Wilcoxon's signed-rank statistic: what null hypothesis and why it matters.

Area of Science:

• Statistics
• Statistical inference
• Hypothesis testing

Background:

• The term "signed-rank test" (Wilcoxon signed-rank test) is used for two distinct statistical tests.
• This ambiguity can lead to confusion in statistical analysis and interpretation.

Purpose of the Study:

• To propose distinct terminology for the two tests currently referred to as the "signed-rank test".
• To differentiate between the test for symmetry of distribution and the test for the median of a symmetric distribution.

Main Methods:

• Review of statistical literature concerning signed-rank tests.
• Comparative analysis of the test for symmetry based on signed-rank statistic and the test for median based on signed-rank statistic.
• Examination of the relationship between signed-rank tests, sign test, and one-sample t-test.

Main Results:

• The signed-rank statistic is utilized in two separate statistical tests: one for distribution symmetry and another for the median of a symmetric distribution.
• The proposed distinct names are "test for symmetry based on signed-rank statistic" and "test for median based on signed-rank statistic".

Conclusions:

• Adopting distinct terminology for these two tests will enhance clarity and reduce ambiguity in statistical practice.
• The proposed nomenclature facilitates a better understanding of the tests' connections and contrasts with related statistical methods like the sign test and one-sample t-test.