1 Power Fifteen Analysis of Variance ANOVA Analysis of Variance One Way ANOVA Tabular Regression Two Way ANOVA Tabular Regression 2 One Way ANOVA Apple Juice Concentrate Example Data File xm 15 01 New product Try 3 different advertising strategies one in each of three cities City 1 convenience of use City 2 quality of product City 3 price Record Weekly Sales 3 Advertising Strategies Weekly Sales for 20 Weeks Convenience Quality Price 529 804 672 658 630 531 793 774 443 614 624 532 Mean 577 5 Mean 653 0 Mean 608 65 4 Is There a Significant Difference in Average Sales Figure 1 Mean Apple Juice Sales By Advertising Strategy 660 640 620 600 580 560 540 520 convenience quality price A dvertising Strategy Null Hypothesis H0 2 Alternative Hypothesis 1 2 1 3 2 3 5 Table Way ANOVA of Apple Juice Sales By Advertising Strategy Source of Variation Sum of Squares k Explained between ESS n j x j 1 Degrees of Mean Freedom Square x 2 k ESS k xij x j 2 n k USS n k xij x 2 n j treatments k Unexplained withi n USS j n j i treatments k Total TSS j n j i Fk 1 n k ESS k 1 USS n k 6 Apple Juice Concentrate ANOVA Source of Variation Explained Between Treatments Unexplained Within Treatments Total Sum of Degrees of Squares Freedom ESS k 1 2 57 512 23 Mean Square ESS k 1 28 756 12 USS 506 984 n k 57 USS n k 8894 45 TSS 564 496 n 1 59 F2 57 28 756 12 8894 45 3 23 7 F Distribution Test of the Null Hypothesis of No Difference in Mean Sales with Advertising Strategy Figure 2 F D istribution Density For 2 DOF 57 DOF 1 0 DENSITY 0 8 0 6 F2 60 critical 5 3 15 0 4 0 2 0 0 0 2 4 6 8 10 F Variable 8 One Way ANOVA and Regression 9 Regression Set Up y 1 is column of 20 sales observations For city 1 1 is a column of 20 ones 0 is a column of 20 Zeros Regression of a quantitative variable on three dummies y 1 1 y 2 0 y 3 0 0 0 1 0 0 1 Y C 1 Dummy city 1 C 2 Dummy city 2 C 3 Dummy city 3 e 10 11 One Way ANOVA and Regression Table 5 One Way ANOVA Estimated Using Regression Dependent Variable SALESAJ Method Least Squares Sample 1 60 Included observations 60 Variable Coefficient Std Error t Statistic Prob CONVENIENCE QUALITY PRICE 577 5500 653 0000 608 6500 21 08844 21 08844 21 08844 27 38704 30 96483 28 86178 0 0000 0 0000 0 0000 R squared Adjusted Rsquared S E of regression Sum squared resid Log likelihood Durbin Watson stat 0 101882 0 070370 Mean dependent var S D dependent var 613 0667 97 81474 94 31038 506983 5 Akaike info criterion Schwarz criterion 11 97977 12 08448 F statistic Prob F statistic 3 233041 0 046773 356 3930 1 525930 Regression Coefficients are the City Means F statistic Dependent Variable SALESAJ Method Least Squares Included observations 60 Variable Coefficient CONVENIENCE 577 5500 QUALITY 653 0000 PRICE 608 6500 Std Error 21 08844 21 08844 21 08844 Sample 1 60 t Statistic 27 38704 30 96483 28 86178 R squared 0 101882 Mean dependent var Adjusted R squared 0 070370 S D dependent var S E of regression 94 31038 Akaike info criterion Sum squared resid 506983 5 Schwarz criterion Log likelihood 356 3930 Durbin Watson stat Regression Coefficients are the City Means F statistic Prob 0 0000 0 0000 0 0000 613 0667 97 81474 11 97977 12 08448 1 525930 14 Table 6 Test of the Null Hypothesis All Treatment Means Are Equal Wald Test Equation Untitled Null Hypothesis C 1 C 3 C 2 C 3 F statistic Chi square 3 233041 6 466083 Probability Probability 0 046773 0 039437 15 Anova and Regression One Way Interpretation Salesaj c 1 convenience c 2 quality c 3 price e E salesaj convenience 1 quality 0 price 0 c 1 mean for city 1 c 1 mean for city 1 convenience c 2 mean for city 2 quality c 3 mean for city 3 price Test the null hypothesis that the means are equal using a Wald test c 1 c 2 c 3 16 One Way ANOVA and Regression Table 5 One Way ANOVA Estimated Using Regression Dependent Variable SALESAJ Method Least Squares Sample 1 60 Included observations 60 Variable Coefficient Std Error t Statistic Prob CONVENIENCE QUALITY PRICE 577 5500 653 0000 608 6500 21 08844 21 08844 21 08844 27 38704 30 96483 28 86178 0 0000 0 0000 0 0000 R squared Adjusted Rsquared S E of regression Sum squared resid Log likelihood Durbin Watson stat 0 101882 0 070370 Mean dependent var S D dependent var 613 0667 97 81474 94 31038 506983 5 Akaike info criterion Schwarz criterion 11 97977 12 08448 F statistic Prob F statistic 3 233041 0 046773 356 3930 1 525930 Regression Coefficients are the City Means F statistic Anova and Regression One Way Alternative Specification Drop Price Salesaj c 1 c 2 convenience c 3 quality e E Salesaj convenience 0 quality 0 c 1 mean for city 3 price the omitted one E Salesaj convenience 1 quality 0 c 1 c 2 mean for city 1 convenience so mean for city 1 c 1 c 2 so mean for city 1 mean for city 3 c 2 and so c 2 mean for city 1 mean for city 3 18 19 Anova and Regression One Way Alternative Specification Drop Price Salesaj c 1 c 2 convenience c 3 quality e E Salesaj convenience 0 quality 0 c 1 mean for city 3 price the omitted one E Salesaj convenience 1 quality 0 c 1 c 2 mean for city 1 convenience so mean for city 1 c 1 c 2 so mean for city 1 mean for city 3 c 2 and so c 2 mean for city 1 mean for city 3 20 Anova and Regression One Way Alternative Specification Salesaj c 1 c 2 convenience c 3 quality e Test that the mean for city 1 mean for city 3 Using the t statistic for c 2 H 0 x1 x3 i e H 0 x1 x3 0 21 Anova and Regression One Way Alternative Specification Drop Quality Salesaj c 1 c 2 convenience c 3 price e E Salesaj convenience 0 price 0 c 1 mean for city 2 quality the omitted one E Salesaj convenience 1 price 0 c 1 c 2 mean for city 1 convenience so mean for city 1 c 1 c 2 and so mean for city 1 mean for city 2 c 2 so c 2 mean for city 1 mean for city 2 22 Anova and Regression One Way Alternative Specification Drop Quality Salesaj c 1 c 2 convenience c 3 price e Test that the mean for city 1 mean for …
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