How to Report Mauchly's Test of Sphericity in APA 7

You run the repeated measures ANOVA, open the output and find two tables nobody warned you about: one with a Mauchly's W and a chi-square, and one that prints the same effect four times over, two of those rows carrying degrees of freedom with decimal points in them. The question stops being statistical and becomes editorial: which row goes into the manuscript, how it is written, and what to do when the p value crosses .05 once you correct it. The full write-up of the test itself, with its means, its F and its table, is in how to report a repeated measures ANOVA in APA 7; this article is only about sphericity and the corrections.

What sphericity assumes, and what breaks without it

Sphericity says nothing about the variances of your measurements. It is about the variances of the differences between them. With four levels of a within-subjects factor, every participant generates six possible difference scores, and the assumption requires all of them to have the same variance in the population. That is easy to violate: in a working memory task with four levels of cognitive load, the gap between the easiest and the hardest condition varies enormously across people, while the gap between two adjacent conditions is small and similar for almost everyone.

When those variances differ, the error term of a repeated measures ANOVA is too small and the F comes out larger than it should. The consequence is a Type I error rate above the nominal 5%, and with a serious violation it can approach double the rate you think you are working at. The correction is not cosmetic and it does not weaken your analysis — it hands back the precision the model had helped itself to, by cutting the degrees of freedom.

With two measurements there is nothing to test. A single difference score cannot have unequal variances with itself, so sphericity holds by definition and SPSS returns W = 1.000 with an empty p cell. If your design is pre-post and a reviewer asks for Mauchly's test, the answer is that it does not apply, and writing that line into the manuscript is cheaper than arguing it later. The assumption bites from three levels of the within-subjects factor upwards, including when that factor sits inside an interaction with a between-subjects factor.

Mauchly's test checks the assumption, but it checks it badly. With small samples it rarely detects a genuine violation; with large samples it flags a trivial departure as significant. That is the standard failure of assumption tests, which respond to sample size more than to the size of the violation (the wider argument is in how to check statistical assumptions). This is why many methodologists no longer treat it as a switch and simply apply the Greenhouse-Geisser correction every time. APA 7 does not forbid that, as long as you state it in the analysis section and did not decide it after seeing the result.

The Mauchly sentence, written out in full

Here is what goes in the text: "Mauchly's test indicated that the assumption of sphericity had been violated, W = .58, χ²(5, N = 45) = 23.27, p < .001, ε = .78." That line carries four formatting decisions. W is italicised because it is a statistic written with a Latin letter, and it takes no leading zero because it is bounded between 0 and 1. The chi-square is set upright, because chi is a Greek letter and APA 7 reserves italics for the Latin alphabet. Epsilon follows the same rule: upright, and no zero in front of the decimal point either.

The degrees of freedom of that chi-square are not the ANOVA's, and this is where most manuscripts slip. They come from k(k − 1)/2 − 1, where k is the number of repeated measurements: three measurements give 2, four give 5, five give 9. They have nothing to do with the effect's df, which are 3 and 132 in this example — they count the covariances the test is comparing, not groups or people. As with every chi-square in APA 7, the sample size goes inside the brackets, after the df and separated by a comma.

A non-significant result gets reported too, and the same sentence says what you did next: "Mauchly's test did not indicate a violation of sphericity, W = .93, χ²(5, N = 45) = 3.10, p = .685; uncorrected degrees of freedom are reported." A manuscript that mentions Mauchly and never says which row of the table it ended up copying forces the reviewer to guess, and reviewers guess unkindly.

This sentence belongs in the first paragraph of the Results section or in the data analysis paragraph of the Method, never in a footnote or the supplementary file. It takes one line and answers, at once, the three questions a methods reviewer will ask anyway: whether you checked the assumption, what you found, and what you decided to do about it.

Corrected degrees of freedom come out decimal, and stay that way

Epsilon is a multiplier for the degrees of freedom and nothing else. It equals 1 when sphericity is perfect and falls towards 1/(k − 1) as things get worse. With ε = .78 and original df of (3, 132), the corrected values are 3 × .78 = 2.34 and 132 × .78 = 102.96. The F is not recalculated: it is still 12.87, from the same sums of squares. The only thing that changes is the reference distribution you compare it against, and therefore the p value.

