Adjusted R 2 R 2 never decreases when a new x variable is added to

Adjusted r 2 r 2 never decreases when a new x

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What is the net effect of adding a new variable? x Did the new x variable add enough explanatory power to offset the loss of one degree of freedom?
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Chap 15-26 Shows the proportion of variation in y explained by all x variables adjusted for the number of x variables used (where n = sample size, k = number of independent variables) Penalize excessive use of unimportant independent variables Smaller than R 2 Useful in comparing among models Adjusted R 2 (continued) - - - - - = 1 k n 1 n ) R 1 ( 1 R 2 2 A
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Chap 15-27 Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 .44172 R 2 A = 44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables Multiple Coefficient of Determination (continued)
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Chap 15-28 Is the Model Significant? F-Test for Overall Significance of the Model Shows if there is a linear relationship between all of the x variables considered together and y Use F test statistic Hypotheses: H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship) H A : at least one β i ≠ 0 (at least one independent variable affects y)
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Chap 15-29 F-Test for Overall Significance Test statistic: where F has (numerator) D 1 = k and (denominator) D 2 = (n – k – 1) degrees of freedom (continued) MSE MSR 1 k n SSE k SSR F = - - =
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Chap 15-30 6.5386 2252.8 14730.0 MSE MSR F = = = Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 (continued) F-Test for Overall Significance With 2 and 12 degrees of freedom P-value for the F-Test
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Chap 15-31 H 0 : β 1 = β 2 = 0 H A : β 1 and β 2 not both zero α = .05 df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Reject H 0 at α = 0.05 The regression model does explain a significant portion of the variation in pie sales (There is evidence that at least one independent variable affects y ) 0 α = .05 F .05 = 3.885 Reject H 0 Do not reject H 0 6.5386 MSE MSR F = = Critical Value: F α = 3.885 F-Test for Overall Significance (continued) F
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Chap 15-32 Are Individual Variables Significant? Use t-tests of individual variable slopes Shows if there is a linear relationship between the variable x i and y Hypotheses: H 0 : β i = 0 (no linear relationship) H A : β i ≠ 0 (linear relationship does exist between x i and y)
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Chap 15-33 Are Individual Variables Significant? H 0 : β i = 0 (no linear relationship) H A : β i ≠ 0 (linear relationship does exist between x i and y ) Test Statistic: ( df = n – k – 1) i b i s 0 b t - = (continued)
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Chap 15-34 Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 t-value for Price is t = -2.306, with p-value .0398 t-value for Advertising is t = 2.855, with p-value .0145 (continued) Are Individual Variables Significant?
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Chap 15-35 d.f. = 15-2-1 = 12 α = .05 t α /2 = 2.1788 Inferences about the Slope: t Test Example H 0 : β i = 0 H A : β i 0 The test statistic for each variable falls in the rejection region (p-values < .05)
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