ANOVAs and MANOVAs

ANOVA (Analysis of Variance) and MANOVA (Multivariate Analysis of Variance) are statistical methods of determining the significance of mean differences across multiple groups. These techniques are especially useful in contexts where multiple factors or variables are present, such as in complex experimental settings.

ANOVA is utilized when examining a single dependent variable across various independent groups. Its core objective is to compare the means of these groups to establish whether any significant differences exist. The primary statistical output of ANOVA is the F-statistic, which represents the ratio of between-group variance to within-group variance. A significant F-statistic suggests that at least one group differs in its mean value.

MANOVA extends the capabilities of ANOVA by enabling the simultaneous analysis of multiple dependent variables. This is particularly advantageous when the dependent variables are intercorrelated, allowing MANOVA to more accurately assess the impact of independent groups while considering these correlations. Like ANOVA, MANOVA yields an F-statistic, yet it incorporates variance-covariance relationships among the variables.

Both ANOVA and MANOVA require meeting certain assumptions: normality, homogeneity of variances (for ANOVA) or covariances (for MANOVA), and the independence of observations. Failure to meet these assumptions may compromise the integrity of the results.

Example in Python

# ANOVA Example
import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

# Sample data for ANOVA
data = {'Group': ['A', 'A', 'B', 'B', 'C', 'C'],
        'Score': [23, 26, 27, 22, 24, 28]}
df = pd.DataFrame(data)

# Fit the model for ANOVA
model = ols('Score ~ C(Group)', data=df).fit()
anova_result = sm.stats.anova_lm(model, typ=2)
print(anova_result)

# MANOVA Example
from statsmodels.multivariate.manova import MANOVA

# Sample data for MANOVA
data = {'Group': ['A', 'A', 'B', 'B', 'C', 'C'],
        'Metric1': [23, 26, 27, 22, 24, 28],
        'Metric2': [15, 18, 19, 14, 16, 20]}
df = pd.DataFrame(data)

# Fit the MANOVA model
manova = MANOVA.from_formula('Metric1 + Metric2 ~ C(Group)', data=df)
manova_result = manova.mv_test()
print(manova_result)

In the provided code, ANOVA is used to compare one dependent variable ('Score') across different groups. In contrast, MANOVA evaluates two dependent variables ('Metric1' and 'Metric2') to ascertain the impact of group differences while considering correlations between the dependent variables. These statistical techniques are widely used in fields like psychology, biology, and social sciences, offering invaluable insights into how various independent variables influence outcomes.

Example in R

# Load necessary libraries
library(dplyr)     # For data manipulation
library(car)       # For Anova (Type II)
library(stats)     # For aov
library(MASS)      # For MANOVA

# ANOVA Example
# Sample data for ANOVA
data_anova <- data.frame(Group = factor(c('A', 'A', 'B', 'B', 'C', 'C')),
                         Score = c(23, 26, 27, 22, 24, 28))

# Fit the model for ANOVA
anova_model <- aov(Score ~ Group, data = data_anova)
anova_result <- Anova(anova_model, type = "II")  # Type II ANOVA
print(summary(anova_result))

# MANOVA Example
# Sample data for MANOVA
data_manova <- data.frame(Group = factor(c('A', 'A', 'B', 'B', 'C', 'C')),
                          Metric1 = c(23, 26, 27, 22, 24, 28),
                          Metric2 = c(15, 18, 19, 14, 16, 20))

# Fit the MANOVA model
manova_model <- manova(cbind(Metric1, Metric2) ~ Group, data = data_manova)
manova_result <- summary(manova_model, test = "Pillai")  # Pillai's trace test
print(manova_result)

ANOVA: The aov function in R fits an ANOVA model. Here, we use ANOVA from the car package to specify Type II ANOVA (similar to typ=2 in Python’s anova_lm).

MANOVA: The manova function fits the model in R, and summary (with test = "Pillai") performs Pillai's trace test, equivalent to Python’s mv_test in statsmodels.

MANOVA and ANOVA compare variance between groups and are important statistical methods to understand. For more detail on how this approach works, check out the following:

• https://www.youtube.com/watch?v=EFdlFoHI_0I&list=PL7FCBC940F3067FD0 
• https://statisticsbyjim.com/anova/multivariate-anova-manova-benefits-use/ 
• https://statistics.laerd.com/spss-tutorials/one-way-manova-using-spss-statistics.php 
• https://www.geeksforgeeks.org/manova-multivariate-analysis-of-variance/