How to Report a MANOVA in APA 7

You measured three outcomes on every participant — depression, anxiety and stress — somebody warned you that three separate ANOVAs inflate the error rate, and you ran a MANOVA. Then the output arrived: four multivariate criteria whose values look nothing like one another, an F whose degrees of freedom match nothing in your design, a Box's M with a p below .001, and three univariate tables underneath. Most manuscripts settle that by reporting one line of it and hoping nobody asks. Here is what each of those numbers is, which one you report, and what to do with the ANOVAs that follow. If your design has one outcome rather than several, start at how to report an ANOVA in APA 7.

Which numbers go in

Four things: the multivariate criterion with its value, its F with both degrees of freedom, the p value and a multivariate effect size. The full template is Wilks' Λ = .78, F(6, 230) = 5.07, p < .001, η²_p = .12. Underneath it goes a second layer: one univariate ANOVA per dependent variable, each with its own F, p and effect size. A MANOVA reported without that second layer tells the reader that something differs between the groups without saying what.

The degrees of freedom are where this analysis goes wrong most often. The first one is easy: number of dependent variables × (number of groups − 1), so three outcomes and three groups give 6. The second is not the N − k = 117 of your error term; it is built by Rao's transformation out of the sample size, the number of outcomes and the number of groups, and here it comes out at 230. Copying 117 into that slot because it is the number you recognize is the most repeated error in this whole family of tests.

That F is called approximate because it usually is, but it is worth knowing when it is not. The transformation returns an exact F in four special cases — one dependent variable, two dependent variables, two groups or three groups — and an approximation everywhere else. The example in this article lands in one of the exact cases because it has three groups, and its error df still looks nothing like yours: exact does not mean familiar. Pillai's F, which is exact only when there is a single dimension to test, genuinely is an approximation here.

Italics follow the usual rule, letter by letter. Λ is Greek, so it stays upright; Pillai's V, Hotelling's T, the F and the p are Latin letters and are italicized; η²_p is Greek and upright. Leading zeros split the same way, and here they split in a way that catches people out. Wilks' Λ runs from 0 to 1, so it cannot exceed 1 and drops the zero: Λ = .78. Pillai's V runs from 0 to s, the number of dimensions being tested, so with two or more it can exceed 1 and keeps it: V = 0.23. Hotelling's trace has no upper bound at all. Every univariate result also carries its own partial eta squared, written the same way as in any ANOVA: upright and with no leading zero.

Wilks' lambda or Pillai's trace

The four criteria in your output are four ways of squeezing the same set of eigenvalues into a single number. A MANOVA looks for up to s = min(number of outcomes, number of groups − 1) linear combinations of your dependent variables that separate the groups, each with its own eigenvalue. Wilks' Λ is a ratio of determinants and behaves like an inverted effect size: the smaller it gets, the more the groups differ. Pillai's trace adds up λ/(1 + λ) across the dimensions, the Hotelling-Lawley trace adds up the raw eigenvalues, and Roy's largest root keeps only the biggest one and throws the rest away.

Olson (1974) compared six multivariate tests under violations of the assumptions, and Pillai's trace held its nominal alpha best when the covariance matrices were unequal and the cells were of different sizes. Roy's largest root behaved worst, which follows from what it does: resting the whole test on one dimension makes it powerful when the group differences really do lie along a single dimension, and badly optimistic when they do not. That is where the standard advice comes from — Pillai when the assumptions look shaky, Wilks when they look fine and you want the number everybody recognizes.

Whichever you choose, report one. Picking the criterion after seeing which one crossed .05 is fishing with four hooks, and the decision costs nothing if you make it before you look. In practice the criteria rarely disagree when the assumptions hold: the same data that give Λ = .78, F(6, 230) = 5.07 give V = 0.23, F(6, 232) = 5.02. When the four do disagree, that is information — usually that one dimension dominates the rest, and it deserves a sentence.

With two groups the question disappears. There is only one dimension to find, s = 1, and all four criteria give the same F: with 80 participants, two groups and three outcomes, Λ = .85, F(3, 76) = 4.47, p = .006, η²_p = .15. Notice that the error df is now N − k − (number of outcomes) + 1 = 76, and that the four criteria agree to the last decimal, which stops happening as soon as there are more than two groups. This is Hotelling's T² under another name, the multivariate t test.

Box's M, and why it comes out significant almost every time

Box's M tests whether the variance-covariance matrices are the same in every group. It is the multivariate cousin of Levene's test, and it asks for a great deal more: Levene compares k variances, while Box's M compares k whole matrices, each holding the variances of every outcome plus every covariance between them. With three outcomes and three groups that is six unique elements per matrix and 12 degrees of freedom for the test.

