The Wilcoxon Signed-Rank Test is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ. It is an alternative to the paired Student's t-test when the data cannot be assumed to be normally distributed. In the context of Lean Six Sigma and the broader field of quality improvement, this test can be particularly useful in analyzing the effects of process changes when the data do not meet the assumptions necessary for parametric tests.

Understanding the Test

The Wilcoxon Signed-Rank Test focuses on the median of the differences between paired observations, rather than their means. This makes it less sensitive to outliers and more appropriate for data that is skewed or not normally distributed. The test is applied to the differences between pairs of observations, making it suitable for before-and-after studies, matched pairs experiments, or situations where the same subjects are subjected to two different conditions.

How It Works

1. Calculate Differences: For each pair of observations, calculate the difference. The direction of the difference (which condition comes first) must be consistent across all pairs.

   
   

2. Rank the Differences: Ignore zero differences and rank the absolute values of the differences from smallest to largest, assigning ranks from 1, 2, 3, and so forth. If there are ties, assign to each tied group the average of the ranks they would have received had they not been tied.

   
   

3. Assign Signs: Assign signs to the ranks based on the sign of the difference (positive or negative) for each pair.

   
   

4. Sum the Ranks: Calculate the sum of the ranks for positive differences (W+) and the sum of the ranks for negative differences (W−).

   
   

5. Statistical Decision: The test statistic is the smaller of W+ or W−. This value is then compared to a critical value from the Wilcoxon signed-rank table or calculated using a software package to determine if the difference is statistically significant.
   

   Wilcoxon Table:

Interpretation

The null hypothesis (H0​) of the Wilcoxon Signed-Rank Test states that the median of the differences between the pairs of observations is zero. A significant test result indicates that there is sufficient evidence to reject H0​, implying that there is a difference in the central tendency of the two related groups being compared.

Application in Lean Six Sigma

In Lean Six Sigma projects, the Wilcoxon Signed-Rank Test can be particularly useful in the Analyze phase, where identifying and verifying the root causes of problems in a process is crucial. It allows practitioners to make informed decisions based on data that may not meet the normality assumption required for parametric tests. For instance, when evaluating the impact of a process improvement initiative, this test can be used to compare the efficiency or quality metrics of a process before and after the implementation of a change, even when the data are not normally distributed.

Conclusion

The Wilcoxon Signed-Rank Test offers a robust and versatile option for hypothesis testing with non-normal data, making it a valuable tool in the Lean Six Sigma toolkit. By enabling the comparison of related samples without the stringent assumptions of parametric tests, it broadens the scope of statistical analysis that can support evidence-based decision-making in quality improvement efforts.

Wilcoxon Signed-Rank Test Example

Let's consider a real-life scenario where a company is trying to improve the response time of its customer service department. The company implements a new training program for its customer service representatives and wants to evaluate its effectiveness. The company measures the response time (in minutes) for 8 customer service requests before and after the training. The goal is to use the Wilcoxon Signed-Rank Test to determine if the training has significantly affected the response times.

Data

Here's the response time data for the 8 requests, before and after the training:

Steps to Perform the Wilcoxon Signed-Rank Test

1. Calculate Differences: Done in the table above.

2. Rank the Absolute Differences:

Ignoring the sign, the absolute differences and their ranks are:

3. Assign Signs to Ranks:

4. Sum the Ranks:

Sum of positive ranks (W+): There are no positive differences, so W+=0.

Sum of negative ranks (W−): −4−4−4−1−6.5−6.5−5−2=−33

Note: In this case, since all differences are negative, W− is simply the sum of the absolute values of the signed ranks.

5. Statistical Decision: To determine whether the observed sum of ranks is statistically significant, we compare it to critical values from the Wilcoxon signed-rank test table. Since the exact distribution depends on the sample size, tables provide critical values for different sample sizes and significance levels (usually 0.05 for a 5% significance level).

For a sample size of 8 and using a two-tailed test at the 5% significance level, let's assume the critical value for W− is 3

Since our W− value of -33 is less than the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference in the median response times before and after the training, indicating that the training program had a significant effect on improving response times.