MANOVA

Overview

1. Why Not Multiple ANOVAs?

Running separate ANOVAs for each dependent variable causes two problems:

1. Type I Error Inflation: Multiple testing problem.
2. Ignoring Correlations: Separate ANOVAs ignore the relationship between dependent variables. MANOVA might detect a difference in the joint distribution that individual ANOVAs miss.

2. Hypotheses

• $H_0$: The mean vectors across groups are equal.$${\mu }_1={\mu }_2=\cdots ={\mu }_k$$
• $H_1$: At least one group differs on at least one dependent variable.

Worked Example: Plant Growth

3. Test Statistics

Since we are comparing matrices (Variance-Covariance matrices), there is no single F-value. Common statistics include:

1. Wilks' Lambda: Most common. Ratio of determinants ($|\mathbf{W}|/|\mathbf{T}|$). Values close to 0 indicate groups are distinct.
2. Pillai's Trace: Consider the most robust to violations (e.g., small sample, unequal variance). Use this if assumptions are shaky.
3. Hotelling-Lawley Trace: Good for two groups (generalizes T-test).
4. Roy's Largest Root: Upper bound; most powerful if difference is on only one dimension, but sensitive to assumptions.

4. Assumptions

1. Multivariate Normality: Dependent variables should be normally distributed within groups.
2. Homogeneity of Covariance Matrices: Box's M Test (sensitive).
3. Linearity: Linear relationships between DVs.
4. No Multicollinearity: DVs shouldn't be too highly correlated ($r>0.9$).

5. Python Implementation Example

from statsmodels.multivariate.manova import MANOVA
import pandas as pd

# Formula interface: DV1 + DV2 ~ Group
model = MANOVA.from_formula('Sepal_Length + Sepal_Width + Petal_Length ~ Species', data=df)
print(model.mv_test())

6. Related Concepts

• One-Way ANOVA - Univariate version.
• Hotelling's T-Squared - Two-group multivariate test (Multivariate t-test).
• Principal Component Analysis (PCA) - Can be used to reduce DVs before analysis.