Two-Way ANOVA: Main Effects, Interaction, APA Reporting

A two-way ANOVA tests how two categorical independent variables (factors) affect one continuous outcome. It answers three questions at once: does factor A matter (main effect of A), does factor B matter (main effect of B), and does the effect of A depend on the level of B (the A × B interaction)? The interaction is usually the most interesting result, and it changes how you interpret everything else.

This guide works through a 2 × 2 example by hand, shows how to read the interaction plot, follows up with simple effects, and gives an APA 7 reporting template. If you are new to ANOVA, start with the one-way ANOVA guide, which covers the F ratio and post hoc tests in more detail.

When to use a two-way ANOVA

• Two categorical independent variables (for example, teaching method and prior knowledge level).
• One continuous dependent variable (for example, exam score).
• Independent groups: each participant appears in exactly one cell (a between-subjects factorial design).

A design with factor A at 2 levels and factor B at 3 levels is called a 2 × 3 design and has 6 cells. If one factor is measured within the same participants (such as pre/post), you need a mixed ANOVA instead.

Assumptions

The assumptions are the same as for one-way ANOVA, applied to the cells:

1. Independence of observations.
2. Normality of residuals within each cell (check Q-Q plots of residuals).
3. Homogeneity of variance across cells (Levene's test on all cells).
4. No extreme outliers.

Balanced designs (equal n per cell) make the analysis more robust and the sums of squares unambiguous. With unequal cell sizes, software must choose a type of sums of squares; Type III is the default in SPSS and is the usual choice when interactions are in the model. State which type you used.

The three effects

• Main effect of A: Do the marginal means of A differ, averaging over B?
• Main effect of B: Do the marginal means of B differ, averaging over A?
• A × B interaction: Is the difference between levels of A the same at every level of B? If not, there is an interaction.

An interaction means "it depends." For example, a new teaching method may help students with low prior knowledge a lot and students with high prior knowledge only a little.

Worked example: a 2 × 2 design

A researcher studies exam scores with two factors: Method (Lecture vs. Flipped) and Prior knowledge (Low vs. High). There are 4 students per cell (N = 16).

Marginal means:

Sums of squares

Each marginal mean is based on 8 students; each cell on 4.

• Method (A): 8 × [(68 − 72)² + (76 − 72)²] = 8 × 32 = 256
• Prior knowledge (B): 8 × [(67 − 72)² + (77 − 72)²] = 8 × 50 = 400
• Cells (between all four cells): 4 × [(61 − 72)² + (75 − 72)² + (73 − 72)² + (79 − 72)²] = 4 × 180 = 720
• Interaction (A × B): SS cells − SS A − SS B = 720 − 256 − 400 = 64
• Error (within cells): the sum of squared deviations from each cell mean = 20 + 14 + 20 + 10 = 64
• Total: 720 + 64 = 784

ANOVA table

Each effect has 1 df (2 − 1 levels; 1 × 1 for the interaction). Error df = N − number of cells = 16 − 4 = 12, so MS error = 64 / 12 = 5.33.

All three effects are significant. Because the interaction is significant, the main effects need careful interpretation.

You can check single-factor comparisons with the ANOVA calculator; for the full factorial model, use statistical software.

Reading the interaction plot

An interaction plot puts one factor on the x-axis (here, Prior knowledge: Low, High), the outcome on the y-axis (mean exam score), and draws a separate line for each level of the other factor (Lecture, Flipped).

In this example:

• The Lecture line rises steeply from 61 (Low) to 75 (High), a 14-point gain.
• The Flipped line rises more gently from 73 (Low) to 79 (High), a 6-point gain.
• The Flipped line sits above the Lecture line at both levels, but the gap narrows from 12 points (Low) to 4 points (High).

Non-parallel lines suggest an interaction. Parallel lines mean the effect of method is the same at every level of prior knowledge (no interaction). Lines that converge, diverge, or cross indicate that the effect of one factor depends on the other. Crossing lines (a "disordinal" interaction) mean the direction of the effect reverses; lines that do not cross, as here, show an "ordinal" interaction where only the size changes.

