Robust One-Way ANOVA under Possibly Non-Regular Conditions

Abstract

Consider the one-way ANOVA problem of comparing the meansSolutions are available based on (i) normal-theory procedures, (ii) linear rank statistics and (iii) M-estimators.The above model presupposes thathave equal variances ( homoscedasticity). However practising statisticans content that homoscedasticity is often violated in practice. Hence a more realistic problem to consider iswhere F is symmetric about the origin and s 1 Y F F F Y s c are unknown and possibly unequal ( heteroscedasticity). Now we have to compare m 1 Y m 2 Y F F F Y m c . At present, nonparametric tests of the equality of m 1 Y m 2 Y F F F Y m c are available. However, simultaneous tests for paired comparisons and contrasts and do not seem to be available.This paper begins by proposing a solution applicable to both the homoscedastic and the heteroscedastic situations, assuming F to be symmetric. Then the assumptions of symmetry and the identical Biometrical Journal 41 (1999) 3, 321±339 * Research supported in part by NSF grants DMS-9007717 and DMS-9208066.shapes of F 1 Y F F F Y F c are progressively relaxed and solutions are proposed for these cases as well. The procedures are all based on either the 15% trimmed means or the sample medians, whose quantiles are estimated by means of the bootstrap. Monte Carlo studies show that these procedures tend to be superior to the Wilcoxon procedure and Dunnett's normal theory procedure. A rigorous justification of the bootstrap is also presented. The methodology is illustrated by a comparison of mean effects of cocaine administration in pregnant female Sprague-Dawley rats, where skewness and heteroscedascity are known to be present.

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

“…Appendix A SAS/IML Program for Q-Statistics *Checking for symmetry using the Q2 and Q1 indices presented in Babu, Padmanabhan and Puri (1999); *This program details all the steps in obtaining the Q2 and Q1 indices; OPTIONS NOCENTER; PROC IML; RESET NONAME; *Although the Q2 and Q1 calculations differ, both share common steps; *Hence, they are incorporated into one module QMOD with the variable QCHOICE being the switch that activates Q2 or Q1: 1 activates Q1 and 2 activates Q2; START QMOD(QCHOICE,Y,OSY,GINFO,Q) GLOBAL(NY,WOBS,BOBS,PER); G = INT(PER#NY); NYPRIME = NY -2#G; NPRIME = SUM(NYPRIME); …”

“…Work by Hogg et al (1975) and Babu et al (1999), however, may provide a successful solution to this problem. The details of this method are presented in Othman, Keselman, Wilcox, and Fradette (2003).…”

“…As well, the 20% rule is not universally recommended; others have had success with values other than 20%. For example, Babu et al (1999) obtained good Type I error control, for the procedures they investigated, with 15% symmetric trimming. Indeed, as Huber (1993) argues, an estimator should have a breakdown point of at least .1; thus, even 10% trimming might provide effective Type I error control.…”

Vol. 1, No. 2 (Full Issue)

“…As indicated in our introduction, Keselman et al (2002) found that by first applying the Babu et al (1999) procedure prior to testing for treatment group equality with sample symmetrically or asymmetrically determined trimmed means one could achieve excellent control over Type I errors even though data were obtained from very heterogenous distributions that were extremely nonnormal in form. Accordingly, they recommended that users adopt the Babu et al (1999) test for symmetry.…”

“…(1993) Appendix A SAS/IML Program for Q-Statistics *Checking for symmetry using the Q2 and Q1 indices presented in Babu, Padmanabhan and Puri (1999); *This program details all the steps in obtaining the Q2 and Q1 indices; OPTIONS NOCENTER; PROC IML; RESET NONAME; *Although the Q2 and Q1 calculations differ, both share common steps; *Hence, they are incorporated into one module QMOD with the variable QCHOICE being the switch that activates Q2 or Q1: 1 activates Q1 and 2 activates Q2; START QMOD(QCHOICE,Y,OSY,GINFO,Q) GLOBAL(NY,WOBS,BOBS,PER); G = INT(PER#NY); NYPRIME = NY -2#G; NPRIME = SUM(NYPRIME); *Initialize group information matrix; IF QCHOICE = 1 THEN GINFO = J(BOBS, 1:3,8]`#NYPRIME)/NPRIME; ELSE IF QCHOICE = 2 THEN Q = SUM(GINFO [1:3,9]`#NYPRIME)/NPRIME; FINISH; *QMOD; START SHOWGRP(X, GINFO); X1 = X[GINFO [1,3]:GINFO [1,4] , 40, 32, 48, 32, 52, 41, 35, 30, 99, 40, 35, 34, 39, 50, 49, 35, 43, 36, 40, 56, 41, 40, 64, 42, 48, 51, 63, 51, 60, 51, 83, 55, 55, 48}; *Group sizes are entries in the following 1x3 row vector; NY = {15 10 10}; *WOBS and BOBS are variable names carried over from past programs; *WOBS = within subjects groups; WOBS = NCOL(Y); *BOBS = between subject groups; BOBS = NCOL(NY); Wilcox, Othman and Fradette (2002) found that by utilizing a test for symmetry prior to testing for equality of trimmed means they were able to achieve excellent Type I error control even though data were extremely heterogeneous and very nonnormal in form. In particular, they used a test for symmetry first proposed by Hogg, Fisher, and Randles (1975) and subsequently modified by Babu, Padmanaban and Puri (1999) in order to determine whether data should be trimmed symmetrically or asymmetrically.…”