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.

Comparing measures of the ‘typical’ score across treatment groups

“…2.1. Babu et al's (1999) procedure Consider the problem of comparing distributions F 1 ¼ F 2 ¼ · · · ¼ F J : One way of approaching this problem is to consider the one-way ANOVA problem of comparing means m 1 ¼ m 2 ¼ · · · ¼ m J from J distributions F 1 ð yÞ ¼ Fð y 2 m 1 Þ; F 2 ð yÞ ¼ Fð y 2 m 2 Þ; : : : ; F J ð yÞ ¼ Fð y 2 m J Þ: When the distributions are unknown and one cannot assume that they are normal with equal variances, Babu et al (1999) suggested the following procedure:…”

“…In addition, other authors have developed procedures that use other measures of central tendency-e.g. sample medians-to assess the effects of a treatment variable across groups (see Babu, Padmanabhan, & Puri, 1999;Wilcox & Keselman, 2003a, 2003b. Keselman (2003a, 2003b) introduced a modified one-step M (MOM) estimator which empirically determines whether to trim symmetrically or asymmetrically and by what amount, or whether no trimming at all is appropriate.…”

“…Keselman (2003a, 2003b) introduced a modified one-step M (MOM) estimator which empirically determines whether to trim symmetrically or asymmetrically and by what amount, or whether no trimming at all is appropriate. And yet another tactic that has been advocated is the use of a preliminary test-a test for distributional shape-to determine what measure of central tendency should be used in a test statistic when comparing groups (see, for example, Babu et al, 1999).…”

“…To date, the adaptive procedures due to Babu et al (1999) have not been examined. Accordingly, the purpose of our investigation was to compare these new approaches, as well as some modifications of them, to the methods of Wilcox (1997Wilcox ( , 2003.…”

Attentional bias modification for acute experimental pain: A randomized controlled trial of retraining early versus later attention on pain severity, threshold and tolerance

“…In view of these findings, extreme outliers that fell more than three standard deviations from the group mean were replaced with the next extreme plus one (four data points; one case in each condition) (Tabachnick and Fidell, 2001). The main analyses were a series of one-way analyses of variance (ANOVAs) conducted on this dataset (Glass et al, 1972;Lix et al, 1996;Babu et al, 1999;Tabachnick and Fidell, 2001).…”