Conduct and Interpret a Wilcoxon Sign Test

What is the Wilcoxon Sign Test?

The Wilcoxon Sign test is a statistical comparison of the average of two dependent samples.  The Wilcoxon sign test is a sibling of the t-tests.  It is, in fact, a non-paracontinuous-level alternative to the dependent samples t-test.  Thus the Wilcoxon signed rank test is used in similar situations as the Mann-Whitney U-test.  The main difference is that the Mann-Whitney U-test tests two independent samples, whereas the Wilcox sign test tests two dependent samples.

The Wilcoxon Sign test is a test of dependency.  All dependence tests assume that the variables in the analysis can be split into independent and dependent variables.  A dependence tests that compares the averages of an independent and a dependent variable assumes that differences in the average of the dependent variable are caused by the independent variable.  Sometimes the independent variable is also called factor because the factor splits the sample in two or more groups, also called factor steps.

Dependence tests analyze whether there is a significant difference between the factor levels.  The t-test family uses mean scores as the average to compare the differences, the Mann-Whitney U-test uses mean ranks as the average, and the Wilcoxon Sign test uses signed ranks.

Unlike the t-test and F-test the Wilcoxon sign test is a non-paracontinuous-level test.  That means that the test does not assume any properties regarding the distribution of the underlying variables in the analysis.  This makes the Wilcoxon sign test the analysis to conduct when analyzing variables of ordinal scale or variables that are not multivariate normal.

The Wilcoxon sign test is mathematically similar to conducting a Mann-Whitney U-test (which is sometimes also called Wilcoxon 2-sample t-test).  It is also similar to the basic principle of the dependent samples t-test, because just like the dependent samples t-test the Wilcoxon sign test, tests the difference of observations.

However, the Wilcoxon signed rank test pools all differences, ranks them and applies a negative sign to all the ranks where the difference between the two observations is negative.  This is called the signed rank.  The Wilcoxon signed rank test is a non-paracontinuous-level test, in contrast to the dependent samples t-tests.  Whereas the dependent samples t-test tests whether the average difference between two observations is 0, the Wilcoxon test tests whether the difference between two observations has a mean signed rank of 0.  Thus it is much more robust against outliers and heavy tail distributions.  Because the Wilcoxon sign test is a non-paracontinuous-level test it does not require a special distribution of the dependent variable in the analysis.  Therefore it is the best test to compare mean scores when the dependent variable is not normally distributed and at least of ordinal scale.

For the test of significance of Wilcoxon signed rank test it is assumed that with at least ten paired observations the distribution of the W-value approximates a normal distribution.  Thus we can normalize the empirical W-statistics and compare this to the tabulated z-ratio of the normal distribution to calculate the confidence level.