R help - Aug 2006 - removing intercept from lm() results in oddly high Rsquared

Can anyone help me understand why an lm model summary would return an 
r.squared of ~0.18 with an intercept term, and an r.squared of ~0.98 
without the intercept?   The fit is NOT that much better, according to 
plot.lm: residuals are similar between the two models, and a plot of 
observed x predicted is almost identical.

Thanks,

-Joseph

-- 
************************************
Joseph P. LeBouton
Forest Ecology PhD Candidate
Department of Forestry
Michigan State University
East Lansing, Michigan 48824

Office phone: 517-355-7744
email: lebouton at msu.edu

Dear Joseph

Have a look at the questions and answers in the two links 
below. There the topic has been discussed.

http://finzi.psych.upenn.edu/R/Rhelp02a/archive/68905.html

http://finzi.psych.upenn.edu/R/Rhelp02a/archive/6943.html

Best regards,

Christoph Buser

--------------------------------------------------------------
Christoph Buser <buser at stat.math.ethz.ch>
Seminar fuer Statistik, LEO C13
ETH Zurich	8092 Zurich	 SWITZERLAND
phone: x-41-44-632-4673		fax: 632-1228
http://stat.ethz.ch/~buser/
--------------------------------------------------------------

Joseph LeBouton writes:
 > Can anyone help me understand why an lm model summary would return an 
 > r.squared of ~0.18 with an intercept term, and an r.squared of ~0.98 
 > without the intercept?   The fit is NOT that much better, according to 
 > plot.lm: residuals are similar between the two models, and a plot of 
 > observed x predicted is almost identical.
 > 
 > Thanks,
 > 
 > -Joseph
 > 
 > -- 
 > ************************************
 > Joseph P. LeBouton
 > Forest Ecology PhD Candidate
 > Department of Forestry
 > Michigan State University
 > East Lansing, Michigan 48824
 > 
 > Office phone: 517-355-7744
 > email: lebouton at msu.edu
 > 
 > ______________________________________________
 > R-help at stat.math.ethz.ch mailing list
 > https://stat.ethz.ch/mailman/listinfo/r-help
 > PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
 > and provide commented, minimal, self-contained, reproducible code.

Joseph LeBouton <lebouton <at> msu.edu> writes:
> 
> Can anyone help me understand why an lm model summary would return an 
> r.squared of ~0.18 with an intercept term, and an r.squared of ~0.98 
> without the intercept?   The fit is NOT that much better, according to 
> plot.lm: residuals are similar between the two models, and a plot of 
> observed x predicted is almost identical.
There are reasons why the standard textbooks and Bill Venables

http://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf

tell you that removing Intercepts can be dangerous for your health.

Dieter

##
set.seed(10)
x = runif(20,5,10)
y = 2 * x + rnorm(20,0,0.3)

# a fit with good data
summary(lm(y~x))$r.squared
# 0.98

# add one outlier at 0
x = c(x,0)
y = c(y,20)
summary(lm(y~x))$r.squared
# 0.00008

# removing the intercept: perfect correlation again
summary(lm(y~x-1))$r.squared
# 0.91

#... because it is similar to adding MANY data points
# at (0,0)
x = c(x,rep(0,1000))
y = c(y,rep(0,1000))
summary(lm(y~x))$r.squared
# 0.90