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

From: Millo Giovanni <Giovanni_Millo_at_generali.com>
Date: Thu 12 Jan 2006 - 02:45:40 EST


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." <Bernhard_Pfaff@fra.invesco.com> Subject: Re: [R] Obtaining the adjusted r-square given the regression

        coef ficients
To: "'Alexandra R. M. de Almeida'" <alexandrarma@yahoo.com.br>,

        r-help@stat.math.ethz.ch
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:alexandrarma@yahoo.com.br] Gesendet: Mittwoch, 11. Januar 2006 03:48 An: r-help@stat.math.ethz.ch
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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