1. 1 Power Fifteen Analysis of Variance (ANOVA)

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

3. 3 One-Way ANOVA w Apple Juice Concentrate Example, Data File xm 15-01 w New product w Try 3 different advertising strategies, one in each of three cities City 1: convenience of use City 2: quality of product City 3: price w Record Weekly Sales

4. 4 Advertising Strategies & Weekly Sales for 20 Weeks

5. 5 Is There a Significant Difference in Average Sales? Null Hypothesis, H 0 :       Alternative Hypothesis:

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

7. 7 Apple Juice Concentrate ANOVA F 2, 57 = 28,756.12/8894.45 = 3.23

8. 8 F-Distribution Test of the Null Hypothesis of No Difference in Mean Sales with Advertising Strategy F 2, 60 (critical) @ 5% =3.15

9. 9 One-Way ANOVA and Regression

10. 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

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

13. Dependent Variable: SALESAJ Method: Least SquaresSample: 1 60 Included observations: 60 VariableCoefficientStd. Errort-StatisticProb. CONVENIENCE 577.550021.0884427.387040.0000 QUALITY653.000021.0884430.964830.0000 PRICE608.650021.0884428.861780.0000 R-squared0.101882 Mean dependent var613.0667 Adjusted R-squared0.070370 S.D. dependent var97.81474 S.E. of regression 94.31038 Akaike info criterion11.97977 Sum squared resid506983.5 Schwarz criterion12.08448 Log likelihood-356.3930 Durbin-Watson stat1.525930 Regression Coefficients are the City Means; F statistic (?)

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15. 15

16. 16 Anova and Regression: One-Way Interpretation w Salesaj = c(1)*convenience+c(2)*quality+c(3)*price+ e w 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)

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

18. 18 Anova and Regression: One-Way Alternative Specification: Drop Price w Salesaj = c(1) + c(2)*convenience+c(3)*quality+e w E[Salesaj/(convenience=0, quality=0)] = c(1) = mean for city(3) (price, the omitted one) w 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)

19. 19

20. 20 Anova and Regression: One-Way Alternative Specification: Drop Price w Salesaj = c(1) + c(2)*convenience+c(3)*quality+e w E[Salesaj/(convenience=0, quality=0)] = c(1) = mean for city(3) (price, the omitted one) w 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)

21. 21 Anova and Regression: One-Way Alternative Specification w 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)

22. 22 Anova and Regression: One-Way Alternative Specification, Drop Quality w Salesaj = c(1) + c(2)*convenience+c(3)*price+e w E[Salesaj/(convenience=0, price=0)] = c(1) = mean for city(2) (quality, the omitted one) w 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)

23. 23 Anova and Regression: One-Way Alternative Specification, Drop Quality w 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)

24. 24 Two-Way ANOVA w Apple Juice Concentrate w Two Factors 3 advertising strategies 2 advertising media: TV & Newspapers w 6 cities City 1: convenience on TV City 2: convenience in Newspapers City 3: quality on TV Etc.

25. 25 Advertising Strategies In Two Media: Weekly Sales

26. 26 Mean Weekly Sales By Strategy and Medium

27. 27

28. price

29. 29 Is There Any Difference In Mean Sales Among the Six Cities?

30. 30 Table of ANOVA for Two-Way

31. 31 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:

32. 32 Formulas For Sums of Squares (Cont.) Where the mean for treatment i, strategy, is:

33. 33 Mean Weekly Sales By Strategy and Medium

34. 34 Formulas For Sums of Squares (Cont.) Where the mean for treatment j, medium, is:

35. 35 Formulas For Sums of Squares (Cont.) Where is the mean for each city

36. 36 Table of Two-Way ANOVA for Apple Juice Sales

37. 37 F-Distribution Tests Test for Interaction: Test for Advertising Medium: Test for Advertising Strategy:

38. 38 Two-Way ANOVA and Regression

39. 39 Two-Way ANOVA and Regression w 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. w You need to drop one of the strategy variables and drop one of the media varibles and include the constant.

