The Wilcoxon Signed-Ranks Test and the One-Sample Sign Test are non-parametric analyses used to test whether a sample of n subjects (or objects) comes from a population where the median value (θ) is equal to a certain value. If the test result is significant, it is concluded that there is a high likelihood that the sample comes from a population with a median value different from θ. This test serves as an alternative to the One-Sample T-test when the assumption of normality is not met.
Case Example
Taken from: Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition by David J. Sheskin
Example 6.1
A doctor claims that the median number of visits his patients make each year is five times. To evaluate the validity of this statement, he randomly selects ten of his patients and determines the number of clinic visits made by each patient over the past year. He obtained the following values for the ten patients in his sample: 9, 10, 8, 4, 8, 3, 0, 10, 15, 9. Do these data support his claim that the median number of patient visits is five?
Analysis Steps
Here are the steps to conduct the Wilcoxon Signed-Ranks Test and the One-Sample Sign Test using SmartstatXL, an Excel Add-in:
- Activate the worksheet (Sheet) to be analyzed.
- Place the cursor on the dataset (to create a dataset, see Data Preparation method).
- If the active cell is not on the dataset, SmartstatXL will automatically try to determine the dataset.
- Activate the SmartstatXL Tab
- Click on the Non-Parametric Menu. SmartstatXL will display a dialog box to confirm whether the dataset is correct or not (usually the cell address of the dataset is automatically selected correctly).
- If it is correct, click on the Next Button
- A dialog box for the Non-Parametric Test will appear next:
- If using the Sign Test
- Next, press the "OK" button
Analysis Results
Below are the Output Analysis of the Wilcoxon Signed-Ranks Test and the One-Sample Sign Test:
Wilcoxon Signed-Ranks Test Statistical Summary
Based on the analysis using the Wilcoxon Signed-Ranks Test for one sample, it was found that the median of patient visits is not equal to 5. The obtained test statistic T is 11.000 with a p-value of 0.046. Since this p-value is less than 0.05, we reject the null hypothesis (H0) stating that the median number of visits is 5. Therefore, there is strong evidence to suggest that the median number of patient visits per year is not 5.
One-Sample Sign Test Statistical Summary
Meanwhile, based on the analysis using the One-Sample Sign Test, there is not enough evidence to reject the null hypothesis (H0). With a test statistic T of 3.000 and a p-value of 0.344 (which is greater than 0.05), we do not have sufficient evidence to say that the median number of patient visits differs from 5 per year.
Conclusion
Of the two tests, only the Wilcoxon Signed-Ranks Test provides sufficient evidence to reject the doctor's claim that the median number of patient visits is 5 times per year. However, the Sign Test does not support this conclusion. Therefore, it is essential to consider both test results when drawing the final conclusion about the doctor's claim.
In this case, the difference in conclusions from the two tests occurred due to the following factors:
- Calculation Method:
- Wilcoxon Signed-Ranks Test: This test considers the difference between each observation in the sample and the hypothesized median value. These differences are ranked by their absolute values, and the test statistic is calculated based on the signs and ranks of these differences.
- Sign Test: This test only considers the sign of the difference between each observation in the sample and the hypothesized median value, without taking into account the magnitude of the difference.
- Test Power: Because the Wilcoxon Signed-Ranks Test takes into account the magnitude of the difference between observations and the hypothesized median value, this test often has greater statistical power compared to the Sign Test.
- Data Distribution: If most of the differences between observations and the hypothesized median value are close to zero, or if there are some very extreme differences, then the Wilcoxon Test may be more sensitive in detecting these differences compared to the Sign Test.
- Sample Size: With a small sample size, sampling randomness can affect the results of both tests.
- Ties: In the Wilcoxon Test, ties (when there are two or more differences with the same value) can affect the ranks and test statistics. The Sign Test is not affected by ties as it only considers the sign of the difference.
