Nonparametric Tests for the Umbrella Alternative with Unknown Peak in a Mixed Design

Abstract

Aims: Introducing and comparing 4 different tests for the unknown umbrella alternative in a mixed design. Study Design: Simulation study consisting of a randomized complete block portion and a completely randomized design portion for various underlying distributions. Place and Duration of Study: Simulation Study – conducted at North Dakota State University from September 2018 through December 2019. Methodology: This paper proposes four non-parametric tests for testing the umbrella alternative with unknown peak when the data are mixture of a randomized complete block and a completely randomized design. The proposed tests are various combinations of a modified (unmodified) Mack-Wolfe’s test and a modified (unmodified) Kim-Kim’s test, respectively. In this paper, the proposed tests are an extension of Magel et al. (2010) and Hassan and Magel (2020) peak known tests to the unknown peak setting. The four proposed test statistics are compared to each other. Results: When there were 3 populations, the unmodified versions of the test statistics did better than the modified versions.  When there were 4 and 5 populations, the results varied. Conclusion: All of the test statistics reached their asymptotic distributions quickly.  The standardize first versions of the test statistics were generally better than the standardized last version of the test statistics, which meant that it was better to place equal weights on the RCBD portion and the CRD portion.