IntroductionThe Regression IdentitySums of Squares on the TI-83Explaining VariationThe Coefficient of DeterminationTheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationThe Coefficient of DeterminationLecture 49Section 13.9Robb T. KoetherHampden-Sydney CollegeTue, Apr 22, 2008TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationOutline1Introduction2The Regression Identity3Sums of Squares on the TI-834Explaining Variation5The Coefficient of DeterminationTheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationIntroductionOne goal of regression is to produce a model that willpredict values of the response variable, given values ofthe explanatory variable.We have already done that.Another goal is to “explain” the variability in theresponse variable as a consequence of the variability inthe explanatory variable.That is where we now turn out attention.TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationResidual Sum of SquaresRecall that the line of “best” fit was that line with thesmallest sum of squared residuals.This is also called the residual sum of squares or sumof squared errors:SSE =X(y −ˆy)2.TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationOther Sums of SquaresThere are two other sums of squares associated with y.The regression sum of squares:SSR =X(ˆy − y)2.The total sum of squares:SST = SSY =X(y − y)2.TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExample - SST, SSR, and SSEPlot the following data.x y1 83 124 95 148 169 2011 1715 24TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExample - SST, SSR, and SSEThe regression line129242118152 141210864 166TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExample - SST, SSR, and SSEThe deviations of y from y129242118152 141210864 166TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExample - SST, SSR, and SSEThe deviations ofˆy from y129242118152 141210864 166TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExample - SST, SSR, and SSEThe deviations of y fromˆy129242118152 141210864 166TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationThe Regression IdentityIt turns out thatSST = SSR + SSE.This is the regression identity.It also turns out thatr2=SSRSST.Consequently,r2=SSRSST=SST − SSESST= 1 −SSESST.TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SST.x y y − y (y − y)21 83 124 95 148 169 2011 1715 24TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SST.x y y − y (y − y)21 8 -73 12 -34 9 -65 14 -18 16 19 20 511 17 215 24 9TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SST.x y y − y (y − y)21 8 -7 493 12 -3 94 9 -6 365 14 -1 18 16 1 19 20 5 2511 17 2 415 24 9 81TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SST.x y y − y (y − y)21 8 -7 493 12 -3 94 9 -6 365 14 -1 18 16 1 19 20 5 2511 17 2 415 24 9 81206TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSR.x yˆyˆy − y (ˆy − y)21 8 8.43 12 10.64 9 11.75 14 12.88 16 16.19 20 17.211 17 19.415 24 23.8TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSR.x yˆyˆy − y (ˆy − y)21 8 8.4 -6.63 12 10.6 -4.44 9 11.7 -3.35 14 12.8 -2.28 16 16.1 1.19 20 17.2 2.211 17 19.4 4.415 24 23.8 8.8TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSR.x yˆyˆy − y (ˆy − y)21 8 8.4 -6.6 43.563 12 10.6 -4.4 19.364 9 11.7 -3.3 10.895 14 12.8 -2.2 4.848 16 16.1 1.1 1.219 20 17.2 2.2 4.8411 17 19.4 4.4 19.3615 24 23.8 8.8 77.44TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSR.x yˆyˆy − y (ˆy − y)21 8 8.4 -6.6 43.563 12 10.6 -4.4 19.364 9 11.7 -3.3 10.895 14 12.8 -2.2 4.848 16 16.1 1.1 1.219 20 17.2 2.2 4.8411 17 19.4 4.4 19.3615 24 23.8 8.8 77.44181.50TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSE.x yˆy y −ˆy (y −ˆy)21 8 8.43 12 10.64 9 11.75 14 12.88 16 16.19 20 17.211 17 19.415 24 23.8TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSE.x yˆy y −ˆy (y −ˆy)21 8 8.4 -0.43 12 10.6 1.44 9 11.7 -1.35 14 12.8 1.28 16 16.1 -0.19 20 17.2 2.811 17 19.4 -2.415 24 23.8 0.2TheCoefficient ofDeterminationRobb T.KoetherIntroductionTheRegressionIdentitySums ofSquares onthe TI-83ExplainingVariationTheCoefficient ofDeterminationExampleCompute SSE.x yˆy y −ˆy (y −ˆy)21 8 8.4 -0.4 0.163 12 10.6 1.4 1.964 9 11.7 -1.3 1.695 14 12.8 1.2 1.448 16 16.1 -0.1 0.019 20 17.2 2.8 7.8411 17 19.4
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