Written out: "Cognitive load had a significant effect on task accuracy, F(2.34, 102.96) = 12.87, p < .001, η²p = .23, with the Greenhouse-Geisser correction (ε = .78)." Two decimal places on the df, the F untouched, the name of the correction spelled out, and epsilon on display so that any reader can reconstruct the original degrees of freedom without asking you for the data.

Rounding those df to whole numbers is the most common mistake here and the most expensive. F(2, 103) is not a tidier version of F(2.34, 102.96); it is a different test, against a different distribution, returning a different p. Degrees of freedom for an F do not have to be integers — they are integers in between-subjects designs because they count groups and people, whereas here they are a scale multiplied by a decimal factor. Leave them with two decimals even if your table template complains, and never collapse the numerator df to 1 just because it looks odd.

The effect size does not change when you correct. SPSS prints the same η²p = .23 in all four rows, because the sums of squares are identical: the variance partition is what it always was, and only the significance threshold has moved. If the partial eta squared in your corrected row differs from the one in the sphericity-assumed row, you have not applied a correction — you have copied two different effects.

Greenhouse-Geisser, Huynh-Feldt or lower bound

Greenhouse-Geisser. The conservative option and the one you see in most papers. Its epsilon never exceeds 1 and never drops below the lower bound, so it overcorrects when the violation is mild; in exchange, it does not let false positives through. Huynh-Feldt. It adjusts the Greenhouse-Geisser epsilon upwards, because that one underestimates when sphericity is nearly met. Its value can exceed 1, in which case it is truncated to 1, which amounts to no correction at all. In the example above, a Greenhouse-Geisser epsilon of .78 corresponds to a Huynh-Feldt epsilon of .83, giving df of (2.49, 109.56). Lower bound. It assumes the worst case, ε = 1/(k − 1), which with four measurements is .33 and leaves the df at (1, 44). Almost nobody reports it; it appears in your output because SPSS prints it, not because you need it.

The most cited decision rule is Girden's (1992): report Greenhouse-Geisser when its epsilon is below .75, and Huynh-Feldt when it is above, because Greenhouse-Geisser overcorrects there and costs you power for nothing. Our example sits at ε = .78, so the convention sends you to Huynh-Feldt — and here the conclusion is identical either way, p < .001 with both. It is a convention, not a law: some methodologists report Greenhouse-Geisser as standard, which is equally defensible. What does not survive scrutiny is computing both and keeping whichever one crosses .05.

Whatever your rule is, put it in the Method before you look at the results: "Where Mauchly's test indicated a violation of sphericity, the Greenhouse-Geisser correction was applied." Twenty words turn a decision that looks retrospective into a declared one, and that difference is what separates a defensible analysis from one that smells of fishing.

Two routes avoid the assumption altogether. The multivariate approach — Pillai's trace, printed in the table just above in SPSS — makes no sphericity assumption and often has more power when epsilon is low and the sample is not tiny. Mixed models go further: they model the covariance structure instead of patching it afterwards, they handle missing data without deleting whole participants, and they become the sensible default once you have four or five measurement occasions. The full comparison, drawbacks included, is in mixed models versus repeated measures ANOVA.

When the correction changes the decision

This is the part that hurts. A study with 28 participants measured across four sessions returns this Mauchly's test: W = .40, χ²(5, N = 28) = 23.57, p < .001, with a Greenhouse-Geisser epsilon of .62. Uncorrected, the session effect is F(3, 81) = 2.83, p = .044. With Greenhouse-Geisser, F(1.86, 50.22) = 2.83, p = .072. With Huynh-Feldt (ε = .66), p = .068. With the lower bound, .104. The effect is identical in all four rows; the only thing that moves is the bar you hold it against.

The sentence you owe the reader does not say "marginally significant" or "approaching significance". It says this: "After applying the Greenhouse-Geisser correction (ε = .62), the effect of session did not reach significance, F(1.86, 50.22) = 2.83, p = .072, η²p = .09." If your threshold was .05, then .072 is outside it, and no adverb moves it back in.

The effect size is still there and it is what carries the honest reading. An η²p of .09 with 28 participants is not the absence of an effect: it is a study that could not separate that effect from noise with the design it had. Write exactly that in the discussion, add the confidence interval for the mean difference, and whoever replicates the study will know how many participants to bring.

Copying the sphericity-assumed row after a significant Mauchly's test is the repeated measures version of p-hacking, and it is detectable from the outside with no access to your data: whole-number degrees of freedom sitting next to a significant Mauchly's test tell the story on their own. If you report the uncorrected row for some deliberate reason, say so in the same sentence and justify it.

