Larry E. Toothaker

Linearly Independent, Orthogonal, and Uncorrelated Variables

Abstract Linearly independent, orthogonal, and uncorrelated are three terms used to indicate lack of relationship between variables. This short didactic article compares these three terms in both an algebraic and a geometric framework. An example is used to illustrate the differences.

An Empirical Comparison of the ANOVA F-test, Normal Scores Test and Kruskal-Wallis Test under Violation of Assumptions.

The present research compares the ANOVA F-test, the Kruskal-Wallis test, and the normal scores test in terms of empirical alpha and empirical power with samples from the normal distribution and two exponential distributions. Empirical evidence supports the use of the ANOVA F-test even under violation of assumptions when testing hypotheses about means. If the researcher is willing to test hypotheses about medians, the Kruskal-Wallis test was found to be competitive to the F-test. However, in the cases investigated, the normal scores test was not consistently better than the F-test or the Kruskal-Wallis test and could not be recommended on the basis of this research.

N = 1 Designs: The Failure of ANOVA-Based Tests

Several methods have been proposed for the analysis of data from single-subject research settings. This research focuses on the modifications of ANOVA-based tests proposed by Shine and Bower, a procedure that precedes the ANOVAF test by preliminary testing of within-phase lag one serial correlation and the one-way ANOVA as presented by Gentile, Roden and Klein. Monte Carlo simulation is used to investigate these tests with respect to robustness and power. Each test was analyzed under various patterns of serial correlation, various patterns of phase and trial means, normal and exponential distributions, and equal and unequal phase variances. The findings indicate that the probability of a Type I error for these ANOVA-based tests is seriously inflated by nonzero serial correlation. These tests, therefore, cannot be recommended for use with data that have nonzero serial correlation.

Nonparametric Competitors to the Two-Way ANOVA:

The ANOVA F and several nonparametric competitors for two-way designs were compared for empirical α and power. Simulation of 2 × 2, 2 × 4, and 4 × 4 designs was done with cell sizes of 5 and 10 when sampling from normal, exponential, and mixed normal distributions. Conservatism of both α and power in the presence of other nonnull effects was seen in the tests due to Puri and Sen (1985) and, to a lesser degree, in the rank transform tests (Conover & Iman, 1981). Tests by McSweeney (1967) and Hettmansperger (1984) had liberal α for some designs and distributions, especially for small n. The ANOVA F suffers from conservative α and power for the mixed normal distribution, but it is generally recommended.

Bootstrapping to test for nonzero population correlation coefficients using univariate sampling.

This article proposes 2 new approaches to test a nonzero population correlation (rho): the hypothesis-imposed univariate sampling bootstrap (HI) and the observed-imposed univariate sampling bootstrap (OI). The authors simulated correlated populations with various combinations of normal and skewed variates. With alpha set=.05, N> or =10, and rho< or =0.4, empirical Type I error rates of the parametric r and the conventional bivariate sampling bootstrap reached .168 and .081, respectively, whereas the largest error rates of the HI and the OI were .079 and .062. On the basis of these results, the authors suggest that the OI is preferable in alpha control to parametric approaches if the researcher believes the population is nonnormal and wishes to test for nonzero rhos of moderate size.

Comparison of Tukey's T-Method and Scheffe's S-Method for Various Numbers of All Possible Differences of Averages Contrasts under Violation of Assumptions.

Empirical .05 and .01 rates of Type I error were compared for the Tukey and Scheffé multiple comparison techniques. The experimentwise error rate was defined over five sets of the all possible 25 differences of averages contrasts. The robustness of the Tukey and Scheffé statistics was not only related to the type of assumption violation, but also to the sets containing different numbers of contrasts. The Tukey method could be judged as robust a statistic as the Scheffé method.

ON "THE ANALYSIS OF RANKED DATA DERIVED FROM COMPLETELY RANDOMIZED FACTORIAL DESIGNS"

Extensions of the Kruskal-Wallis procedure for a factorial design are reviewed and researched under various degrees and kinds of nonnullity. It was found that the distributions of these test statistics are a Function of effects other than those being tested except under the completely null situation and their use is discouraged.

An Evaluation of Two Unequal nk Forms of the Tukey Multiple Comparison Statistic

Abstract The harmonic mean and Kramer [13] unequal nk forms of the Tukey multiple comparison test were compared for observed Type I error and correct decision rates. Sensitivity was evaluated under the restriction that the analysis of variance F-test be significant at α = 0.05, under numerous parametric specifications which represent behavioral research data. The data indicates that both procedures are adversely affected by combining unequal variances with unequal sample sizes and have the same sensitivity for detecting real mean differences.

Book Review : Nonparametric Statistics for the Behavioral Sciences (Second Edition): Sidney Siegel and N. John Castellan, Jr. New York: McGraw-Hill, 1988, 399 pp., approx.

to 312; it has the same nine-chapter format as the old edition (with the same or similar chapter titles); and it adds an author index and an example author index as well as a small set of BASIC programs. In other respects, it is similar to the old edition except as indicated in the following chapter-by-chapter notes. My comments in these notes will refer to changes made by Castellan, as he did all of the rewriting for the deceased Siegel. Chapter 1 (Introduction): On page 3, there is an emphasis on ease of interpretation of nonparametric tests for data with less than interval scale of measurement. The old edition stated that &dquo;it is permissible to use the parametric techniques only with scores which are truly numerical,&dquo; so it appears that the stance with respect to scale of measurement has been softened, as promised in the preface. Also, a short paragraph

47.95

Abstract The Bartlett-Kendall test was investigated to determine the optimum: (1) subsample size (M) for equal nj where nj is evenly divisible; (2) strategy for equal nj where nj is not evenly divisible by number of observations per subsamples, K, K > 1; (3) strategy for unequal nj including two modifications. Results for (1) indicated that a choice of M which minimizes |M — K|, where KM = nj , would typically yield highest power. For (2), highest power was obtained by keeping K as constant as possible without throwing away observations. For (3), the unmodified test gave controlled α and adequate power for constant K. The modifications yielded good α control and high power for both K and M varied but yielded liberal α for constant M (varied K).