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.

Analysis of Variance Table Response: Y Df Sum Sq Mean Sq F value Pr(>F) X 1 252378 252378 105.88 4.449e-10 *** Residuals 23 54825 2384 Suppose we need to testH0: β1 = 0 with significant level 0.01, based on the calculation, the p-value is 4.449×10−10 <0.01, we should reject H0. Equivalence of F-test and t-test We have two methods to test ...

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  Analysis, Methods, Regression

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