Factorial ANOVA in APA 7: main effects, the interaction and the order you read them in

A factorial ANOVA gives you three results and most manuscripts report two and a half. The usual shape is a long paragraph on the main effect that confirms the hypothesis, a shorter one on the second factor, and a closing line of housekeeping —"the interaction was significant"— that never says in which direction, between which levels, or what that does to the two claims you just made. This is not a writing problem: when the interaction is significant, the main effects you have just described stop meaning what they appear to mean. Here is exactly what gets reported, in what order it is read, how the interaction is broken into simple effects, what the plot does and does not do, and how to put all of it into words without asserting things your data cannot support. For the wider picture of the section, start with how to report results in APA 7.

The three effects, and the order you read them in

A factorial ANOVA does not answer one question, it answers three. The main effect of the first factor: what happens to the outcome when you move between levels of A, averaging over all levels of B. The main effect of the second: the same the other way round. And the interaction: whether the effect of A is the same inside every level of B or changes depending on where you look. All three come out of one model, are tested against the same error term, and are written in that order, which is the order of the table. The only thing that does not follow that order is the interpretation.

The example running through this article is a 2 × 3 between-subjects design with 180 adults referred from primary care for anxiety symptoms, 30 per cell. The first factor is the intervention (brief CBT or structured psychoeducation) and the second is the delivery format (face-to-face one-to-one, face-to-face group, or online). The outcome is the GAD-7 score at the end of the program, 0 to 21, where higher is worse. The six means: CBT one-to-one, M = 6.90 (SD = 3.78); CBT group, M = 7.40 (SD = 4.11); CBT online, M = 10.70 (SD = 3.92); psychoeducation one-to-one, M = 11.10 (SD = 4.24); psychoeducation group, M = 10.80 (SD = 3.85); psychoeducation online, M = 10.60 (SD = 4.05).

The three tests are written with one template: the main effect of intervention, F(1, 174) = 17.58, p < .001, η²_p = .09; the main effect of format, F(2, 174) = 3.21, p = .043, η²_p = .04; and the intervention × format interaction, F(2, 174) = 4.90, p = .009, η²_p = .05. Those 174 error degrees of freedom are not a number you copy without looking: they are 180 participants minus the six cells, and they are identical on all three lines precisely because all three effects are tested against the same error. If the second degree of freedom changes from line to line in your write-up, you have either applied a sphericity correction —which has no business in a between-subjects design— or fitted three different models.

Three test statistics are not enough for a reviewer to reconstruct the analysis. You also need the total N and the n per cell, means and standard deviations for the six cells and for the marginals, an effect size for every term, and a table carrying it: source, SS, df, MS, F, p and η²_p, with the error and total rows. Building that table by hand in Word is an afternoon lost to cell borders; the universal APA table generator takes the output as it comes out of SPSS, R or jamovi and returns it with the three horizontal rules in place. If what you need is the mechanics of the one-way test, that is in how to report an ANOVA in APA 7.

Why the interaction is read first

The main effect of intervention says CBT finished 2.50 points below psychoeducation (M = 8.33 against M = 10.83). That is a real number and it is an average: it comes from pooling the three formats and taking the mean. When the advantage is not the same across all three, that average describes no group in the study. Here CBT wins by 4.20 points one-to-one, 3.40 in groups, and loses by 0.10 online. None of the three resembles 2.50, and the third says the opposite of the headline. That is the whole reason the interaction is interpreted first: it does not invalidate the main effects, it changes what they mean.

The useful distinction is between ordinal and disordinal interactions. In an ordinal one the direction holds across every level and only the size changes: CBT always wins, sometimes by a lot and sometimes barely. There the main effect can still be read, with a caveat attached about the magnitude depending on the other factor. In a disordinal one the rank order reverses: the lines cross and the group that was ahead ends up behind, and the main effect becomes an artifact of averaging two things that are not the same thing. The one in our example sits right on the boundary between the two: it does not cross dramatically, it switches off, and online the two conditions are indistinguishable (10.70 against 10.60). That is enough to make "CBT is superior" a claim that has to be qualified in the same sentence. Rosnow and Rosenthal (1989) put it the other way round, and it is still the clearest way to hold it: the interaction is what remains of the pattern of means after the main effects have been removed, not a label you stick on the whole pattern.

