[R] Average R-squared of model1 to model n

[Liaw, Andy]

The Y1, Y2, etc. that Kan mentioned are predicted values of a test set data
from models that supposedly were fitted to the same (or similar) data.  It's
hard for me to imagine the outcome would be as `severe' as Y1 = -Y2.

That said, I do not think that the R-squared (or q-squared as some call it)
of the aggregate model is necessarily larger or equal to the average
R-squared of the component models.  It obviously depends on how the
component models are generated.  As a hypothetical example (because I
haven't acutally tried it, just speculating):  Suppose the data are
generated from a step function, the sort that would be perfect for
regression trees.  If one grows several well-pruned trees, I'd guess that
the average R-squared of the individual trees has a chance of being larger
than the R-squared of the averaged model.

Best,
Andy

> From: Gabor Grothendieck
> 
> Suppose m=2, Y1=Y and Y2= -Y.  Then (b) is zero so (a) must be
> greater or equal to (b).  Thus (b) is not necessarily greater 
> than (a).
> 
> 
> kan Liu <kan_liu1 <at> yahoo.com> writes:
> 
> : 
> : Hi,
> : 
> : We got a question about interpretating R-suqared.
> : 
> : The actual outputs for a test dataset is X=(x1,x2, ..., xn).
> : model 1 predicted the outputs as Y1=(y11,y12,..., y1n)
> : model n predicted the outputs as Y2=(y21,y22,..., y2n)
> : 
> : ... 
> : model m predicted the outputs as Ym=(ym1,ym2,..., ymn)
> : 
> : Now we have two ways to calculate R squared to evaluate the average 
> performance of committee model.
> : 
> : (a) Calculate R squared between (X, Y1), (X, Y2), ..., 
> (X,Ym), and then 
> averaging the R squared
> : (b) Calculate average Y=(Y1+Y2, + ... Ym)/m, and then 
> calculate the R 
> squared between (X, Y). 
> : 
> : We found it seemed that R squared calculated in (b) is 
> 'always' higher than 
> that in (a).
> : 
> : Does this result depends on the test dataset or this 
> happened by chance?Can 
> you advise me any reference for
> : this issue? 
> : 
> : Many thanks in advance!
> : 
> : Kan
> : 
> : 
> : 		
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> : 
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