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Coefficient of Determination Section 4.3 Alan Craig

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2 Objectives Compute and interpret the coefficient of determination.

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3 Coefficient of Determination The coefficient of determination, R 2, measures the percentage of the total variation in the response variable that is explained by the least-squares regression. R 2 is calculated by squaring the linear correlation coefficient, r.

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4 On the Calculator r and R 2 are part of the calculator output for a linear regression with DiagnosticsOn.

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5 Least-Squares Regression Recall that the least-squares regression line minimizes the sum of the squared errors (residuals) = residuals 2 Error or residual = actual - predicted

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6 Estimating y If I have no information about values of the predictor variable x, then my best guess for y is the mean of y: and the deviation is the actual value minus the mean.

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7 Actual Deviation Mean Total Deviation

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8 Estimating y However, if I have additional data on the values of x and corresponding values of y, I can often do better by calculating the regression of y on x. Part of the total deviation is now explained by the regression equation although some of the deviation is still unexplained (unless there is a perfect linear correlation).

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9 Actual Deviation Mean Predicted Unexplained Deviation Deviation explained by the regression

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10 Deviation Note that That is, y = mean of y + explained deviation + unexplained deviation

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11 Deviation Or That is, Total deviation = explained deviation + unexplained deviation

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12 Variation The total variation of y is The explained variation is The unexplained variation is

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13 Variation R2R2

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14 Interpreting R 2 Thus, R 2 is the percentage of variation in the response variable, y, that is explained by the predictor variable x.

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15 Interpreting R 2 Using our example from Sections 4.1 and 4.2 (problem 10, p. 172), R 2 =0.9835=98.35%, so the predictor variable, Carats, explains 98.35% of the variation in the response variable, Price.

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16 Questions ???????????????

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