Introduction to Non-Parametric Statistical Tests

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In statistical analysis, non-parametric tests are used when the data does not meet the assumptions required for parametric tests. One of the most commonly used non-parametric tests is the Wilcoxon signed-rank test, also known as the Wilcoxon test. This test is used to compare two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ.

Background: The Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is based on the concept of ranking and summing the absolute values of the differences between paired observations. This test is particularly useful when data points are not normally distributed, as it does not rely on assumptions about the distribution of the data.

For a pair of observations (x_i, y_i) where i = 1,…,n, the Wilcoxon signed-rank test considers the difference d_i = x_i - y_i. The ranks of these differences are then used to calculate the sum of absolute values of the positive and negative rank orders.

Key Concepts

• Pairwise Comparison: This refers to the comparison between two samples or repeated measurements on a single sample.
• Weighted Data: When each pair has an associated weight, the traditional Wilcoxon signed-rank test cannot be applied. In this case, we need to modify the test to account for the weights.

Pairing and Weighted Wilcoxon Signed-Rank Test in R

R provides various functions for non-parametric tests. However, when it comes to paired data with weights, there is no built-in function that directly performs the weighted version of the Wilcoxon signed-rank test. We will implement a solution using the wilcox.test() function.

R Code

# Required Libraries
library(factoextra)

# Set seed for reproducibility
set.seed(9)

# Sample data
x <- sample(x = c(1:100), size = 20, replace = TRUE)
y <- sample(x = c(1:100), size = 20, replace = TRUE)
weight <- runif(n = 20)

data <- data.frame(x = x, y = y, weight = weight)

# Perform Wilcoxon Signed-Rank Test
wilcox.test(data$x, data$y, paired = TRUE, method = "sumless")

Explanation

The method="sumless" parameter is used to perform the weighted version of the test. The sumless method does not work as expected if we just add weights to our data and pass it directly into the wilcox.test() function.

Instead, we need to calculate the weighted difference vector by adding up the product of corresponding elements from each pair, and their respective weight.

Weighted Wilcoxon Signed-Rank Test

# Calculate Weighted Differences
difference_vector <- (data$x - data$y) * data$weight

# Rank Difference Vector
rank_difference_vector <- rank(difference_vector, ties.method = "average")

# Sum Positive and Negative Rank Orders
sum_positive_ranks <- sum(rank_difference_vector[rank_difference_vector > 0])
sum_negative_ranks <- sum(rank_difference_vector[rank_difference_vector < 0])

# Calculate Weighted Wilcoxon Signed-Rank Test Statistic
weighted_wilcox_statistic <- ifelse(sum_positive_ranks >= sum_negative_ranks, 
                                    (2 * (sum_positive_ranks - sum_negative_ranks)) / length(difference_vector), 
                                    (2 * (sum_negative_ranks - sum_positive_ranks)) / length(difference_vector))

# Calculate p-value
p_value <- 0.5

Explanation of Code Blocks

• Weighted Differences: We calculate the weighted differences between pairs by multiplying corresponding elements from each pair and their respective weight.
• Rank Difference Vector: The rank() function is used to rank all positive and negative values in the difference vector, with the average value assigned in case of ties.
• Sum Positive and Negative Rank Orders: These are calculated separately for the ranked differences vector. We then calculate the weighted Wilcoxon signed-rank test statistic based on these sums.
• p-value Calculation: The p-value is typically estimated from a standard normal distribution using a random permutation.

Interpretation of Results

After calculating the p-value, we can compare it to our chosen significance level (commonly 0.05) and make conclusions about whether our data suggests significant differences between our sample pairs or not.

The results will be an estimate of the true probability that our observed test statistic would occur under the null hypothesis. If this is less than our chosen significance level, we can conclude that there is statistical evidence against our null hypothesis at a certain confidence interval.

Final Result

The test statistic shows that the observed sum of positive ranks is larger than the negative rank sum.

Let’s calculate our p-value and conclude whether there are significant differences between our samples or not.

# Calculate z-score for Wilcoxon Signed-Rank Test
wilcox_z_score <- (weighted_wilcox_statistic - 0) / sqrt((sum_positive_ranks * (n-1)) + (sum_negative_ranks * (n-1)))

# p-value Calculation from z-score
p_value <- 2 * pnorm(-abs(wilcox_z_score))

print(p_value)

This will print the calculated p-value.

The final answer is not provided here, as this would require a numerical computation of the test statistic and its associated p-value.

Last modified on 2024-07-05