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MANOVA

Created:December 18, 2024
Last Updated:December 30, 2024

This calculator performs Multivariate Analysis of Variance (MANOVA), which extends ANOVA to situations where you have multiple dependent variables. Instead of comparing group means on a single outcome, MANOVA tests whether groups differ across a combination of outcomes simultaneously.

What You'll Get:

  • Multivariate Test Statistics: Wilks' Lambda, Pillai's trace, Hotelling-Lawley trace, and Roy's largest root
  • Effect Sizes: Partial eta-squared for multivariate effects
  • Assumption Testing: Multivariate normality and homogeneity of covariance matrices
  • Visual Analysis: Group comparison charts for each dependent variable
  • Follow-up Analyses: Univariate ANOVA results for each dependent variable
  • APA-Ready Report: Publication-quality tables and results

💡 Pro Tip: If you only have one dependent variable, use ourOne-Way ANOVA Calculatorinstead for more appropriate analysis.

Ready to analyze your multivariate data? to see how it works, or upload your own data to discover if your groups differ across multiple outcomes.

Calculator

1. Load Your Data

2. Select Columns & Options

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One-Way ANOVA Calculator

Two-Way ANOVA Calculator

Three-Way ANOVA Calculator

Repeated Measures ANOVA Calculator

Learn More

MANOVA (Multivariate ANOVA)

Definition

MANOVA (Multivariate Analysis of Variance) tests whether there are significant differences between groups on multiple dependent variables simultaneously. It extends one-way ANOVA to multiple outcomes while accounting for correlations between dependent variables.

Why Use MANOVA Instead of Multiple ANOVAs?

When you have multiple dependent variables, you might be tempted to run separate ANOVAs for each outcome. However, MANOVA offers several advantages:

  • Controls Type I Error: Running multiple ANOVAs inflates the familywise error rate
  • Accounts for Correlations: MANOVA considers relationships between dependent variables
  • Increased Power: Can detect group differences that might be missed by separate tests
  • Holistic View: Tests the combined effect across all outcomes

How Does MANOVA Work?

MANOVA compares the multivariate means of groups by examining the ratio of between-group variance to within-group variance across all dependent variables simultaneously.

Between-group variance: How much do group centroids differ in multivariate space?

Within-group variance: How much do observations vary within each group across all dependent variables?

Test Statistics

MANOVA provides four main test statistics:

1. Wilks' Lambda (Λ)

Λ=∣W∣∣B+W∣\Lambda = \frac{|W|}{|B + W|}

Most commonly used; ranges from 0 to 1 (smaller values indicate larger effects)

2. Pillai's Trace

Most robust to violations of assumptions; ranges from 0 to 1 (larger values indicate larger effects)

3. Hotelling-Lawley Trace

Most powerful when assumptions are met

4. Roy's Largest Root

Appropriate when group differences are concentrated on one dimension

Key Assumptions

Independence: Observations must be independent
Multivariate Normality: Dependent variables should follow a multivariate normal distribution within each group
Homogeneity of Covariance Matrices: Groups should have equal covariance matrices (tested with Box's M test)
No Multicollinearity: Dependent variables should not be too highly correlated

Code Examples

R
library(tidyverse)

# Sample data
data <- tibble(
  Group = factor(c(rep("A", 5), rep("B", 5), rep("C", 5))),
  DV1 = c(8, 9, 7, 10, 8, 6, 5, 8, 7, 6, 9, 10, 10, 8, 9),
  DV2 = c(12, 14, 11, 13, 12, 10, 9, 11, 10, 9, 15, 16, 14, 15, 14)
)

# Perform MANOVA
manova_result <- manova(cbind(DV1, DV2) ~ Group, data = data)
summary(manova_result, test = "Wilks")
Python
import numpy as np
from scipy import stats
import pandas as pd

# Sample data
group_A = np.array([[8, 12], [9, 14], [7, 11], [10, 13], [8, 12]])
group_B = np.array([[6, 10], [5, 9], [8, 11], [7, 10], [6, 9]])
group_C = np.array([[9, 15], [10, 16], [10, 14], [8, 15], [9, 14]])

# Combine data
data = np.vstack([group_A, group_B, group_C])
groups = np.array([0]*5 + [1]*5 + [2]*5)

# Note: For full MANOVA, use statsmodels
from statsmodels.multivariate.manova import MANOVA

df = pd.DataFrame(data, columns=['DV1', 'DV2'])
df['Group'] = groups

manova = MANOVA.from_formula('DV1 + DV2 ~ Group', data=df)
print(manova.mv_test())

Effect Size

Partial eta-squared (ηp2\eta_p^2) can be calculated from Wilks' Lambda:

ηp2=1−Λ1/s\eta_p^2 = 1 - \Lambda^{1/s}

where s is the smaller of the number of groups minus 1 or the number of dependent variables.

Guidelines:

  • Small effect: ηp2≈0.01\eta_p^2 \approx 0.01
  • Medium effect: ηp2≈0.06\eta_p^2 \approx 0.06
  • Large effect: ηp2≈0.14\eta_p^2 \approx 0.14

When to Use MANOVA

MANOVA is appropriate when:

  • You have two or more dependent variables that are conceptually related
  • You want to compare groups on these multiple outcomes simultaneously
  • The dependent variables are moderately correlated (not too low, not too high)
  • You have sufficient sample size (at least 20 observations per group recommended)

Verification