GR5205 p40-p50.pdf - Caution u2022 A statistically...

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CautionA statistically significant slope does not always imply a strong correlation.2.7Analysis of Variance Approach to Regression AnalysisPartitioningSSTYi-¯Y=ˆYi-¯Y+Yi-ˆYin-1=1+n-2Define thesums of squares regressionbySSR=nXi=1(ˆYi-¯Y)2.40
Notice thatPROPOSITION2.4Total variationSSTcan be partitioned into two sources of variationSSRandSSE. This can be represented with the additive identitySST=SSR+SSEPROOF:2Equivalent testing procedure for null hypothesisH0:β1= 0Consider testingH0:β1= 0(2.11)The F-statistic for testing the above hypothesis isfcalc=SSR/1SSE/(n-2)=SSRMSE(2.12)Alternative HypothesisRejection region for a leveltestHA:β16= 0fcalcf,1,n-2(two-tailed)41
ANOVA tableSourcedfSum of SquaresMean SquareF-valueRegression1SSRSSRfcalcErrorn-2SSEMSETotaln-1SSTExpected mean squaresNext weinformallymotivate how the F-statistic tests the hypothesis stated in (2.11).Firstconsider Proposition 2.5PROPOSITION2.5The expected value ofSSRisE[SSR] =σ2+β21SxxPROOF:2Motivation of theF–statistic42

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