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

2. Introduction 1. Analysis of Variance (ANOVA) is an inferential statistical technique

3. Developer 1.Developed by Sir Ronald Fisher in the 1920’s 1.Agricultural geneticist

4. Relationship Between ANOVA and Independent t-Test 1. Actually, Independent t-Test is really a special case of ANOVA

5. Similarities With Other Parametric Inferential Procedures 1. Like all parametric inferential procedures

6. Purpose of ANOVA 1. Determine whether differences between the means of the groups are due to chance (sampling error)

7. ANOVA and Research Designs 1. Can be used with both experimental and ex post facto research designs

8. Experimental Research Designs 1. Researcher manipulates levels of Independent Variable to determine its effect on a Dependent Variable

9. Example of an Experimental Research Design Using ANOVA 1. 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 1.Independent Variable 1.Different dosages of new drug 1.0 mg (placebo) 2.100 mg 3.200 mg

11. Ex Post Facto Research Designs 1. 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 1. 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 1.Independent Variable 1.Political Party Affiliation 1.Democrat 2.Independent 3.Republican

14. Null Hypothesis in ANOVA 1. No differences among the population means

15. Alternative Hypothesis in ANOVA 1. At least one population mean is different from one other population mean

16. Example of Pairwise Comparisons 1.Dr. Mildred wants to determine whether birth order has an effect on number of self-reported delinquent acts 2.Independent Variable 1.Birth Order 1.First Born (or only child) 2.Middle Born (if three or more children) 3.Last Born

17. Example of Pairwise Comparisons -- continued 3.Dependent Variable 1.Number of self-reported delinquent acts 4.Possible pairwise comparisons 1.FB ≠ MB 2.FB ≠ LB 3.MB ≠ LB 5.It is possible for this particular analysis that: 1.Any one of the pairwise comparisons could be statistically significant 2.Any two of the pairwise comparisons could be statistically significant 3.All three of the pairwise comparisons could be statistically significant

18. Types of ANOVA 1.One-Way ANOVA 1.One Independent Variable 2.Groups are independent

19. Types of ANOVA -- continued 3.Repeated-Measures ANOVA 1.Groups are dependent 2.Measure the dependent variable at more than two points in time

20. ANOVA and Multiple t- Tests 1. Testwise alpha

21. The Logic of ANOVA 1.Total variability of the DV can be analyzed by dividing it into its component parts

22. Components of Total Variability 1. Between-Groups 2. Measure of the overall differences between treatment conditions (groups, samples)

23. Within-Groups Variability 1. Measure of the amount of variability inside of each treatment condition (group, sample) 2. There will always be variability within a group

24. Between-Group (BG) Variability 1. Treatment Effect (TE)

25. Within-Group (WG) Variability 1. Individual Differences (ID) 2. 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 1. Obtained test statistic for ANOVA Is

27. The F-Ratio -- continued

29. 1. If H 0 is true, TE = 0, F = 1

30. The F-Ratio -- continued

31. 1. If H 0 is false, TE > 0, F > 1

32. The F-Ratio -- continued

33. 1.F = Systematic Variability 1.divided by

34. Systematic Variability 1. Due to treatment

35. Unsystematic Variability 1. Uncontrolled or unexplained

36. ANOVA Vocabulary 1. Factor

37. Factor 1.Independent variable

38. Level 1. Different values of a factor

39. Notation for ANOVA 1.k: number of levels of a factor 1.Also the number of different samples

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

41. F-Distribution 1. Always positive

42. Example 1. A police psychologist wants to determine whether caffeine has an effect on learning and memory 2. Randomly assigns 120 police officers to one of five groups:

43. Experimental Groups 1. 0 mg (placebo) 2. 50 mg 3. 100 mg 4. 150 mg 5. 200 mg

44. Example -- continued 3. Records how many “nonsense” words each police officer recalls after studying a 20-word list for 2 minutes 4. CVC, dif, zup

45. ANOVA Summary Table

46. Example of ANOVA 1.Number of Samples: 5 2.Nature of Samples: 1. 3.  Known:

47. Example of ANOVA -- continued 4.Independent Variable: caffeine 5.Dependent Variable and its Level of Measurement: number of syllables remembered—interval/ratio

48. Example of ANOVA -- continued 6. Target Population: 7. Appropriate Inferential Statistical Technique: one way analysis of variance 8. Null Hypothesis: no differences in memory between the groups

49. Example of ANOVA -- continued 9.Alternative Hypothesis: Caffeine does have an effect on memory and there will be differences among the groups 10.Decision Rule: 1.If the p-value of the obtained test statistic is less than.05, reject the null hypothesis

50. Example of ANOVA -- continued 11.Obtained Test Statistic: F 12.Decision: accept or reject the null hypothesis

51. Results 1. The results of the One-way ANOVA involving caffeine as the independent variable and number of nonsense words recalled as the dependent variable were statistically significant, F (4, 115) = 5.14, p <.01. The means and standard deviations for the five groups are contained in Table 1.

52. Discussion 1. It appears that the ingesting small to moderate amounts of caffeine results in better retention of nonsense syllables, but that ingesting moderate to large amounts of caffeine interferes with the ability to retain nonsense syllables

53. Assumptions of ANOVA 1. Observations are independent

54. SPSS Procedure Oneway Analyze, Compare Means, One-Way ANOVA Move DV into Depdent List Move IV into Factor Options Descriptives Homogeniety of Variance

55. Sample Printout: ANOVA

56. Sample Printout: Post Hoc Tests

57. SPSS Procedure One- Way Output Descriptives Levels of IV N Mean Standard Deviation Standard Error of the Mean 95% Confidence Interval Lower Bound Upper Bound

58. SPSS Procedure One- Way Output -- continued Test of Homogeneity of Variance ANOVA Summary Table Sum of Squares df Mean Square F Sig