} \leq 1, $$ by the arithmetic mean–geometric mean inequality. Equality holds exactly when all the eigenvalues are equal, which for a symmetric covariance matrix means $S$ is a scalar multiple of $I_q$. $\square$

What Epsilon Adds

The Greenhouse–Geisser epsilon summarizes how close the contrast covariance is to spherical for the purpose of the univariate degrees-of-freedom adjustment. For an estimated covariance matrix $S$, its estimate is

\hat{\epsilon}_{GG} =\frac{\{\operatorname{tr}(S)\}^2}{q\,\operatorname{tr}(S^2)}.

If $\lambda_1,\ldots,\lambda_q$ are the eigenvalues of $S$, the same quantity is $\hat{\epsilon}_{GG}=(\sum_j\lambda_j)^2/[q\sum_j\lambda_j^2]$. Equal eigenvalues give the largest value, 1; increasingly unequal eigenvalues give smaller values. The lower bound is $1/q$.

Result: For positive eigenvalues, $1/q\leq\hat{\epsilon}_{GG}\leq 1$. Indeed, $(\sum_j\lambda_j)^2\geq\sum_j\lambda_j^2$, so the ratio is at least $1/q$. Cauchy–Schwarz gives $(\sum_j\lambda_j)^2\leq q\sum_j\lambda_j^2$, so the ratio is at most 1. The upper equality occurs when all eigenvalues are equal. $\square$

Huynh–Feldt epsilon is another estimated adjustment, often less conservative than Greenhouse–Geisser. Both estimates are commonly included in software output. Their values are not p-values: they describe the estimated correction, not the probability that sphericity is true. A smaller epsilon means a stronger reduction in the degrees of freedom for the univariate test.

For a within-subject effect with uncorrected numerator and error degrees of freedom $d_1$ and $d_2$, an epsilon correction uses

d_{1,\text{adj}}=\hat{\epsilon}d_1, \qquad d_{2,\text{adj}}=\hat{\epsilon}d_2.

The observed $F$ statistic is unchanged; the reference $F$ distribution changes because both degrees of freedom are adjusted. For a one-group repeated-measures effect with $k$ occasions, the usual uncorrected degrees of freedom are $k-1$ and $(n-1)(k-1)$. In other designs, use the uncorrected degrees of freedom appropriate to that model and multiply both by the selected epsilon.

Worked Example: A Nonsignificant Test with a Visible Epsilon Adjustment

Suppose $k=4$, $n=12$, and the estimated covariance matrix in orthonormal contrast coordinates has eigenvalues $4,1,1$. Thus $q=3$, $\operatorname{tr}(S)=6$, $|S|=4$, and $\operatorname{tr}(S^2)=4^2+1^2+1^2=18$.

Mauchly’s statistic is $W=4/(6/3)^3=4/8=0.5$. The correction term in the chi-square formula is $(2q^2+q+2)/(6q)=(18+3+2)/18=23/18$. Therefore $\chi^2_M=-[11-23/18]\log(0.5)\approx 6.74$, with $3(4)/2-1=5$ degrees of freedom. The approximate p-value is about $0.24$, so the test is not significant at the 0.05 level.

The Greenhouse–Geisser estimate is $\hat{\epsilon}_{GG}=6^2/[3(18)]=36/54=2/3$. The uncorrected degrees of freedom are $3$ and $11(3)=33$; the Greenhouse–Geisser corrected degrees of freedom are $2$ and $22$. Thus a nonsignificant Mauchly test has not removed the practical effect of using a correction: the estimated epsilon is below 1, and the corrected reference distribution uses fewer degrees of freedom.

Choosing and Reporting a Correction

A defensible report states the test result and the corrected analysis rather than presenting the diagnostic p-value as a verdict. Greenhouse–Geisser is a standard choice when a correction is warranted, particularly when its estimated epsilon indicates appreciable departure from 1. Huynh–Feldt is also used; its less conservative adjustment may be reasonable when the estimate is near 1 and the analysis plan supports that choice. State which estimate was used. Do not silently switch corrections to obtain a preferred significance result.

The numerical threshold sometimes used to guide the choice between Greenhouse–Geisser and Huynh–Feldt is a convention, not a theorem. The choice should be made and described transparently, ideally according to the analysis plan and the scientific context. When the conclusion is important or the diagnostic is uncertain, reporting the corrected results under both methods can show whether the interpretation depends on the correction.

