[R] MANOVA proportion of variance explained

[Michael Friendly]

Sam Brown wrote:
> Hi Michael
>  
> Thank you very much for the intel regarding eta^2. It is pretty much the sort of thing that I am wanting.
>   
The latest developer version of the heplots package on R-Forge now 
includes an initial implementation of etasq() for
multivariate linear models.  Note that for s>1 dimensional tests, the 
values of eta^2 differ according to the test statistic:
Pillai trace (default), Hotelling-Lawley trace, Wilks' Lambda, Roy 
maximum root test.  See ?heplots:::etasq for details.

 > # install.packages("heplots",repos="http://R-Forge.R-project.org")
 > library(heplots)
 > data(Soils)  # from car package
 > soils.mod <- lm(cbind(pH,N,Dens,P,Ca,Mg,K,Na,Conduc) ~ Block + 
Contour*Depth, data=Soils)
 > etasq(Anova(soils.mod))
                  eta^2
Block         0.5585973
Contour       0.6692989
Depth         0.5983772
Contour:Depth 0.2058495
 > etasq(soils.mod) # same
                  eta^2
Block         0.5585973
Contour       0.6692989
Depth         0.5983772
Contour:Depth 0.2058495
 > etasq(Anova(soils.mod), anova=TRUE)

Type II MANOVA Tests: Pillai test statistic
                eta^2 Df test stat approx F num Df den Df    Pr(>F)   
Block         0.55860  3    1.6758   3.7965     27     81 1.777e-06 ***
Contour       0.66930  2    1.3386   5.8468     18     52 2.730e-07 ***
Depth         0.59838  3    1.7951   4.4697     27     81 8.777e-08 ***
Contour:Depth 0.20585  6    1.2351   0.8640     54    180    0.7311   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 >

>  
> 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
> http://www.informaworld.com/smpp/content~db=all~content=a908623057~frm=titlelink
>   
This paper may be more confusing than helpful, since the emphasis is on 
the use of eta^2 as measures of multivariate
'effect size' and I think they try to blend in too many different 
threads from the effect-size literature.  And they don't
discuss what happens in designs with more than one factor or regressor.  
In the general case, etasq() calculates
measures of partial eta^2, reflecting the *additional* proportion of 
variance associated with a given term in the
full model that includes it, relative to the reduced model that excludes 
it, analogous to partial R^2 in univariate
regression models.

>   
>> 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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-- 
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