The plot is a visual aid. Whether the lines are "different enough" is decided by the interaction F test.

Interpreting main effects when there is an interaction

The main effect of Method says that, averaged over prior knowledge, Flipped students scored 8 points higher (76 vs. 68). That average hides the fact that the advantage is 12 points for low-prior students and only 4 points for high-prior students. When an interaction is significant, report and interpret the main effects cautiously and focus on the simple effects.

Simple effects

Simple effects test the effect of one factor at each level of the other. Here, the natural question is: does Method matter at each level of Prior knowledge? Using the pooled error term (MS error = 5.33, df = 12):

• Low prior knowledge: Flipped (73.0) vs. Lecture (61.0), difference = 12.0, F(1, 12) = 54.00, p less than .001.
• High prior knowledge: Flipped (79.0) vs. Lecture (75.0), difference = 4.0, F(1, 12) = 6.00, p = .031.

The flipped classroom helps both groups, but the benefit is three times larger for students with low prior knowledge.

A note on multiple testing. You ran two simple-effects tests. With a Bonferroni correction (α = .05 / 2 = .025), the high-prior effect (p = .031) would no longer be significant. Decide your correction before running the analysis and state it in the methods. With more than two levels in a factor, a significant simple effect is followed by pairwise comparisons (for example, Tukey or Bonferroni-adjusted).

Effect size: partial eta squared

In factorial designs, the standard effect size is partial eta squared:

partial η² = SS effect / (SS effect + SS error)

• Method: 256 / (256 + 64) = .80
• Prior knowledge: 400 / (400 + 64) = .86
• Interaction: 64 / (64 + 64) = .50

Partial η² values within one model do not sum to 1, because each uses its own denominator. Classic eta squared (SS effect / SS total) gives smaller values here: .33, .51, and .08. SPSS reports partial eta squared by default, so say which one you report. These values are large because the example data have little within-cell noise; real studies usually show much smaller effects.

How to report a two-way ANOVA in APA 7

Report each effect with F, both df, the exact p, and partial η², then describe the interaction with simple effects and cell means.

Also include a table of cell means and SDs, and an interaction plot with error bars (state whether bars show SE or 95% CI).

Common mistakes

• Interpreting main effects alone when the interaction is significant. The phrase "qualified by a significant interaction" exists for a reason.
• Reading the plot without the test. Slightly non-parallel lines are expected from sampling error.
• Running separate one-way ANOVAs for each factor. That ignores the interaction and uses the wrong error term.
• Not stating the sums-of-squares type with unbalanced cells.
• Calling partial η² "eta squared." They are different statistics.
• Tiny cells. With many factor levels, cells can become very small, which weakens every test. Plan the sample size with an a priori power analysis.

FAQ

What is the difference between one-way and two-way ANOVA?

One-way ANOVA has one factor; two-way ANOVA has two factors and also tests their interaction. See the one-way ANOVA guide for the single-factor case.

What does a significant interaction mean?

The effect of one factor depends on the level of the other. On a plot, the lines are not parallel. Follow it up with simple effects.

What if the interaction is not significant?

Interpret the main effects directly. Some researchers then drop the interaction term and refit, but many keep the full model as planned; either way, report what you did.

Do I need post hoc tests in a 2 × 2 ANOVA?

Not for the main effects, because each factor has only two levels. You do need follow-up tests (simple effects) to unpack a significant interaction. For factors with three or more levels, add post hoc comparisons.

Should I report partial eta squared or eta squared?

Partial eta squared is the most common in factorial ANOVA and is what SPSS reports. Label it clearly. Generalized eta squared is another option when comparing effects across different designs.

From data to a draft paper

If you have a complete dataset with your factors and outcome, Data2Paper's Generate Paper can produce a paper draft with the statistical analysis, tables, figures, and references. The first half is a free preview; the full draft is a one-time $19.90. It is AI-generated, so verify every number and interpretation before you use it.