Name: comparing two groups with rank sums
Uploaded: 2025-01-09T15:14:08.000Z
Duration: 1 min 21 s
Channel: JoVE

Wilcoxon rank-sum, or the Mann-Whitney U, is a nonparametric test used to determine the difference between two populations by comparing their medians.
It strictly applies to two independent simple random samples with equal or unequal sample sizes.
Consider the prey capture response time in two different spider species.
Here, the null hypothesis states that the median response time of the two species is the same. The alternative hypothesis states otherwise.
The values in these two samples are ranked, considering them as a single pool of data points. However, the rank sum is calculated independently.
When high or low ranks are primarily found in one sample than the other, the two samples may have different medians.
The test statistic z is calculated using the following equations to test the hypothesis.
The Wilcoxon rank-sum test is two-tailed. So, the test statistic should be compared with the positive and negative critical values, typically at 5%.
Since, in the present example, the test statistic is beyond the range of these critical values, the medians of the two samples are significantly different.

wilcoxon rank-sum test

Wilcoxon Rank-Sum Test

The Wilcoxon rank-sum test, also known as the Mann-Whitney U test, is a nonparametric test used to determine if there is a significant difference between the distributions of two independent samples. This test is designed specifically for two independent populations and has the following key requirements:
<ol>
	<li>The samples must be randomly drawn.</li>
	<li>The data should be ordinal or capable of being converted to an ordinal scale, allowing the values to be ordered and ranked.</li>
</ol>
The null hypothesis is that the median of the distributions of the two populations are identical, and the alternative hypothesis is that the distributions of the two populations are different. In addition, there is no assumption that the samples are normally distributed. In this test, when two samples are drawn from two identical populations and ranked as one single pool of data points, their medians are expected to be different if smaller or larger ranks predominantly fall into one of the samples.
To perform the Wilcoxon rank-sum test, also known as the Mann-Whitney U test, data from both samples are combined into a single ranked list, where each value is assigned a rank from smallest to largest. If any values are tied, they are given the average of the ranks for those positions. The ranks are then separated back into their respective groups, and the sum of ranks is calculated for each group.
The test statistic (denoted U&#160;in the Mann-Whitney U version) is derived from these rank sums, and its significance is evaluated to determine if there is a difference between the two samples. A significant result suggests that one sample has systematically higher or lower ranks, indicating a difference in the underlying distributions of the two populations. For small sample sizes (typically n&#60;20), critical values for U are used from a table, while for larger samples, a z-score approximation is applied, assuming a normal distribution. This test is particularly useful when the assumptions for a two-sample t-test are not met, such as with ordinal or non-normally distributed data, offering a robust alternative to assess differences between two independent groups.
When the sample size is sufficiently large, this test is generally more efficient than its parametric counterpart and, therefore, more preferred for the data analysis. The significance tested using the Wilcoxon rank-sum test is usually reliable despite having outliers in the data. However, the test is also prone to higher type-I error when the data are either biased, heteroscedastic (having different variance), or when data/sample distributions are extremely far from the normal distribution.

Wilcoxon rank-sum, or the Mann-Whitney U, is a nonparametric test used to determine the difference between two populations by comparing their medians.
It ...

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Statistics

Unit 1

Nonparametric Statistics

KEY TERMS AND CONCEPTS

comparing two groups with rank sums

The Wilcoxon rank-sum test compares two independent groups by using ranks instead of raw numbers. It is also called the Mann-Whitney U test. This nonparametric test helps show whether the two groups come from different distributions.

The test is meant for two independent populations. The samples should be randomly drawn. The data must be ordinal, or it must be possible to put the data in order and rank it. The null hypothesis says the medians of the two populations are the same. The alternative hypothesis says the distributions are different. The test does not require the samples to follow a normal distribution.

To run the test, the data from both samples are combined into one list and ranked from smallest to largest. If values are tied, each tied value gets the average of the ranks for those positions. The ranks are then split back into the original groups, and the rank sum is found for each group. The test statistic, called U in the Mann-Whitney U version, comes from these rank sums and is used to test for a difference between the groups.

A significant result suggests that one sample has mostly higher or lower ranks than the other. That pattern points to a difference in the underlying distributions of the two populations. For small sample sizes, usually n less than 20, critical U values are taken from a table. For larger samples, a z-score approximation is used and a normal distribution is assumed.

This test is useful when the assumptions for a two-sample t-test are not met. It works well with ordinal data and with data that are not normally distributed. With large enough samples, it is often more efficient than the parametric version and may be preferred for analysis. It is usually reliable even when outliers are present. However, it can have a higher type-I error when the data are biased, heteroscedastic, or very far from normal.

Comparing Two Groups with Rank Sums

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key terms and concepts

Learn how the Wilcoxon rank-sum test compares two independent groups using ranks, U values, and z-score methods when t-test assumptions are not met.

Comparing Two Groups with Rank Sums | JoVE K12

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Comparing Two Groups with Rank Sums

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