# [R] MANOVA proportion of variance explained

Sam Brown s_d_j_brown at hotmail.com
Mon Jun 21 03:11:16 CEST 2010

```Hi Michael

Thank you very much for the intel regarding eta^2. It is pretty much the sort of thing that I am wanting.

Found a good paper regarding all this:

Estimating an Effect Size in One-Way Multivariate Analysis of Variance (MANOVA)
H. S. Steyn Jr; S. M. Ellisa
Multivariate Behavioral Research
2009 44: 1, 106 — 129

While the interpretation of univariate eta^2 is pretty intuitive, I'm still having a little bit of trouble getting my head around the exact interpretation of the statistic in the multivariate case. Will have to continue reading around!

Thanks again!

Sam

> Date: Wed, 16 Jun 2010 10:11:05 -0400
> From: friendly at yorku.ca
> To: s_d_j_brown at hotmail.com
> CC: r-help at r-project.org
> Subject: Re: MANOVA proportion of variance explained
>
> I think you are looking for a multivariate measure of association,
> analogous to R^2 for a univariate linear model. If so, there are
> extensions of eta^2 from univariate ANOVAs for each of the multivariate
> test statistics, e.g.,
>
> for Pillai (-Bartlett) trace and Hotelling-Lawley trace and a given
> effect tested on p response measures
>
> eta2(Pillai) = Pillai / s
> eta2(HLT) = HLT / (HLT+s)
> where s = min(df_h, p)
>
> Alternatively, you could look at the candisc package which, for an
> s-dimensional effect, gives a breakdown of the variance reflected in
> each dimension of the latents roots of HE^{-1}
>
>
> Sam Brown wrote:
>> Hello everybody
>>
>> After doing a MANOVA on a bunch of data, I want to be able to make some comment on the amount of variation in the data that is explained by the factor of interest. I want to say this in the following way: XX% of the data is explained by A.
>>
>> I can acheive something like what I want by doing the following:
>>
>> X <- structure(c(9, 6, 9, 3, 2, 7), .Dim = as.integer(c(3, 2)))
>> Y <- structure(c(0, 2, 4, 0), .Dim = as.integer(c(2, 2)))
>> Z <- structure(c(3, 1, 2, 8, 9, 7), .Dim = as.integer(c(3, 2)))
>> U <- rbind(X,Y,Z)
>> m <- manova(U~as.factor(rep(1:3, c(3, 2, 3))))
>> summary(m,test="Wilks")
>> SS<-summary(m)\$SS
>> (a<-mean(SS[[1]]/(SS[[1]]+SS[[2]])))
>>
>> and concluding that 94% of variation is explained.
>>
>> Is my desire misguided? If it is a worthy aim, is this a valid way of acheiving it?
>>
>> Thanks a lot!
>>
>> Sam
>>
>> Samuel Brown
>> Research assistant
>> Bio-Protection Research Centre
>> PO Box 84
>> Lincoln University
>> Lincoln 7647
>> Canterbury
>> New Zealand
>> sam.brown at lincolnuni.ac.nz
>> http://www.the-praise-of-insects.blogspot.com
>>

> --
> Michael Friendly Email: friendly AT yorku DOT ca
> Professor, Psychology Dept.
> York University Voice: 416 736-5115 x66249 Fax: 416 736-5814
> 4700 Keele Street Web: http://www.datavis.ca
> Toronto, ONT M3J 1P3 CANADA
>
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