Two-Way ANOVA Calculator

A two-way ANOVA tests the effect of two independent variables (factors) on a continuous dependent variable simultaneously. Use it when you want to examine the main effect of Factor A, the main effect of Factor B, and whether the two factors interact — for example, whether the effect of a teaching method differs by gender.

An interaction effect (A×B) occurs when the influence of one factor on the outcome depends on the level of the other factor. For example, a drug might work better for younger patients but worse for older ones. A significant interaction p-value (below α) indicates such dependency exists and should be interpreted before main effects.

The significance level α is the threshold probability for rejecting the null hypothesis. The most common choice is 0.05, meaning you accept a 5% chance of a false positive (Type I error). Use 0.01 for high-stakes or confirmatory research, and 0.10 for exploratory or pilot studies.

One-way ANOVA tests the effect of a single factor on a dependent variable across multiple groups. Two-way ANOVA extends this to two factors simultaneously, allowing you to detect main effects of each factor and whether they interact. Two-way ANOVA is more efficient and informative than running two separate one-way tests.

Two-way ANOVA assumes: (1) the dependent variable is continuous, (2) observations are independent, (3) residuals are approximately normally distributed within each cell, and (4) variances are roughly equal across groups (homogeneity of variance, tested by Levene's test). Violations of normality are less critical with larger sample sizes.

The F-statistic is the ratio of the variance explained by a factor (Mean Square) to the unexplained within-group variance (Mean Square Error). A larger F indicates a stronger effect relative to random noise. If the corresponding p-value is less than your α level, you reject the null hypothesis — meaning that factor significantly affects the outcome.

Sum of squares quantifies the total variability in your data attributed to each source. SSA measures how much of the variation is due to Factor A, SSB to Factor B, SSAB to the interaction, and SSE to random error. Together they sum to the total SST. Larger SS for a factor relative to SSE suggests a meaningful effect.

Eta-squared (η²) measures the proportion of total variance explained by a factor. Values around 0.01 indicate a small effect, 0.06 a medium effect, and 0.14 or above a large effect. It helps you understand practical significance beyond just statistical significance — a result can be statistically significant but explain only a tiny fraction of variance.