MANOVA

Multivariate analysis of variance: testing group differences across multiple correlated outcomes jointly

Introduction

MANOVA tests whether group means differ across several dependent variables simultaneously, accounting for their correlation. It is more powerful than separate ANOVAs when outcomes are correlated.

Prerequisites

ANOVA, multivariate normal.

Theory

Four standard multivariate test statistics of the between-to-within variance ratio:

• Wilks’ lambda: \(|E|/|H + E|\), most common.
• Pillai’s trace: robust to covariance heterogeneity.
• Hotelling-Lawley trace.
• Roy’s greatest root.

All test the same null but with slightly different operating characteristics.

Assumptions

• Multivariate normal within groups.
• Homogeneous covariance matrices (Box’s M).
• Independent observations.

R Implementation

fit <- manova(cbind(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) ~
              Species, data = iris)
summary(fit)                     # Pillai default
summary(fit, test = "Wilks")
summary(fit, test = "Hotelling-Lawley")

# Follow-up univariate ANOVAs
summary.aov(fit)

# car Anova for Type II/III
library(car)
Manova(lm(cbind(Sepal.Length, Sepal.Width) ~ Species, data = iris))

Output & Results

Test statistic, approximate F, df, p-value for each test; univariate follow-ups.

Interpretation

“MANOVA showed a significant multivariate species effect (Wilks’ lambda = 0.02, F(8, 288) = 199, p < 0.001). Univariate follow-up ANOVAs confirmed significant species differences on each flower dimension.”

Practical Tips

• Pillai is robust to covariance heterogeneity; Wilks is the classical choice.
• Always follow a significant MANOVA with univariate ANOVAs or discriminant analysis.
• MANOVA is sensitive to departures from multivariate normality; consider robust MANOVA.
• For few groups with many outcomes, MANOVA is more powerful than Bonferroni-corrected univariate tests.
• Report effect size: partial eta-squared or multivariate eta.