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<p>The Kruskal–Wallis test is widely recommended as a nonparametric counterpart to the ANOVA <i>F</i> procedure for comparing more than two treatments. However, the associated asymptotic chi-square distribution tends to be conservative, especially for small sample sizes. It was shown that the Kruskal–Wallis <i>H</i> statistic has a noncentral chi-square distribution under the local alternative with small location shifts. This paper describes a new and simplified <i>F</i> test procedure and the associated approximate <i>F</i> distribution has only a noninteger denominator degrees of freedom. For the purpose of power and sample size determination, approximate noncentral <i>F</i> distributions are proposed for the <i>F</i>-transformation of <i>H</i> when there are differences in location among populations. Theoretical examination and empirical assessment are presented to justify the validity and usefulness of the proposed test in maintaining Type I error performance. The features of the suggested power and sample size calculations are also explicated for several important distributions. Simulation results showed that the presented approach outperforms the current method based on the noncentral chi-square approximation. This study extends the utility of Kruskal–Wallis statistic as a feasible nonparametric method for actual applications.</p>

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Alternative Nonparametric Test and Sample Size Procedures for the Comparison of Several Location Shifts

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  • 发表日期：

    2025年4月7日

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  • Show-Li Jan,
  • Gwowen Shieh

摘要

The Kruskal–Wallis test is widely recommended as a nonparametric counterpart to the ANOVA F procedure for comparing more than two treatments. However, the associated asymptotic chi-square distribution tends to be conservative, especially for small sample sizes. It was shown that the Kruskal–Wallis H statistic has a noncentral chi-square distribution under the local alternative with small location shifts. This paper describes a new and simplified F test procedure and the associated approximate F distribution has only a noninteger denominator degrees of freedom. For the purpose of power and sample size determination, approximate noncentral F distributions are proposed for the F-transformation of H when there are differences in location among populations. Theoretical examination and empirical assessment are presented to justify the validity and usefulness of the proposed test in maintaining Type I error performance. The features of the suggested power and sample size calculations are also explicated for several important distributions. Simulation results showed that the presented approach outperforms the current method based on the noncentral chi-square approximation. This study extends the utility of Kruskal–Wallis statistic as a feasible nonparametric method for actual applications.