Wilcoxon Signed Rank Test

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Wilcoxon Signed Rank Test

Definition

Wilcoxon Signed Rank Test is a non-parametric alternative to the paired t-test. It tests the null hypothesis that the differences between pairs of observations come from a distribution with zero median, without requiring normality.

Formula

Test Statistic:

W = \min(W^+, W^-)

Where:

$W^+$ = sum of positive ranks
$W^-$ = sum of negative ranks

Key Assumptions

Paired Observations: Data must be paired

Independence: Pairs must be independent

Continuous Data: Differences should be from continuous distribution

Symmetry: Distribution of differences should be symmetric

Practical Example

Step 1: State the Data

Weight measurements before and after treatment (kg):

Subject	Before	After	Difference
1	70	68	+2
2	80	78	+2
3	90	91	-1
4	60	58	+2
5	85	85	0 (ignored)

Step 2: State Hypotheses

$H_0$ : median difference = 0
$H_a$ : median difference ≠ 0
$\alpha = 0.05$

Step 3: Assign Ranks

1. Order absolute differences in ascending order: $|-1|, |2|, |2|, |2|$
2. Assign ranks:
- $|-1| = 1 \rightarrow \text{rank } 1$
- $|2| = 2 \rightarrow \text{mean rank } 3 \; (\text{ranks } 2,3,4)$

Step 4: Calculate Sums

$W^+$ = 3 + 3 + 3 = 9 (positive differences)
$W^-$ = 1 (negative differences)

Step 5: Calculate Test Statistic

$W = \min(W^+, W^-) = \min(9, 1) = 1$

Step 6: Draw Conclusion

For $n = 4$ (excluding zero difference), at $\alpha = 0.05$ , the critical value is $0$ . Since $W = 1 \gt 0$ , we fail to reject $H_0$ . There is insufficient evidence to conclude that the treatment had a significant effect on weight.

Code Examples

1# Sample data
2before <- c(70, 80, 90, 60, 85)
3after <- c(68, 78, 91, 58, 85)
4
5# Perform the test
6result <- wilcox.test(before, after, paired = TRUE)
7
8# Print results
9print(result)

Python

1from scipy.stats import wilcoxon
2
3# Sample data
4before = [70, 80, 90, 60, 85]
5after = [68, 78, 91, 58, 85]
6
7# Calculate the test
8stat, p_value = wilcoxon(before, after)
9
10# Print results
11print(f"Wilcoxon Signed Rank Test Statistic: {stat}")
12print(f"P-value: {p_value}")

Software Implementation Differences

Different statistical software packages report the Wilcoxon Signed-Rank test statistic in distinct ways:

R(wilcox.test): Reports $V$ , which is the sum of all positive ranks.
Python(scipy.stats.wilcoxon): Reports $W = \text{min}(W^+, W^-)$ , where:
- $W^+$ is the sum of positive ranks
- $W^-$ is the sum of negative ranks

While these approaches use different test statistics, they lead to equivalent results:

Both methods yield identical p-values
The statistics can be converted between formats if needed
The critical values in this table are based on R's implementation

Alternative Tests

Consider these alternatives:

Paired t-test: When data is normally distributed
Sign Test: When only direction of difference matters

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Wilcoxon Signed Rank Test

Calculator

1. Load Your Data

2. Select Columns & Options

Learn More

Wilcoxon Signed Rank Test

Definition

Formula

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: State Hypotheses

Step 3: Assign Ranks

Step 4: Calculate Sums

Step 5: Calculate Test Statistic

Step 6: Draw Conclusion

Code Examples

Software Implementation Differences

Alternative Tests

Related Calculators

Paired T-Test Calculator

Mann-Whitney U Test Calculator

One Sample T-Test Calculator

Effect Size Calculator

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