Two-way ANOVA
Models and regression · reference distribution: F
When to use it
Assess at once the effect of two factors on a numeric response, and whether they interact, such as dose and sex in a trial.
Null hypothesis
For each factor, the level means are equal; for the interaction, the effect of one factor does not depend on the level of the other.
Assumptions
- Independent observations
- Normal residuals
- Equal variances across cells
Test statistic
F_A = \dfrac{MS_A}{MS_{\text{error}}}, \quad F_B = \dfrac{MS_B}{MS_{\text{error}}}, \quad F_{AB} = \dfrac{MS_{AB}}{MS_{\text{error}}}Effect size
Partial η² of each effect: factor A, factor B and the interaction.
\eta^2_p = \dfrac{SS_A}{SS_A + SS_{\text{error}}}How to report it
Dose: F(2, 54) = 7.9, p = .001, η²p = .23. Interaction: F(2, 54) = 1.1, p = .34
In R and Python
R
summary(aov(y ~ dose * sex, data = df))Python
import statsmodels.api as sm
import statsmodels.formula.api as smf
m = smf.ols("y ~ C(dose) * C(sex)", data=df).fit()
sm.stats.anova_lm(m, typ=2)In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- One-way ANOVA, with a single factor
- ANCOVA, when there is also a numeric covariate
Where it sits in the catalog
Models and regression. Tests run inside a fitted model: whether a coefficient matters, whether the model explains anything and whether the residuals meet the assumptions.
In the decision tree
- What do you want to do? Compare groups, or one group with a reference value
- What kind of response did you measure? Numeric: a measurement, such as weight, time or score
- How many groups or measurements? Two factors at once, such as dose and sex