On Power and Sample Size Calculations of the Jonckheere–Terpstra and Tryon–Hettmansperger Tests for the Ordered Alternative Problem

The Jonckheere–Terpstra procedure is the most common nonparametric test for homogeneity against ordered alternatives in the multisample location problem. The asymptotic performance and related results of the null distribution have been well documented. However, the associated nonnull distribution and power function are seldom discussed and have received relatively limited attention in the literature. This paper derives the exact mean and variance of the Jonckheere–Terpstra test and the closely related Tryon–Hettmansperger test. Moreover, using the asymptotic normality property, the exact variance large-sample power functions are developed for power and sample size calculations. Even though the two nonparametric tests are distribution-free, the corresponding power and sample size results for a specified alternative are generally not distribution-free. Explicit formulas for required probability evaluations are described for the normal, uniform, Laplace, and exponential shift population distributions. Numerical examples and Monte Carlo simulation studies are presented to illustrate the usefulness and accuracy of the suggested power and sample size procedures.