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- 1. Group Difference Methods By Rama Krishna Kompella
- 2. The basic ANOVA situationTwo variables: 1 Categorical (IV), 1 Continuous (DV)Main Question: Do the (means of) the quantitative variablesdepend on which group (given by categorical variable) theindividual is in?If categorical variable has only 2 values: • 2-sample t-testANOVA allows for 3 or more groups
- 3. ANOVA - Analysis of Variance• Extends independent-samples t test• Compares the means of groups of independent observations – Don’t be fooled by the name. ANOVA does not compare variances.• Can compare more than two groups
- 4. ANOVA – Null and Alternative HypothesesSay the sample contains K independent groups• ANOVA tests the null hypothesis H0: μ1 = μ2 = … = μK – That is, “the group means are all equal”• The alternative hypothesis is H1: μi ≠ μj for some i, j – or, “the group means are not all equal”
- 5. Assumptions• Homogeneity of variance σ21 = σ22 = ... = σ2k – Moderate departures are not problematic, unless sample sizes are very unbalanced• Normality – Scores with in each group are normally distributed around their group mean – Moderate departures are not problematic• Independence of observations – Observations are independent of one another – Violations are very serious -- do not violate• If assumptions violated, may need alternative statistics
- 6. The Logic of ANOVA t = difference between sample means difference expected by chance (error) F= variance (differences) between sample means variance (difference) expected by chance (error) Concerned with variance: variance = differences between scores
- 7. The Logic of ANOVATwo sources of variance:Between group variance: Differences between group meansWithin group variance: Differences among people within the same group
- 8. The Logic of ANOVA
- 9. The Logic of ANOVA• If H0 True: – F= 0 + Chance ≈ 1 Chance• If H0 False: – F= Treatment Effect + Chance > 1 Chance
- 10. The F statistic• F is a statistic that represents ratio of two variance estimates• Denominator of F is called “error term” • When no treatment effect, F ≈ 1If treatment effect, observed F will be > 1• How large does F have to be to conclude there is a treatment effect (to reject H0)?• Compare observed F to critical values based on sampling distribution of F
- 11. Computing ANOVA(1) Compute SS (sums of squares) (2) Compute df (3) Compute MS (mean squares) (4) Compute F
- 12. Computing ANOVA
- 13. Computing ANOVA
- 14. Computing ANOVA
- 15. Computing ANOVA
- 16. Example• Does presence of offer during festival season affect sales?IV = Number of offers presentDV = Sales (in units)• Three conditions: No offer, Only one offer on a product, Multiple offers on a product• Is there a significant difference among these means? MO SO NO 10 6 1 13 8 3 5 10 4 9 4 5 8 12 2 X2 = 9 X 1= 8 X 0= 3
- 17. Computing ANOVA MO SO NO 10 6 1 13 8 3 5 10 4 9 4 5 8 12 2n 5 5 5 N = 15Xj 9 8 3 X .. = 6.67
- 18. Computing ANOVA
- 19. Computing ANOVA
- 20. Computing ANOVA
- 21. Computing ANOVACritical Value:• We need two df to find our critical F value from Table (Note E.3 α =.05; E.4 α =.01)• “Numerator” df: dfG “Denominator” df: dfE• df = 2,12 and α = .05 Fcritical= 3.89 Decision: Reject H0 because observed F (7.38) exceeds critical value (3.89)Interpret findings: • At least two of the means are significantly different from each other.• “The amount of sales generated is influenced by the number of offers present on the product, F(2,12) = 7.38, p ≤ .05.”
- 22. Types of ANOVA• One-way ANOVA, is used to test for differences among two or more independent groups.• Factorial ANOVA, is used in the study of the interaction effects among treatments.• Repeated measures ANOVA, is used when the same subject is used for each treatment.• Multivariate analysis of variance (MANOVA), is used when there is more than one response variable
- 23. Questions?

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