What Is the Wilcoxon Test?
The Wilcoxon test, which consists of the rank sum test and the signed rank test, is a nonparametric statistical tool designed to compare two paired groups. This test calculates the differences between sets of paired data to determine if they are statistically significantly different from one another.
Key Takeaways
- The Wilcoxon test compares two paired groups and has two versions: the rank sum test and the signed rank test.
- The primary objective is to determine if two or more sets of pairs are statistically significantly different.
- Both tests assume that the data come from dependent populations, such as the same individuals or entities studied across different time points or conditions.
Understanding the Wilcoxon Test
The Wilcoxon rank sum and signed rank tests were introduced by the renowned statistician Frank Wilcoxon in a 1945 research paper. These tests are foundational in hypothesis testing for nonparametric statistics, which are used when population data can be ranked but do not necessarily have numerical values. Examples include customer satisfaction ratings or music reviews. Nonparametric distributions lack parameters and cannot be precisely described by equations, as opposed to parametric distributions.
Practical Questions Addressed by the Wilcoxon Test
These nonparametric models can help answer questions such as:
- Are student test scores significantly different between the 5th and 6th grades?
- Does a medical treatment show an effect on the same individuals over consecutive tests?
These tests assume that the data come from two matched or dependent populations and are continuous rather than discrete. Due to its nonparametric nature, the Wilcoxon test does not require a specific probability distribution for the dependent variable.
Types of the Wilcoxon Test
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Wilcoxon Rank Sum Test: This test evaluates the null hypothesis that two populations share the same continuous distribution. It is crucial that the data are randomly and independently paired, measurable on an interval scale, and derive from the same population.
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Wilcoxon Signed Rank Test: This test utilizes the magnitudes and signs of differences between paired observations. It serves as a nonparametric alternative to the paired Student’s t-test, particularly useful when population data deviate from a normal distribution.
Calculating a Wilcoxon Test Statistic
To arrive at a Wilcoxon signed rank test statistic (W), follow these steps:
- *Calculate the difference score (D aw_subscript{i} aw): For each item in a sample of n items, find the difference between two measurements (subtract one from the other).
- Absolute differences (|D aw_subscript{i}| aw*):** Ignore the positive or negative signs to obtain a set of n absolute differences.
- Omitting zeros: Remove difference scores of zero, resulting in n’ <= n* non-zero absolute differences, where n’ becomes the actual sample size.
- *Assign ranks (R aw): Rank each absolute difference from 1 to *n *, with the smallest absolute difference getting rank 1. If two or more differences are equal, assign them the averaged rank.
- Reassign signs: Reassign the original positive or negative signs to the ranks based on whether D aw_subscript{i}|raw was originally positive or negative.
- Sum of positive ranks (W): Finally, calculate the Wilcoxon test statistic W as the sum of the positive ranks.
Typically, this test is conducted using statistical software or spreadsheet tools.
Related Terms: Rank Sum Test, Signed Rank Test, Nonparametric Statistics, T-Test, Null Hypothesis, Hypothesis Testing.
References
- Wilcoxon, Frank. “Individual Comparisons by Ranking Methods”. *International Biometric Society,*Vol. 1, No. 6, Dec. 1945, pp. 80-83.
