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MANOVA proportion of variance explained

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Sam Brown

MANOVA proportion of variance explained

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

[hidden email]

http://www.the-praise-of-insects.blogspot.com

_________________________________________________________________

ws Live

[[alternative HTML version deleted]]

______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

Michael Friendly

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
>
> [hidden email]
>
> http://www.the-praise-of-insects.blogspot.com
>
>
>
>
>
>
>
>
>
>
>
>
>
> _________________________________________________________________
>
> ws Live
>
> [[alternative HTML version deleted]]
>

--
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

______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

Sam Brown

Re: MANOVA proportion of variance explained

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
http://www.informaworld.com/smpp/content~db=all~content=a908623057~frm=titlelink

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: [hidden email]
> To: [hidden email]
> CC: [hidden email]
> 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
>> [hidden email]
>> 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
>
_________________________________________________________________
[[elided Hotmail spam]]

______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.

Michael Friendly

Re: MANOVA proportion of variance explained

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: [hidden email]
>> To: [hidden email]
>> CC: [hidden email]
>> 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
>>> [hidden email]
>>> 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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