The Coefficient of Determination Robb T Koether Introduction The Coefficient of Determination The Regression Identity Lecture 49 Section 13 9 Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Robb T Koether Hampden Sydney College Tue Apr 22 2008 Outline The Coefficient of Determination Robb T Koether 1 Introduction 2 The Regression Identity 3 Sums of Squares on the TI 83 4 Explaining Variation 5 The Coefficient of Determination Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Introduction The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination One goal of regression is to produce a model that will predict values of the response variable given values of the explanatory variable We have already done that Another goal is to explain the variability in the response variable as a consequence of the variability in the explanatory variable That is where we now turn out attention Residual Sum of Squares The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Recall that the line of best fit was that line with the smallest sum of squared residuals This is also called the residual sum of squares or sum of squared errors X SSE y y 2 Other Sums of Squares The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination There are two other sums of squares associated with y The regression sum of squares X y y 2 SSR The total sum of squares SST SSY X y y 2 Example SST SSR and SSE The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Plot the following data x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 Example SST SSR and SSE The Coefficient of Determination The regression line Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination 24 21 18 15 12 9 6 2 4 6 8 10 12 14 16 Example SST SSR and SSE The Coefficient of Determination The deviations of y from y Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination 24 21 18 15 12 9 6 2 4 6 8 10 12 14 16 Example SST SSR and SSE The Coefficient of Determination The deviations of y from y Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination 24 21 18 15 12 9 6 2 4 6 8 10 12 14 16 Example SST SSR and SSE The Coefficient of Determination The deviations of y from y Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination 24 21 18 15 12 9 6 2 4 6 8 10 12 14 16 The Regression Identity The Coefficient of Determination Robb T Koether It turns out that SST SSR SSE Introduction The Regression Identity Sums of Squares on the TI 83 This is the regression identity It also turns out that r2 Explaining Variation The Coefficient of Determination SSR SST Consequently r2 SSR SST SSE SSE 1 SST SST SST Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SST x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y y y y 2 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SST x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y y 7 3 6 1 1 5 2 9 y y 2 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SST x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y y 7 3 6 1 1 5 2 9 y y 2 49 9 36 1 1 25 4 81 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SST x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y y 7 3 6 1 1 5 2 9 y y 2 49 9 36 1 1 25 4 81 206 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SSR x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y 8 4 10 6 11 7 12 8 16 1 17 2 19 4 23 8 y y y y 2 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SSR x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y 8 4 10 6 11 7 12 8 16 1 17 2 19 4 23 8 y y 6 6 4 4 3 3 2 2 1 1 2 2 4 4 8 8 y y 2 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SSR x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y 8 4 10 6 11 7 12 8 16 1 17 2 19 4 23 8 y y 6 6 4 4 3 3 2 2 1 1 2 2 4 4 8 8 y y 2 43 56 19 36 10 89 4 84 1 21 4 84 19 36 77 44 Example The Coefficient of Determination Robb T Koether Introduction The Regression Identity Sums of Squares on the TI 83 Explaining Variation The Coefficient of Determination Compute SSR x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 y 8 4 10 6 11 7 12 8 16 1 17 2 19 4 23 8 y y 6 6 4 4 3 3 2 2 1 1 2 2 4 4 8 8 y y 2 43 56 19 36 10 89 4 84 1 21 4 84 19 36 77 44 181 50 Example The Coefficient of Determination Robb T Koether Introduction …
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