# How to Use the Wilcoxon Rank Sum Test in R: From Basics to Advanced Techniques

Statistics is the backbone of empirical research. It provides researchers, scientists, and analysts with tools to decipher patterns, relationships, and differences in collected data. Among the myriad statistical tests available, the non-parametric tests stand out for their versatility in handling data that don't necessarily fit the "normal" mold. These tests, which don't rely on stringent distributional assumptions, offer a robust alternative to their parametric counterparts.

The Wilcoxon Rank Sum Test, popularly known as the Mann-Whitney U test, is one such non-parametric method. Designed to assess if there's a significant difference between the distributions of two independent samples, this test comes in handy when the data under scrutiny doesn't adhere to a normal distribution. In this article, we embark on a journey to understand its nuances and explore its application in R, a premier software in the world of statistics and data analysis.

## Introduction to the Wilcoxon Rank Sum Test

Statistical testing provides a structured way for researchers to draw conclusions from data. When it comes to comparing two independent samples, many initially turn to the well-known Student's t-test. However, this parametric test assumes that the data are normally distributed and that the variances of the two populations are equal. In real-world scenarios, these assumptions are not always met, necessitating the use of non-parametric tests.

Enter the Wilcoxon Rank Sum Test.

The Wilcoxon Rank Sum Test, which is also referred to as the Mann-Whitney U test, offers a non-parametric alternative to the t-test. Instead of focusing on mean values and assuming specific data distributions, the Wilcoxon test works with the ranks of the data. By focusing on ranks, this test avoids making strong assumptions about the shape of the data distribution.

The fundamental principle behind the Wilcoxon Rank Sum Test is straightforward. Imagine you combine the two independent samples you have into a single dataset and then rank the combined data from the smallest to the largest value. If the two original samples come from identical populations, then the ranks should be evenly distributed between the two groups. On the other hand, if one sample consistently has higher (or lower) values than the other, the ranks will reflect this difference.

In practice, the test involves several steps:

1. Pool all the data from the two samples together.
2. Rank the data from the smallest to the largest value. In the case of ties, assign the average rank.
3. Sum the ranks for each of the original samples.
4. The smaller of these rank sums is then used as the test statistic, denoted as $$U$$.

The Mann-Whitney U test then compares this $$U$$ value to a distribution of $$U$$ values expected by chance to determine if the observed difference between the groups is statistically significant.

The Wilcoxon Rank Sum Test is particularly useful because it's less sensitive to outliers compared to parametric tests. It's also versatile, applicable to both ordinal data (e.g., Likert scale responses) and continuous data.

Tthe Wilcoxon Rank Sum Test offers researchers a robust tool to compare two independent samples without getting entangled in strict distributional assumptions. This makes it a valuable asset, especially in exploratory research phases where the nature of data distribution might be unknown.

## Basic Application in R

R, being a versatile statistical software, offers an easy-to-use function for the Wilcoxon Rank Sum Test: wilcox.test(). With a simple command, researchers and analysts can quickly evaluate the differences between two independent samples. Here, we will delve into the application of this test in R with two illustrative examples.

Official Documentation: For further details and variations, refer to the official R documentation

#### Example 1: Comparing Test Scores of Two Groups of Students

Consider two groups of students: Group A and Group B, who took a math test. We wish to determine if there's a significant difference in their test score distributions.

Group A Scores Group B Scores
78 82
80 85
77 84
79 86
81 83

In R, we can use the following code:

group_a <- c(78, 80, 77, 79, 81)
group_b <- c(82, 85, 84, 86, 83)

result <- wilcox.test(group_a, group_b)
print(result)

Wilcoxon rank sum exact test

data:  group_a and group_b
W = 0, p-value = 0.007937
alternative hypothesis: true location shift is not equal to 0

We can observe a p-value less than 0.05, suggesting a significant difference between the test scores of Group A and Group B.

#### Example 2: Comparing Satisfaction Ratings of Two Products

Imagine a scenario where customers rated their satisfaction with two products, X and Y, on a scale of 1 to 5. We are interested in understanding if there's a significant difference in the satisfaction ratings between the two products.

Product X Ratings Product Y Ratings
5 4
4 3
5 4
4 5
3 2

To test this in R:

product_x <- c(5, 4, 5, 4, 3)
product_y <- c(4, 3, 4, 5, 2)

result <- wilcox.test(product_x, product_y)
print(result)

Warning message:
In wilcox.test.default(product_x, product_y) :
cannot compute exact p-value with ties

Wilcoxon rank sum test with continuity correction

data:  product_x and product_y
W = 16.5, p-value = 0.4432
alternative hypothesis: true location shift is not equal to 0

Again, we can see a p-value greater than 0.05, suggesting no significant difference in satisfaction ratings between Product X and Product Y.

In both examples, it's vital to interpret the results in context and consider the practical significance of the findings, not just the statistical significance.

