Introduction to Analysis of Variance

1. Introduction to Analysis of Variance CJ 526 Statistical Analysis in Criminal Justice

2. Introduction • Analysis of Variance (ANOVA) is an inferential statistical technique

3. Developer • Developed by Sir Ronald Fisher in the 1920’s • Agricultural geneticist

4. Relationship Between ANOVA and Independent t-Test • Actually, Independent t-Test is really a special case of ANOVA

5. Similarities With Other Parametric Inferential Procedures • Like all parametric inferential procedures

6. Purpose of ANOVA • Determine whether differences between the means of the groups are due to chance (sampling error)

7. ANOVA and Research Designs • Can be used with both experimental and ex post facto research designs

8. Experimental Research Designs • Researcher manipulates levels of Independent Variable to determine its effect on a Dependent Variable

9. Example of an Experimental Research Design Using ANOVA • Dr. Sophie studies the effect of different dosages of a new drug on impulsivity among children at-risk of becoming delinquent

10. Example of an Experimental Research Design Using ANOVA -- continued • Independent Variable • Different dosages of new drug • 0 mg (placebo) • 100 mg • 200 mg

11. Ex Post Facto Research Designs • Researcher investigates effects of pre-existing levels of an Independent Variable on a Dependent Variable

12. Example of an Ex Post Facto Research Design Using ANOVA • Dr. Horace wants to determine whether political party affiliation has an effect on attitudes toward the death penalty

13. Example of an Ex Post Facto Research Design Using ANOVA -- continued • Independent Variable • Political Party Affiliation • Democrat • Independent • Republican

14. Null Hypothesis in ANOVA • No differences among the population means

15. Alternative Hypothesis in ANOVA • At least one population mean is different from one other population mean

16. Example of Pairwise Comparisons • Dr. Mildred wants to determine whether birth order has an effect on number of self-reported delinquent acts • Independent Variable • Birth Order • First Born (or only child) • Middle Born (if three or more children) • Last Born

17. Example of Pairwise Comparisons -- continued • Dependent Variable • Number of self-reported delinquent acts • Possible pairwise comparisons • FB ≠ MB • FB ≠ LB • MB ≠ LB • It is possible for this particular analysis that: • Any one of the pairwise comparisons could be statistically significant • Any two of the pairwise comparisons could be statistically significant • All three of the pairwise comparisons could be statistically significant

18. Types of ANOVA • One-Way ANOVA • One Independent Variable • Groups are independent

19. Types of ANOVA -- continued • Repeated-Measures ANOVA • Groups are dependent • Measure the dependent variable at more than two points in time

20. ANOVA and Multiple t-Tests • Testwise alpha

21. The Logic of ANOVA • Total variability of the DV can be analyzed by dividing it into its component parts

22. Components of Total Variability • Between-Groups • Measure of the overall differences between treatment conditions (groups, samples)

23. Within-Groups Variability • Measure of the amount of variability inside of each treatment condition (group, sample) • There will always be variability within a group

24. Between-Group (BG) Variability • Treatment Effect (TE)

25. Within-Group (WG) Variability • Individual Differences (ID) • Example: for race, there is more within group variability than between group variability (more genetic variation among white, or Asians, etc, than between the races

26. The F-Ratio • Obtained test statistic for ANOVA Is

27. The F-Ratio -- continued

28. The F-Ratio -- continued

29. The F-Ratio -- continued • If H0 is true, TE = 0, F = 1

30. The F-Ratio -- continued

31. The F-Ratio -- continued • If H0 is false, TE > 0, F > 1

32. The F-Ratio -- continued

33. The F-Ratio -- continued • F = Systematic Variability • divided by

34. Systematic Variability • Due to treatment

35. Unsystematic Variability • Uncontrolled or unexplained

36. ANOVA Vocabulary • Factor

37. Factor • Independent variable

38. Level • Different values of a factor

39. Notation for ANOVA • k: number of levels of a factor • Also the number of different samples

40. Degrees of Freedom • Between Groups • k - 1

41. F-Distribution • Always positive

42. Example • A police psychologist wants to determine whether caffeine has an effect on learning and memory • Randomly assigns 120 police officers to one of five groups:

43. Experimental Groups • 0 mg (placebo) • 50 mg • 100 mg • 150 mg • 200 mg

44. Example -- continued • Records how many “nonsense” words each police officer recalls after studying a 20-word list for 2 minutes • CVC, dif, zup

45. ANOVA Summary Table

46. Example of ANOVA • Number of Samples: 5 • Nature of Samples: •  Known:

47. Example of ANOVA --continued • Independent Variable: caffeine • Dependent Variable and its Level of Measurement: number of syllables remembered—interval/ratio

48. Example of ANOVA -- continued • Target Population: • Appropriate Inferential Statistical Technique: one way analysis of variance • Null Hypothesis: no differences in memory between the groups

49. Example of ANOVA -- continued • Alternative Hypothesis: Caffeine does have an effect on memory and there will be differences among the groups • Decision Rule: • If the p-value of the obtained test statistic is less than .05, reject the null hypothesis

50. Example of ANOVA -- continued • Obtained Test Statistic: F • Decision: accept or reject the null hypothesis