Definition of Wilk's Lambda in MANOVA and relation to eta squared

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?

• So you have one between-subjects factor with two levels (two groups?) and one (?) within-subject factor (repeated measures factor) - with how many levels? I am just clarifying. – amoeba Oct 19 '16 at 20:49
• Hi Amoeba, I have one between-subjects factor with two levels, yes, and a set of commensurate DVs that are treated as repeated measures because they were all filled out by the same participants. This is what Tabachnick and Fidell refer to as 'profile analysis' and is a repeated measures MANOVA. As such, there is no within-subjects factor. Does that help? – Kristi Lo Oct 20 '16 at 8:28
• It does clarify your situation, but this is not a "repeated measures MANOVA". You do not have any "repeated measures", you have several (nine) dependent variables. So what you have is a one-way MANOVA with a single two-level factor, which is the simplest form of MANOVA. This explains your confusion: when there is only one factor (predictor), partial eta squared is the same as the normal eta squared: If you only have one predictor variable, then partial eta squared is equivalent to eta squared Does it make sense? – amoeba Oct 20 '16 at 12:57
• Yes, that makes sense, thank you! I was wondering - are there benchmarks for multivariate effect sizes (eta squared and partial eta squared)? Tabachnick and Fidell write that eta squared tends to be much larger in the multivaraite than the univariate case.. – Kristi Lo Oct 21 '16 at 9:30
• This I don't know. – amoeba Oct 21 '16 at 9:33