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Between-group variance from ANOVA

5 messages · tedtoal, Ista Zahn, arun +2 more

tedtoal
# 
I'm trying also to understand how to get the between-group variance out of a
one-way ANOVA, but I'm beginning to think that in a sense, the variance does
not exist.  Emma said:

*The model is response(i,j)= group(i)+ error(i,j)*

Yes, if by group(i) you mean intercept + coefficient[i].

*we assume that group~N(0,P^2) and error~N(0,sigma^2) *

Only the error is assumed to be a random variable.  Group is a fixed effect,
not a random variable, and therefore it has no variance associated with it. 
The model does not predict a variance for it.  One could compute the
variance of the coefficients and call this a group variance, but it seems to
me that isn't the right way to think about it.

I'm trying to calculate a heritability value for a trait in an organism,
defined as Vg/Vp, where Vg = variance due to genotype and Vp = total
variance.  The model is p~g,  or p[i,j] = intercept + g_coefficient[i] +
error[i,j].  But to get Vg, I think it is actually necessary to use a
different model, where g is modelled as a random variable (a random effect),
so the model can estimate a variance associated with it.

If anyone can add something to this, please do.
ted




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Ista Zahn
# 
There is nothing about R in your question, hence it is not appropriate
for this list. Please consult with a local statistician, or post on a
stats help list such as http://stats.stackexchange.com/
On Tue, Jul 24, 2012 at 8:55 PM, tedtoal <twtoal at ucdavis.edu> wrote:
S Ellison
# 
The classical calculations in a one way anova table make no assumptions about the origin or distribution of the between-group differences. Nor does the F test commonly applied (because the F test assumes the null hypothesis, which is that there is no group effect - so we don't need to make assumptions about it to calculate a p-value). 

For one way anova you are therefore free to think of the between group effects, if hypothesised to be present, as fixed or random. If the experiment tests controlled changes it usually makes more sense to think of them as fixed, and one tends to worry about the size of individual effects; If you're thinking of them as drawn randomly from a larger population of possible effects (ie random) it is usually sensible to calculate a variance. 

The classical calcuilations of the between-group variance are given in practically every textbook on the topic. For a slightly more modern take on it you'd probably go for REML solutions which you can get from lme in the nlme package, among others. To do that, assuming data y with a grouping factor g, you would do something like
library(nlme)
l <- lme(y~1, random=~1|g)
summary(l) #for the whole picture
VarCorr(l) #for just variances

... and that will give you estimates of within- and between-group variance components

S Ellison

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Peter Dalgaard
#
On Jul 25, 2012, at 14:56 , arun wrote:

            

      
      
      
    

        
Beware, though, that this works for balanced designs only (identical group sizes). For unequal replication, you need to go the lme/lmer route.

-pd