Development of an optimal test based on weighted rankings

Abstract

The aim of this article is to construct a distribution–free test based on the method of weighted rankings for testing the equality of treatment effects in a randomized block design which, in terms of efficiency, is superior to both the Quade and Friedman tests under normality, being almost as efficient as the variance-ratio test, but whose performance remains quite reasonable under nonnormality. A conjecture which, according to Tardif (1987, JASA 82,,637-644), should ameliorate the performance of the Quade test relative to the Friedman test is first examined but it is seen that it does not provide a satisfactory solution. On the other hand, a maximization of the asymptotic efficiency for a fixed set of within-block scores is considered and, by allowing the between-block scores to depend on the number of observations per block, it gives rise to the formulation of an optimal test outperforming both the Quade and Friedman tests under normality. A Monte Carlo investigation finally examines the performance of this optimal test in small samples under normal, Laplace or Cauchy errors. Based on this study, its Fisher approximation is recommended when the number of observations per block takes small to moderate values

Keywords:

• distribution-free test
• randomized blocks
• Friedman test
• Quade test
• asymptotic efficiency
• Monte Carlo study