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st: RE: eivreg, deming, and R^2

From   "Nick Cox" <[email protected]>
To   <[email protected]>
Subject   st: RE: eivreg, deming, and R^2
Date   Sun, 5 Oct 2008 18:07:43 +0100

Here, as indeed elsewhere, the answer depends on what you mean by R^2 --
in terms of algebraic definition and in terms of statistical

Suppose you have variables y and x. Let corr(,) mean Pearson
correlation. Then one definition of R^2 is the square of corr(y, x) and
this does not depend on any assumptions about whether x or y or both is
measured with error or how that error behaves. 

Another definition would be the square of corr(x, predicted x) and
another the square of corr(y, predicted y) where the predictions come
from taking one variable as response and predicting it from the other by
plain flavour linear regression. These definitions have distinct
meanings but in practice give the same numerical result. 

Yet other definitions can of course be found. And, more importantly,
once you move away from that plain regression territory the different
definitions typically disagree in terms of numerical result. 

In the case of -deming- and -eivreg-, my take is as follows. (Note
incidentally that -deming- is a user-written command: use -findit
deming- to locate it. As a matter of fact, the user concerned is a Stata
developer, but -deming- is not an official command.) 

-eivreg- can take multiple predictors, so R^2 in the first sense is not
defined uniquely. 

I guess that you are in practice concerned with a single predictor as
well as a single response. 

In that situation, you can use any of the definitions above, but it will
be important to say which sense you are using and to realise that R^2
will depend on assumptions insofar as predictions do. R^2 doesn't I
think play any formal part in either analysis; it's just a descriptive
statistic that may seem convenient or attractive. 

[email protected] 

David Airey

Is R^2 as interpretable in regression models that account for errors  
in both y and x, like -deming- or -eivreg-? Can I interpret R^2 from - 
eivreg- just like in -regress-?

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