1 Power Fifteen Analysis of Variance (ANOVA)

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

3 One-Way ANOVA w Apple Juice Concentrate Example, Data File xm 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 Advertising Strategies & Weekly Sales for 20 Weeks

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

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

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

8 F-Distribution Test of the Null Hypothesis of No Difference in Mean Sales with Advertising Strategy F 2, 60 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

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

Dependent Variable: SALESAJ Method: Least SquaresSample: 1 60 Included observations: 60 VariableCoefficientStd. Errort-StatisticProb. CONVENIENCE QUALITY PRICE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Regression Coefficients are the City Means; F statistic (?)

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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)

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

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)

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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 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 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 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 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 Advertising Strategies In Two Media: Weekly Sales

26 Mean Weekly Sales By Strategy and Medium

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price

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

30 Table of ANOVA for Two-Way

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

33 Mean Weekly Sales By Strategy and Medium

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

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

36 Table of Two-Way ANOVA for Apple Juice Sales

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

38 Two-Way ANOVA and Regression

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

SALESAPJCONVENIENCEQUALITYPRICE TELEVISIONNEWSPAPERS

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 = [( )/2]/( /54) = (1609.6/2)/ = 0.09

Table of Two-Way ANOVA for Apple Juice Sales

Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 VariableCoefficientStd. Errort-StatisticProb. CONVENIENCE QUALITY TELEVISION C CONVENIENCE*TELEVISION QUALITY*TELEVISION

R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 Variable CoefficientStd. Errort-StatisticProb. CONVENIENCE QUALITY TELEVISION C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

47 ANOVA By Difference w Regression with interaction terms, USS = 501,136.7 w Regression dropping interaction terms< USS = 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 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 = [( )/1]/( /54) = 13172/ = 1.42 w Salesaj = c(1) +e, SSR = w Test for strategy effect: F 2, 54 = [( )/2]/( /54) = ( /2)/(9280.3) = 5.32

Dependent Variable: SALESAPJ Method: Least Squares Sample: 1 60 Included observations: 60 VariableCoefficientStd. Errort-StatisticProb. CONVENIENCE QUALITY C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

50 Wald Test: Equation: Untitled Null Hypothesis:C(2)=C(3) F-statistic Probability Chi-square Probability