The use of nonparametric contrasts in one-way layouts and random block designs

Abstract

Nonparametric simple-contrast estimates for one-way layouts based on Hodges–Lehmann estimators for two samples and confidence intervals for all contrasts involving only two treatments are found in the literature.

Tests for such contrasts are performed from the distribution of the maximum of the rank sum between two treatments. For random block designs, simple contrast estimates based on Hodges–Lehmann estimators for one sample are presented. However, discussions concerning the significance levels of more complex contrast tests in nonparametric statistics are not well outlined.

This work aims at presenting a methodology to obtain p-values for any contrast types based on the construction of the permutations required by each design model using a C-language program for each design type. For small samples, all possible treatment configurations are performed in order to obtain the desired p-value. For large samples, a fixed number of random configurations are used. The program prompts the input of contrast coefficients, but does not assume the existence or orthogonality among them.

In orthogonal contrasts, the decomposition of the value of the suitable statistic for each case is performed and it is observed that the same procedure used in the parametric analysis of variance can be applied in the nonparametric case, that is, each of the orthogonal contrasts has a χ² distribution with one degree of freedom. Also, the similarities between the p-values obtained for nonparametric contrasts and those obtained through approximations suggested in the literature are discussed.

Keywords:

• Permutation test
• Orthogonal contrasts
• Nonorthogonal contrasts
• Design of experiments