Does a new teaching method raise scores equally for every group of students, or does it help some groups more than others? A one-way comparison cannot answer the second half of that question. A two-way ANOVA can, because it tests two factors together and asks whether they interact. That interaction is often the most interesting finding in a thesis, and also the one most often misread.
This article is part of our guide How to Write a Master’s Thesis by the Mastermind PhD team and relates to our Statistical Analysis service. Sources last checked: November 2026.
This guide explains the three effects a two-way ANOVA tests, its assumptions, the SPSS steps, how to read the output and the interaction plot, what to do when the interaction is significant, and how to report everything in APA style.
What is a two-way ANOVA? It is a parametric test that examines the effect of two categorical independent variables (factors) on one continuous dependent variable. It produces three tests: a main effect for each factor and an interaction effect, which shows whether the effect of one factor changes across the levels of the other. A design with two levels of each factor is called a 2 × 2 factorial design.
What are the three effects in a two-way ANOVA?
| Effect | Question it answers | Example (method × gender on test score) |
|---|---|---|
| Main effect of factor A | Do the levels of A differ, averaging over B? | Do blended and traditional classes differ overall? |
| Main effect of factor B | Do the levels of B differ, averaging over A? | Do female and male students differ overall? |
| A × B interaction | Does the effect of A depend on the level of B? | Is the gain from the blended method larger for one gender? |
When the interaction is significant, the main effects describe averages that may not hold for any real group. Interpret the interaction first and treat the main effects with caution.
When should you use it?
Use a two-way ANOVA when you have one continuous outcome (a scale score, a test score) and two categorical factors, each with two or more independent groups, and every participant appears in only one cell. Typical thesis questions combine an intervention with a demographic variable, or two demographic variables such as sector and experience band.
If the same people are measured more than once, you need repeated measures or mixed ANOVA. If you want to control for a continuous covariate, consider ANCOVA. If the factors are continuous, regression with an interaction term is the better tool.
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What assumptions must be checked?
- Continuous dependent variable measured at interval or ratio level.
- Independent observations: each participant contributes one score to one cell.
- No extreme outliers in any cell, checked with boxplots in Explore using both factors.
- Approximate normality of the residuals or of each cell, using Shapiro-Wilk or Q-Q plots. With roughly 20 or more cases per cell, the test is fairly robust.
- Homogeneity of variance across cells, using Levene’s test. A non-significant result supports the assumption.
Unequal cell sizes are common in surveys. SPSS uses Type III sums of squares by default, which is the usual choice for unbalanced designs, but very small cells (fewer than about 10) make every estimate unstable. Merge sparse categories before analysis if it is theoretically sensible.
How do you run a two-way ANOVA in SPSS?
- Analyze, then General Linear Model, then Univariate.
- Move the outcome to Dependent Variable and the two factors to Fixed Factor(s).
- Under Plots, put one factor on the Horizontal Axis and the other under Separate Lines, then click Add. Tick Include error bars if you want confidence intervals in the plot.
- Under EM Means, move both factors and their interaction to Display Means for. Tick Compare main effects only if a factor has three or more levels.
- Under Options, tick Descriptive statistics, Estimates of effect size and Homogeneity tests.
- Under Post Hoc, request Tukey or Games-Howell for factors with three or more levels.
- Click OK.
How do you read the output?
Levene’s Test of Equality of Error Variances comes first. If Sig. is above .05, variances are treated as equal.
Tests of Between-Subjects Effects is the central table. Ignore the Corrected Model and Intercept rows for interpretation and read three rows: factor A, factor B and A * B. For each, report F, the effect degrees of freedom, the error degrees of freedom, p and partial eta squared. The error row gives the denominator degrees of freedom.
Partial eta squared (η²p) is the proportion of variance explained by an effect after removing other effects. Common benchmarks are about .01 small, .06 medium and .14 large (Cohen, 1988), but values from different designs are not directly comparable.
