Sample Size Determinations for the Wilcoxon–Mann–Whitney Test: A Comprehensive Review

Rahardja, Dewi; Zhao, Yan; Qu, Yongming

doi:10.1198/sbr.2009.0016

Cited by 18 publications

(15 citation statements)

References 12 publications

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“…The power of the Wilcoxon–Mann–Whitney test has also been described under specific models, such as the proportional odds model. Rahardja et al presented a comprehensive review.…”

Section: Introductionmentioning

confidence: 99%

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Power and sample size evaluation for the Cochran–Mantel–Haenszel mean score (Wilcoxon rank sum) test and the Cochran–Armitage test for trend

Lachin

2011

Statistics in Medicine

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The power of a chi-square test, and thus the required sample size, are a function of the noncentrality parameter that can be obtained as the limiting expectation of the test statistic under an alternative hypothesis specification. Herein, we apply this principle to derive simple expressions for two tests that are commonly applied to discrete ordinal data. The Wilcoxon rank sum test for the equality of distributions in two groups is algebraically equivalent to the Mann-Whitney test. The Kruskal-Wallis test applies to multiple groups. These tests are equivalent to a Cochran-Mantel-Haenszel mean score test using rank scores for a set of C-discrete categories. Although various authors have assessed the power function of the Wilcoxon and Mann-Whitney tests, herein it is shown that the power of these tests with discrete observations, that is, with tied ranks, is readily provided by the power function of the corresponding Cochran-Mantel-Haenszel mean scores test for two and R > 2 groups. These expressions yield results virtually identical to those derived previously for rank scores and also apply to other score functions. The Cochran-Armitage test for trend assesses whether there is an monotonically increasing or decreasing trend in the proportions with a positive outcome or response over the C-ordered categories of an ordinal independent variable, for example, dose. Herein, it is shown that the power of the test is a function of the slope of the response probabilities over the ordinal scores assigned to the groups that yields simple expressions for the power of the test.

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“…The power of the Wilcoxon–Mann–Whitney test has also been described under specific models, such as the proportional odds model. Rahardja et al presented a comprehensive review.…”

Section: Introductionmentioning

confidence: 99%

“…Kruskal and Wallis presented a generalization of the Wilcoxon test to R > 2 groups that is also equivalent to the CMH mean score test. Some authors have also described bootstrap procedures on the basis of sample data that can be used to estimate the power of the Wilcoxon or Kruskal–Wallis tests .…”

Section: Introductionmentioning

confidence: 99%

Power and sample size evaluation for the Cochran–Mantel–Haenszel mean score (Wilcoxon rank sum) test and the Cochran–Armitage test for trend

Lachin

2011

Statistics in Medicine

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“…For the standard WMW test applied to a continuous outcome, methods A and B are used when data are available from previous pilot studies, while methods C and D are used when data from both groups are limited to summary statistics [37]. Method D can also be used for data not satisfying the location shift alternative or for an exploratory or a pilot study [29].…”

Section: Methods Amentioning

confidence: 99%

Power and sample size calculations for the Wilcoxon-Mann-Whitney test in the presence of death-censored observations

Matsouaka

Betensky

2014

Statist. Med.

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We consider a clinical trial of a potentially lethal disease in which patients are randomly assigned to two treatment groups and are followed for a fixed period of time; a continuous endpoint is measured at the end of follow-up. For some patients, however, death (or severe disease progression) may preclude measurement of the endpoint. A statistical analysis that includes only patients with endpoint measurements may be biased. An alternative analysis includes all randomized patients, with rank scores assigned to the patients who are available for the endpoint measurement based on the magnitude of their responses and with "worst-rank" scores assigned to those patients whose death precluded the measurement of the continuous endpoint. The worst-rank scores are worse than all observed rank scores. The treatment effect is then evaluated using the Wilcoxon-Mann-Whitney (WMW) test. In this paper, we derive closed-form formulae for the power and sample size of the WMW test when missing measurements of the continuous endpoints due to death are replaced by worst-rank scores. We distinguish two approaches for assigning the worst-rank scores. In the tied worst-rank approach, all deaths are weighted equally and the worstrank scores are set to a single value that is worse than all measured responses. In the untied worstrank approach, the worst-rank scores further rank patients according to their time of death, so that an earlier death is considered worse than a later death, which in turn is worse than all measured responses. In addition, we propose four methods for implementation of the sample size formulae for a trial with expected early death. We conduct Monte Carlo simulation studies to evaluate the accuracy of our power and sample size formulae and to compare the four sample size estimation methods.

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“…However, most of them require special situations, eg, either continuous data as used in the works of Bürkner et al, Wang et al, or Noether, or they require ordered categorical data as in the works of Fan, Tang, Lachin, Hilton and Mehta, or Whitehead . For a review of different methods, we refer to the work of Rahardja et al A rather well‐known method for sample size calculation in case of continuous data is given by Noether who approximated the variance under alternative by the variance under the null hypothesis. A similar approximation was also used by Zhao et al who generalized Noether's formula to allow for ties.…”

Section: Introductionmentioning

confidence: 99%

Optimal sample size planning for the Wilcoxon‐Mann‐Whitney test

Happ

Bathke

Brunner

2018

Statistics in Medicine

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There are many different proposed procedures for sample size planning for the Wilcoxon‐Mann‐Whitney test at given type‐I and type‐II error rates α and β, respectively. Most methods assume very specific models or types of data to simplify calculations (eg, ordered categorical or metric data, location shift alternatives, etc). We present a unified approach that covers metric data with and without ties, count data, ordered categorical data, and even dichotomous data. For that, we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following “synthetic data” approach. We evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities. Then, well‐known relations for the ranks of the data are used for the calculations. In addition to computing the necessary sample size N for a fixed allocation proportion t = n1/N, where n1 is the sample size in the first group and N = n1 + n2 is the total sample size, we provide an interval for the optimal allocation rate t, which minimizes the total sample size N. It turns out that, for certain distributions, a balanced design is optimal. We give a characterization of such distributions. Furthermore, we show that the optimal choice of t depends on the ratio of the two variances, which determine the variance of the Wilcoxon‐Mann‐Whitney statistic under the alternative. This is different from an optimal sample size allocation in case of the normal distribution model.

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Sample Size Determinations for the Wilcoxon–Mann–Whitney Test: A Comprehensive Review

Cited by 18 publications

References 12 publications

Power and sample size evaluation for the Cochran–Mantel–Haenszel mean score (Wilcoxon rank sum) test and the Cochran–Armitage test for trend

Power and sample size evaluation for the Cochran–Mantel–Haenszel mean score (Wilcoxon rank sum) test and the Cochran–Armitage test for trend

Power and sample size calculations for the Wilcoxon-Mann-Whitney test in the presence of death-censored observations

Optimal sample size planning for the Wilcoxon‐Mann‐Whitney test

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