🪟 Two-Way ANOVA

1. The big idea — two factors, three tests

Back in Unit 13, one-way ANOVA let you compare three or more groups on one grouping variable — that variable is called a factor. A two-way ANOVA takes the natural next step: it studies two factors at the same time. The word "two-way" literally means "two factors." A 2×2 design (say it "two-by-two") is the simplest case — two factors, each with two levels. Two levels times two levels gives four groups, one for every combination.

Here's the payoff and the twist. Because you're crossing two factors, you don't get one answer — you get three, and each one gets its own F test:

Two main effects plus one interaction — three separate F ratios, each tested for significance on its own. That third one, the interaction, is new this unit, and it's the star of the show. Crucially, the interaction is not "the two main effects added together" — it's a genuinely separate question that the first two tests can't answer.

2. Cell means and marginal means — the raw material

Everything in a two-way ANOVA is built from averages of a 2×2 grid. Two kinds of averages matter, and keeping them straight is most of the battle.

• A cell mean is the average of one combination of the two factors — one inner box of the grid. A 2×2 has four cell means.
• A marginal mean sits in the margin — the edge — of the table. A row marginal averages across a whole row (collapsing the column factor); a column marginal averages down a whole column (collapsing the row factor). The marginals are exactly what the main effects compare.

Here's the layout. The four inner numbers are cell means; the bold edge numbers are marginal means:

Read it like a map. The main effect of Factor A compares the two row marginals (far-right column). The main effect of Factor B compares the two column marginals (bottom row). The interaction lives inside the four cells — it asks whether the gap between cells changes from one row to the next. Same four numbers, three different questions.

3. Main effects — the "overall" effects

A main effect is the effect of one factor, averaging over the other. It's the marginal-means comparison, in words:

• Main effect of Factor A (the rows): ignore Factor B for a second. Lump everyone together by Factor A only — that's the two row marginals. Do they differ? If yes, Factor A moved scores overall.
• Main effect of Factor B (the columns): ignore Factor A. Lump everyone together by Factor B only — the two column marginals. Do those differ? If yes, Factor B moved scores overall.

Main effects are the big picture — one factor at a time, the other one blurred out. Each gets its own F ratio (the same logic as Unit 13: between-groups variance over within-groups variance), and you decide each one with the familiar rule — if p < .05, reject the null for that effect; otherwise fail to reject it. "Fail to reject" — never "accept" — because absence of evidence isn't evidence of absence.

4. The interaction — the "it depends" effect

An interaction means the effect of one factor depends on the level of the other. That's the whole definition: it depends. The factors aren't acting alone — they're teaming up. In the table, you spot it by asking whether the gap between the two cells in one row is different from the gap in the other row. If Factor B helps a lot in Row 1 but barely at all in Row 2, B's effect depends on A — that's an interaction.

And here's the subtle part that trips people up: an interaction can hide a main effect. If one group's scores rise while another's fall by about the same amount, the two changes cancel when you average them — so the main effect can look flat (non-significant) even though something dramatic is happening. The story isn't "nothing happened." The story is "it depends," and only the interaction term tells you that.

5. ★ Reading the interaction graph — the key skill

Here's the trick that makes all three results pop out of one picture. Put one factor on the x-axis and draw one line per group of the other factor. Each plotted point is a cell mean. Now read it in three quick looks:

1. Main effect of the x-axis factor? Do the lines, on average, slope up or down as you move across the x-axis? If they slope, that factor moved scores overall.
2. Main effect of the "lines" factor? Is one line generally higher than the other? If yes, that group scored higher overall.
3. Interaction? Are the lines non-parallel? Parallel lines = no interaction. Lines that cross or fan apart = an interaction.

Here's the picture itself. On the left, the two lines cross — the effect of Factor B flips direction depending on Factor A. That non-parallelness is the interaction. On the right, for contrast, two perfectly parallel lines: there's a gap (a main effect) but the lines stay in step, so no interaction.

6. The centerpiece — the Two-Way ANOVA calculator

Here's the grand-finale calculator, embedded right here. Hit "Load class data (gender × threat)" to drop in our stereotype-threat study, then Run. You'll get all three F tests at once — the two main effects and, the one to watch, the interaction. The worked example below walks you through exactly what you'll see.

