I conducted a repeated measures MANOVA with a two-level between-subejcts factor and nine DVs, using SPSS 23. The interaction effect I was interested in was significant. Looking at what SPSS labels to be a partial eta square and saw that it was .423 (the same as the Pillai's trace statistic, .423), while wilk's lambda amounted to .577 - essentially, thus, 1 - .423 (partial eta square). I was confused by this because the text book I am using (Tabachnick & Fidell) states that for MANOVA, subtracting the wilk's lambda from 1 will yield eta squared -- not partial eta squared, they are very explicit about this. They also provide a formula for converting that multivariate eta squared to a partial eta squared (1 - lambda^(1/s), with s as min (p, df-effect).). Funnily enough, however, in the book, the authors also say that IBM SPSS will produce the partial eta squared values. If I apply their equations to my data, however, this is not true - so either SPSS labels the effect size incorrectly, or the equations in the book are wrong.
In my search for alternative text books the confusion just got worse - I have even found some sources that state that subtracting wilk's lambda from 1 will produce partial eta square. I have even checked basic texts on how wilk's lambda is defined and such, but those articles typically do not even mention how this relates to any effect size one might want to calculate.
Does anybody know what is happening here?