Why is Pillai's trace equal to partial eta squared in a repeated measures MANOVA (profile analysis)?

Question

I conducted a repeated measures MANOVA using the GLM command in SPSS, with one between-subjects IV (two levels) and 10 DVs (repeated measures). According to Tabachnick and Fidell (Book: Using Multivariate Statistics, 6th edition), this type of design is referred to as "profile analysis". Essentially, I am looking to see whether two independent samples differ on a set of commensurate measures.

I am curious why Pillai's trace is equal to the partial eta squared that SPSS computes for the effect I am interested in. I have seen this in many sample SPSS outputs shown on the web, but nobody ever taps into this!

On a related note, I was wondering whether in this specific case that I am looking at (essentially, I have one IV only, with two levels), the partial eta square produced is actually equal to (a full, i.e., non-partial) eta square? I have read that this is the case for one-way MANOVA, but I wonder whether this applies to the specific case I used MANOVA for as well.

Answer 1

The short answer to your question is...

... in a multivariate context, there is much less of a distinction between hypothesis test statistics (e.g. F in a univariate context) and effect size estimates (e.g. eta-squared in a univariate context), so it often works out that they reflect each other, as you note. This is like noticing that R-squared an eta-squared agree; it's because they're both using the same information. In a MANOVA, Pillai's trace and multivariate partial eta squared (depending on how it's calculated) may just reflect the same information.

The longer answer:

As a review, here's a nice summary of some measures of effect size for univariate ANOVAs, inclduing both eta-squared and partial eta-squared: http://www.theanalysisfactor.com/effect-size/ The most important take-home for this particular question is that eta squared is (roughly) the variance explained divided by total varianace, while partial eta squared is the variance explained by a given predictor divided by the variance explained by that predictor plus the unexplained variance. Another way to articulate it is the variance explained by a predictor divided by the total variance minus variance explained by any of the other predictors. Jumping the gun a bit and answering your second question first, you're correct that "partial eta-squared" is the same as just "eta squared" when the only predictor in your model is a single categorical variable with 2 levels. But interpreting that statistic is still a little tricky since (partial) eta squared in a multivariate context is a little different from what you're used to thinking about with univariate ANOVAs.

There are several common statistics that summarize a MANOVA, one of which is the one you mention, Pillai's trace (which is itself based on Roy's test statistic). Two other common ones are Wilk's lambda and the Hotelling-Lawley Trace. They are all designed to roughly fill the role of an F test --- i.e. they should yield extreme values when the null hypothesis is violated --- but extended to matrices. It is not at all straightforward to calculate the multivariate sums of squares for the model and the error, though, so what we end up with isn't really an F test in any of these cases (although there are transformations to covert them to more or less fit F distributions, to allow the calculation of a p value). It just so happens that one way to reasonably substitute for an F test is with a measure of effect size, like eta-squared, so several of these measures are actually closer to eta-squared than F in terms of interpretation (e.g. Wilk's lambda can be interpreted as the proportion of variance attributable to error, i.e. 1 - eta-squared).

Just as there is no way to calculate an actual F test for a MANOVA (just several options for statistics that approximate it), there's no obvious way to calculate (partial) eta-squared. You can calculate something like an eta-squared for a MANOVA several different ways (see this article for several examples, as well as details in Rencher 2002, Kline 2004, Huberty & Olejnik 2006, and Cohen 1988). Here is a list of effect size statistics SPSS calculates for a MANOVA (although the equations as presented there are not at all easy to parse, since the notation is incomplete).

Since IBM doesn't releatse the SPSS code, it can be difficult to tell exactly what calculations are being used under the hood; in this case, I'm not sure from the documentation how the statistic you see as partial eta-squared is being calculated, but I'm not surprised it turns out the same as Pillai's trace, given how closely related all of these measures are. In fact, one common multivariate measure of effect size is just Roy's statistic itself. Pillai's trace is Roy's calculated on each eigenvalue (instead of just the largest one), so if one eigenvalue can capture close to all of the variance in the data Pillai's and Roy's will be redundant.

Additional reading:

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.
Huberty, C. J., & Olejnik, S. (2006). Applied MANOVA and discriminant analysis. New York: Wiley.
Rencher, A. C. (2002). Methods of multivariate analysis (2nd ed.). New York: Wiley.

Answer 2

Let - considering MANOVA with one factor with $k$ groups, for simplicity - $\bf B$ be the between-groups scatter matrix and $\bf W$ be the pooled within-group scatter matrix. (Please see "Bonus" caption here for how to compute both these scatter matrices efficiently with matrix algebra.)

Formulas of MANOVA effects. Wilks' lambda $\Lambda = \mathbf {\frac {\det W} {\det (B+W)}} = \prod_i \frac {1}{1+\lambda_i} $, where $\lambda_1,\lambda_2,...,\lambda_k$ are the eigenvalues of $\bf W^{-1}B$. Corresponding Eta-squared of the effect $\eta^2 = 1-\Lambda^{1/s}$, where $s=\min(k-1,\text{rank of} ~\mathbf {W})$. Wilks' lambda expresses the "within-to-total ratio" idea.

Hotelling's trace $T= \text{tr} \mathbf{(W^{-1}B)} = \sum_i \lambda_i$. Corresponding Eta-squared $\eta^2 = \frac{T/s}{1+T/s}$. Hotelling's expresses the "between-to-within ratio" idea.

Pillai's trace $V= \text{tr} \mathbf{(B(B+W)^{-1})} = \sum_i \frac{\lambda_i}{1+\lambda_i}$. Corresponding Eta-squared $\eta^2 = V/s$. Pillai's trace expresses the "between-to-total ratio" idea and is the sum of all the squared canonical correlations (of the discriminant latent functions implied).

Roy's largest root $\Theta = \lambda_1$. Corresponding Eta-squared $\eta^2 = \frac{\Theta}{1+\Theta}$.

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Links with univariate ANOVA. In univariate ANOVA, Fisher's F statistic is the "standardized" eigenvalue $\lambda = \frac {B}{W}$; $F= \frac {B/df_b}{W/df_w}= \frac {B}{W} \cdot \frac {df_w}{df_b}$.

where $B$ and $W$ are the between and the within sums-of -squares, and two $df$s are their corresponding degrees of freedom.

Because $B+W=T$ (total SS) holds, we might use equivalently other statistics to convey the magnitude of the effect (they, like $\lambda$, are "standardizable" into the same value F necessary to obtain the p-value). These are: Eta-squared (aka R-square) $\eta^2 = \frac B T$; and Wilks' lambda $\Lambda = \frac W T$.

In MANOVA, the idea is the same, only in place of the three principal scalars $B+W=T$ we have three matrices, $\bf B+W=T$. Because of the matrices, the formulas for the three interchangeable statistics that still express the same ideas of "B/W", "B/T", "W/T" become more complex : Hotelling's trace for "B/W", Pillai's trace for "B/T", and Wilks' lambda for "W/T". In the simplest case of one-way MANOVA with 2 groups all the three statistics are standardizable into the same F value (with exact F distribution). But in more complex instances of factorial designs the statistics are standardized into different F values (and those "F"s often follow the F distribution only approximately). That is why the three statistics may yield us different p-values. They thus represent in such situations not quite identical tests and, one might say, not completely equivalent hypotheses. Standard expression of effect size in a form of Eta will then - computed for each of the three statistics separately - be different. Just because multivariate data is more complex.

Texts say Wilks' lambda is most convenient and is related to the likelihood-ratio criterion. Pillai's trace, however, is the most robust of the three to violations to MANOVA assumptions of multivariate normality and homogeneity of variance-covariance matrices in cells of between-subject design.