Aligned Rank Tests for Interactions in Split-Plot Designs: Distributional Assumptions and Stochastic Heterogeneity

Three aligned rank methods for transforming data from multiple group repeated measures (split-plot) designs are reviewed. Univariate and multivariate statistics for testing the interaction in split-plot designs are elaborated. Computational examples are presented to provide a context for performing these ranking procedures and statistical tests. SAS/IML and SPSS syntax code to perform the procedures is included in the Appendix

New Nonparametric Rank Tests for Interactions in Factorial Designs with Repeated Measures

New rank tests for interactions in factorial designs are summarily presented and applied to some common factorial designs with repeated measures. The resulting p‑values of these tests are compared among each other, along with those obtained by parametric and randomization tests

Aligned Rank Tests As Robust Alternatives For Testing Interactions In Multiple Group Repeated Measures Designs With Heterogeneous Covariances

Data simulation was used to investigate whether tests performed on aligned ranks (Beasley, 2002) could be used as robust alternatives to parametric methods for testing a split-plot interaction with non-normal data and heterogeneous covariance matrices. Results indicated the aligned rank method do not have any distinct advantage over parametric methods in this situation

The Aligned Rank Transform and discrete Variables - a Warning

For two-way layouts in a between subjects anova design the aligned rank transform (ART) is compared with the parametric F-test as well as six other nonparametric methods: rank transform (RT), inverse normal transform (INT), a combination of ART and INT, Puri & Sen‘s L statistic, van der Waerden and Akritas & Brunners ATS. The type I error rates are computed for the uniform and the exponential distributions, both as continuous and in several variations as discrete distribution. The computations had been performed for balanced and unbalanced designs as well as for several effect models. The aim of this study is to analyze the impact of discrete distributions on the error rate. And it is shown that this scaling impact is restricted to the ART as well as the combination of ART- and INT-method. There are two effects: first with increasing cell counts their error rates rise beyond any acceptable limit up to 20 percent and more. And secondly their rates rise when the number of distinct values of the dependent variable decreases. This behaviour is more severe for underlying exponential distributions than for uniform distributions. Therefore there is a recommendation not to apply the ART if the mean cell frequencies exceed 10

Generalizations of the Tests by Kruskal-Wallis, Friedman and van der Waerden for Split-plot Designs

Generalizations of the 1-factorial tests by Kruskal-Wallis and Friedman, as well as of the van der Waerden test are proposed for factorial split-plot designs, both allowing interactions. They are compared in regard to the type I error control and the power with the parametric F test, including the Huynh-Feldt adjustment, the inverse normal transform (INT), the ANOVA type statistic by Brunner et al. (ATS), the aligned rank transform (ART), the L statistic by Puri \& Sen and a procedure by Koch. The two methods proposed show a perfect type I error control, except for two situations, and an attractive power, particularly in case of nonnormal distributions. The charm and advantage of these procedures are the possibility to apply them with statistical standard tools using only variable transformations and data management, and to receive results from well-known methods which are easy to understand

JMASM24: Numerical Computing for Third-Order Power Method Polynomials (Excel)

The power method polynomial transformation is a popular procedure used for simulating univariate and multivariate non-normal distributions. It requires software that solves simultaneous nonlinear equations. Potential users of the power method may not have access to commercial software packages (e.g., Mathematica, Fortran). Therefore, algorithms are presented in the more commonly available Excel 2003 spreadsheets. The algorithms solve for (1) coefficients for polynomials of order three, (2) intermediate correlations and Cholesky factorizations for multivariate data generation, and (3) the values of skew and kurtosis for determining if a transformation will produce a valid power method probability density function (pdf). The Excel files are available at http://www.siu.edu/~epse1/headrick/PowerMethod3rd/ or can be requested from the author at [email protected]

Numerical Computing and Graphics for the Power Method Transformation Using Mathematica

This paper provides the requisite information and description of software that perform numerical computations and graphics for the power method polynomial transformation. The software developed is written in the Mathematica 5.2 package PowerMethod.m and is associated with fifth-order polynomials that are used for simulating univariate and multivariate non-normal distributions. The package is flexible enough to allow a user the choice to model theoretical pdfs, empirical data, or a user's own selected distribution(s). The primary functions perform the following (a) compute standardized cumulants and polynomial coefficients, (b) ensure that polynomial transformations yield valid pdfs, and (c) graph power method pdfs and cdfs. Other functions compute cumulative probabilities, modes, trimmed means, intermediate correlations, or perform the graphics associated with fitting power method pdfs to either empirical or theoretical distributions. Numerical examples and Monte Carlo results are provided to demonstrate and validate the use of the software package. The notebook Demo.nb is also provided as a guide for user of the power method.

An aligned rank test for a nonparametric analysis of the two way interaction

Problemas que requieren un análisis no paramétrico de la interacción surgen con cierta frecuencia en estudios del comportamiento. Hay, sin embargo, una ausencia de procedimientos en paquetes estadísticos usuales como SAS o SPSS. El objetivo del presente trabajo es revisar los fundamentos del contraste de rangos alineados en dos diseños de investigación de dos factores de amplio uso en Psicología. El procedimiento ha mostrado buenas propiedades en numerosas distribuciones no normales tanto en control del Error Tipo I como en potencia o sensibilidad estadística. Es fácilmente ejecutable mediante programas estadísticos comunes y se ilustra paso a paso con el módulo del modelo lineal general de Spss. Se aplica a dos estudios de caso donde aparecen diferencias de género en habilidades lingüísticas en niños ciegos no detectadas con otros procedimientos. Limitaciones al contraste de rangos alineados se han observado en situaciones de heterogeneidad de covarianzas entre grupos, y también con tamaños muestrales inferiores a diez participantes por condición. Su uso adecuado tras el diagnóstico de supuestos puede aumentar la sensibilidad de detección de efectos comportamentales de interés teórico o aplicadoResearch problems that require nonparametric analysis of the interaction frequently arise in the behavioral sciences. There is, however, a lack of available procedures in commonly used statistical packages such as SAS or SPSS. The purpose of the present study is to review the fundamentals of the aligned rank test for two widely used two-way research designs in psychology. The procedure has shown good properties in nonnormal distributions in terms of Type I Error control and statistical power. It is easily conducted using common statistical packages. It is applied to two case studies which result in gender differences in linguistic abilities in blind children not revealed by other procedures. Limitations of the aligned rank test have been observed in situations of covariance heterogeneity across groups, and also with sample sizes smaller than ten participants per condition. Its adequate use after model diagnostics can, however, increase sensitivity to detect behavioral effects of theoretical or practical interes