Comparing One-Step M-Estimators of Location Corresponding to Two Independent Groups | Psychometrika | Cambridge Core (original) (raw)

Abstract

Experience with real data indicates that psychometric measures often have heavy-tailed distributions. This is known to be a serious problem when comparing the means of two independent groups because heavy-tailed distributions can have a serious effect on power. Another problem that is common in some areas is outliers. This paper suggests an approach to these problems based on the one-step M-estimator of location. Simulations indicate that the new procedure provides very good control over the probability of a Type I error even when distributions are skewed, have different shapes, and the variances are unequal. Moreover, the new procedure has considerably more power than Welch's method when distributions have heavy tails, and it compares well to Yuen's method for comparing trimmed means. Wilcox's median procedure has about the same power as the proposed procedure, but Wilcox's method is based on a statistic that has a finite sample breakdown point of only 1/n, where n is the sample size. Comments on other methods for comparing groups are also included.

References

Brown, B. M. (1982). Robustness against inequality of variances. Australian Journal of Statistics, 24, 283–295.CrossRefGoogle Scholar

Cressie, N. A. C., Whitford, A. B. (1986). How to use the two sample t-test. Biometrical Journal, 28, 131–148.CrossRefGoogle Scholar

Dana, E. (1988). Salience of the self and the salience of a standard: Attempts to match self to a standard. Unpublished doctoral dissertation, University of Southern California, Department of Psychology.Google Scholar

DiCiccio, T. J., Romano, J. P. (1988). A review of bootstrap confidence intervals. Journal of the Royal Statistical Society, Series B, 50, 338–370.CrossRefGoogle Scholar

Donoho, D. L., Huber, P. J. (1988). The notion of breakdown point. In Bickel, P. J., Doksum, K. A., Hodges, J. L. Jr. (Eds.), A Festschrift for Erich Lehmann (pp. 157–184). Belmont, CA: Wadsworth.Google Scholar

Efron, B. (1982). The Jackknife, the bootstrap and other resampling methods, Philadelphia, PA: Society for Industrial and Applied Mathematics.Google Scholar

Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82, 171–185.CrossRefGoogle Scholar

Hall, P. (1986). On the number of bootstrap simulations required to construct a confidence interval. Annals of Statistics, 14, 1453–1462.Google Scholar

Hall, P. (1988). On symmetric bootstrap confidence intervals. Journal of the Royal Statistical Society, Series B, 50, 35–45.CrossRefGoogle Scholar

Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., Stahel, W. A. (1986). Robust statistics, New York: Wiley.Google Scholar

Harrell, F. E., Davis, C. E. (1982). A new distribution-free quantile estimator. Biometrika, 69, 635–640.CrossRefGoogle Scholar

He, X., Simpson, D. G., Portnoy, S. L. (1990). Breakdown robustness of tests. Journal of the American Statistical Association, 85, 446–452.CrossRefGoogle Scholar

Huber, P. (1964). Robust estimation of a location parameter. Annals of Mathematical Statistics, 35, 73–101.CrossRefGoogle Scholar

Huber, P. (1970). Studentizing robust estimates. In Puri, M. L. (Eds.), Nonparametric techniques in statistical inference (pp. 453–463). Cambridge, England: Cambridge University Press.Google Scholar

Kafadar, K. (1982). Using biweight M-estimates in the two-sample problem Part 1: Symmetric populations. Communications in Statistics—Theory and Methods, 11, 1883–1901.CrossRefGoogle Scholar

Kendall, M. G., Stuart, A. (1973). The advanced theory of statistics, Vol. 2, New York: Hafner.Google Scholar

Markatou, M., Hettmansperger, T. P. (1990). Robust bounded-influence tests. Journal of the American Statistical Association, 85, 187–190.CrossRefGoogle Scholar

Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166.CrossRefGoogle Scholar

Parrish, R. S. (1990). Comparison of quantile estimators in normal sampling. Biometrics, 46, 247–258.CrossRefGoogle Scholar

Patel, K. R., Mudholkar, S., Fernando, J. L. I. (1988). Student's t approximations for three simple robust estimators. Journal of the American Statistical Association, 83, 1203–1210.Google Scholar

Ramberg, J. S., Tadikamalla, P. R., Dudewicz, E. J., Mykytka, E. F. (1979). A probability distribution and its uses in fitting data. Technometrics, 21, 201–214.CrossRefGoogle Scholar

Sawilowsky, S. S., Blair, R. C. (1990). A more realistic look at the robustness of the independent and dependent samples t tests to departure from population normality, Detroit: Wayne State University, College of Education.Google Scholar

Schrader, R. M., Hettmansperger, T. P. (1980). Robust analysis of variance. Biometrika, 67, 93–101.CrossRefGoogle Scholar

Sen, P. K. (1982). On M tests in linear models. Biometrika, 69, 245–248.Google Scholar

Sheather, S. J., Marron, J. S. (1990). Kernel quantile estimators. Journal of the American Statistical Association, 85, 410–416.CrossRefGoogle Scholar

Shoemaker, L. H., Hettmansperger, T. P. (1982). Robust estimates and tests for the one- and two-sample scale models. Biometrika, 69, 47–54.CrossRefGoogle Scholar

Shorack, G. R. (1976). Robust studentization of location estimates. Statistical Neerlandica, 30, 119–141.CrossRefGoogle Scholar

Tan, W. Y. (1982). Sampling distributions and robustness of t, F and variance-ratio in two samples and ANOVA models with respect to departure from normality. Communications in Statistics—Theory and Methods, 11, 2485–2511.Google Scholar

Tiku, M. L. (1982). Robust statistics for testing equality of means or variances. Communications in Statistics—Theory and Methods, 11, 2543–2558.CrossRefGoogle Scholar

Tukey, J. W. (1960). A survey of sampling from contaminated distributions. In Olkin, I., Ghurye, S., Hoeffding, W., Madow, W., Mann, H. (Eds.), Contributions to probability and statistics (pp. 448–485). Stanford, CA: Stanford University Press.Google Scholar

Welch, B. (1937). The significance of the difference between two means when the population variances are unequal. Biometrika, 29, 350–362.CrossRefGoogle Scholar

Wilcox, R. R. (1987). New designs in analysis of variance. Annual Review of Psychology, 38, 29–60.CrossRefGoogle Scholar

Wilcox, R. R. (1990). Comparing variances and means when distributions have non-identical shapes. Communications in Statistics—Simulation and Computation, 19, 155–173.CrossRefGoogle Scholar

Wilcox, R. R. (1990). Comparing the means of two independent groups. Biometrical Journal, 32, 771–780.CrossRefGoogle Scholar

Wilcox, R. R. (1990). Comparing biweight measures of location in the two-sample problem. Communications in Statistics—Simulation and Computation, 19, 1231–1246.CrossRefGoogle Scholar

Wilcox, R. R. (1991). Testing whether independent groups have identical medians. Psychometrika, 56, 381–395.CrossRefGoogle Scholar

Yoshizawa, C. N., Sen, P. K., Davis, C. E. (1985). Asymptotic equivalance of the Harrell-Davis median estimator and the sample median. Communications in Statistics—Theory and Methods, 14, 2129–2136.CrossRefGoogle Scholar

Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61, 165–170.CrossRefGoogle Scholar

Zarembra, S. K. (1962). A generalization of Wilcoxon's test. Monatshefte fur Mathematik, 66, 359–370.CrossRefGoogle Scholar