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." <

[hidden email]>

Subject: Re: [R] Obtaining the adjusted r-square given the regression

coef ficients

To: "'Alexandra R. M. de Almeida'" <

[hidden email]>,

[hidden email]
Message-ID: <25D1C2585277D311B9A20000F6CCC71B077C0389@DEFRAEX02>

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:

[hidden email]]

Gesendet: Mittwoch, 11. Januar 2006 03:48

An:

[hidden email]
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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