40. 40 = Regression Set-Up Convenience dummy Quality dummy TV dummy constant

41. SALESAPJCONVENIENCEQUALITYPRICE TELEVISIONNEWSPAPERS 49110010 71210010 55810010 44710010 47910010 62410010 54610010 44410010 58210010 67210010 46410001 55910001 75910001 55710001 52810001 67010001 53410001 65710001 55710001 47410001 67701010 62701010

42. 42 ANOVA and Regression: Two-Way Series of Regressions; Compare to Table 11, Lecture 15 w 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 w Salesaj = c(1) + c(2)*convenience + c(3)* quality + c(4)*television + e, SSR=502,746.3 w Test for interaction effect: F 2, 54 = [(502746.3-501136.7)/2]/(501136.7/54) = (1609.6/2)/9280.3 = 0.09

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

44. Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 VariableCoefficientStd. Errort-StatisticProb. CONVENIENCE -48.5000043.08204-1.1257590.2652 QUALITY 62.7000043.082041.4553630.1514 TELEVISION -24.4000043.08204-0.5663610.5735 C 624.400030.4636020.496590.0000 CONVENIENCE*TELEVISION 4.00000060.92720 0.065652 0.9479 QUALITY*TELEVISION -19.70000 60.92720 -0.3233370.7477

45. R-squared0.184821 Mean dependent var614.3167 Adjusted R-squared0.109342 S.D. dependent var102.0765 S.E. of regression96.33436 Akaike info criterion12.06817 Sum squared resid501136.7 Schwarz criterion12.27760 Log likelihood-356.0450 F-statistic2.448631 Durbin-Watson stat2.452725 Prob(F-statistic)0.045165

46. Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 Variable CoefficientStd. Errort-StatisticProb. CONVENIENCE -46.5000029.96267 -1.5519310.1263 QUALITY 52.8500029.96267 1.7638620.0832 TELEVISION-29.6333324.46441 -1.2112830.2309 C627.016724.4644125.629740.0000 R-squared 0.182203 Mean dependent var614.31 Adjusted R-squared0.138393 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-statistic4.158888 Durbin-Watson stat2.456222 Prob(F-statistic) 0.009921

47. 47 ANOVA By Difference w Regression with interaction terms, USS = 501,136.7 w Regression dropping interaction terms< USS = 502746.3 w Difference is 1,609.6 and is the sum of squares explained by interaction terms w F-test of the interaction terms: F 2, 54 = [1609.6/2]/[501,136.7/54]

48. 48 ANOVA and Regression: Two-Way Series of Regressions w Salesaj = c(1) + c(2)*convenience + c(3)* quality + e, SSR=515,918.3 w Test for media effect: F 1, 54 = [(515918.3- 502746.3)/1]/(501136.7/54) = 13172/9280.3 = 1.42 w Salesaj = c(1) +e, SSR = 614757 w Test for strategy effect: F 2, 54 = [(614757- 515918.3)/2]/(501136.7/54) = (98838.7/2)/(9280.3) = 5.32

49. Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 VariableCoefficientStd. Errort-StatisticProb. CONVENIENCE -46.5000030.08521-1.5456100.1277 QUALITY52.8500030.085211.7566770.0843 C612.200021.2734628.777650.0000 R-squared0.160777 Mean dependent var614.31 Adjusted R-squared0.131330 S.D. dependent var102.07 S.E. of regression95.13779 Akaike info criterion 11.99724 Sum squared resid515918.3 Schwarz criterion12.101 Log likelihood-356.9171 F-statistic5.459975 Durbin-Watson stat2.379774 Prob(F-statistic) 0.006769

50. 50 Wald Test: Equation: Untitled Null Hypothesis:C(2)=C(3) F-statistic138.2678Probability0.000000 Chi-square138.2678Probability0.000000