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Robust One-Way ANOVA under Possibly Non-Regular Conditions
Authors:Gutti Jogesh Babu  A R Padmanabhan  Madan L Puri
Abstract:Consider the one-way ANOVA problem of comparing the means m1, m2, …, mc of c distributions F1(x) = F(xm1), …, Fc(x) = F(xmc). Solutions are available based on (i) normal-theory procedures, (ii) linear rank statistics and (iii) M-estimators. The above model presupposes that F1, F2, …, Fc have equal variances (= homoscedasticity). However practising statisticans content that homoscedasticity is often violated in practice. Hence a more realistic problem to consider is F1(x) = F((xm1)/σ1), …, Fc(x) = F((xmc)/σc), where F is symmetric about the origin and σ1, …, σc are unknown and possibly unequal (= heteroscedasticity). Now we have to compare m1, m2, …, mc. At present, nonparametric tests of the equality of m1, m2, …, mc 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 shapes of F1, …, Fc 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.
Keywords:Bias-corrected bootstrap  15% trimmed means  Sample medians  Smooth bootstrap  Sprague-Dawley rats
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