Extended ANOVA and Rank Transform Procedures

The rank transform procedure is often used in the analysis of variance when observations are not consistent with normality. The data are ranked and the analysis of variance is applied to the ranked data. Often the rank residuals will be consistent with normality and a valid analysis results. Here we find that the rank transform procedure is equivalent to applying the intended analysis of variance to first order orthonormal polynomials on the rank proportions. Using higher order orthonormal polynomials extends the analysis to higher order effects, roughly detecting dispersion, skewness etc. differences between treatment ranks. Using orthonormal polynomials on the original observations yields the usual analysis of variance for the first order polynomial, and higher order extensions for subsequent polynomials. Again first order reflects location differences, while higher orders roughly detect dispersion, skewness etc. differences between the treatments.     

Some exact tables for the squared ranks test

Since the squared ranks test was first proposed by Taha in 1964 it has been mentioned by several authors as a test that is easy to use, with good power in many situations. It is almost as easy to use as the Wilcoxon rank sum test, and has greater power when two populations differ in their scale parameters rather than in their location parameters. This paper discuss the versatility of the squared ranks test, introduces a test which uses squared ranks, and presents some exact tables     

Detection of concordance for extreme ranks

When analysing ranking data from one or more groups of judges one may wish to allow for the possibility that the judges have paid more attention to the allocation of the extreme ranks, rather than to the intermediate ranks. In some cases they may have only worried about assigning the top ranks (1 and 2, say) while randomly allocating the remaining ones.

In another context, the analystmay wish to only take account of the agreement among judges with respect to extreme ranks (top or bottom, or both),

In such situations an analysis of concordance within, and between groups, if appropriate, should be able to deal with extreme ranks specifically. We propose a data analyticapproach, related to an analysis of diversity,which actyally permits an analysis of concordance for each rank separately.     

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n?1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n?1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.     

Generalized sequential ranks and tests of randomness

Generalized sequential ranks are defined and are proved to be independent and uniformly distributed under the hypothesis of randomness. By comparison with a Spear-MAsr-type statistic it is shown that in certain cases the test based on the sum of all sequential ranks is an asymptotically optimum test for trend against logistic regression. The test is equivalent to tests proposed by Jonckheere and Terpstra and has a high efficiency when compared with the optimal parametric test for normal regression alternatives.     

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.     

Comparing Nonparametric Tests of Equality of Means for Randomized Block Designs

A number of nonparametric tests are compared empirically for a randomized block layout. We assess tests appropriate for when the data are not consistent with normality or when outliers invalidate traditional analysis of variance (ANOVA) tests. The objective is to assess, within this setting, tests that use ranks within blocks, the rank transform procedure that ranks the complete sample and continuous analogs of the Cochran–Mantel–Haenszel tests. The usual linear model is assumed, and our primary foci are tests of equality of means and component tests that assess linear and quadratic trends in the means. These tests include the traditional Page and Friedman tests. We conclude that the rank transform tests have competitive power and warrant greater use than is currently apparent.     

A continuously adaptive nonparametric two–sample test

A two–sample test statistic for detecting shifts in location is developed for a broad range of underlying distributions using adaptive techniques. The test statistic is a linear rank statistics which uses a simple modification of the Wilcoxon test; the scores are Winsorized ranks where the upper and lower Winsorinzing proportions are estimated in the first stage of the adaptive procedure using sample the first stage of the adaptive procedure using sample measures of the distribution's skewness and tailweight. An empirical relationship between the Winsorizing proportions and the sample skewness and tailweight allows for a ‘continuous’ adaptation of the test statistic to the data. The test has good asymptotic properties, and the small sample results are compared with other populatr parametric, nonparametric, and two–stage tests using Monte Carlo methods. Based on these results, this proposed test procedure is recommended for moderate and larger sample sizes.     

Multivaritate two stage test based on ranks

A distribution free two stage test based on ranks for the multivariate two sample location problem is presented. The asymptotic distribution of the first and second stage test statistics is derived. Results of a Monte Carlo power study are used to compare the two stage test with the usual one stage test. A brief table of critical values is also presented. The test is illustrated by using data from an exercise study conducted by the Multipurpose Arthritis center.     

