Example

For example, if you want to determine if the height of individuals varies across different age groups, ANOVA would be suitable since you have a single dependent variable (height).

However, if you have multiple dependent variables that are related to the same set of independent variables, then using MANOVA would be advantageous. For instance, suppose you have data on the height, weight, and body fat percentage of individuals, and you want to investigate if these variables differ based on different exercise routines. In this scenario, MANOVA would be appropriate as you have multiple dependent variables (height, weight, and body fat percentage) that are of interest and are potentially influenced by the independent variable (exercise routine).

Data Description

The data is from an experiment in which the optimal conditions for growth and product formation were determined for a bacterial strain in a broth with a certain carbon source. Two different nitrogen sources (yeast extract and ammonium chloride) and three different incubation temperatures (30, 35, 37) were evaluated. Bacterial growth was evaluated after 24 hours using dry cell weight (in mg/ml) and optical density at 600 nm. The yield of a desired fermentation product was determined using gas chromatography and expressed in mM. \(^4\)

# reading data
data <- read.table("data/MANOVA.txt", header = T, sep = "\t")

library(tidyverse)
data <- data %>% mutate(across(c("Temperature","N.source"), factor))
summary(data)

#   Experiment        Temperature              N.source     Replica     
#  Length:120         30:40       ammonium chloride:60   Min.   : 1.00  
#  Class :character   35:40       yeast extract    :60   1st Qu.: 5.75  
#  Mode  :character   37:40                              Median :10.50  
#                                                        Mean   :10.50  
#                                                        3rd Qu.:15.25  
#                                                        Max.   :20.00  
#    Dry.weight     Optical.density Product.yield  
#  Min.   : 1.670   Min.   :0.210   Min.   : 8.40  
#  1st Qu.: 4.805   1st Qu.:1.770   1st Qu.:41.90  
#  Median : 6.070   Median :1.910   Median :57.65  
#  Mean   : 6.024   Mean   :1.976   Mean   :57.10  
#  3rd Qu.: 7.080   3rd Qu.:2.223   3rd Qu.:72.70  
#  Max.   :10.300   Max.   :2.480   Max.   :93.10

data %>% 
  group_by(Temperature) %>% 
  summarise(Mean_dry_weight = mean(Dry.weight), 
            Mean_optical_density = mean(Optical.density),
            Mean_product_yield = mean(Product.yield))

# # A tibble: 3 × 4
#   Temperature Mean_dry_weight Mean_optical_density Mean_product_yield
#   <fct>                 <dbl>                <dbl>              <dbl>
# 1 30                     5.92                 1.92               56.6
# 2 35                     5.93                 1.99               56.5
# 3 37                     6.22                 2.01               58.3

Hypothesis

\(H_0:\) There is no significant difference in the average levels of dry weight, optical density, and product yield among two treatment groups.

MANOVA

manova_result <- manova(cbind(Dry.weight, Optical.density, Product.yield) ~ Temperature,
                        data = data)
summary(manova_result, test = "Pillai")

#              Df   Pillai approx F num Df den Df Pr(>F)
# Temperature   2 0.030774  0.60427      6    232 0.7268
# Residuals   117

summary(manova_result, test = "Wilks")

#              Df   Wilks approx F num Df den Df Pr(>F)
# Temperature   2 0.96939  0.60053      6    230 0.7298
# Residuals   117

summary(manova_result, test = "Hotelling-Lawley")

#              Df Hotelling-Lawley approx F num Df den Df Pr(>F)
# Temperature   2         0.031409  0.59677      6    228 0.7328
# Residuals   117

summary(manova_result, test = "Roy")

#              Df      Roy approx F num Df den Df Pr(>F)
# Temperature   2 0.024543  0.94899      3    116 0.4195
# Residuals   117

A p-value lower than 0.05 means the null hypothesis can be rejected.

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

# Load the data
data = pd.read_table("data/MANOVA.txt", sep="\t")

# Perform MANOVA analysis
# Change variable names
new_column_names = {
    'Dry weight': 'Weight',
    'Optical density': 'Density',
    'Product yield': 'Yield'
}
data = data.rename(columns=new_column_names)
manova_result = MANOVA.from_formula('Weight + Density + Yield ~ Temperature', data)
print(manova_result.mv_test()) 

#                   Multivariate linear model
# =============================================================
#                                                              
# -------------------------------------------------------------
#        Intercept        Value  Num DF  Den DF  F Value Pr > F
# -------------------------------------------------------------
#           Wilks' lambda 0.8059 3.0000 116.0000  9.3151 0.0000
#          Pillai's trace 0.1941 3.0000 116.0000  9.3151 0.0000
#  Hotelling-Lawley trace 0.2409 3.0000 116.0000  9.3151 0.0000
#     Roy's greatest root 0.2409 3.0000 116.0000  9.3151 0.0000
# -------------------------------------------------------------
#                                                              
# -------------------------------------------------------------
#       Temperature       Value  Num DF  Den DF  F Value Pr > F
# -------------------------------------------------------------
#           Wilks' lambda 0.9771 3.0000 116.0000  0.9079 0.4396
#          Pillai's trace 0.0229 3.0000 116.0000  0.9079 0.4396
#  Hotelling-Lawley trace 0.0235 3.0000 116.0000  0.9079 0.4396
#     Roy's greatest root 0.0235 3.0000 116.0000  0.9079 0.4396
# =============================================================

Refererences:

1. Stat PSU - Lesson 8: Multivariate Analysis of Variance (MANOVA)
   
2. MANOVA, towards data science
3. P.K. Bhattacharya, Prabir Burman - Theory and Methods of Statistics
   
4. Data description
   
5. Data source
   
6. Applied MANOVA and Discriminant Analysis-Wiley-Interscience (2006)
   
7. Srivastava Multivariate Statistics - page 187