Partial eta-squared are very often used in psychological litterature. As underlined by some authors (e.g., Baguley, 2009; Bakeman, 2005; Olejnik & Algina, 2003), this standardized measure of effect size can be misleading when comparing it across studies with various research designs especially because of its design-sensitive nature. Olejnik and Algina (2003) proposed the generalized eta-squared which is invariant across different research designs. Following this original work, Bakeman (2005) described how to easily compute the generalized eta-squared for various research designs including manipulated or measured and within- and between-suject independent variables. Lakens (2013) recently proposed a spreadsheet to compute generalized eta-squared for some of the research designs mentionned by Bakeman.
The Lakens (2013)'s spreadsheet does not allow to compute generalized eta-squared for designs including three within-subject independent variables. Unless I read Bakeman (2005) too quickly, his paper does not seem to mention this particular design. Thus, I suppose that computing the generalized eta-squared for three-way within-subject designs is simply a generalization of Olejnik and Algina (2003)'s formulas.
Lakens (2013)'S spreadsheet allows to compute a generalized eta-squared for a P*Q within-subject design where P and Q are two within-subject independent variable. The underlying formula for computing generalized eta-squared for P is:
This formula is a generalization of the following formula proposed by Olejnik and Algina (2003, p. 440) for a within-subject design with only one independent variable (mentionned as A in the formula):
Then, if we consider a PQT within-subject design where P, Q, and T are within-subject independent variables, I guess we can compute a generalized eta-squared for P with the following generalization of the previous formulas:
Do you agree with my generalization to compute the generalized eta-squared in within-subject designs with three independent variables? If I am wrong, can you explain why, please?
Thanks for your comments
Baguley, T. (2009). Standardized or simple effect size: What should be reported? British Journal of Psychology, 100(3), 603–617. doi:10.1348/000712608X377117
Bakeman, R. (2005). Recommended effect size statistics for repeated measures designs. Behavior Research Methods, 37(3), 379–384. doi:10.3758/BF03192707
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4(November), 863. doi:10.3389/fpsyg.2013.00863
Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics: Measures of effect size for some common research designs. Psychological Methods, 8(4), 434–447. doi:10.1037/1082-989X.8.4.434