What you do not do is delete the main effects from the section. They are reported with their numbers and qualified immediately afterwards: "the effect of intervention was qualified by delivery format". In this example the most contaminated of the three is the second one. The main effect of format is significant, F(2, 174) = 3.21, p = .043, and it suggests online works worse (M = 10.65 against 9.00 one-to-one and 9.10 in groups). But that difference comes entirely from the CBT arm: within psychoeducation, format moved nothing at all, F(2, 174) = 0.12, p = .888, η²_p = .00. Writing "online delivery produced poorer outcomes" and stopping there attributes to all six groups something that happened in three.

The opposite case also has to be told. If the interaction is non-significant, you report it with its numbers —including the effect size, which is usually the first thing a reviewer asks for— and from there the main effects are read as they stand, no caveats. What is not acceptable is quietly dropping the interaction term from the model, because that changes the error term and the main effects you publish are no longer the ones you computed. Nor is calling an interaction with p = .065 "marginally significant": either you declare it non-significant, or you argue in advance why your study uses a different threshold.

Simple effects: unpacking the interaction without cheating

A simple effect is the effect of one factor within a fixed level of the other: the CBT-versus-psychoeducation difference among the one-to-one participants only. A post hoc test is a different animal: it compares levels of one factor with the other collapsed, and when the interaction is significant that collapsed comparison is exactly the one you no longer care about. What nobody should ever do is run t tests on all six cells pairwise: that is fifteen comparisons, most of them answering no hypothesis at all, with an error rate nobody is watching.

In this design the simple effects that answer the question are three, one per format, with a Bonferroni correction over that family of three. One-to-one, CBT finished 4.20 points lower, F(1, 174) = 16.54, p < .001, d = 1.05. In groups, 3.40 points, F(1, 174) = 10.84, p = .004 after correction, d = 0.85. Online the difference vanishes, F(1, 174) = 0.01, p > .999 after correction (.920 uncorrected), d = −0.03, with d defined as the difference favoring CBT divided by the square root of the model's MS error (4.00). The same data read the other way give the other half of the story: format mattered within CBT, F(2, 174) = 7.99, p < .001, η²_p = .08, and did not matter within psychoeducation, F(2, 174) = 0.12, p = .888.

Here is the trap that gets flagged most often in review. An effect being significant in one group and non-significant in another is not an interaction: it is a comparison of two p values, and the difference between significant and not significant is not itself significant (Gelman and Stern, 2006). What the interaction tests is the difference between differences, and that is tested directly. In the example, the CBT advantage averaged across the two face-to-face formats (3.80 points) against the online advantage (−0.10) gives a one-degree-of-freedom contrast, F(1, 174) = 9.51, p = .002. That is the number holding up the sentence "face-to-face and online do not behave the same way"; the simple effects only describe it.

Two technical decisions worth declaring. The first is the error term: simple effects can use the MS error of the full model, which carries 174 degrees of freedom and therefore more power, or you can split the file and run separate analyses, which costs df and gives slightly different numbers. With homogeneous variances the first is preferable, and you say so. The second is the correction and the family it applies to: three comparisons are not fifteen, and Bonferroni over three is cheap. An effect size goes with every term, and if you are unsure which index to use and how to typeset it, that is settled in how to report partial eta squared in APA 7.

The interaction plot: which axis, which bars, and what to avoid

An interaction plot puts the outcome on the vertical axis, the levels of one factor along the horizontal, and one line (or one set of bars) per level of the other. Which factor goes along the bottom is not a matter of taste: it decides the sentence the reader will take from the picture. Put format on the horizontal axis with one line per intervention and the plot says "CBT pulls away face-to-face and converges with psychoeducation online", which is the study hypothesis. Swap them and it says the same thing in a way that takes longer to see. Choose the axis that answers your question, and do not print both versions.

Error bars are compulsory and you have to declare what they are. Mean ± 1 standard error, mean ± 1 standard deviation and a 95% confidence interval produce pictures of very different sizes from identical data, and the figure note has to say which one is on the page. It is also worth being clear about what can and cannot be read off them. With 95% confidence intervals for two independent groups, overlap does not mean the test is non-significant —that is the standard misreading— while non-overlap does imply significance at .05, and by a stricter margin than it looks. With ±1 standard error bars neither inference holds: two bars that fail to touch can sit on a test with p = .12. And in a factorial the test runs against the MS error pooled over all six cells, not against the standard deviations of two of them, so the eyeball rule is a rough guide at best. The test is already in the text; the bars are there to convey precision, not to replace it.

