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Power Fifteen

Power Fifteen. Analysis of Variance (ANOVA). Analysis of Variance. One-Way ANOVA Tabular Regression Two-Way ANOVA Tabular Regression. 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

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Power Fifteen

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  1. Power Fifteen Analysis of Variance (ANOVA)

  2. Analysis of Variance • One-Way ANOVA • Tabular • Regression • Two-Way ANOVA • Tabular • Regression

  3. 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

  4. Advertising Strategies & Weekly Sales for 20 Weeks

  5. Is There a Significant Difference in Average Sales? Null Hypothesis, H0 : m1 = m2 = m3 Alternative Hypothesis:

  6. Fk-1, n-k = [ESS/(k-1)]/[USS/(n-k)]

  7. Apple Juice Concentrate ANOVA F2, 57 = 28,756.12/8894.45 = 3.23

  8. F-Distribution Test of the Null Hypothesis of No Difference in Mean Sales with Advertising Strategy F2, 60 (critical) @ 5% =3.15

  9. One-Way ANOVA and Regression

  10. 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 = C(1)*Dummy(city 1) + C(2)*Dummy(city 2) + C(3)*Dummy(city 3) + e

  11. One-Way ANOVA and Regression Regression Coefficients are the City Means; F statistic

  12. Dependent Variable: SALESAJ Method: Least Squares Sample: 1 60 Included observations: 60 Variable Coefficient Std. Error t-Statistic Prob. CONVENIENCE 577.5500 21.08844 27.38704 0.0000 QUALITY 653.0000 21.08844 30.96483 0.0000 PRICE 608.6500 21.08844 28.86178 0.0000 R-squared 0.101882 Mean dependent var 613.0667 Adjusted R-squared 0.070370 S.D. dependent var 97.81474 S.E. of regression 94.31038 Akaike info criterion 11.97977 Sum squared resid 506983.5 Schwarz criterion 12.08448 Log likelihood -356.3930 Durbin-Watson stat 1.525930 Regression Coefficients are the City Means; F statistic (?)

  13. Anova and Regression: One-WayInterpretation • 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)

  14. One-Way ANOVA and Regression Regression Coefficients are the City Means; F statistic

  15. Anova and Regression: One-WayAlternative 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)

  16. Anova and Regression: One-WayAlternative 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)

  17. Anova and Regression: One-WayAlternative 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)

  18. Anova and Regression: One-WayAlternative 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)

  19. Anova and Regression: One-WayAlternative Specification, Drop Quality • Salesaj = c(1) + c(2)*convenience+c(3)*price+e • Test that the mean for city(1) = mean for city(2) • Using the t-statistic for c(2)

  20. Two-Way ANOVA • Apple Juice Concentrate • Two Factors • 3 advertising strategies • 2 advertising media: TV & Newspapers • 6 cities • City 1: convenience on TV • City 2: convenience in Newspapers • City 3: quality on TV • Etc.

  21. Advertising Strategies In Two Media: Weekly Sales

  22. Mean Weekly Sales By Strategy and Medium

  23. price

  24. Is There Any Difference In Mean Sales Among the Six Cities?

  25. Table of ANOVA for Two-Way

  26. Formulas For Sums of Squares a is the # of treatments for strategies =3 b is the # of treatments for media =2 r is the # of replicates or observations =10 The Grand Mean:

  27. Formulas For Sums of Squares (Cont.) Where the mean for treatment i, strategy, is:

  28. Mean Weekly Sales By Strategy and Medium

  29. Formulas For Sums of Squares (Cont.) Where the mean for treatment j, medium, is:

  30. Formulas For Sums of Squares (Cont.) Where is the mean for each city

  31. Table of Two-Way ANOVA for Apple Juice Sales

  32. F-Distribution Tests Test for Interaction: Test for Advertising Medium: Test for Advertising Strategy:

  33. Two-Way ANOVA and Regression

  34. Two-Way ANOVA and Regression • With Two-Way ANOVA you cannot include both 3 dummy variables for strategy and two dummy variables for media, without a constant, so a different specification is needed. • You need to drop one of the strategy variables and drop one of the media varibles and include the constant.

