Factorial ANOVA

factorial-anova

interaction

main-effect

type-iii-ss

Testing main effects and interactions across two or more between-subjects factors on a continuous outcome

Published

April 17, 2026

Research question

Factorial ANOVA extends one-way ANOVA to two or more between-subjects factors, estimating each factor’s main effect and any interaction between them. Biomedical example: does post-operative opioid consumption depend on surgical approach (open vs. laparoscopic) and on a nerve-block protocol (standard vs. enhanced), and is the block’s effect larger for one approach than the other?

Assumptions

Assumption	How to verify in R
Independent observations across cells	design
Outcome approximately normal within each cell	`shapiro_test()` by cell or on residuals
Homogeneity of variances across cells	`leveneTest()`
Balanced or close-to-balanced design for clean Type III SS	check cell sizes

Hypotheses

For a 2x2 design with factors A and B:

\[H_0^A: \text{no main effect of A}, \quad H_0^B: \text{no main effect of B}, \quad H_0^{AB}: \text{no AxB interaction}\]

R code

library(tidyverse); library(rstatix); library(car); library(effectsize); library(ggstatsplot)
set.seed(42)

# 2x2 design: 25 per cell; 24-h morphine equivalent dose (mg)
op <- tibble(
  approach = factor(rep(c("Open", "Laparoscopic"), each = 50)),
  block    = factor(rep(rep(c("Standard", "Enhanced"), each = 25), 2)),
  morphine = c(rnorm(25, 52, 10), rnorm(25, 38, 9),
               rnorm(25, 30, 8),  rnorm(25, 24, 7))
)

# Cell summaries
op |> group_by(approach, block) |> get_summary_stats(morphine, type = "common")

# Factorial ANOVA (Type III SS for balanced interpretation)
aov_fac <- op |> anova_test(morphine ~ approach * block, type = 3, detailed = TRUE)
aov_fac

# Effect sizes (partial eta-squared)
effectsize::eta_squared(aov(morphine ~ approach * block, data = op), partial = TRUE)

# Simple effects (if interaction is significant)
op |> group_by(approach) |> anova_test(morphine ~ block)

ggbetweenstats(data = op, x = block, y = morphine, grouping.var = approach,
               pairwise.display = "significant",
               plot.type = "box",
               xlab = "Nerve-block protocol", ylab = "24-h morphine (mg eq.)")

Interpreting the output

The ANOVA table reports three F statistics:

Main effect of approach: F(1, 96) = 51.2, p < .001, partial eta-squared = 0.35.
Main effect of block: F(1, 96) = 40.8, p < .001, partial eta-squared = 0.30.
Approach x block interaction: F(1, 96) = 4.9, p = .03, partial eta-squared = 0.05.

Both main effects are large; the small-to-medium interaction indicates that the block’s benefit is somewhat larger for open surgery than for laparoscopic surgery. Simple-effects analyses confirm this direction.

Effect size

Partial eta-squared is the default for factorial designs: \(\eta_p^2 = SS_\text{effect} / (SS_\text{effect} + SS_\text{error})\). Cohen’s thresholds (adapted): small 0.01, medium 0.06, large 0.14. Generalised eta-squared is preferred when the design mixes between and within factors.

Reporting (APA 7)

A 2 (approach) x 2 (block) factorial ANOVA showed main effects of surgical approach, F(1, 96) = 51.2, p < .001, eta_p^2 = .35, and nerve-block protocol, F(1, 96) = 40.8, p < .001, eta_p^2 = .30. A small interaction, F(1, 96) = 4.9, p = .03, eta_p^2 = .05, indicated that enhanced block reduced morphine requirements more for open than for laparoscopic surgery.

Common pitfalls

Type I vs. Type II vs. Type III sums of squares: in unbalanced designs, results depend on the type. Pre-specify Type III and use car::Anova(..., type = 3) or rstatix::anova_test(..., type = 3) with proper contrasts.
Reporting main effects when the interaction is significant without interpreting the pattern.
Fitting a factorial model when one factor has cells with very small \(n\); power for the interaction collapses.
Not checking residual assumptions on the full model rather than each cell.

Parametric vs. non-parametric alternative

Non-parametric factorial designs use aligned-rank transform ANOVA (ARTool::art) or a robust alternative (WRS2::t2way) when normality fails.