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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA). What is ANOVA?. 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):

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Analysis of Variance (ANOVA)

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  1. Analysis of Variance (ANOVA)

  2. What is ANOVA? • 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. • 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. How Do We State The Null and Alternative Hypotheses? • H0: The means for all groups are the same (equal). • Ha: The means are different for at least one pair of groups. • H0: 1 = 2 = ………. = k • Ha: 1  2  ……….  k

  4. How do you determine which means are significantly different? • 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. What multiple comparison tests are available in SPSS? Scheffe recommended Games-Howell recommended

  6. What assumptions need to be examined? • 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. • 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. • In general ANOVA is a fairly robust procedure.

  7. Application: One-way ANOVA

  8. Description of Customer Survey Variables GINO'S Samouel's Restaurant VS. Variable DescriptionVariable Type Restaurant Perceptions X1 Excellent Food Quality Metric X2 Attractive Interior Metric X3 Generous Portions Metric X4 Excellent Food Taste Metric X5 Good Value for the Money Metric X6 Friendly Employees Metric X7 Appears Clean & Neat Metric X8 Fun Place to Go Metric X9 Wide Variety of menu Items Metric X10 Reasonable Prices Metric X11 Courteous Employees Metric X12 Competent Employees Metric Selection Factor Rankings X13 Food Quality Nonmetric X14 Atmosphere Nonmetric X15 Prices Nonmetric X16 Employees Nonmetric Relationship Variables X17 Satisfaction Metric X18 Likely to Return in Future Metric X19 Recommend to Friend Metric X20 Frequency of Patronage Nonmetric X21 Length of Time a Customer Nonmetric Classification Variables X22 Gender Nonmetric X23 Age Nonmetric X24 Income Nonmetric X25 Competitor Nonmetric X26 Which AD Viewed (#1, 2 or 3) Nonmetric X27 AD Rating Metric X28 Respondents that Viewed Ads Nonmetric

  9. The Samouel’s Research Problem • Dependent variable is: X27 – AD Rating • Independent variable is X26 – Which AD Viewed (e.g., # 1, 2 or 3): • ‘1’ – AD #1 • ‘2’ – AD #2 • ‘3’ – AD #3 • Research question is: • Are there differences in the mean ratings of the ADS based on which AD was viewed?

  10. USING SPSS TO DO A ONE-WAY ANOVA 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 X28 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 X27 — AD Rating and the nonmetric independent variable is X26 — 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 X26 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. Initial Considerations – Descriptives & Levene’s Test of Equal Variances There is not a significant difference in the variances of the three groups.

  12. The Restaurant Problem: Tests of Between Subjects Effects There are significant differences between ratings for the ads, but we are not sure where the difference are based on this test.

  13. AD Evaluations: Post Hoc Tests There are significant differences between ratings for all three ads.

  14. The Restaurant Problem – Profile Plot • Mean Ratings of Ads: • Ad #1 = 39.79 • Ad #2 = 68.03 • Ad #3 = 51.50

  15. Two-way ANOVA

  16. What Is Two-way ANOVA? • 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 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).

  17. What are the hypotheses in a two-way ANOVA? • Three hypotheses are tested simultaneously: • The effect of independent variable #1 on the dependent variable (main effect). • The effect of independent variable #2 on the dependent variable (main effect). • The combined (joint) effect of independent variables #1 and #2 on the dependent variable (interaction effect).

  18. ANOVA Terms 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. Samouel’s Restaurant: The Problem And The Design • 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: • No differences in ad ratings based on which ad was viewed; • No differences in ad ratings based on gender; • No differences in ad ratings based on the combined effects of which ad viewed and gender. • The metric dependent variable for these hypotheses is X27 — AD Rating and the nonmetric independent variables are X26 — AD Viewed (# 1, 2 or 3) and X22 — Gender.

  20. Using SPSS To Execute Two-way ANOVA 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 X28 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 X27 — AD Rating by clicking on it and move it to the Dependent variable box. Next, highlight X26 — AD Viewed and X22 — Gender, and move them to the box labelled “Fixed Factors.” Now click on the Post Hoc box and highlight X26 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 X22 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 X26 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. SPSS Output For The Two-way ANOVA – Initial Data Sample sizes for each of the groups. Mean ratings of ads by which ad viewed and gender.

  22. SPSS Output For Two-way ANOVA • AD Rating main effect significant (X26). • Gender main effect not significant (X22). • Interaction effect significant (X26 * X22). 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 X22 raises a red flag.

  23. Post Hoc Tests For Two-way ANOVA All comparisons significantly different.

  24. Two-way ANOVA – Profile Plot AD Viewed 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. Two-way ANOVA – Profile Plot - Gender 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. AD Rating By “Which AD Viewed” and “Gender” There is a significant difference between AD Ratings by males and females for ads #1 and #3, but not for ad #2.

  27. AD Rating By “Viewed #1” and “Gender”

  28. AD Rating By “Viewed AD #2” and “Gender” 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.

  29. AD Rating By “Viewed AD #3” and “Gender”

  30. ANOVALearning Checkpoint 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?

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