R help - Dec 2008 - r2 for lm() with zero intercept

Hello list
I'm a little confused about the R2 and adjusted R2 values reported by lm()
when I try to fix an intercept. When using +0 or -1 in the formula I have found
that the standard error generally increases (as I would expect) but the R2 also
increases
(which seems counter intuitive). I've pasted a short test script below to
illustrate. I do realise that many will say I shouldn't be fixing the
intercept anyway but I'd appreciate knowing if this is a problem in the
software or with my own logic.
> x=1:100
> y= 20 + 0.8*(x+20*rnorm(100))
> mod1 = lm(y ~ x)
> summary(mod1)
Call:
lm(formula = y ~ x)
Residuals:
Min 1Q Median 3Q Max
-41.332 -9.885 1.191 12.842 34.067
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 19.30668 3.02193 6.389 5.64e-09 ***
x 0.82630 0.05195 15.905 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
'.' 0.1 ' ' 1
Residual standard error: 15 on 98 degrees of freedom
Multiple R-Squared: 0.7208, Adjusted R-squared: 0.7179
F-statistic: 253 on 1 and 98 DF, p-value: < 2.2e-16
> mod2 = lm(y ~ 0 + x)
> summary(mod2)
Call:
lm(formula = y ~ 0 + x)
Residuals:
Min 1Q Median 3Q Max
-34.049 -6.728 6.364 18.292 47.323
Coefficients:
Estimate Std. Error t value Pr(>|t|)
x 1.11446 0.03053 36.51 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
'.' 0.1 ' ' 1
Residual standard error: 17.76 on 99 degrees of freedom
Multiple R-Squared: 0.9308, Adjusted R-squared: 0.9301
F-statistic: 1333 on 1 and 99 DF, p-value: < 2.2e-16


I'm running R on Windows XP and have been rolling back from version 2.8 to
see if it is a version issue but I'm back to 2.1 and I am still getting the
same output.

Cheers, Glenn

______________________________________________
Glenn Newnham
CSIRO Sustainable Ecosystems
Private Bag 10
Clayton South, VIC 3169, Australia
______________________________________________

	[[alternative HTML version deleted]]

G'day Glenn,

On Tue, 2 Dec 2008 12:53:44 +1100
<Glenn.Newnham at csiro.au> wrote:
> I'm a little confused about the R2 and adjusted R2 values reported by
> lm() when I try to fix an intercept. When using +0 or -1 in the
> formula I have found that the standard error generally increases (as
> I would expect) but the R2 also increases (which seems counter
> intuitive). 
?summary.lm

In particular the part:

r.squared: R^2, the 'fraction of variance explained by the model',

              R^2 = 1 - Sum(R[i]^2) / Sum((y[i]- y*)^2),

          where y* is the mean of y[i] if there is an intercept and
          zero otherwise.

> I do realise that many will say I shouldn't be fixing the intercept
> anyway 
Quite true; accept if there are very good reasons.  I have seen
intercept through the origin being misused to obtain a large R^2 and
significant coefficient when there were none.

Cheers,

	Berwin

=========================== Full address ============================Berwin A
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