Sample size calculation for the Wilcoxon-Mann-Whitney test adjusting for ties

Based on [Zhao et al., 2008]

1. When to use Wilcoxon-Mann-Whitney (WMW) test [1]?

• Non-parametric test for comparing two groups of observations on a continuous or ordered categorical (ordinal) variable when there is no underlying distributional assumption imposed on the data.

• Data representation

Group	\(C_1\)	\(C_2\)	\(\cdots\)	\(C_D\)	Total
A	\(m_1(p_1)\)	\(m_2(p_2)\)	\(\cdots\)	\(m_D(p_D)\)	\(m\)
B	\(m_1(q_1)\)	\(m_2(q_2)\)	\(\cdots\)	\(m_D(q_D)\)	\(n\)
Total	\(M_1\)	\(M_2\)	\(\cdots\)	\(M_D\)	\(N\)

where

• \(N\) is the total sample size; \(m\) and \(n\) are sample sizes for Group A and B, respectively

• \(M_D\) is the column total for Column \(D\), which stands for distinct outcome value

• \(p_i\) = \(m_i/m\), enclosed in the parenthese; \(q_i\) is defined similarly

2. Details about WMW

WMW uses the competing probability \(\pi = Pr(X > Y) + 0.5Pr(X = Y)\) to quantify the difference between two groups under comparison, where X and Y are random variables with CDF \(F_X\) and \(F_Y\), respectively. The null hypothesis is

\[H_0: \pi = 0.5\]

An unbiased estimator \(\hat{\pi}\) of the competing probability \(\pi\) is

\[\hat{\pi} = (mn)^{-1}\sum_{i=1}^m\sum_{j=1}^n\delta(X_i - Y_j)\]

where \(\delta(t) = 1\) if \(t> 0\), 0.5 if \(t=0\), and 0 if \(t<0\).

The variance of \(\hat{\pi}\) under the null hypothesis can be calculated as

\[\hat{\sigma}^2_0 = V_0[\hat{\pi}] = \frac{N+1}{12mn} - \frac{1}{12N(N-1)mn}\sum_{c=1}^D(M_c^3 - M_c),\]

where \(m\) and \(n\) are sample sizes for each group, \(N=m+n\) and \(M_c\) is the marginal total for a unique group defined by the outcome value. Where there is no ties, it reduces to \((N+1)/(12mn)\).

In WMW test, under null the z-statistic is construct as follows:

\[z_0 = \frac{\hat{\pi} - 0.5}{\hat{\sigma}_0}\sim N(0, 1)\]

In general, e.g. Sigel and Castellan recommended when \(m=3\) or 4 and \(n>12\); \(m>4\) and \(n>10\), above normal approximation can be made.

3. Sample size formula

• To compute the total sample size \(N\) , we assume the treatment fraction \(t = n/N\) is known.

• In addition, we assume the proportions \(p_1,\cdots, p_D\) and \(q_1,\cdots,q_D\) in above table are known.

Derived from the fact that

\[\left(\frac{\mu_1 - \mu_0}{\sigma_0}\right)^2 = \left(Z_{\alpha/2} + \frac{\sigma_1}{\sigma_0}Z_{\beta}\right)^2\]

where \(\mu_0\), \(\sigma_0\) and \(\mu_1\), \(\sigma_1\) are the means and sd of \(\hat{\pi}\) under the null and the alternative hypothesis, and \(Z_{\alpha/2}\) is the upper \((\alpha/2)\)th quantile of the standard normal distribution.

Further assume \(\sigma_0 = \sigma_1\) along with the fact that \(\mu_0 = 0.5\), the sample size formula simplifies to

\[\left(\frac{\mu_1 - \mu_0}{\sigma_0}\right)^2 = \left(Z_{\alpha/2} + Z_{\beta}\right)^2\]

The final sample size formula for the WMW test is derived as follows:

\[N = \frac{(Z_{\alpha/2}+Z_{\beta})^2(1-\sum_{c=1}^D((1-t)p_c+tq_c)^3)}{12t(1-t)(\sum_{c=2}^Dp_c\sum_{d=1}^{c-1}q_d + 0.5\sum_{c=1}^Dp_cq_c-0.5)^2}\]

When there is no ties, the sample size formula simplifies to

\[N = \frac{(Z_{\alpha/2}+Z_{\beta})^2}{12t(1-t)(\hat{\mu}_1 -0.5)^2} = \frac{(Z_{\alpha/2}+Z_{\beta})^2}{12t(1-t)(\sum_{c=2}^Dp_c\sum_{d=1}^{c-1}q_d + 0.5\sum_{c=1}^Dp_cq_c-0.5)^2}\]

[ZRQ08]

Yan D Zhao, Dewi Rahardja, and Yongming Qu. Sample size calculation for the wilcoxon–mann–whitney test adjusting for ties. Statistics in medicine, 27(3):462–468, 2008.