Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test is a non-parametric statistical hypothesis test used to compare two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ. It is named after Frank Wilcoxon, who introduced the test in 1945 in his paper "Individual Comparisons by Ranking Methods."

History and Development

Frank Wilcoxon, while working at the American Cyanamid Company, developed this test to analyze paired differences in experiments where the normality assumption of the data could not be met. His work was initially published in Biometrics Bulletin, and it provided an alternative to the Paired T-Test for non-normal data. The test was later expanded upon by others, including Sidney Siegel, who formalized the test's use in his book "Nonparametric Statistics for the Behavioral Sciences."

Context and Use

• Non-parametric Test: The test does not assume that the differences between pairs follow a normal distribution, making it particularly useful for small sample sizes or when the data might be ordinal or not normally distributed.
• Applications: It's commonly used in fields like psychology, education, and medicine where data might not meet parametric test assumptions. Examples include:
  • Before-and-after studies to assess the effect of a treatment.
  • Comparing the performance of two groups on a matched-pairs design.
  • Evaluating changes in scores on tests or scales over time.
• Procedure:
  1. Calculate the differences between paired observations.
  2. Ignore any pairs where the difference is zero.
  3. Rank the absolute differences from smallest to largest, giving tied values the average rank.
  4. Assign the signs of the original differences to these ranks.
  5. Sum the ranks for positive and negative differences separately.
  6. Use the smaller of these two sums (T) for the test statistic.
  7. Compare this value against critical values from Wilcoxon tables or use an approximation to the normal distribution for larger samples.
• Assumptions:
  • The observations are paired or matched.
  • The distribution of the differences should be symmetric.
  • Observations are independent of each other.

Limitations

While the Wilcoxon Signed-Rank Test is robust against violations of normality, it does have limitations:

• It assumes that the distribution of differences is symmetric, which might not always be the case.
• The test can be less powerful than parametric tests when the data is actually normally distributed.

Sources:

• NIST/SEMATECH e-Handbook of Statistical Methods
• StatsDirect - Wilcoxon Signed Rank Test
• Statistics How To - Wilcoxon Signed-Rank Test
• ScienceDirect - Wilcoxon Signed-Rank Test

Related Topics:

• Mann-Whitney U Test
• Paired T-Test
• Non-Parametric Tests
• Sign Test