It comes out significant in almost any real psychological dataset, and that is mostly the test's fault rather than yours. Box's M is acutely sensitive to departures from multivariate normality and, like any significance test, it gains power with the sample, so past a few hundred participants a difference between matrices that would not change a single conclusion gets flagged at p < .001. The error degrees of freedom give the game away: 120 participants produce Box's M = 42.37, F(12, 66339.00) = 3.40, p < .001. An error df of 66,339 from 120 people is a large-sample approximation, not a count of anything.

The usual convention is to read Box's M at α = .001 instead of .05, which Tabachnick and Fidell (2019) recommend precisely because of that sensitivity. Two things then matter more than the p value. The first is whether your cells are equal in size: with equal or near-equal n, MANOVA is robust to heterogeneous covariance matrices, and noting the result and carrying on is enough. The second, when the cells are unequal, is the direction of the imbalance: larger variances in the smaller group make the test liberal, so the significance you find may not be there at all; larger variances in the larger group make it conservative, which costs power but does not invent effects.

So it is reported, not hidden, and it is reported with its consequence attached: “Box's M was significant, M = 42.37, F(12, 66339.00) = 3.40, p < .001; because the cells were equal in size (n = 40) and Pillai's trace is the more robust criterion under heterogeneous covariance matrices, it is the one reported below.” What you cannot write is “the assumption of homogeneity of covariance matrices was met” with no numbers, and what makes no sense is abandoning the MANOVA for a rank-based test because Box's M came out significant. The rest of the assumptions live in the assumptions checklist.

The multivariate effect size

The multivariate partial eta squared comes straight out of the criterion you reported. From Wilks it is 1 − Λ^1/s, with s = min(number of outcomes, number of groups − 1); here s = 2, so 1 − .78^1/2 = .12. From Pillai it is V/s = 0.23/2 = .115, which rounds to the same .12. They are two different quantities and they do not have to agree — they coincide exactly only when every dimension contributes equally — so do not use one to check the other: report the one belonging to the criterion you chose.

What it means is not what most people assume. It is the proportion of variance accounted for in the linear combinations of your dependent variables, averaged across the s dimensions — not the mean of the univariate effects, and certainly not their sum. In the worked example the multivariate η²_p of .12 sits below the univariate .14 for depression, and there is no contradiction in that: an average across the dimensions that separate the groups is not the same thing as the single outcome that separates them best.

Cohen's (1988) conventions — .01 small, .06 medium, .14 large — get pasted next to multivariate criteria all the time, and they were never derived for them. Treat them as what they are: rules of thumb for the univariate case, useful for orienting a reader who has no other reference. The comparison that carries weight is against the effects reported in your own literature, where an η²_p of .12 for a three-session intervention may be unremarkable or remarkable depending on the field.

Report an effect size for the multivariate test and one for every univariate test, without exception. A MANOVA table with three F values, three p values and no effect sizes is one of the most common reasons a reviewer sends this analysis back. If you are building that table by hand, the universal APA table generator lays out the rows and the note for you.

The follow-up ANOVAs, and their correction

The standard route after a significant multivariate test is one univariate ANOVA per outcome, each corrected. With three dependent variables, Bonferroni divides .05 by three and puts the bar at .017. In the worked example condition affected depression, F(2, 117) = 9.84, p < .001, η²_p = .14, and anxiety, F(2, 117) = 4.61, p = .012, η²_p = .07, but not stress, F(2, 117) = 1.32, p = .271, η²_p = .02. Notice how tight that middle one is: with five outcomes the bar would sit at .010 and anxiety would fall on the wrong side of it.

Bonferroni is the blunt version and Holm's sequential procedure is the sharp one: it controls the same familywise error rate with more power, by ordering the p values and testing them against .05/3, then .05/2, then .05. Either is defensible. What is not defensible is choosing the correction after seeing which outcomes it saves, so name the procedure in your analysis plan and stay with it.

There is a widespread idea that a significant MANOVA protects the univariate tests and licenses them uncorrected. It does not. The multivariate test asks whether the groups differ on some combination of the outcomes, the univariate tests ask about each outcome separately, and the first does not control the familywise error rate of the second (Bray and Maxwell, 1985; Huberty and Morris, 1989). The deeper version of the point is worth sitting with: if your question really was three separate questions, three corrected ANOVAs are the honest analysis and the MANOVA in front of them is decoration.

The follow-up that genuinely answers the multivariate question is a descriptive discriminant analysis, which tells you which combination of outcomes separates the groups and how much each one contributes (Huberty and Morris, 1989). It is the right tool when the outcomes are conceptually a set — three subscales of one construct — rather than three independent questions that happen to share a sample. And whichever route you take, a third layer is still waiting: within each significant outcome, the pairwise comparisons between groups need a correction of their own, usually Tukey.

