Alexandra,
some additional remarks taken from my past struggles with R2 :^) Without
intercept the definition is indeed problematic, as Bernhard notes.
First, to estimate a model omitting the intercept you simply have to
specify "-1" in the model formula (example on an in-built dataset, for
data description see help(mtcars)):
> data(mtcars)
> attach(mtcars)
> mod<-lm(mpg~hp+wt+qsec) # with intercept
> summary(mod)
and
> mod0<-lm(mpg~hp+wt+qsec-1) # without
> summary(mod0)
The reported R2s are different not only in value (which is obvious) but
also in the definition.
In fact, there are 2 definitions of R2. With reference to the usual
analysis of variance in OLS regression (see e.g. Ch.3 in Greene 2003,
Econometric Analysis, and 3.5.2. in particular), let, in our example,
> SST<-sum(mpg^2) # total sum of squares
> SSR<-sum(fitted(mod)^2) # regression sum of squares
> SSE<-sum(resid(mod)^2) # error sum of squares
where (a) SST=SSR+SSE, as you may readily check,
then the *uncentered* R2 is defined as
> uR2<-SSR/SST
while the *centered* R2 as
> cSST<-sum((mpg-mean(mpg))^2)
> cSSR<-sum((fitted(mod)-mean(mpg))^2) # as 1) mean(y)=mean(y_hat)
> cSSE<-sum(resid(mod)^2) # as 2) mean(e)=0
> cR2<-cSSR/cSST
and (b) cSST=cSSR+cSSE.
The problem is that the meaning of R2 derives from decompositions (a)
and (b), but while (a) always holds for OLS models, (b) only holds for
models with an intercept (as do (1-2) above, on which it is based). Thus
*centered R2 is meaningless in models without intercept*. People are
used to cR2, though, so R reports cR2 for models with intercept, uR2 for
those without (EViews, e.g., reports cR2 for both).
Adjusted R2s are the same, adjusted by a factor penalizing for df. See
Greene, who gives
adjR2 = 1-(n-1)/(n-K)(1-R2) for n obs. and K regressors.
Finally, it is of course feasible to calculate the model coefficients on
your own, but it would be inefficient (R has an optimized routine for
OLS, so you'd better use coef(lm(y~X))). Anyway, if you like,
> y<-mpg # just for notational simplicity..
> X<-cbind(hp,wt,qsec) # add rep(1,length(hp)) to this data matrix
# if you want an intercept
> b<-solve(crossprod(X),crossprod(X,y)) # the coefficients for mod0
> y_hat<-X%*%b # fitted values for y
> e<-y-y_hat # model residuals
from which you can obtain anything you need.
Cheers
Giovanni
Giovanni Millo
Ufficio Studi
Assicurazioni Generali SpA
Via Machiavelli 4, 34131 Trieste (I)
tel. +39 040 671184
fax +39 040 671160
*****************
Original message:
Date: Wed, 11 Jan 2006 09:16:46 -0000
From: "Pfaff, Bernhard Dr." <[EMAIL PROTECTED]>
Subject: Re: [R] Obtaining the adjusted r-square given the regression
coef ficients
To: "'Alexandra R. M. de Almeida'" <[EMAIL PROTECTED]>,
[email protected]
Message-ID: <[EMAIL PROTECTED]>
Content-Type: text/plain; charset="iso-8859-1"
Hello Alexandra,
R2 is only defined for regressions with intercept. See a decent
econometrics
textbook for its derivation.
HTH,
Bernhard
-----Urspr?ngliche Nachricht-----
Von: Alexandra R. M. de Almeida [mailto:[EMAIL PROTECTED]
Gesendet: Mittwoch, 11. Januar 2006 03:48
An: [email protected]
Betreff: [R] Obtaining the adjusted r-square given the regression
coefficients
Dear list
I want to obtain the adjusted r-square given a set of coefficients
(without
the intercept), and I don't know if there is a function that does it.
Exist????????????????
I know that if you make a linear regression, you enter the dataset and
have
in "summary" the adjusted r-square. But this is calculated using the
coefficients that R obtained,and I want other coefficients that i
calculated
separately and differently (without the intercept term too).
I have made a function based in the equations of the book "Linear
Regression
Analisys" (Wiley Series in probability and mathematical statistics), but
it
doesn't return values between 0 and 1. What is wrong????
The functions is given by:
adjustedR2<-function(Y,X,saM)
{
if(is.matrix(Y)==F) (Y<-as.matrix(Y))
if(is.matrix(X)==F) (X<-as.matrix(X))
if(is.matrix(saM)==F) (saM<-as.matrix(saM))
RX<-rent.matrix(X,1)$Rentabilidade.tipo
RY<-rent.matrix(Y,1)$Rentabilidade.tipo
r2m<-matrix(0,nrow=ncol(Y),ncol=1)
RSS<-matrix(0,ncol=ncol(Y),nrow=1)
SYY<-matrix(0,ncol=ncol(Y),nrow=1)
for (i in 1:ncol(RY))
{
RSS[,i]<-(t(RY[,i])%*%RY[,i])-(saM[i,]%*%(t(RX)%*%RX)%*%t(saM)[,i])
SYY[,i]<-sum((RY[,i]-mean(RY[,i]))^2)
r2m[i,]<-1-(RSS[,i]/SYY[,i])*((nrow(RY))/(nrow(RY)-ncol(saM)-1))
}
dimnames(r2m)<-list(colnames(Y),c("Adjusted R-square"))
return(r2m)
}
Thanks!
Alexandra
Alexandra R. Mendes de Almeida
---------------------------------
Ai sensi del D.Lgs. 196/2003 si precisa che le informazioni ...{{dropped}}
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