Transcription of Chapter 1 Simple Linear Regression (part 4)
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Chapter 1 Simple Linear Regression (part 4)
Chapter 1 Simple Linear Regression (part 4)1 Analysis of Variance (ANOVA) approach to regressionanalysisRecall the model againYi= 0+ 1Xi+ i,i=1, .., nThe observations can be written deviation of eachYifrom the mean Y,Yi YThe fitted Yi=b0+b1Xi,i=1, .., nare from the Regression and determined mean is Y=1nn i=1Yi= YThus the deviation of Yifrom its mean is Yi YThe residualsei=Yi Yi,withmeanis e=0(why?)Thus the deviation ofeifrom its mean isei=Yi Yi1 WriteYi Y Total deviation= Yi Y Deviationdue the Regression +ei Deviationdue to the errorobsdeviation ofdeviation ofdeviation ofYi Yi=b0+b1 Xiei=Yi Yi1Y1 Y Y1 Ye1 e=e12Y2 Y Y2 Ye2 e= Y Yn Yen e=enSum of ni=1(Yi Y)2 ni=1( Yi Y)2 ni=1e2isquaresTotal SumSum ofSum ofof squaressquares due tosquares ofregressionerror/residuals(SST)(SSR)(SS E)We haven i=1(Yi Y)2 SST=n i=1( Yi Y)2 SSR+n i=1e2i SSEP roof:n i=1(Yi Y)2=n i=1( Yi Y+Yi Yi)2=n i=1{( Yi Y)2+(Yi Yi)2+2( Yi Y)(Yi Yi)}=SSR+SSE+2n i=1( Yi Y)(Yi Yi)=SSR+SSE+2n i=1( Yi Y)ei=SSR+SSE+2n i=1(b0+b1Xi Y)ei=SSR+SSE+2b0n i=1ei+2b1n i=1 Xiei 2 Yn i=1ei=SSR+SSEIt is also
Chapter 1 Simple Linear Regression (part 4) 1 Analysis of Variance (ANOVA) approach to regression analysis Recall the model again Yi = β0 +β1Xi +εi,i=1,...,n The observations can be written as obs YX 1 Y1 X1 2 Y2 X2..... n Yn Xn The deviation of each Yi from the mean Y¯, Yi −Y¯ The fitted Yˆ i = b0 + b1Xi,i=1,...,n are from the ...
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