Wilcoxon Test
- Wilcoxon’s Signed-Rank Test (WSR)
- Signed-rank test
Wilcoxon test analyzes whether there is a difference between 2 dependent samples (alt hypothesis) or not (null hypothesis).
T-Test
- Parametric test for 2 dependent groups ()
- T-Test tests whether the average difference between two observations is 0
- Requires normally distributed data
- Answers the question: “Is there a difference in mean?”
Wilcoxon Test
- Wilcoxon test is the nonparametric counterpart to the T-Test.
- Tests whether the difference between two observations has a mean signed rank of 0
- Does not require normally distributed data
- Compares the differences in ranks
Wilcoxon rank-sum test is a popular nonparametric test for comparing two independent populations (groups)1
One important assumption for applicability of Wilcoxon ranksum test is that all the observations under the study are independent1
Samples
Dependent samples have measured values that come in pairs. Often, the pairs of data come from repeated samples of the same participant.
When to apply
- You should always use the T-Test when possible, unless the data is not normally distributed.
- Only 2 dependent samples with at least ordinally scaled characteristics need to be available.
How the test Works
The Wilcoxon Test determines whether there is a difference in rank totals between 2 dependent groups.
Calculation
Rank Differences
The ranks are based on the magnitude of difference between one pair. The pair with the difference closest to 0 gets rank 1. A (+) is assigned to all positive values and a (-) is assigned to all negative values.
Determine Rank Sums
Next you sum the positive ranks \(T^+\) and the negative ranks \(T^-\).
\[ T^+ = \sum{\textbf{All positive ranks}} \]
\[ T^- = \sum{\textbf{All Negative ranks}} \]
Calculate Test Statistic W
\[ W = \textbf{min}(T^+, T^-) \]
Expected value of W (\(\mu_W\)) refers to the value of W we would expect if there were no difference between within the pairs. If the null hypothesis is true, we would expect \(W\) to be equal to \(\mu_W\)
\[ \mu_W = \frac{n(n+1)}{4} \]
Standard deviation
Standard deviation of
\[ \sigma_W = \sqrt{\frac{n(n+1)(2n+1)-\sum\frac{t^3_i-t_i}{2}}{24}} \]
Z-Value
Z-Value (\(Z\)) calculates the value that we would expect if there was no difference (\(\mu_W\)) with the value that actually occured (\(W\)) divided by the standard deviation (\(\sigma_W\)) \[ Z = \frac{W - \mu_W}{\sigma_W} \]
If there are more than 25 cases, normal distribution is assumed and you can use the method above.
If there are less than 25 cases, you should use a table.
Hypothesis
Null Hypothesis
Both rank sums are equal.
Interpreting the results
- Z-value:
- A negative value indicates that group 1 had lower values than group 2.
- If the groups are evenly distributed, then the z-score will be closer to 0.
- p-value: indicates whether the difference was statistically significant.