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A One-Way Random Effects Model for Trimmed Means

Published online by Cambridge University Press:  01 January 2025

Rand R. Wilcox
Rand R. Wilcox*
Affiliation:
University of Southern California
*
Requests for reprints should be sent to Rand R. Wilcox, Department of Psychology, University of Southern California, Los Angeles, CA 90089-1061.
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Abstract

The random effects ANOVA model plays an important role in many psychological studies, but the usual model suffers from at least two serious problems. The first is that even under normality, violating the assumption of equal variances can have serious consequences in terms of Type I errors or significance levels, and it can affect power as well. The second and perhaps more serious concern is that even slight departures from normality can result in a substantial loss of power when testing hypotheses. Jeyaratnam and Othman (1985) proposed a method for handling unequal variances, under the assumption of normality, but no results were given on how their procedure performs when distributions are nonnormal. A secondary goal in this paper is to address this issue via simulations. As will be seen, problems arise with both Type I errors and power. Another secondary goal is to provide new simulation results on the Rust-Fligner modification of the Kruskal-Wallis test. The primary goal is to propose a generalization of the usual random effects model based on trimmed means. The resulting test of no differences among J randomly sampled groups has certain advantages in terms of Type I errors, and it can yield substantial gains in power when distributions have heavy tails and outliers. This last feature is very important in applied work because recent investigations indicate that heavy-tailed distributions are common. Included is a suggestion for a heteroscedastic Winsorized analog of the usual intraclass correlation coefficient.

Keywords

Winsorizingcontaminationresistanceintraclass correlationKruskal-WallisRust-Fligner method

Information

Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 3 , September 1994 , pp. 289 - 306
Copyright
Copyright © 1994 The Psychometric Society

