2. A statistical method for testing whether two or more dependent variable means are equal (i.e., the probability that any differences in means across several groups are due solely to sampling error). Variables in ANOVA (Analysis of Variance): Dependent variable is metric. Dependent variable is metric. Independent variable(s) is nominal with two or more levels – also called treatment, manipulation, or factor. Independent variable(s) is nominal with two or more levels – also called treatment, manipulation, or factor. One-way ANOVA: only one independent variable with two or more levels. Two-way ANOVA: two independent variables each with two or more levels. With ANOVA, a single metric dependent variable is tested as the outcome of a treatment or manipulation. With MANOVA (Multiple Analysis of Variance), two or more metric dependent variables are tested as the outcome of a treatment(s).

3. H 0 : The means for all groups are the same (equal). H a : The means are different for at least one pair of groups. H 0 :  1 =  2 = ………. =  k H a :  1   2  ……….   k

4. The F-statistic assesses whether you can conclude that statistical differences are present somewhere between the group means. But to identify where the differences are you must use follow-up tests called “multiple comparison tests”. Many multiple comparison tests are available in SPSS.

5. Scheffe recommended Games-Howell recommended

6. Samples are independent. Samples are independent. Dependent variable is normally distributed for each of the samples – with larger sample sizes ( > 20/group) not a serious problem should this be violated somewhat. Dependent variable is normally distributed for each of the samples – with larger sample sizes ( > 20/group) not a serious problem should this be violated somewhat. Whether the sample sizes for the groups are very different (ratio of 1.5 or higher may be a problem). Whether the sample sizes for the groups are very different (ratio of 1.5 or higher may be a problem). The variances for the different populations from which the samples are drawn are equal – possibly a problem if they are not equal or at least comparable. The variances for the different populations from which the samples are drawn are equal – possibly a problem if they are not equal or at least comparable. In general ANOVA is a fairly robust procedure. In general ANOVA is a fairly robust procedure.

7. Application: One-way ANOVA

8. Variable Description Variable Type Restaurant Perceptions X 1 Excellent Food Quality Metric X 2 Attractive Interior Metric X 3 Generous Portions Metric X 4 Excellent Food Taste Metric X 5 Good Value for the Money Metric X 6 Friendly Employees Metric X 7 Appears Clean & Neat Metric X 8 Fun Place to Go Metric X 9 Wide Variety of menu Items Metric X 10 Reasonable Prices Metric X 11 Courteous Employees Metric X 12 Competent Employees Metric Selection Factor Rankings X 13 Food Quality Nonmetric X 14 Atmosphere Nonmetric X 15 Prices Nonmetric X 16 Employees Nonmetric Relationship Variables X 17 Satisfaction Metric X 18 Likely to Return in Future Metric X 19 Recommend to Friend Metric X 20 Frequency of Patronage Nonmetric X 21 Length of Time a Customer Nonmetric Classification Variables X 22 Gender Nonmetric X 23 Age Nonmetric X 24 Income Nonmetric X 25 Competitor Nonmetric X 26 Which AD Viewed (#1, 2 or 3) Nonmetric X 27 AD Rating Metric X 28 Respondents that Viewed Ads Nonmetric Description of Customer Survey Variables VS.

9.  Dependent variable is: X 27 – AD Rating  Independent variable is X 26 – Which AD Viewed (e.g., # 1, 2 or 3): ‘1’ – AD #1 ‘1’ – AD #1 ‘2’ – AD #2 ‘2’ – AD #2 ‘3’ – AD #3 ‘3’ – AD #3  Research question is: Are there differences in the mean ratings of the ADS based on which AD was viewed? Are there differences in the mean ratings of the ADS based on which AD was viewed?

10. Phil Samouel asked the researcher to test the effectiveness of three different ads. If the mean ratings of the ads are statistically different he would like to select the highest rated ad and run an advertising campaign for his restaurant. Not all of the 200 customer respondents agreed to look at and evaluate the ads. To identify the respondents who viewed the ads we go to the Data pull down menu and click on “Select Cases”, then on “If condition satisfied,” then on If. Next highlight variable X 28 and click on the Arrow box to move it to the box. Now click on the equal sign (=) below and then one (1) to select only the respondents who viewed the ADS (a zero = did not agree to view ads). Finally click on Continue and then OK. The metric dependent variable for these hypotheses is X 27 — AD Rating and the nonmetric independent variable is X 26 — AD Viewed (# 1, 2 or 3). The click-through sequence to run the one-way ANOVA is: ANALYZE  GENERAL LINEAR MODEL  UNIVARIATE. Click on X 27 — AD Rating to highlight it and then on the arrow box to move it into the Dependent Variable box. Click on X 26 — Which AD Viewed to highlight it and then on the arrow box to move it to the box labelled “Fixed Factors.” Click on the Post Hoc box and highlight X 26 in the Factor(s) box and then click on the Arrow box to move this variable to the box for Post Hoc Tests. Now look to the lower left side of the screen and click on Scheffe test and Games-Howell and then Continue. Now go to the Options box and click on Descriptive statistics and Homogeneity Tests (Levene test of equal variances) and then Continue, and then click on the Plots box and highlight X26 and move it to the Horizontal Axis box and then under Plots below click Add. Finally, click on Continue and then OK to execute the program.