  35. Regression Set-Up Convenience dummy Quality dummy constant TV dummy =

  36. SALESAPJ CONVENIENCE QUALITY PRICE TELEVISION NEWSPAPERS 491 1 0 0 1 0 712 1 0 0 1 0 558 1 0 0 1 0 447 1 0 0 1 0 479 1 0 0 1 0 624 1 0 0 1 0 546 1 0 0 1 0 444 1 0 0 1 0 582 1 0 0 1 0 672 1 0 0 1 0 464 1 0 0 0 1 559 1 0 0 0 1 759 1 0 0 0 1 557 1 0 0 0 1 528 1 0 0 0 1 670 1 0 0 0 1 534 1 0 0 0 1 657 1 0 0 0 1 557 1 0 0 0 1 474 1 0 0 0 1 677 0 1 0 1 0 627 0 1 0 1 0

  37. ANOVA and Regression: Two-WaySeries of Regressions; Compare to Table 11, Lecture 15 • Salesaj = c(1) + c(2)*convenience + c(3)* quality + c(4)*television + c(5)*convenience*television + c(6)*quality*television + e, SSR=501,136.7 • Salesaj = c(1) + c(2)*convenience + c(3)* quality + c(4)*television + e, SSR=502,746.3 • Test for interaction effect: F2, 54 = [(502746.3-501136.7)/2]/(501136.7/54) = (1609.6/2)/9280.3 = 0.09

  38. Table of Two-Way ANOVA for Apple Juice Sales

  39. Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 Variable Coefficient Std. Error t-Statistic Prob. CONVENIENCE -48.50000 43.08204 -1.125759 0.2652 QUALITY 62.70000 43.08204 1.455363 0.1514 TELEVISION -24.40000 43.08204 -0.566361 0.5735 C 624.4000 30.46360 20.49659 0.0000 CONVENIENCE*TELEVISION 4.000000 60.92720 0.065652 0.9479 QUALITY*TELEVISION -19.70000 60.92720 -0.323337 0.7477

  40. R-squared 0.184821 Mean dependent var 614.3167 Adjusted R-squared 0.109342 S.D. dependent var 102.0765 S.E. of regression 96.33436 Akaike info criterion 12.06817 Sum squared resid 501136.7 Schwarz criterion 12.27760 Log likelihood -356.0450 F-statistic 2.448631 Durbin-Watson stat 2.452725 Prob(F-statistic) 0.045165

  41. Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 Variable Coefficient Std. Error t-Statistic Prob. CONVENIENCE -46.50000 29.96267 -1.551931 0.1263 QUALITY 52.85000 29.96267 1.763862 0.0832 TELEVISION -29.63333 24.46441 -1.211283 0.2309 C 627.0167 24.46441 25.62974 0.0000 R-squared 0.182203 Mean dependent var 614.31 Adjusted R-squared 0.138393 S.D. dependent var 102.0765 S.E. of regression 94.75027 Akaike info criterion 12.00471 Sum squared resid 502746.3 Schwarz criterion 12.14433 Log likelihood -356.1412 F-statistic 4.158888 Durbin-Watson stat 2.456222 Prob(F-statistic) 0.009921

  42. ANOVA By Difference • Regression with interaction terms, USS = 501,136.7 • Regression dropping interaction terms< USS = 502746.3 • Difference is 1,609.6 and is the sum of squares explained by interaction terms • F-test of the interaction terms: F2, 54 = [1609.6/2]/[501,136.7/54]

  43. ANOVA and Regression: Two-WaySeries of Regressions • Salesaj = c(1) + c(2)*convenience + c(3)* quality + e, SSR=515,918.3 • Test for media effect: F1, 54 = [(515918.3-502746.3)/1]/(501136.7/54) = 13172/9280.3 = 1.42 • Salesaj = c(1) +e, SSR = 614757 • Test for strategy effect: F2, 54 = [(614757-515918.3)/2]/(501136.7/54) = (98838.7/2)/(9280.3) = 5.32

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