While the basic application of the Wilcoxon Rank Sum Test in R is straightforward, there are variations and advanced techniques that can be employed to cater to specific research questions and data scenarios. Here, we'll explore some of these advanced methodologies and how they can be applied using R.

### Paired Samples: Wilcoxon Signed Rank Test

Sometimes, the data isn't from two independent samples but rather from paired or matched samples. For instance, you might measure a parameter before and after a specific treatment on the same subjects. In such cases, the Wilcoxon Signed Rank Test is the appropriate non-parametric test to use.

Example: Comparing Blood Pressure Before and After a Treatment

Suppose we have ten patients, and we measure their blood pressure before and after administering a new drug.

Before Treatment After Treatment
140 135
150 145
138 132
145 140
152 148
... ...

To test the paired data in R:

bp_before <- c(140, 150, 138, 145, 152, 142, 155, 143, 146, 151)
bp_after <- c(135, 145, 132, 140, 148, 137, 150, 139, 142, 147)

# Wilcoxon Signed Rank Test
result_paired <- wilcox.test(bp_before, bp_after, paired = TRUE)
print(result_paired)
	Wilcoxon signed rank test with continuity correction

data:  bp_before and bp_after
V = 55, p-value = 0.004995
alternative hypothesis: true location shift is not equal to 0

The p-value below 0.05 would suggest the drug had a significant effect on reducing blood pressure.

### Handling Ties: Adjusting for Tied Ranks

In some datasets, you might have tied values, leading to tied ranks. While R's wilcox.test() function automatically handles ties by assigning the average rank, there are other methods to adjust for them.

Example: Comparing Sales of Two Salespeople Over Several Months with Tied Values

Suppose we're comparing sales figures of two salespeople, Alice and Bob, over multiple months. Some months, they made identical sales.

Alice's Sales Bob's Sales
5000 5000
5100 5150
5200 5200
5050 5075
... ...

To test this in R:

Warning message:
In wilcox.test.default(sales_alice, sales_bob) :
cannot compute exact p-value with ties

Wilcoxon rank sum test with continuity correction

data:  sales_alice and sales_bob
W = 46.5, p-value = 0.8199
alternative hypothesis: true location shift is not equal to 0


R will handle the tied ranks (like the first and third month) by assigning average ranks. The p-value indicates that there's a significant difference in sales distributions between Alice and Bob.

### Exact Method vs. Approximation

When dealing with the Wilcoxon Rank Sum Test (or its paired counterpart, the Wilcoxon Signed Rank Test), there are two computational approaches to determine the p-value: the exact method and the approximation method.

### Why Two Methods?

For small sample sizes, it's feasible to compute the exact distribution of the test statistic, which allows us to derive the exact p-value. However, as sample sizes grow, computing this exact distribution becomes computationally intensive, making it impractical. In these cases, an approximation using the normal distribution is employed.

#### Exact Method

The exact method calculates the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the data, given the null hypothesis. It involves evaluating all possible distributions of ranks and determining where the observed test statistic lies within this distribution.

• It provides the precise p-value.
• Suitable for small sample sizes.

• Computationally intensive for larger sample sizes.

#### Approximation Method

For larger sample sizes, R defaults to an approximation method based on the central limit theorem. This method assumes that the test statistic follows a normal distribution.

• Computationally efficient, even for large sample sizes.
• Provides results that are close to the exact method for large samples.

• Might not be as accurate as the exact method for smaller sample sizes.

#### How to Choose and Apply in R?

By default, R will choose the appropriate method based on the sample size. For small samples, R will use the exact method, while for larger samples, it will use the approximation. However, you can explicitly specify which method you want to use.

Example:

Suppose we're comparing the scores of two small groups of students.

Group A Scores Group B Scores
78 82
80 85

To force the exact method:

group_a <- c(78, 80)
group_b <- c(82, 85)

result_exact <- wilcox.test(group_a, group_b, exact = TRUE)
print(result_exact)


On the other hand, to use the approximation:

result_approx <- wilcox.test(group_a, group_b, exact = FALSE)
print(result_approx)


In practice, for most real-world scenarios with moderate to large sample sizes, the difference in p-values obtained from the exact and approximation methods is negligible. However, for small sample sizes or when precision is paramount, researchers might opt for the exact method.

## Conclusion

The world of statistical testing is vast, often presenting analysts and researchers with a variety of methods to choose from based on the data's characteristics. The Wilcoxon Rank Sum Test emerges as a beacon for those navigating through non-normally distributed data, offering a reliable tool to discern differences between two independent samples. Its non-parametric nature ensures it remains resilient against common violations of assumptions, making it a favored choice for many.

In mastering this test within the R environment, one not only expands their statistical toolkit but also ensures they are equipped to handle diverse datasets that don't fit traditional molds. As always, while the Wilcoxon Rank Sum Test is powerful, it's imperative to approach its results with caution, ensuring a comprehensive understanding of its underlying assumptions and context. Pairing this knowledge with R's capabilities, analysts can confidently explore, interpret, and present their findings.