Which row to copy in each package

SPSS. You get two tables. "Mauchly's Test of Sphericity" holds W, the approximate chi-square, its df, the p value and the three epsilon columns (Greenhouse-Geisser, Huynh-Feldt, lower bound). "Tests of Within-Subjects Effects" holds the same effect four times over, one row per correction. Copy one whole row — F, effect df, error df and p — and make it the row you announced. Pairing the df from one row with the p value from another happens more often than you would think when this is transcribed by hand.

jamovi and JASP. The correction is a tick box in the assumptions panel, offering "None", "Greenhouse-Geisser" and "Huynh-Feldt". Tick only the one you intend to report: leave all three on and your table prints three rows per effect, which puts you straight back into choosing by p value.

R. Watch this one, because it changes what you are copying without telling you. The aov_ez function in the afex package applies Greenhouse-Geisser by default and prints degrees of freedom that are already corrected, so pasting its output means you have applied the correction whether or not you asked for it, and you have to declare it. The ezANOVA function in the ez package does the opposite: it gives you the uncorrected table plus a separate sphericity corrections table with GGe, p[GG], HFe and p[HF], where epsilon is handed to you but the multiplication of the df is yours to do.

In the table itself, the note is one line: "Degrees of freedom were corrected using the Greenhouse-Geisser estimate of sphericity (ε = .78)." Before you submit, read the full sentence aloud and check three things: that the correction you name matches the df you wrote, that epsilon is visible, and that W and epsilon carry no leading zero. The APA 7 results formatter hands the line back with the italics and the decimals in place, and the full write-up of the test, table and effect size is in how to report an ANOVA in APA 7.

Frequently asked questions

How do you report Mauchly's test of sphericity in APA 7?

In one sentence in the text, with W italicised and no leading zero, the chi-square upright with its df and the sample size inside the brackets, the p value and epsilon: "Mauchly's test indicated that the assumption of sphericity had been violated, W = .58, χ²(5, N = 45) = 23.27, p < .001, ε = .78." Immediately after it, name the correction you applied, so the reader knows where the degrees of freedom in your table came from.

What do I do if Mauchly's test is significant?

You do not rerun anything or switch tests: you correct the degrees of freedom and copy that row. The F statistic and the effect size are unchanged; what changes is the df pair, multiplied by epsilon, and with it the p value. With ε = .78, df of (3, 132) become (2.34, 102.96), and the sentence reads F(2.34, 102.96) = 12.87, p < .001, η²p = .23, with the Greenhouse-Geisser correction.

Why are my corrected degrees of freedom decimals, and should I round them?

They are decimal because they are the original df multiplied by epsilon, a number between 1/(k − 1) and 1. Do not round them to integers: F(2, 103) is a different test against a different reference distribution, and the p value it returns is not the one you computed. The convention is to report them to two decimal places, exactly as your software prints them.

Greenhouse-Geisser or Huynh-Feldt?

Girden's (1992) convention, which most people follow, is Greenhouse-Geisser when epsilon is below .75 and Huynh-Feldt when it is above, because Greenhouse-Geisser overcorrects in that range and costs power. Reporting Greenhouse-Geisser as standard is also defensible, provided you state it in advance. What is not acceptable is computing both and keeping the one that crosses .05.

Do I need Mauchly's test with only two measurements?

No, and it cannot be computed. With two measurements there is a single difference score per person, so sphericity holds by definition: SPSS returns W = 1.000 and leaves the p value blank. The assumption only applies from three levels of the within-subjects factor upwards. If your design is pre-post, add one line saying sphericity does not apply and you have pre-empted the question.

Does the effect size change when I correct for sphericity?

No. The sums of squares are the same before and after, so partial eta squared does not move: SPSS prints the same η²p in all four rows of the table. The correction only alters the degrees of freedom and the p value. If the effect size in your corrected row differs from the sphericity-assumed row, you have copied two different effects.

With W italicised, chi and epsilon upright, the corrected degrees of freedom carrying their two decimals and the correction named in full, your repeated measures ANOVA stops handing anyone an easy reason to send it back. If the harder question is the earlier one — how many measurement occasions your design can carry, whether to correct or move to a mixed model, what belongs in the table — that is far cheaper to settle before data collection than after: in my statistical consulting we go through the analysis plan and write down what gets reported where.