Wilcoxon Signed-Rank Test Sample Size

Methodology

The Wilcoxon signed-rank test is a nonparametric alternative to the one-sample t-test. For sample-size planning, this method/the software uses the one-sample t-test power calculation and adjusts the resulting sample size for the assumed underlying distribution, following Al-Sunduqchi and Guenther (1990).

Wilcoxon adjustment

The calculator first finds the smallest integer t-test-equivalent sample size n′ whose calculated power reaches the requested target. It then multiplies n′ by W and rounds up to the next whole subject. This is the convention used in the this method validation examples.

Power calculation

For a two-sided test, the t-test power is the probability of falling below the lower critical value or above the upper critical value under the noncentral t distribution. For a one-sided test, the corresponding single tail is used. The noncentrality parameter is λ = |μ₁ − μ₀|√n′/σ and the degrees of freedom are df = n′ − 1.

Assumptions

• The Wilcoxon signed-rank differences are continuous and symmetrically distributed.
• The differences are mutually independent and have a common median.
• The measurement scale is at least interval.
• The population is treated as infinite; no finite-population correction is applied.
• The calculation is a planning approximation based on the this method/the software method rather than an exact enumeration of the finite-sample signed-rank statistic.

Validation example

This worked example uses μ₀ = 1.5, μ₁ = 2.0, σ = 1.0, α = 0.05, and target power = 0.80 with a two-sided alternative. The corresponding one-sample t-test requires n′ = 34. Under a Normal distribution, W = π/3, giving 34 × π/3 = 35.605, which rounds up to N = 36. This yields achieved power 0.80778 for this validated example.

References

Al-Sunduqchi, M. S. (1990). Determining the Appropriate Sample Size for Inferences Based on the Wilcoxon Statistics. Ph.D. dissertation under the direction of William C. Guenther, Department of Statistics, University of Wyoming, Laramie, Wyoming.

Chow, S. C., Shao, J., Wang, H., & Lokhnygina, Y. (2018). Sample Size Calculations in Clinical Research, 3rd ed. Taylor & Francis/CRC Press. this method cites the two-sided one-sample t-test example on pages 45–46 as the validation basis for its Wilcoxon sample-size procedure.

the software, LLC. this method Sample Size Software: Wilcoxon Signed-Rank Tests, Chapter 411. The documentation gives the Wilcoxon adjustment factors, power equations, worked examples, and validation results used by this calculator.