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:   ≠    or   ≠    or   ≠  

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 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

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

11

12 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.

13 Advertising Strategies In Two Media: Weekly Sales

14 Mean Weekly Sales By Strategy and Medium

15

price

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

18 Table of ANOVA for Two-Way

19 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:

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

21 Mean Weekly Sales By Strategy and Medium

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

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

24 Table of Two-Way ANOVA for Apple Juice Sales

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

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

SALESAPJCONVENIENCEQUALITYPRICE TELEVISIONNEWSPAPERS

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)

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)

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

33 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]