A Welch-type test for comparing K linear contrasts under heteroscedasticity with applications to meta-analysis

A test for comparing linear contrasts with heteroscedasticity, both across components of each contrast and between the contrasts, is developed under assumption of normality. The test is based on the weighted sums of squares. This is an extension of methods for weighted one-way ANOVA developed in Welch (1951) under the null, and in Kulinskaya et al. (2000) [12] underthe alternatives. We provide very accurate approximations to the null distribution and to the distribution under alter natives. The quality of these approximations is studied by simulation. The main application is the I by 2 layout which is widespread in meta-analysis. Our methods allow the homo-geneity of effect sizes across I studies to be tested, without the assumption of equal variances in the treatment and the control groups.