Kruskal-Wallis Test in R

Introduction

The Kruskal-Wallis test is a non-parametric alternative to ANOVA. It tests the null hypothesis that the medians of multiple independent groups are equal. This test is useful when the assumptions of ANOVA (normality and equal variances) are violated.

Load necessary libraries

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(tidyr)

Simulate data (percentage for seven groups)

The dataset consists of percentages from seven different groups, as described in the transcript

dat <- data.frame(
  Judge = rep(c("Judge_A", "Judge_B", "Judge_C", "Judge_D", "Judge_E", "Judge_F", "Judge_Spock"), each = 5),
  Proportion = c(
    6.4, 8.7, 13.3, 17.7, 17.7,  # Judge_A
    24.5, 30.1, 35.6, 40.2, 45.0, # Judge_B
    20.1, 25.3, 27.7, 29.4, 31.2, # Judge_C
    15.8, 18.3, 21.9, 22.5, 26.0, # Judge_D
    10.0, 12.1, 13.7, 15.5, 16.9, # Judge_E
    7.8, 8.5, 9.3, 10.2, 11.0,    # Judge_F
    48.9, 50.1, 55.3, 60.4, 64.8  # Judge_S
  )
)

# Display first few rows
head(dat)

##     Judge Proportion
## 1 Judge_A        6.4
## 2 Judge_A        8.7
## 3 Judge_A       13.3
## 4 Judge_A       17.7
## 5 Judge_A       17.7
## 6 Judge_B       24.5

Rank Data

To perform the Kruskal-Wallis test, the data is ranked across all groups.

# Rank the Proportions
dat <- dat %>%
  mutate(Rank = rank(Proportion))

# Display ranked data
head(dat)

##     Judge Proportion Rank
## 1 Judge_A        6.4  1.0
## 2 Judge_A        8.7  4.0
## 3 Judge_A       13.3 10.0
## 4 Judge_A       17.7 15.5
## 5 Judge_A       17.7 15.5
## 6 Judge_B       24.5 21.0

Calculate Group-wise Ranks

The average rank for each group is calculated.

# Calculate average rank for each group
group_ranks <- dat %>%
  group_by(Judge) %>%
  summarise(Avg_Rank = mean(Rank), .groups = "drop")

# Display group-wise ranks
group_ranks

## # A tibble: 7 × 2
##   Judge       Avg_Rank
##   <chr>          <dbl>
## 1 Judge_A          9.2
## 2 Judge_B         26.8
## 3 Judge_C         23.2
## 4 Judge_D         18.4
## 5 Judge_E         10.4
## 6 Judge_F          5  
## 7 Judge_Spock     33

Perform the Kruskal-Wallis Test

The kruskal.test() function is used to perform the test.

# Perform the Kruskal-Wallis test
kruskal_result <- kruskal.test(Proportion ~ Judge, data = dat)

# Display the test results
kruskal_result

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Proportion by Judge
## Kruskal-Wallis chi-squared = 30.187, df = 6, p-value = 3.622e-05

Interpret Results

The test statistic H is compared to a chi-squared distribution with k-1 degrees of freedom (where k is the number of groups).

# Interpret results
if (kruskal_result$p.value < 0.05) {
  cat("Reject the null hypothesis: Significant differences exist between the group medians.\n")
} else {
  cat("Fail to reject the null hypothesis: No significant differences between the group medians.\n")
}

## Reject the null hypothesis: Significant differences exist between the group medians.