Abstract
We propose to use the squared multiple correlation coefficient as an effect size measure for experimental analysis‐of‐variance designs and to use Bayesian methods to estimate its posterior distribution. We provide the expressions for the squared multiple, semipartial, and partial correlation coefficients corresponding to four commonly used analysis‐of‐variance designs and illustrate our contribution with two worked examples.
Highlights
In the empirical sciences, researchers are commonly advised to report effect size (ES) interval estimates to complement the use of statistical hypothesis testing (e.g., Loftus, 1996; Cohen, 1992, 1994; Cumming, 2014)
This is true for psychology, where the need for reporting ES estimates is stressed in publication guidelines of several organizations and scientific journals, but it applies to other fields of science, such as biology (Nakagawa & Cuthill, 2007), genetics (Park et al, 2010), and marketing research (Fern & Monroe, 1996)
We have shown that the theory on squared multiple correlation coefficients provides a useful framework for deriving ES measures for the analysis of variance (ANOVA) model, and we have analyzed several aspects that are commonly encountered in experimental designs
Summary
Researchers are commonly advised to report effect size (ES) interval estimates to complement the use of statistical hypothesis testing (e.g., Loftus, 1996; Cohen, 1992, 1994; Cumming, 2014). Credible intervals do not have the anomalies that beset some of the classical ES intervals for ANOVA designs Despite their added value, Bayesian methods of ES estimation for ANOVA models have so far received little attention. The connection between ES measures such as ω2 and η2 for the ANOVA model and the squared multiple correlations ρ2 for the linear regression model has been noted before (e.g., Keren & Lewis, 1979; Maxwell, Camp, & Arvey, 1981; Sechrest & Yeaton, 1982; Cohen, Cohen, West, & Aiken, 2003), little is known about the assumptions that underpin this relation.
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