# Two-sample paired (Wilcoxon) signed rank test

The Wilcoxon two-sample paired signed rank test is used to test the null hypothesis that the population median of the paired differences of the two samples is 0.

### Assumptions:

• The paired differences are independent.
• Each paired difference comes from a continuous distribution that is symmetric, with the same center of symmetry. Strictly speaking, the population distributions need not be the same for all the paired differences. However, if we want a consistent test, we assume that the paired differences all come from the same continuous, symmetric distribution. (The Wilcoxon signed rank test is a nonparametric test. We need not specify or know what the distribution is, only that all the paired difference follow the same one.)
• The paired differences all have the same median. (This median will also be the center of symmetry for the population distribution associated with each paired difference. Moreover, since the mean of a continuous symmetric distribution is equal to its median, this means that the paired differences will also all have the same mean.)
• Because the test statistic for the Wilcoxon signed rank is based only on the ranks of the paired differences, the test can be performed when the only data available are those relative ranks for the paired differences.

Note that it is not assumed that the two samples are independent of each other. In fact, they should be related to each other such that they create pairs of data points, such as the measurements on two matched people in a case/control study, or before- and after-treatment measurements on the same person.

The two-sample paired signed rank test is equivalent to performing a one-sample signed rank test on the paired differences.

### Guidance:

• Ways to detect before performing the signed rank test whether your data violate any assumptions.
• Ways to examine signed rank test results to detect assumption violations.
• Possible alternatives if your data or signed rank test results indicate assumption violations.

To properly analyze and interpret results of the two-sample paired signed rank test, you should be familiar with the following terms and concepts:

If you are not familiar with these terms and concepts, you are advised to consult with a statistician. Failure to understand and properly apply the two-sample paired signed rank test may result in drawing erroneous conclusions from your data. Additionally, you may want to consult the following references:

• Brownlee, K. A. 1965. Statistical Theory and Methodology in Science and Engineering. New York: John Wiley & Sons.
• Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd ed. New York: John Wiley & Sons.
• Daniel, Wayne W. 1978. Applied Nonparametric Statistics. Boston: Houghton Mifflin.
• Daniel, Wayne W. 1995. Biostatistics. 6th ed. New York: John Wiley & Sons.
• Hollander, M. and Wolfe, D. A. 1973. Nonparametric Statistical Methods. New York: John Wiley & Sons.
• Lehmann, E. L. 1975. Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
• Miller, Rupert G. Jr. 1996. Beyond ANOVA, Basics of Applied Statistics. 2nd. ed. London: Chapman & Hall.
• Rosner, Bernard. 1995. Fundamentals of Biostatistics. 4th ed. Belmont, California: Duxbury Press.
• Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed. New York: W. H. Freeman and Co.
• Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ: Prentice-Hall./li>