A Monte Carlo study of rank tests for block designs

Tests based on ranks and the F-test are compared for block designs with n observations per block-treatment combination. Com-parisons are made on level of significance and on power. Rank tests examined include the Friedman as well as those using aligned ranks, weighted ranks, and the rank transformation. It is seen that the performance of these tests in relationship to each other depends on sample size, distribution of the random error term, and the severity of the block effects.     

Correction methods for ties in rank correlations

Equal values are common when rank methods are applied to rounded data or data consisting solely of small integers. A popular technique for resolving ties in rank correlation is the mid-rank method: the mean of the rankings remains unaltered, but the variance is reduced and modified according to the number and location of ties. Although other methods for breaking ties were proposed in the literature as early as 1939, no such procedure has gained such wide acceptance as mid-ranks. This research analyses various techniques for assigning ranks to tied values, with two objectives: (1) to enable the computation of rank correlation coefficients, such as those of Spearman, Kendall and Gini, by using the usual definition applied in the absence of ties, and (2) to determine whether it really makes a difference which of the various techniques is selected and, if so, which technique is most appropriate for a given application.     

Nearly exact tests in factorial experiments using the aligned rank transform

A procedure is studied that uses rank-transformed data to perform exact and estimated exact tests, which is an alternative to the commonly used F-ratio test procedure. First, a common parametric test statistic is computed using rank-transformed data, where two methods of ranking-ranks taken for the original observations and ranks taken after aligning the observations-are studied. Significance is then determined using either the exact permutation distribution of the statistic or an estimate of this distribution based on a random sample of all possible permutations. Simulation studies compare the performance of this method with the normal theory parametric F-test and the traditional rank transform procedure. Power and nominal type I error rates are compared under conditions when normal theory assumptions are satisfied, as well as when these assumptions are violated. The method is studied for a two-factor factorial arrangement of treatments in a completely randomized design and for a split-unit experiment. The power of the tests rivals the parametric F-test when normal theory assumptions are satisfied, and is usually superior when normal theory assumptions are not satisfied. Based on the evidence of this study, the exact aligned rank procedure appears to be the overall best choice for performing tests in a general factorial experiment.     

Analysis of Means Using Ranks

A nonparametric procedure, called the analysis of means using ranks (ANOMR), is proposed for testing the equality of several population means. The ANOMR procedure may be used graphically in the form of a Shewhart control chart and so has the advantage of pinpointing which population mean, if any, is significantly different from the others. Exact and asymptotic critical values are given for the implementation of ANOMR. Results from a Monte Carlo power study are presented which indicate that for light-tailed distributions such as the uniform and the normal, ANOMR is only slightly less powerful than the parametric competitive procedures based on analysis of variance and analysis of means. For heavy-tailed distributions such as the Cauchy, ANOMR is shown to provide greater power than the parametric procedures. The results also indicate that for both light and heavy-tailed distributions the use of the ANOMR test instead of the Kruskal-Wallis test leads to only a small loss of power for a range of alternatives.     

Analysis of ranked data in randomized blocks when there are missing values

Data consisting of ranks within blocks are considered for randomized block designs when there are missing values. Tied ranks are possible. Such data can be analysed using the Skillings–Mack test. Here we suggest a new approach based on carrying out an ANOVA on the ranks using the general linear model platform available in many statistical packages. Such a platform allows an ANOVA to be calculated when there are missing values. Indicative sizes and powers show the ANOVA approach performs better than the Skillings–Mack test.     

The 'improved' brown and forsythe test for mean equality: some things can't be fixed

Mehrotra (1997) presented an ‘;improved’ Brown and Forsythe (1974) statistic which is designed to provide a valid test of mean equality in independent groups designs when variances are heterogeneous. In particular, the usual Brown and Fosythe procedure was modified by using a Satterthwaite approximation for numerator degrees of freedom instead of the usual value of number of groups minus one. Mehrotra then, through Monte Carlo methods, demonstrated that the ‘improved’ method resulted in a robust test of significance in cases where the usual Brown and Forsythe method did not. Accordingly, this ‘improved’ procedure was recommended. We show that under conditions likely to be encountered in applied settings, that is, conditions involving heterogeneous variances as well as nonnormal data, the ‘improved’ Brown and Forsythe procedure results in depressed or inflated rates of Type I error in unbalanced designs. Previous findings indicate, however, that one can obtain a robust test by adopting a heteroscedastic statistic with the robust estimators, rather than the usual least squares estimators, and further improvement can be expected when critical significance values are obtained through bootstrapping methods.     

