A modified Friedman test for randomized complete block designs

The Friedman test is often used for a randomized complete block design when the normality assumption is not satisfied or the data are ordinal. The Friedman test can be viewed as an extension of the sign test for multiple measurements within each subject or block. We propose a modified Friedman test based on the Wilcoxon sign rank approach. Coincidentally, Tukey proposed a test statistic similar to our proposed test, but blocks are ranked by the minimum difference within each block. In the proposed test, we use the variance of block to rank the blocks, with the least variance being ranked the smallest. In both Tukey test and the modified Friedman test, linear ranks are used for blocks and treatments. The Tukey test belongs to the family of weighted-ranking test from Quade (1979 Quade, D. (1979). Using weighted rankings in the analysis of complete blocks with additive block effects. Journal of the American Statistical Association 74(367):680–683. [Google Scholar]).The modified Friedman test, the Friedman test and the Tukey test are compared under various conditions and the results indicate that the proposed test is generally more powerful than the Friedman test and the Tukey test when the number of groups is small.     

On the asymptotic efficiency of aligned-rank tests in randomized block designs

This paper is concerned with the ranking-after-alignment procedure, the alignment being made on the mean, in randomized block designs. The asymptotic efficiencies, as the number of blocks goes to infinity, of a class of aligned-rank tests, relative to the maximin most powerful test based on aligned observations, are established and studied. Some asymptotic efficiencies under doubleexponentiality are also obtained using Monte Carlo methods.     

Approximate and exact distributions of rank tests for balanced incomplete block designs

Judges rank k out of t objects according to m replic ations of abasic balanced incomplete block design with bblocks. In Alvo and Cabilio(1991),it is shown that the Durbin test, which is the usual test in this situation, can be written in terms of Spearman correlations between the blocks, and using a Kendall correlation, they generated a new statistic for this situation.This Kendall tau based statistic has a richer support than the Durbin statistic, and is at least as efficient.In the present paper,exact and simulation based tables are generated for both statistics, and various approximations to these null distributions are considered and compared.     

Some useful nonparametric tests for ordered alternatives in randomized block experiments

Testing for ordered alternatives in randomized block designs has been a problem of interest for almost three decades (Jonckheere (1954)). Three classes of rank tests have evolved—tests based on “within-blocks” rankings (W-tests), tests based on “ranking after alignment” within blocks (RAA-tests), and tests based on “among-blocks” rankings (A-Tests). This paper focuses on the latter. A simplified version of the Skillings-Wolfe generalized Purl test (1977) is suggested and two very useful A-tests—a generalized Johnson-Mehrotra “Optimal contrast” procedure and a generalized Tryon-Hettmansperger rank test—are developed. These procedures are compared and contrasted with other recent competitors presented by Skllllngs and Wolfe (1978) and by Salama and Quade (1981).     

Analysis of means using ranks for the randomized complete block design

A nonparametric test procedure is proposed for the analysis of randomized complete block designs. Such a procedure may be carried out graphically in the form of a Shewhart control chart. Exact and asymptotic critical values are given for the implementation of the proposed procedure. A Monte Carlo study is made to compare the powers of the proposed procedure to those of analysis of variance, the analysis of means, and the Friedman procedures. Results of the study indicate that the proposed procedure has superior power performance when testing against slippage alternative hypotheses under heavy-tailed distributions such as the Cauchy distribution. However, when testing against symmetric alternatives under light-tailed distributions, the proposed procedure does not perform well     

A monte carlo study of the f test and three tests based on ranks of treatment effects in randomized block designs

The F test is compared with three procedures based on ranks for testing treatment effects in the randomized complete block, fixed effects, model with one observation per cell.     

On the asymptotic distribution of a class of aligned rank order test statistics in randomized block designs

In the present investigation, the unconditional asymptotic distribution of a class of aligned rank order test statistics for randomized block designs is derived under the null hypothesis and for nearby alternatives, as the number of blocks tends to infinity. The proofs of these results are based on the asymptotic equivalence in quadratic mean between aligned observations and their ranks and thus are quite similar to the Hájek and SKidák (1967) approach.     

A study of several useful tests for ordered alternatives in the randomized block design

This study investigates the small sample powers of several tests designed against ordered location alternatives in randomized block experiments. The results are intended to aid the researcher in the selection process. Toward this end the small sample powers of three classes of rank tests — tests based on ‘within-blocks’ rankings (W-tests), ‘among-b locks’ rankings (A-tests), and ‘ranking after alignment’ within blocks (RAA-tests)— are compared and contrasted with the asymptotic properties given by Pirie (1974) as well as with the empirical powers of competing parametric procedures.     

Choosing appropriate covariance matrices in a nonparametric analysis of factorials in block designs

The standard nonparametric, rank-based approach to the analysis of dependent data from factorial designs is based on an estimated unstructured (UN) variance–covariance matrix, but the large number of variance–covariance terms in many designs can seriously affect test performance. In a simulation study for a factorial arranged in blocks, we compared estimates of type-I error probability and power based on the UN structure with the estimates obtained with a more parsimonious heterogeneous-compound-symmetry structure (CSH). Although tests based on the UN structure were anti-conservative with small number of factor levels, especially with four or six blocks, they became conservative at higher number of factor levels. Tests based on the CSH structure were anti-conservative, and results did not depend on the number of factor levels. When both tests were anti-conservative, tests based on the CSH structure were less so. Although use of the CSH structure is concluded to be more suitable than use of the UN structure for the small number of blocks typical in agricultural experiments, results suggest that further improvement of test statistics is needed for such situations.     

Linear rank tests for homogeneity against ordered alternatives in anova and anocova

A unified method of constructing rank tests for homogeneity against ordered alternatives in unbalanced analysis of variance and analysis of covariance is considered. The relationship between these tests with some of the existing methods are studied. The normal theory likelihood ratio tests are also derived and the asymptotic relative efficiency comparisons, in Pitman sense, of the rank tests with respect to the likelihood ratio tests are carried out.     

Efficiency of tests based on weighted rankings for equality of treatment effects in a complete randomized blocks layout

Quade (1972,1979) proposed a family of nonparametric tests based on a method of weighted within-block rankings, for testing the hypothesis of no treatment effects in a complete randomized blocks layout.

In this paper we obtain an expression for the Pitman asymp-totic relative efficiency of these tests with respect to the Friedman test.     

Comparison of exact tests for association in unordered contingency tables using standard,mid-p,and randomized test versions

Pearson’s chi-square (Pe), likelihood ratio (LR), and Fisher (Fi)–Freeman–Halton test statistics are commonly used to test the association of an unordered r×c contingency table. Asymptotically, these test statistics follow a chi-square distribution. For small sample cases, the asymptotic chi-square approximations are unreliable. Therefore, the exact p-value is frequently computed conditional on the row- and column-sums. One drawback of the exact p-value is that it is conservative. Different adjustments have been suggested, such as Lancaster’s mid-p version and randomized tests. In this paper, we have considered 3×2, 2×3, and 3×3 tables and compared the exact power and significance level of these test’s standard, mid-p, and randomized versions. The mid-p and randomized test versions have approximately the same power and higher power than that of the standard test versions. The mid-p type-I error probability seldom exceeds the nominal level. For a given set of parameters, the power of Pe, LR, and Fi differs approximately the same way for standard, mid-p, and randomized test versions. Although there is no general ranking of these tests, in some situations, especially when averaged over the parameter space, Pe and Fi have the same power and slightly higher power than LR. When the sample sizes (i.e., the row sums) are equal, the differences are small, otherwise the observed differences can be 10% or more. In some cases, perhaps characterized by poorly balanced designs, LR has the highest power.