Re: [R] Calculating R2 for a unit slope regression

From: Rolf Turner <r.turner_at_auckland.ac.nz>
Date: Tue, 04 Nov 2008 09:15:14 +1300

On 4/11/2008, at 4:30 AM, J. Sebastian Tello wrote:

> Does anyone know of a literature reference, or a piece of code that
> can help me
> calculate the amount of variation explained (R2 value), in a
> regression constrained
> to have a slope of 1 and an intercept of 0?

The question is ``wrong''. The idea of ``amount of variation explained''
depends on decomposing the ``total sum of squares'' into two pieces --- the sum of
squares of the residuals what is left over which is the sum of squares ``explained
by the model''. In the usual regression setting this is

        sum((y_i - ybar)^2) = sum((y_i - yhat_i)^2) + sum((yhat_i - ybar)^2)

or

        SST = SSE + SSR (T for total, E for error, R for regression)

where yhat_i results from fitting the model by least squares.

The R-squared value is SSR/SST or 1 - SSE/SST. (Or this quantity time 100%.)

However if you constrain the slope to be 1 and the intercept to be 0 then
yhat_i = x_i and the forgoing identity does not hold. The problem is that
the ``sum of squares left over'' can be negative (and hence not a sum of squares).

I.e. in this case you have

        SST = SSE + something

where ``something'' is not necessarily a sum of squares.

Thus you can have the ``amount of variation explained'' being negative!

E.g. x_1 = -1, x_2 = 1, y_1 = 1, y_2 = -1. In this setting the ``total sum of squares'' is 2 and the ``residual sum of squares'' is 4, so the ``amount of variation explained by the model'' is -2, or you could say that R-squared is -100%. (!!!)

Bottom line --- the R-squared concept makes no sense in this context.

The R-squared concept is at best dubious, and should be used, if at all, only in the completely orthodox setting.

        cheers,

                Rolf Turner

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