11. There is not a significant difference in the variances of the three groups. Initial Considerations – Descriptives & Levene’s Test of Equal Variances

12. There are significant differences between ratings for the ads, but we are not sure where the difference are based on this test.

13. There are significant differences between ratings for all three ads.

14. Mean Ratings of Ads: 1.Ad #1= 39.79 2.Ad #2= 68.03 3.Ad #3= 51.50

15. Two-way ANOVA

16.  Examines the effect (if any) of two or more nonmetric independent variables on a single metric dependent variable.  Total variation is examined for: Variation due to each of the independent variables (main effects). Variation due to each of the independent variables (main effects). Variation due to the interaction of the independent variables – that is their possible combined effect on the dependent variable beyond the separate influence of each (interaction effect). Variation due to the interaction of the independent variables – that is their possible combined effect on the dependent variable beyond the separate influence of each (interaction effect). Variation that remains unexplained (error). Variation that remains unexplained (error).

17.  Three hypotheses are tested simultaneously: 1. The effect of independent variable #1 on the dependent variable (main effect). 2. The effect of independent variable #2 on the dependent variable (main effect). 3. The combined (joint) effect of independent variables #1 and #2 on the dependent variable (interaction effect).

18. Main Effect = the impact any single experimental variable has on a response (dependent) variable. Interaction Effect = the combined impact of multiple independent variables on a response variable; i.e., is the difference in the mean ratings of the ads (response variable) the same when we compare males and females? Blocking Variable = a grouping variable the researcher doesn’t manipulate or control in any way, such as gender.

19. Phil Samouel asked the researcher to test three different ads for their effectiveness. If the ratings of the ads are statistically different he would like to use that information to attract more customers. He also would like to know how various demographic characteristics are related to ad ratings. In this case, we use gender, which is referred to as a blocking variable. The null hypotheses are: (1)No differences in ad ratings based on which ad was viewed; (2)No differences in ad ratings based on gender; (3)No differences in ad ratings based on the combined effects of which ad viewed and gender. The metric dependent variable for these hypotheses is X 27 — AD Rating and the nonmetric independent variables are X 26 — AD Viewed (# 1, 2 or 3) and X 22 — Gender.

20. Recall that not all of the 200 customer respondents agreed to look at and evaluate the ads. To identify the respondents who viewed the ads we go to the Data pull down menu and click on “Select Cases”, then “If condition satisfied,” then on If. Next highlight variable X 28 and click on the Arrow box to move it to the box. Now click on the equal sign (=) below and then one (1) to select only the respondents who viewed the ADS (a zero = did not agree to view ads). Finally click on Continue and then OK. The click through sequence is: ANALYZE  GENERAL LINEAR MODEL  UNIVARIATE. Highlight the dependent variable X 27 — AD Rating by clicking on it and move it to the Dependent variable box. Next, highlight X 26 — AD Viewed and X 22 — Gender, and move them to the box labelled “Fixed Factors.” Now click on the Post Hoc box and highlight X 26 in the Factor(s) box and then click on the Arrow box to move this variable to the box for Post Hoc Tests. We do not move X 22 because it has only two groups and not three. Look to the lower left side of the screen and click on Scheffe test and then Continue. Now go to the Options box and click on Descriptive statistics and then Continue, and then click on the Plots box and highlight X 26 and move it to the Horizontal Axis box and then click the Add button above the Plots box below. Finally, click on Continue and then OK to execute the program.

21. Mean ratings of ads by which ad viewed and gender. Sample sizes for each of the groups.

22. AD Rating main effect significant (X 26 ). Gender main effect not significant (X 22 ). Interaction effect significant (X 26 * X 22 ). If the interaction effect is not significant, the main effects of the treatments are independent and can be interpreted directly. If the interaction effect is significant, then the type of interaction must be determined. The significant interaction and nonsignificant main effect for X 22 raises a red flag.

23. All comparisons significantly different.

24. The three ads are rated differently, with ad #1 rated lowest at 39.79, #3 somewhat higher at 51.50, and #2 the highest at 68.03.

25. There is a difference in ratings by gender across all three ads, with female ratings overall slightly more favorable (55.55 vs. 54.56). But remember overall there was not a statistically significant difference.

26. There is a significant difference between AD Ratings by males and females for ads #1 and #3, but not for ad #2.

28. Note: do not be fooled by the slope of the line – the mean rating for males is 68.5 and for females is 67.8.

30. 1. When should ANOVA be used? 2. What is the difference between one-way and two-way ANOVA? 3. What are “multiple comparison tests” and why are they used? 4. What is the difference between a main effect and an interaction effect?