🪟 Worked example — read a 2×2 table cold

Before the famous study, let's build the skill with clean numbers. Imagine a memory experiment crossing Factor A = study method (Massed vs. Spaced) with Factor B = test delay (Immediate vs. Delayed). The outcome is the number of words recalled. Here are the four cell means — predict each step before you reveal it.

🪟 The textbook interaction — stereotype threat

Now the example to remember for the exam. In a classic line of research (Spencer, Steele, & Aronson-style), people take a hard math test. Half are quietly reminded beforehand of a negative stereotype about their group's math ability — that's the threat condition — and half are not — the control condition. The two factors are gender (Woman, Man) — our row factor, shown as the two lines — and threat condition (Control, Threat) — our column factor, on the x-axis; the outcome is the math score.

Here are the four cell means — the average math score for each combination — from our N = 200, with the marginal means added on the edges:

Now picture the line graph — put condition on the x-axis (Control → Threat) and draw one line per gender:

• The Women's line drops sharply (76.4 → 68.9). Under threat, women's scores fall about 7.5 points.
• The Men's line stays flat — even ticks up a hair (79.2 → 80.6). The reminder does nothing to men.

The two lines are not parallel — they fan apart. That non-parallelness is the interaction. The effect of the threat reminder depends on gender: it hurts women's scores but leaves men's untouched. The marginals tell the "overall" story (men higher overall, 79.9 vs 72.7; a slight dip under threat overall, 77.8 vs 74.8) — but if you'd only looked at those marginals and ignored the cells, you'd badly miss the real headline.

📏 How big? Partial eta-squared

A two-way ANOVA reports three effect sizes — one per test. The standard choice is partial η² (partial eta-squared): each effect's SS divided by that SS plus the error SS. Read it just like Unit 13's η² — the share of variance an effect accounts for, with the other effects set aside:

partial η² = SS_effect ÷ (SS_effect + SS_within)

For our stereotype-threat data: gender partial η² ≈ .13, condition ≈ .03, interaction ≈ .05. Common benchmarks: ~.01 small, ~.06 medium, ~.14 large. So the gender gap is a large effect, the overall threat dip is small, and the interaction — the headline — lands at the small-to-medium boundary: real and meaningful, but subtler than the marginal gender gap. The calculator above prints partial η² for all three effects.

✍️ Report it in APA style

A two-way ANOVA gets three APA sentences — one per test — and the interaction, when significant, leads the story. Same house rules as Units 10–13 (italicized letters, two decimals, no leading zero on p, effect sizes included):

The anatomy: interaction first (it's the finding), immediately unpacked with the cell means that show what depends on what, then the main effects in a supporting sentence. When an interaction is significant, the main effects are footnotes to the "it depends" story — the write-up's order should say so.

7. Why the graph beats the table

You could stare at the four numbers and eventually work it out. But the line graph hands you all three results in a single glance. Train your eye on the three questions, in order, and they become automatic:

💻 In Google Sheets

Sheets has no one-click two-way ANOVA, so we do the part that matters: the four cell means and the interaction graph. Use =AVERAGEIFS, which says "average the score column, but only the rows where gender matches and condition matches":

=AVERAGEIFS(score, gender, "Woman", cond, "Control")  → 76.4
=AVERAGEIFS(score, gender, "Woman", cond, "Threat")   → 68.9
=AVERAGEIFS(score, gender, "Man",   cond, "Control")  → 79.2
=AVERAGEIFS(score, gender, "Man",   cond, "Threat")   → 80.6

For the marginal means, drop one condition and average by gender alone (and vice versa):

=AVERAGEIF(gender, "Woman", score)   → 72.7   (Woman row marginal)
=AVERAGEIF(cond,   "Threat", score)  → 74.8   (Threat column marginal)

Lay the four cell means in a little 2×2 table, then Insert ▸ Chart ▸ Line chart with condition on the x-axis and one line per gender. That picture is your interaction graph — you'll watch the Women's line dive while the Men's line holds flat. Read the two main effects and the interaction right off it. The three F-values come from the provided template (or this page's calculator above).

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