The rank transformation as a method of discrimination with some examples

The procedure of statistical discrimination Is simple in theory but so simple in practice. An observation x₀possibly uiultivariate, is to be classified into one of several populations π₁,…,π_k which have respectively, the density functions f₁(x), ? ? ? , f_k(x). The decision procedure is to evaluate each density function at X₀ to see which function gives the largest value f_i(X₀) , and then to declare that X₀ belongs to the population corresponding to the largest value. If these den-sities can be assumed to be normal with equal covariance matricesthen the decision procedure is known as Fisher’s linear discrimi-nant function (LDF) method. In the case of unequal covariance matrices the procedure is called the quadratic discriminant func-tion (QDF) method. If the densities cannot be assumed to be nor-mal then the LDF and QDF might not perform well. Several different procedures have appeared in the literature which offer discriminant procedures for nonnormal data. However, these pro-cedures are generally difficult to use and are not readily available as canned statistical programs.

Another approach to discriminant analysis is to use some sortof mathematical trans format ion on the samples so that their distribution function is approximately normal, and then use the convenient LDF and QDF methods. One transformation that:applies to all distributions equally well is the rank transformation. The result of this transformation is that a very simple and easy to use procedure is made available. This procedure is quite robust as is evidenced by comparisons of the rank transform results with several published simulation studies.     

Probability proportional to size (πps) sampling using ranks

There can be gains in estimation efficiency over equal probability samplin methods when one makes use of auxiliary information for probability proporti onal to size with replacement (πpswr) sampling methods. The usual method is simple to execute, but might lead to more than one appearance in the sampl e for any particular unit. When a suitable variable x is not available, one may know how to rank units reasonably well relative to the unknown y values before sample selection. When such ranking is possible, we introduce a simple and efficient sampling plan using the ranks as the unknown x measures of size. The proposed sampling plan is similar to, has the simplicity of, and has no greater sampling variance than with replacement sampling, but is without replacement.     

Transformations and the Income Tax

Many of the more useful and powerful nonparametric procedures may be presented in a unified manner by treating them as rank transformation procedures. Rank transformation procedures are ones in which the usual parametric procedure is applied to the ranks of the data instead of to the data themselves. This technique should be viewed as a useful tool for developing nonparametric procedures to solve new problems.     

A Non‐parametric Test of Exchangeability for Extreme‐Value and Left‐Tail Decreasing Bivariate Copulas

Abstract. A non‐parametric rank‐based test of exchangeability for bivariate extreme‐value copulas is first proposed. The two key ingredients of the suggested approach are the non‐parametric rank‐based estimators of the Pickands dependence function recently studied by Genest and Segers, and a multiplier technique for obtaining approximate p‐values for the derived statistics. The proposed approach is then extended to left‐tail decreasing dependence structures that are not necessarily extreme‐value copulas. Large‐scale Monte Carlo experiments are used to investigate the level and power of the various versions of the test and show that the proposed procedure can be substantially more powerful than tests of exchangeability derived directly from the empirical copula. The approach is illustrated on well‐known financial data.     

Smoothed ranks for two or multi-sample location problems

Smoothed ranks are proposed for two or multi-sample location problems. The regular ranks in Wilcoxon's two sample test are replaced with smoothed ranks, and the shift parameter is estimated. Asymptotic properties of the smoothed rank estimator are shown and a hypothesis test is proposed. Moreover, the smoothed ranks are applied in the Kruskal–Wallis's r-sample test and the power of the test is computed using regular and smoothed ranks. Examples and Monte Carlo simulations show that the smoothed ranks perform similarly to the traditional rank based estimators under contaminated normal or non-normal populations.