The whole thing, written out

Here is the complete example with nothing left implicit: “Differences between the three conditions on the DASS-21 subscales were tested with a one-way MANOVA (n = 40 per condition). Box's M was significant, M = 42.37, F(12, 66339.00) = 3.40, p < .001; because the cells were equal in size, Pillai's trace is reported. The multivariate effect of condition was significant, V = 0.23, F(6, 232) = 5.02, p < .001, η²_p = .12. Univariate ANOVAs on each subscale were tested against a Bonferroni-corrected alpha of .017. Condition affected depression, F(2, 117) = 9.84, p < .001, η²_p = .14, and anxiety, F(2, 117) = 4.61, p = .012, η²_p = .07, but not stress, F(2, 117) = 1.32, p = .271, η²_p = .02.” Five sentences, every number the reader needs, and no table required for a design this small.

In a larger design that same content becomes a table: one row per dependent variable, columns for the M and SD of each group, then F, df, p and η²_p. The multivariate line goes in the table note or in a first row set apart, never buried among the univariate rows as though it were one more outcome. The corrected alpha belongs in the note as well: a reader looking at p = .012 needs to know it is being judged against .017 rather than .05.

Four things get this analysis sent back again and again. An error df copied from the univariate row, so the multivariate F arrives as F(6, 117) instead of F(6, 230). A multivariate test with no univariate results underneath it. Uncorrected follow-ups justified by the significant MANOVA. And all four criteria reported side by side, which reads as indecision rather than thoroughness. None of them takes more than a minute to fix once you know where to look.

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Frequently asked questions

How do you report a MANOVA in APA 7?

With the multivariate criterion, its F and both degrees of freedom, the p value and a multivariate effect size: Wilks' Λ = .78, F(6, 230) = 5.07, p < .001, η²_p = .12. Underneath go the univariate ANOVAs, one per dependent variable, each with its own F, corrected p and effect size. Λ stays upright because it is a Greek letter, and it drops its leading zero because it cannot exceed 1.

Should I report Wilks' lambda or Pillai's trace?

Pillai's trace when the assumptions look shaky — a significant Box's M, cells of different sizes — because it holds its nominal alpha better under those conditions (Olson, 1974). Wilks' Λ when the assumptions look fine and you want the criterion most readers recognize. Report one, chosen before you see the output: the two rarely disagree, and on the same data Λ = .78, F(6, 230) = 5.07 and V = 0.23, F(6, 232) = 5.02 say exactly the same thing.

What do I do if Box's M is significant?

Usually nothing dramatic. Box's M is very sensitive to non-normality and gains power with the sample, so it is significant in almost any real dataset; the convention is to read it against α = .001 rather than .05. If your cells are equal or nearly equal in size, MANOVA is robust to the violation: report the result, switch to Pillai's trace and carry on. If the cells are unequal, the direction matters — larger variances in the smaller group make the test liberal, larger variances in the larger group make it conservative.

What are the degrees of freedom in a MANOVA?

The first is the number of dependent variables multiplied by the number of groups minus one: three outcomes and three groups give 6. The second is not your error term. It comes out of Rao's transformation and lands somewhere unfamiliar — 230 in a study with 120 participants, where the univariate error df is 117. Copy it from the multivariate row of the output; if you see the same second df on the multivariate line and on the univariate lines, one of them is wrong.

Do the follow-up ANOVAs after a MANOVA need correcting?

Yes. A significant multivariate test does not control the familywise error rate of the univariate tests that follow (Huberty and Morris, 1989), so the idea of a “protected” F does not hold here. With three dependent variables, Bonferroni puts the bar at .05/3 = .017, and Holm's sequential version controls the same rate with more power. Whichever you use, name it in the analysis plan rather than after seeing which outcomes it rescues.

Can I run a MANOVA with only two groups?

Yes, and with two dependent variables as well. What changes is that the multivariate test simplifies: with two groups there is a single dimension, all four criteria give the identical exact F, and the analysis is Hotelling's T² under another name (Λ = .85, F(3, 76) = 4.47, p = .006 with 80 participants and three outcomes). The real question is not how many outcomes you have but whether they belong together conceptually; if they are three unrelated questions, run three corrected ANOVAs and skip the MANOVA.

With the criterion, its F and both degrees of freedom, the p value, a multivariate effect size and the corrected univariate ANOVAs underneath, your MANOVA is reported the way APA 7 asks. When the design gets harder — unequal cells with a significant Box's M, covariates in a MANCOVA, or outcomes that would be better served by a discriminant analysis than by three ANOVAs — that is what my statistical consulting is for.