How do you interpret an interaction in a two-way ANOVA?
Look at the profile plot. Parallel lines suggest no interaction; lines that diverge or cross suggest one. A crossover interaction (lines cross) means the direction of the effect reverses; an ordinal interaction (lines diverge without crossing) means the effect exists in both groups but differs in size.
When the interaction is significant, run simple effects: the effect of one factor at each level of the other. In SPSS this is done with syntax by adding /EMMEANS=TABLES(method*gender) COMPARE(method) ADJ(BONFERRONI) to the pasted Univariate command. The output then tests the method difference separately for each gender.
Worked example (illustrative)
The numbers are an illustrative scenario, not data from a real study. In a 2 (method: traditional, blended) × 2 (gender) design with 120 students, 30 per cell, the outcome is a final test score.
| Source | SS | df | F | p | η²p |
|---|---|---|---|---|---|
| Method | 900 | 1 | 16.57 | < .001 | .13 |
| Gender | 60 | 1 | 1.10 | .295 | .01 |
| Method × Gender | 420 | 1 | 7.73 | .006 | .06 |
| Error | 6,300 | 116 |
Each partial eta squared is the effect SS divided by the effect SS plus the error SS; for the interaction, 420 ÷ 6,720 = .06. Simple effects would then show where the method difference is concentrated.
APA sentence: “A two-way ANOVA showed a significant main effect of teaching method, F(1, 116) = 16.57, p < .001, η²p = .13, no significant main effect of gender, F(1, 116) = 1.10, p = .295, η²p = .01, and a significant method × gender interaction, F(1, 116) = 7.73, p = .006, η²p = .06.” Follow it with the simple effects and a reference to the figure.
What mistakes should you avoid?
- Interpreting main effects without mentioning a significant interaction.
- Running separate one-way ANOVAs or t-tests for each subgroup instead of one two-way ANOVA, which inflates the Type I error rate and cannot test the interaction.
- Reading the interaction from a plot alone without the F test, or the reverse.
- Reporting eta squared and calling it partial eta squared (SPSS gives the partial version).
- Leaving cells with five or six cases and drawing firm conclusions from them.
- Forgetting to report cell means and standard deviations, which readers need to understand the interaction.
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Frequently asked questions
What is the difference between one-way and two-way ANOVA?
One-way ANOVA has one factor and answers whether its groups differ. A two-way ANOVA adds a second factor and tests whether the two factors interact. It needs more cases, because the sample is split into more cells.
What if the interaction is not significant?
Interpret the main effects directly, using post hoc tests for factors with three or more levels. Some authors also refit the model without the interaction term; follow your supervisor’s or field’s convention and state it.
Can I run a two-way ANOVA with unequal group sizes?
Yes. SPSS uses Type III sums of squares by default, which handles unbalanced designs. Very unequal or tiny cells reduce power and make Levene’s test more consequential, so report the cell sizes.
What is the non-parametric alternative?
There is no simple standard equivalent. Options include transforming the outcome, using robust methods such as bootstrapped estimates, or an aligned rank transform. Discuss the choice with your supervisor before collecting data where possible.
How do I write the null hypotheses?
Write three: no difference between the levels of factor A, no difference between the levels of factor B, and no interaction between A and B on the outcome. Each corresponds to one row of the output table.
Sources
- Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE.
- Maxwell, S. E., Delaney, H. D., & Kelley, K. (2018). Designing Experiments and Analyzing Data: A Model Comparison Perspective (3rd ed.). Routledge.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863. https://doi.org/10.3389/fpsyg.2013.00863
- Laerd Statistics. Two-way ANOVA in SPSS Statistics. https://statistics.laerd.com/spss-tutorials/two-way-anova-using-spss-statistics.php
- American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.). APA.
Sources last checked: November 2026. This article is general educational guidance and does not replace your university regulations or your supervisor’s instructions. How we prepare our content.