# Generalized eta squared

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:

where SS P is the effect sum of squares for P and SS PS, SS QS, and SS PQS are the error sum of squares for the two main effects and the interaction effect respectively.

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?