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References

Bickel, P. J., Lehmann, E. L. (1975). Descriptive statistics for nonparametric models II. Location. Annals of Statistics, 3, 1045–1158.Google Scholar
Bowen, R. M., Huang, M. (1990). A comparison of maximum likelihood with method of moment procedures for separating individual and group effects. Journal of Personality and Social Psychology, 58, 90–94.CrossRefGoogle Scholar
Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way model. Annals of Mathematical Statistics, 25, 290–302.CrossRefGoogle Scholar
Bradley, J. V. (1978). Robustness?. British Journal of Mathematical and Statistical Psychology, 31, 144–152.CrossRefGoogle Scholar
Brown, M. B., Forsythe, A. (1974). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16, 129–132.CrossRefGoogle Scholar
Cressie, N. A. C., Whitford, H. J. (1986). How to use the two sample t-test. Biometrical Journal, 28, 131–148.CrossRefGoogle Scholar
Cronbach, L. J., Gleser, G., Rajaratnam, N. (1972). The Dependability of Behavioral Measurements: Theory of Generalizability for Scores and Profiles, New York: Wiley.Google Scholar
Fenstad, G. U. (1983). A comparison between U and V tests in the Behrens-Fisher problem. Biometrika, 70, 300–302.CrossRefGoogle Scholar
Freedman, D. A., Diaconis, P. (1982). On inconsistent M-estimators. Annals of Statistics, 10, 454–461.CrossRefGoogle Scholar
Gleason, J. R. (1993). Understanding elongation: The scale contaminated normal family. Journal of the American Statistical Association, 88, 327–337.CrossRefGoogle Scholar
Gollob, H. F. (1991). Methods of estimating individual- and Group-level correlations. Journal of Personality and Social Psychology, 60, 376–381.CrossRefGoogle Scholar
Hoaglin, D. C. (1985). Summarizing shape numerically: The g-and-h distributions. In Hoaglin, D., Mosteller, F., Tukey, J. (Eds.), Exploring data tables, trends, and shapes, New York: Wiley.Google Scholar
Hogg, R. V. (1974). Adaptive robust procedures: A partial review and some suggestions for future applications and theory. Journal of the American Statistical Association, 69, 909–922.CrossRefGoogle Scholar
Huber, P. J. (1981). Robust statistics, New York: Wiley.CrossRefGoogle Scholar
International Mathematical and Statistical Libraries (1987). Library I, Vol. II, Houston: Author.Google Scholar
Jeyaratnam, S., Othman, A. B. (1985). Test of hypothesis in one-way random effects model with unequal error variances. Journal of Statistical Computation and Simulation, 21, 51–57.CrossRefGoogle Scholar
Kenny, D., La Voie, L. (1985). Separating individual and group effects. Journal of Personality and Social Psychology, 48, 339–348.CrossRefGoogle Scholar
Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166.CrossRefGoogle Scholar
Oshima, T. C., Algina, J. (1992). Type I error rates for James's second-order test and Wilcox's H m test under heterocedasticity and nonnormality. British Journal of Mathematical and Statistical Psychology, 45, 255–264.CrossRefGoogle Scholar
Ramberg, J. S., Tadikamalla, P. R., Dudewicz, E. J., Mykytka, E. F. (1978). A probability distribution and its uses in fitting data. Technometrics, 21, 201–214.CrossRefGoogle Scholar
Randles, R. H., Wolfe, D. A. (1979). Introduction to the Theory of Nonparametric Statistics, New York: Wiley.Google Scholar
Rao, P. S. R. S., Kaplan, J., Cochran, W. G. (1981). Estimators for the one-way random effects model with unequal error variances. Journal of the American Statistical Association, 76, 89–97.CrossRefGoogle Scholar
Rogan, J., Keselman, H. J. (1977). Is the ANOVAF-test robust to variance heterogeneity when sample sizes are equal?: An investigation via a coefficient of variation. American Educational Research Journal, 14, 493–498.CrossRefGoogle Scholar
Rosenberger, J. L., Gasko, M. (1983). Comparing location estimators: Trimmed means, medians and trimean. In Hoaglin, D., Mosteller, F., Tukey, J. (Eds.), Understanding robust and exploratory data analysis (pp. 297–336). New York: Wiley.Google Scholar
Rust, S. W., Fligner, M. A. (1984). A modification of the Kruskal-Wallis statistic for the generalized Behrens-Fisher problem. Communications in Statistics—Theory and Methods, 13, 2013–2027.CrossRefGoogle Scholar
Sawilowsky, S. S., Blair, R. C. (1992). A more realistic look at the robustness and Type II error properties of the t test to departures from normality. Psychological Bulletin, 111, 352–360.CrossRefGoogle Scholar
Scheffé, H. (1959). The Analysis of Variance, New York: Wiley.Google Scholar
Staudte, R. G., Sheather, S. J. (1990). Robust Estimation and Testing, New York: Wiley.CrossRefGoogle Scholar
Tomarken, A., Serlin, R. (1986). Comparison of ANOVA alternatives under variance heterogeneity and specific noncentrality structures. Psychological Bulletin, 99, 90–99.CrossRefGoogle Scholar
Tukey, J. W. et al. (1960). A survey of sampling from contaminated normal distributions. In Olkin, I. et al. (Eds.), Contributions to probability and statistics, Stanford, CA: Stanford University Press.Google Scholar
Welch, B. L. (1938). The significance of the difference between two means when the population variances are unequal. Biometrika, 29, 350–362.CrossRefGoogle Scholar
Westfall, P. (1988). Robustness and power of tests for a null variance ratio. Biometrika, 75, 20–35.CrossRefGoogle Scholar
Wilcox, R. R. (1987). New designs in the analysis of variance. Annual Review of Psychology, 38, 29–60.CrossRefGoogle Scholar
Wilcox, R. R. (1990). Comparing the means of two independent groups. Biometrical Journal, 32, 771–780.CrossRefGoogle Scholar
Wilcox, R. R. (1992). Comparing one-step M-estimators of location corresponding to two independent groups. Psychometrika, 57, 141–154.CrossRefGoogle Scholar
Wilcox, R. R. (1993). Comparing one-step M-estimators of location when there are more than two groups. Psychometrika, 58, 71–78.CrossRefGoogle Scholar
Wilcox, R. R. (1993). Some results on a Winsorized correlation coefficient. British Journal of Mathematical and Statistical Psychology, 46, 339–349.CrossRefGoogle Scholar
Wilcox, R. R. (in press). Some results on the Tukey-McLaughlin and Yuen methods for trimmed means when distributions are skewed. Biometrical Journal.Google Scholar
Wilcox, R. R., Charlin, V., Thompson, K. L. (1986). New Monte Carlo results on the robustness of the ANOVA F, W and F* statistics. Communications in Statistics—Simulation and Computation, 15, 933–944.CrossRefGoogle Scholar
Yuen, K. K. (1974). The two sample trimmed t for unequal population variances. Biometrika, 61, 165–170.CrossRefGoogle Scholar