Wilcoxon Signed-Rank Test Calculator

Understanding the Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test is a non-parametric test used to compare two paired samples. It tests whether the median difference between the pairs is zero, making it a good alternative to the paired t-test when data is not normally distributed.

When to Use the Wilcoxon Signed-Rank Test

This test is appropriate when comparing two related groups, such as before-and-after measurements or matched-pair samples. It does not assume normality, making it useful for skewed or ordinal data.

Considerations and Caveats

It’s essential to consider the following when interpreting the results of the Wilcoxon Signed-Rank Test:

• Sample Size: Small sample sizes can limit the test’s ability to generalize findings to a larger population.
• Non-Normal Distribution: The test does not assume normality, making it suitable for skewed data; however, it works best when differences are symmetrically distributed.
• Paired Data Requirement: The test requires paired data, which means it is valid only when each before-treatment observation has a corresponding after-treatment observation from the same subject.
• Test Type: This example used a two-tailed test. Depending on the research question, a right- or left-tailed test might be more appropriate, which would affect the interpretation of the p-value.

Wilcoxon Signed-Rank Test: Tail Test Variations

The Wilcoxon Signed-Rank Test can be adapted to different research questions by choosing a specific tail type:

• Two-Tailed Test: Tests for any difference between groups.
  \[ \text{p-value} = 2 \times \Pr(|Z| \geq |z|) \]
• Right-Tailed Test: Tests if Group 1 is significantly greater than Group 2.
  \[ \text{p-value} = \Pr(Z \geq z) \]
• Left-Tailed Test: Tests if Group 1 is significantly less than Group 2.
  \[ \text{p-value} = \Pr(Z \leq z) \]

Pr stands for "probability." In these equations, it represents the likelihood of observing a test statistic (Z) that meets a specified condition.

• \(\Pr(Z \geq z)\): The probability that the Z-score (test statistic) is greater than or equal to a specific value \(z\).
• \(\Pr(Z \leq z)\): The probability that the Z-score is less than or equal to \(z\).
• \(\Pr(|Z| \geq |z|)\): The probability that the absolute value of Z is at least as extreme as \(z\), used in two-tailed tests.

Note: The choice of tail type affects which part of the distribution is used to compute the p-value, which in turn influences the outcome of the hypothesis test.

How to Obtain the P-Value from the Wilcoxon Statistic

To calculate the p-value from the Wilcoxon statistic, you can use a statistical library. Note that statistical packages may define the Wilcoxon statistic differently, so ensure consistency with the package’s documentation:

from scipy.stats import wilcoxon
stat, p_value = wilcoxon(data1, data2, alternative="two-sided")

Alternatively, you can consult the Wilcoxon Signed-Rank critical values table. For small sample sizes, compare your W-statistic to the critical value at your chosen significance level:

• If your W-statistic is less than or equal to the critical value in the table, the result is statistically significant at the selected significance level.

Normal Approximation for Large Samples

For larger sample sizes (typically \( n > 20 \)), the Wilcoxon Signed-Rank Test statistic \( W \) can be approximated by a normal distribution:

• Calculate the mean and standard deviation of \( W \): \[ \mu_W = \frac{n(n+1)}{4} \] \[ \sigma_W = \sqrt{\frac{n(n+1)(2n+1)}{24}} \]
• Convert \( W \) to a z-score: \[ Z = \frac{W - \mu_W}{\sigma_W} \]
• Calculate the p-value using the normal distribution:
  • Two-Tailed Test: \( p\text{-value} = 2 \times \Pr(|Z| \geq |z|) \)
  • Right-Tailed Test: \( p\text{-value} = \Pr(Z \geq z) \)
  • Left-Tailed Test: \( p\text{-value} = \Pr(Z \leq z) \)

This normal approximation is accurate for hypothesis testing with larger samples, where exact table values are less practical.