Partial Eta-Squared

Calculation

Partial eta-squared is calculated by dividing the sum of squares for a particular factor, which could be either a main effect or an interaction, by the sum of squares for the factor plus the sums of squares error for that factor as shown in Equation 1:

Partial eta-squared can also be calculated from an F test, as shown in Equation 2:

Equations 1 and 2 are equivalent, and both equal the squared partial correlation when there are two levels of the independent variable.

It is important to distinguish partial eta-squared from eta-squared. Eta-squared is the sum of squares for a particular factor divided by the total sum of squares and reflects the proportion of variance explained. Eta-squared is shown in Equation 3:

In one-way ANOVA, partial eta-squared and eta-squared will always be exactly equal. This is because there are no other factors to partial. In multi-way ANOVA, however, the two values depart, sometimes radically. Partial eta-squared is always greater than or equal to eta-squared, and the discrepancy is a function of the number of factors in the design; the effects of the other factors on the dependent variable; and if the design involves only independent groups factors, or whether some factors are repeated. Partial eta-squared will be increasingly larger than eta-squared as the number of factors in the design increases, as the effect sizes for the other factors increase, and as the proportion of factors that were repeated measures increases.

Whereas cumulated eta-squared values for a particular dependent variable can never sum to more than 1.00, partial eta-squared has no such limitation. That is because eta-squared consistently uses the total sum of squares for the dependent variable in the numerator. Every explanatory factor can account only for unique variance partitioned from the total variance in the dependent variable. This makes sense in experimental designs where components of variation are orthogonal. Partial eta-squared has only the sum of squares for the effect and the error sum of squares in the denominator. Thus, each factor has a unique denominator for the effect size calculation. Each calculation of partial eta-squared has its own upper limit that approaches 1.00 (when there is essentially no variance left as unexplained or error). Thus, the theoretical upper limit for cumulated partial eta-squared values is the number of factors, k. That is, if there are four factors in the design, their eta-squared values can sum to 1.00, whereas their partial eta-squared values can sum to 4.00.

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