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


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,


Prof. Jeffrey Cardille

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Brian D. Ripley,                  [EMAIL PROTECTED]
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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