The vertical axis is where people mislead without meaning to. If the GAD-7 runs from 0 to 21 and you crop the axis to 6–12, a 0.10-point difference fills half the figure and reads as a chasm. Cropping is defensible when the usable range of the scale is narrow, but then you say so in the note. Out go 3D, drop shadows, color gradients and background grids: the figure has to stay legible printed in black and white, so every level gets its own marker (filled circle, open triangle) as well as its own color, and the legend sits inside or beside the plot area, never in the note.

What the plot never does is prove the interaction. In a sample, lines are never exactly parallel: with noise and thirty people per cell there is always some fanning out, and the eye reads it as a pattern. The claim is carried by the test and illustrated by the figure. That is why a figure showing a non-significant interaction is perfectly publishable —and honest— as long as the text says the lines do not diverge more than chance would predict. The figure gets a number, an italicized title beneath it, axes labeled with the scale and its units, and a note giving the n per cell and the content of the bars.

How to describe an interaction in words

The shape that works is always the same: the effect of X depended on Z, followed by the effect of X within each level of Z, with numbers. A model paragraph using our data: "The effect of intervention was qualified by delivery format, F(2, 174) = 4.90, p = .009, η²_p = .05. Participants receiving CBT reported fewer anxiety symptoms than those receiving psychoeducation in the one-to-one format (M = 6.90 against M = 11.10; F(1, 174) = 16.54, p < .001, d = 1.05) and in the group format (M = 7.40 against M = 10.80; F(1, 174) = 10.84, p = .004, d = 0.85), whereas online the two conditions did not differ (M = 10.70 against M = 10.60; F(1, 174) = 0.01, p > .999). A contrast comparing the face-to-face advantage with the online one confirmed the pattern, F(1, 174) = 9.51, p = .002." Three sentences, and the reader can rebuild the whole thing.

Sentences that say nothing. "A significant interaction was found between intervention and format" and full stop: it reports that a number exists, not what happened to anybody. "The two factors interacted with each other" is the same thing with more words. "The interaction was significant, indicating that the two factors are related" confuses an interaction with an association between the factors, which in a balanced design is exactly zero by construction. All three are fixed the same way: put the simple effects and their means directly behind the test.

Sentences that say something false. "The intervention worked face-to-face but not online", resting on two different p values with no test of the difference between differences, is the textbook version of the error Gelman and Stern documented. "The interaction explains 5% of the variance" turns η²_p = .05 into a share of a total it is not: the denominator of partial eta squared excludes the variance of the other effects, which is why the η²_p values of a factorial design can add up to more than 1. "The main effects were canceled out by the interaction" describes something that did not happen: the main effects are still there with their F and their p; what changed is that they can no longer be read alone.

Two further cautions. Reading an interaction is symmetric: saying the effect of intervention depends on format and saying the effect of format depends on intervention are mathematically the same claim. Both are true, so pick the one that answers your research question and tell it once; telling both doubles the paragraph and adds nothing. And causal wording stops where the design stops: if format was randomly allocated, "format modified the effect of the intervention" is legitimate; if one factor is a measured grouping variable —sex, prior diagnosis, education— the correct wording is "the effect differed by", and why it differed goes in the discussion, flagged as a hypothesis.

What slips through when the cells are unequal

With 30 people per cell the design is balanced and the three sums of squares are independent: the order in which the terms enter the model is irrelevant. The moment the cells become unequal —and they always do, through attrition— the factors overlap and order starts to matter. SPSS defaults to Type III sums of squares, which evaluate each effect with the others already in the model. R's aov() uses Type I, sequential, and with unbalanced cells it returns different F values for the same data depending on the order you wrote the factors in the formula. To reproduce Type III in R you need car::Anova(model, type = 3) and, first, orthogonal contrasts via options(contrasts = c("contr.sum", "contr.poly")); without that step the Type III output is wrong and says nothing about it. Whichever you choose, declare it in the method.

Unbalanced cells also change the marginal means. A weighted marginal mean averages people and an unweighted one averages cells, and with 45 participants in one cell and 18 in another they are not the same number. What belongs next to a Type III main effect are the estimated marginal means, unweighted —EMMEANS in SPSS, emmeans in R— because those are the ones matching the test you just ran. Placing a weighted descriptive mean beside a Type III F is the commonest silent inconsistency in factorial results sections.

