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

In 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.