Artificial systematic attenuation in eta squared and some related consequences: attenuation-corrected eta and eta squared, negative values of eta, and their relation to Pearson correlation

Abstract

AbstractIn general linear modeling (GLM), eta squared (η2) is the dominant statistic for the explaining power of an independent variable. This article discusses a less-studied deficiency in η2: its values are seriously deflated, because the estimates by coefficient eta (η) are seriously deflated. Numerical examples show that the deflation in η may be as high as 0.50–0.60 units of correlation and in η2 as high as 0.70–0.80 units of explaining power. A simple mechanism to evaluate and correct the artificial attenuation is proposed. Because the formulae of η and point-biserial correlation are equal, η can also get negative values. While the traditional formulae give us only the magnitude of nonlinear association, a re-considered formula for η gives estimates with both magnitude and direction in binary cases, and a short-cut option is offered for the polytomous ones. Although the negative values of η are not relevant when η2 is of interest, this may be valuable additional information when η is used with non-nominal variables.