Homogeneity of variance is checked across the six cells, not on each factor separately: in the example, Levene's test, F(5, 174) = 0.62, p = .685. With equal cell sizes ANOVA tolerates a fair amount of heterogeneity; with unequal cells it does not, and the direction of the bias depends on whether the larger cells are the more variable ones or the less. The awkward part is that the Welch correction, which solves this for a one-way design, has no factorial equivalent in the usual menus: the sensible outputs are heteroscedasticity-robust standard errors or a trimmed-means ANOVA. Whichever you choose, Levene's test is reported with both its degrees of freedom and its p, even when it comes out non-significant.

Two closing warnings. The first is about power: detecting an interaction takes considerably more sample than detecting a main effect of the same apparent size, because an interaction is estimated on differences between differences and the errors accumulate. A study powered for the main effect is almost never powered for the interaction, and that is why so many "nearly significant" interactions fail to replicate. The second is about restraint: a three-factor design produces seven effects and a three-way interaction that almost nobody can describe in prose. If your hypothesis did not ask for it, do not make it the headline. And if one of your factors is a time point measured in the same people, this is no longer the right model: the mechanics of mixed designs are in repeated-measures ANOVA in APA 7.

Already drafted it? Paste your Results section into Review one section before it goes to your supervisor or the journal. You get the first thing a reviewer would object to, with your own sentence quoted.

The normality check on the residuals comes before any of this; how to read the Shapiro-Wilk test is covered separately.

Frequently asked questions

Can you interpret main effects when the interaction is significant?

You always report them, but you never interpret them alone. A main effect is an average taken across the levels of the other factor, and if the effect changes from level to level that average describes no group in the study. In this article's example CBT beats psychoeducation by 2.50 points on average, but that average pools advantages of 4.20, 3.40 and −0.10. The correct move is to give the main effect with its F and p and qualify it in the same sentence —"qualified by delivery format"— followed by the simple effects.

How do you write an interaction in APA 7?

With the same template as any other ANOVA term, naming both factors with a multiplication sign: intervention × format, F(2, 174) = 4.90, p = .009, η²_p = .05. F is italicized, the two degrees of freedom sit in brackets tight against the letter and separated by a comma, the p value is italic with no leading zero, and partial eta squared is set in roman —because eta is a Greek letter— also with no leading zero. What follows, without exception, is a description of the simple effects.

Which effect size do you report for a factorial ANOVA?

Partial eta squared, η²_p, is the usual choice, one per model term. Two things are worth keeping straight when you write it: it is not the percentage of total variance explained, because its denominator excludes the variance of the other effects, which is why the η²_p values of a factorial design can sum to more than 1 with no arithmetic error anywhere. It also overestimates in small samples, so adding omega squared is a cheap improvement. For two-group simple effects, Cohen's d communicates better than F.

What is the difference between simple effects and post hoc tests?

A simple effect compares levels of one factor within a fixed level of the other: CBT against psychoeducation among the one-to-one participants only. A post hoc test compares levels of one factor with the other collapsed, which is precisely the comparison the interaction has made uninformative. When the interaction is significant you ask for simple effects; when it is not and a main effect with three or more levels is, you ask for post hoc tests. What is never appropriate is the fifteen pairwise comparisons among six cells.

Is an interaction plot compulsory?

The manual does not require it, but reviewers are grateful for it and many journals expect it when the interaction is the headline result, because a pattern of six means is faster to see than to read. The condition is that the figure follows the rules: error bars declared in the note, no silent cropping of the vertical axis, no 3D, legible in black and white, and the n per cell given. It does not replace the text either: the claim is carried by the test, not by the drawing.

What do I do if the interaction is not significant?

Report it anyway, with its degrees of freedom, its p and its effect size, then interpret the main effects normally, since they now describe what they appear to describe. What you cannot do is drop the interaction term from the model without saying so, because that reassigns its degrees of freedom to the error and changes the main effects you are about to publish. And an interaction with p = .065 is not "marginally significant": it is non-significant in a study that probably lacked the power to detect it.

An interaction told properly is worth two studies: it says who an intervention works for and who it does not, which is the question anyone planning to use it actually has. If you are closing the results section of a thesis or a paper and would rather have someone go through the simple effects with the data in front of them, our statistical consultancy works through it with you, cell by cell, with the target journal on the table.