[R] pseudo-R2 or GOF for regression trees?

Prof Brian Ripley ripley at stats.ox.ac.uk
Sat May 5 22:57:00 CEST 2007

On Sat, 5 May 2007, Prof. Jeffrey Cardille wrote:

> Hello,
> Is there an accepted way to convey, for regression trees, something
> akin to R-squared?

Why not use R-squared itself for your purposes?

Just get the fitted values from however you do the fit, and compute 
R-squared from the basic formula (the one which compares with an intercept 
only: all regression trees extend that model).

Now, R-squared has lots of problems of its own (to the extent that it is 
only mentioned as something to avoid in some statistical texts) and these 
are worse here as the number of parameters fitted is unquantifiable. But 
as a factual summary it does mean what you quote.  Whether any model of 
comparable complexity would also explain 42% of the variance is a much 
harder question.

(Small anecdote: one of my first experiences of this was a psychologist 
who had funded a research project to relate personality/intelligence tests 
to 20-odd measurements on facial profiles by (stepwise) linear regression. 
My contribution was to point out that the R^2 produced was less for every 
one of the responses than one would expect on average for the same number 
of random unrelated regressors.  To be systematically worse than such a 
straw man takes some achieving, and I have always suspected a bug in the 
fitting software.)

> I'm developing regression trees for a continuous y variable and I'd
> like to say how well they are doing. In particular, I'm analyzing the
> results of a simulation model having highly non-linear behavior, and
> asking what characteristics of the inputs are related to a particular
> output measure.  I've got a very large number of points: n=4000.  I'm
> not able to do a model sensitivity analysis because of the large
> number of inputs and the model run time.
> I've been googling around both on the archives and on the rest of the
> web for several hours, but I'm still having trouble getting a firm
> sense of the state of the art.  Could someone help me to quickly
> understand what strategy, if any, is acceptable to say something like
> "The regression tree in Figure 3 captures 42% of the variance"?  The
> target audience is readers who will be interested in the subsequent
> verbal explanation of the relationship, but only once they are
> comfortable that the tree really does capture something.  I've run
> across methods to say how well a tree does relative to a set of trees
> on the same data, but that doesn't help much unless I'm sure the
> trees in question are really capturing the essence of the system.
> I'm happy to be pointed to a web site or to a thread I may have
> missed that answers this exact question.
> Thanks very much,
> Jeff
> ------------------------------------------
> Prof. Jeffrey Cardille
> jeffrey.cardille at umontreal.ca
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Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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