```Date: Tue, 6 Dec 2005 19:04:36 -0800 Reply-To: shiling99@YAHOO.COM Sender: "SAS(r) Discussion" From: shiling99@YAHOO.COM Organization: http://groups.google.com Subject: Re: R Square Comments: To: sas-l@uga.edu In-Reply-To: <200512062038.jB6KE12R020379@mailgw.cc.uga.edu> Content-Type: text/plain; charset="iso-8859-1" Suppose y= a+ b1*x + err. err ~ normal(0, sigma) Here is the proof. R**2=SSreg/SStotal=sum over obs (Yhat-Ybar)**2/sum over obs (Y-Ybar)**2 then E(SSreg)=sigma**2 + b1**2*Sxx and E(SSres)=(n-2)*sigma**2 Sxx=sum over obs (X-Xbar)**2 E(SStotal)=E(SSreg) + E(SSres) = )=(n-1)*sigma**2 + b1**2*Sxx So R**2=(sigma**2 + b1**2*Sxx ) /[(n-1)*sigma**2 + b1**2*Sxx ] Let w=b1**2*Sxx /n*sigma**2, then R**2 in terms w will be R**2=[n**(-1) + w] /[1-n**(-1)+w] It is easy to show that the derivertive of R**2 w.r.t w is gt 0. If you deem b1**2*Sxx /n*sigma**2 as a signal to noise ratio, higher the ratio, better result/model. Higher range of X is the same to say bigger Sxx. Here is a simulation pgm. data t1; do i = 1 to 100; x=rannor(123); y=2+2*x+rannor(123); output; end; run; proc reg data=t1; model y=x; where abs(x)<0.5; run; proc reg data=t1; model y=x; where abs(x)>0.5; run; HTH. ```

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