Trimming, Transforming Statistics, And Bootstrapping: Circumventing the Biasing Effects Of Heterescedasticity And Nonnormality

Researchers can adopt different measures of central tendency and test statistics to examine the effect of a treatment variable across groups (e.g., means, trimmed means, M-estimators, & medians. Recently developed statistics are compared with respect to their ability to control Type I errors when data were nonnormal, heterogeneous, and the design was unbalanced: (1) a preliminary test for symmetry which determines whether data should be trimmed symmetrically or asymmetrically, (2) two different transformations to eliminate skewness, (3) the accuracy of assessing statistical significance with a bootstrap methodology was examined, and (4) statistics that use a robust measure of the typical score that empirically determined whether data should be trimmed, and, if so, in which direction, and by what amount were examined. The 56 procedures considered were remarkably robust to extreme forms of heterogeneity and nonnormality. However, we recommend a number of Welch-James heteroscedastic statistics which are preceded by the Babu, Padmanaban, and Puri (1999) test for symmetry that either symmetrically trimmed 10% of the data per group, or asymmetrically trimmed 20% of the data per group, after which either Johnson's (1978) or Hall's (1992) transformation was applied to the statistic and where significance was assessed through bootstrapping. Close competitors to the best methods were found that did not involve a transformation.

[1]  Jiin-Huarng Guo,et al.  A powerful transformation trimmed mean method for one-way fixed effects ANOVA model under non-normality and inequality of variances , 1999 .

[2]  P. Rousseeuw,et al.  Unmasking Multivariate Outliers and Leverage Points , 1990 .

[3]  Regina Y. Liu,et al.  Notions of Limiting P Values Based on Data Depth and Bootstrap , 1997 .

[4]  P. Hall.  On the Number of Bootstrap Simulations Required to Construct a Confidence Interval , 1986 .

[5]  Rand R. Wilcox,et al.  Tests for mean equality that do not require homogeneity of variances: do they really Work? , 2000 .

[6]  Robert P. K Rause.  A New Alternative , 2000 .

[7]  S. Sawilowsky.  A Measure Of Relative Efficiency For Location Of A Single Sample , 2002 .

[8]  R. Wilcox.  ANOVA: A Paradigm for Low Power and Misleading Measures of Effect Size? , 1995 .

[9]  J. Booth,et al.  Resampling-Based Multiple Testing. , 1994 .

[10]  A. R. Othman,et al.  A Test Of Symmetry , 2002 .

[11]  Robert V. Hogg,et al.  A Two-Sample Adaptive Distribution-Free Test , 1975 .

[12]  D. G. Simpson,et al.  Breakdown robustness of tests , 1990 .

[13]  T. Hettmansperger,et al.  Robust analysis of variance based upon a likelihood ratio criterion , 1980 .

[14]  B. L. Welch.  THE SIGNIFICANCE OF THE DIFFERENCE BETWEEN TWO MEANS WHEN THE POPULATION VARIANCES ARE UNEQUAL , 1938 .

[15]  N. J. Johnson,et al.  Modified t Tests and Confidence Intervals for Asymmetrical Populations , 1978 .

[16]  R. Blair,et al.  A more realistic look at the robustness and Type II error properties of the t test to departures from population normality. , 1992 .

[17]  David M. Rocke,et al.  ARE ROBUST ESTIMATORS REALLY NECESSARY , 1982 .

[18]  Lisa M. Lix,et al.  To Trim or Not to Trim: Tests of Location Equality Under Heteroscedasticityand Nonnormality , 1998 .

[19]  Rand R. Wilcox,et al.  Can tests for treatment group equality be improved?: The bootstrap and trimmed means conjecture , 1998 .

[20]  B. L. Welch.  ON THE COMPARISON OF SEVERAL MEAN VALUES: AN ALTERNATIVE APPROACH , 1951 .

[21]  M. Puri,et al.  Robust One‐Way ANOVA under Possibly Non‐Regular Conditions , 1999 .

[22]  S. Johansen,et al.  The Welch - James approximation to the distribution of the residual sum of squares in a weighted lin , 1980 .

[23]  T. Wet,et al.  Efficiency and robustness of hogg’s adaptive trimmed means , 1979 .

[24]  Ronald Schrader,et al.  Robust analysis of variance , 1977 .

[25]  Lisa M. Lix,et al.  Multiple comparison procedures for trimmed means , 1998 .

[26]  H. Keselman,et al.  Robust nonorthogonal analyses revisited: An update based on trimmed means , 1998 .

[27]  R. Wilcox.  A new alternative to the ANOVA F and new results on James's second-order method , 1988 .

[28]  Moti L. Tiku,et al.  Robustness of MML estimators based on censored samples and robust test statistics , 1980 .

[29]  R. Wilcox.  Introduction to Robust Estimation and Hypothesis Testing , 1997 .

[30]  Rand R. Wilcox,et al.  A one-way random effects model for trimmed means , 1994 .

[31]  H. Keselman,et al.  Approximate degrees of freedom tests: A unified perspective on testing for mean equality. , 1995 .

[32]  M. Tiku.  Robust statistics for testing equality of means or variances , 1982 .

[33]  T. Micceri.  The unicorn, the normal curve, and other improbable creatures. , 1989 .

[34]  Peter Hall,et al.  On the Removal of Skewness by Transformation , 1992 .

[35]  H. Keselman,et al.  Repeated measures one-way ANOVA based on a modified one-step M-estimator. , 2003, The British journal of mathematical and statistical psychology.

[36]  Stuart Donnan,et al.  In this number , 1994 .

[37]  An invertible transformation two-sample trimmed t-statistic under heterogeneity and nonnormality , 2000 .