Interval estimates of weighted effect sizes in the one-way heteroscedastic ANOVA E. Kulinskaya 1 and R. G. Staudte 2 * 1 Statistical Advisory Service, Imperial College, London SW7 2AZ, UK 2 Department of Statistical Science, La Trobe University, Melbourne 3068, Australia A framework for comparing normal population means in the presence of heteroscedasticity and outliers is provided. A single number called the weighted effect size summarizes the differences in population means after weighting each according to the difficulty of estimating their respective means, whether the difficulty is due to unknown population variances, unequal sample sizes or the presence of outliers. For an ANOVA weighted for unequal variances, we find interval estimates for the weighted effect size. In addition, the weighted effect size is shown to be a monotone function of a suitably defined weighted coefficient of determination, which means that interval estimates of the former are readily transformed into interval estimates of the latter. Extensive simulations demonstrate the accuracy of the nominal 95% coverage of these intervals for a wide range of parameters. 1. Introduction When heteroscedasticity is present, the reliability of the classical ANOVA is undermined, so Cochran (1937), Welch (1951) and James (1951) suggested weighting the terms appearing in the F statistic to account for the different population variances. The Welch F-test for equal population means is available on some statistical packages, and a reliable estimate of its power under alternatives can now be found in Kulinskaya, Staudte, and Gao (2003). Our purpose here is to estimate the weighted effect size u, defined in (2) below. This is a rescaled version of the non-centrality parameter arising under alternatives to the null hypothesis of equal means in a weighted ANOVA, an approach advocated by Steiger and Fouladi (1997). First we formulate a more general setting which allows one to choose a measure of location (other than the mean) for each population. In this setting we take weights to be inversely proportional to the respective variances of our estimates of population locations. Then the weighted effect size is shown to be a simple monotone * Correspondence should be addressed to Robert G. Staudte, Departmento de Estadı ´stica, Facultad de Ciencias Sociales, Universidad Carlos III de Madrid, Madrid 126, 28903 Getafe, Spain (e-mail: r.staudte@latrobe.edu.au). The British Psychological Society 1 British Journal of Mathematical and Statistical Psychology (2005), 00, 1–16 q 2005 The British Psychological Society www.bpsjournals.co.uk DOI:10.1348/000711005X68174 BJMSP 9717—1/2/2006—THIRUMAL—162088