Worked Example: Reporting a Significant Mauchly Test

In a one-group study with $n=30$ subjects and $k=4$ occasions, suppose software reports Mauchly’s $W=0.62$, an approximate $\chi^2_M=13.25$ on 5 degrees of freedom, and $p\approx0.021$. The uncorrected within-subject test has $F=4.10$, with degrees of freedom $3$ and $87$. Suppose the output gives $\hat{\epsilon}_{GG}=0.58$ and $\hat{\epsilon}_{HF}=0.66$.

The Greenhouse–Geisser degrees of freedom are $0.58(3)=1.74$ and $0.58(87)=50.46$. The Huynh–Feldt degrees of freedom are $0.66(3)=1.98$ and $0.66(87)=57.42$. The value $F=4.10$ stays the same, but its corrected p-value must be obtained from the $F$ distribution with the selected adjusted degrees of freedom.

A clear report could read: “Mauchly’s test indicated evidence against sphericity, $W=0.62$, $\chi^2(5)=13.25$, $p=0.021$. The within-subject effect was evaluated with the Greenhouse–Geisser correction, $\hat{\epsilon}_{GG}=0.58$, $F(1.74,50.46)=4.10$, $p=p_{\text{GG}}$.” In a final report, replace $p_{\text{GG}}$ with the p-value calculated from that corrected $F$ distribution. This makes clear which test was used and avoids confusing the Mauchly p-value with the effect-test p-value.

Why a Nonsignificant Test Does Not Establish Sphericity

A nonsignificant Mauchly test means the data did not provide sufficient evidence to reject the sphericity null hypothesis at the chosen significance level. It does not show that the null is true. A test may have limited power, especially with a small sample, and therefore may fail to detect a meaningful departure. The epsilon estimate provides additional information about the estimated covariance shape, but it too is uncertain and should not be treated as proof.

Worked Example: A Nonsignificant Test Despite a Low Epsilon

Consider $n=6$ subjects at $k=4$ occasions, with estimated contrast-covariance eigenvalues $9,1,1$. Here $q=3$, $\operatorname{tr}(S)=11$, $|S|=9$, and $\operatorname{tr}(S^2)=81+1+1=83$. Therefore $\hat{\epsilon}_{GG}=121/(3\cdot83)=121/249\approx0.486$, well below 1.

Mauchly’s statistic is $W=9/(11/3)^3=243/1331\approx0.183$. The chi-square correction term is $23/18$, so $\chi^2_M=-[5-23/18]\log(243/1331)\approx6.32$, with 5 degrees of freedom. Its approximate p-value is about $0.28$, which is not significant at 0.05. This result does not establish sphericity: the sample is small, and the estimated epsilon indicates that a correction could materially reduce the degrees of freedom.

For the uncorrected degrees of freedom $3$ and $15$, the Greenhouse–Geisser-adjusted values are approximately $1.46$ and $7.29$. The test result should be interpreted using the chosen correction and its adjusted degrees of freedom, not by treating the nonsignificant diagnostic as confirmation of the uncorrected assumptions.

Common Mistakes and a Reporting Checklist

“Mauchly’s test was nonsignificant, so sphericity holds.” A failure to reject is not proof of the null. Consider the epsilon estimates and report the correction used.
Reporting the Mauchly p-value as the effect p-value. These p-values answer different questions. The within-subject effect p-value must come from its $F$ statistic and the selected corrected degrees of freedom.
Correcting only the numerator degrees of freedom. An epsilon correction multiplies both the numerator and denominator degrees of freedom; the observed $F$ statistic is unchanged.
Reporting epsilon without identifying the correction. State whether Greenhouse–Geisser or Huynh–Feldt was used and give the corresponding adjusted degrees of freedom.
Choosing a correction only after seeing which gives significance. Follow a prespecified or otherwise justified rule, and report the decision transparently.

A complete report typically includes the Mauchly statistic, its approximate test statistic and degrees of freedom, the p-value, the selected epsilon estimate, and the corrected $F$ test with adjusted degrees of freedom and its own p-value. If different reasonable corrections lead to different conclusions, disclose that sensitivity rather than presenting only the favorable result.

Check Your Understanding

Use the definitions and reporting principles above to answer the following.

What covariance structure does sphericity require in orthonormal contrast coordinates?
Why does Mauchly’s statistic equal 1 only when the contrast covariance is spherical?
If an effect has uncorrected degrees of freedom $4$ and $60$, what adjusted degrees of freedom result from an epsilon of $0.75$?
What does a nonsignificant Mauchly test allow you to conclude, and what does it not allow you to conclude?
Which quantities should be distinguished when reporting Mauchly’s test alongside a corrected within-subject effect test?