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Robust One‐Way ANOVA under Possibly Non‐Regular Conditions
Author(s) -
Babu Gutti Jogesh,
Padmanabhan A. R.,
Puri Madan L.
Publication year - 1999
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/(sici)1521-4036(199906)41:3<321::aid-bimj321>3.0.co;2-5
Subject(s) - homoscedasticity , heteroscedasticity , mathematics , nonparametric statistics , wilcoxon signed rank test , statistics , estimator , quantile , rank (graph theory) , monte carlo method , combinatorics , mann–whitney u test
Consider the one‐way ANOVA problem of comparing the means m 1 , m 2 , …, m c of c distributions F 1 ( x ) = F ( x — m 1 ), …, F c ( x ) = F ( x — m c ). Solutions are available based on (i) normal‐theory procedures, (ii) linear rank statistics and (iii) M ‐estimators. The above model presupposes that F 1 , F 2 , …, F c have equal variances (= homoscedasticity). However practising statisticans content that homoscedasticity is often violated in practice. Hence a more realistic problem to consider is F 1 ( x ) = F (( x — m 1 )/σ 1 ), …, F c ( x ) = F (( x — m c )/σ c ), where F is symmetric about the origin and σ 1 , …, σ c are unknown and possibly unequal (= heteroscedasticity). Now we have to compare m 1 , m 2 , …, m c . At present, nonparametric tests of the equality of m 1 , m 2 , …, 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 shapes of F 1 , …, 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.