An improved nonparametric test and sample size procedures for the randomized complete block designs

TL;DR

The paper addresses the conservative chi-square approximation of Friedman's statistic in randomized complete block designs by introducing a general class of $F$-transformations of the Friedman statistic with corresponding approximate $F$ distributions. It identifies two key members, $F_R$ and $F_M$, and a new $F_L$ that align the numerator degrees of freedom with the ANOVA $F$ test, and extends to noncentral $F$ forms under heterogeneous location shifts to enable accurate power calculations. Explicit power formulas are derived for four distributions (Uniform, Normal, Laplace, Exponential), and extensive simulations show that the noncentral $F$-based approaches, especially $F_L$ and $F_L$-LB variants, provide improved Type I error control and more reliable power estimates than the standard noncentral chi-square method. The methods are illustrated with a breaking-strength example and offer practical, computation-friendly guidance for sample size planning in RCBD experiments.

Abstract

The Friedman test has been extensively applied as a nonparametric alternative to the conventional F procedure for comparing treatment effects in randomized complete block designs. A chi-square distribution provides a convenient approximation to determining the critical values for the Friedman procedure in hypothesis testing. However, the chi-square approximation is generally conservative and the accuracy declines with increasing number of treatments. This paper describes an alternative transformation of the Friedman statistic along with an approximate F distribution that has the same numerator degrees of freedom as the ANOVA F test. Moreover, two approximate noncentral F distributions are presented for the proposed F-transformation under the alternative hypothesis of heterogeneous location shifts. Explicit power functions are derived when the underlying populations have the uniform, normal, Laplace, and exponential distributions. Theoretical examination and empirical assessment are presented to validate the advantages of the proposed approaches over the existing methods of the Friedman test. The developed test and power procedures are recommended due to their consistently acceptable Type I error rates and accurate power calculations for the location shift structures and population distributions considered here.