Statistics for the Social Sciences

1. Statistics for the Social Sciences Psychology 340 Fall 2013 Thursday, October 17, 2013 Repeated Measures Analysis of Variance (ANOVA)

2. Homework Assignment Due 10/22 Chapter 12: 14, 15, 21, 22 (Use SPSS for #21 & 22. Print out your output, identify the relevant statistics and probabilities on the output, and write out responses to all parts of each question – if you submit electronically, include a word document or similar that identifies all of the relevant information from the output to fully answer each question) Chapter 13: 1, 4, 7, 8

3. Last Time • Brief review of one-way ANOVA • Assumptions in ANOVA • Post-hoc and planned comparisons • Effect sizes in ANOVA • ANOVA in SPSS

4. Today • More practice with SPSS • ANOVA in research articles • ANOVA structural model • Repeated Measures ANOVA

5. One-Way ANOVA in SPSS • Enter the data: similar to independent samples t-test, observations in one column, a second column for group assignment • Analyze: compare means, 1-way ANOVA • Your grouping variable is the “factor” and your continuous (outcome) variable goes in the “dependent list” box • Specify any comparisons or post hocs at this time too • Planned Comparisons (contrasts): are entered with 1, 0, & -1 • Post-hoc tests: make sure that you enter your α-level • Under “options,” you can request descriptive statistics (e.g., to see group means) • SPSS does not give effect size under this procedure (can get it by using analyzeGeneralLineraModelUnivariate), but it’s easy to calculate by hand

6. Demonstration with “Majors” data • Watch the demonstration learn how to use SPSS to find out whether there are significant differences among engineering, computer sciences, and “other sciences” in terms of average SAT scores. • Now you try it using the same data set, but test whether there is a significant difference between majors in college GPA. Use a post-hoc test to see which groups are different. • Save your output and submit it to me using the hand-in link on the class website.

7. ANOVA in Research Articles • F(3, 67) = 5.81, p < .01 • Means given in a table or in the text • Follow-up analyses • Planned comparisons • Using t tests

8. 1 factor ANOVA • Reporting your results • The observed difference • Kind of test • Computed F-ratio • Degrees of freedom for the test • The “p-value” of the test • Any post-hoc or planned comparison results • “The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < 0.05. Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(13) = 5.67, p < 0.05 & t(13) = 6.02, p <0.05). Groups B and C did not differ significantly from one another”

9. Group’s mean’s deviation from grand mean (M-GM) Score’s deviation from group mean (X-M) Score’s deviation from grand mean (X-GM) The structural model and ANOVA • The structural model is all about deviations Score (X) Group mean (M) Grand mean (GM)

10. More than two • Independent & One score per subject • 1 independent variable • The 1 factor between groups ANOVA: Statistical analysis follows design

11. One group • More than 2 scores per subject • The 1 factor within groups ANOVA: Statistical analysis follows design • Repeated measures

12. More than 2 groups • Matched groups • The 1 factor within groups ANOVA: Statistical analysis follows design • Repeated measures • One group • More than 2 scores per subject - OR - • Matched samples

13. Example • Suppose that you want to compare three brand name pain relievers. • Give each person a drug, wait 15 minutes, then ask them to keep their hand in a bucket of cold water as long as they can. The next day, repeat (with a different drug) • Dependent variable: time in ice water • Independent variable: 4 levels, within groups • Drug A • Drug B • Drug C • Placebo

14. MD MA MB MC Within-subjects ANOVA • n = 5 participants • Each participates in every condition (4 of these)

15. Within-subjects ANOVA • Hypothesis testing: a four step program • Step 1: State your hypotheses • Step 2: Set decision criteria • Step 3: Compute your test statistics • Compute your estimated variances (2 steps of partitioning used) • Compute your F-ratio • Step 4: Make a decision about your null hypothesis

16. MD MA MB MC Because we use the same people in each condition, we can figure out how much of the variability comes from the individuals and remove it from the analysis Step 4: Computing the F-ratio • Analyzing the sources of variance • Describe the total variance in the dependent measure • Why are these scores different? • Sources of variability • Between conditions • Within conditions • Individual differences • Left over variance (error)

17. Partitioning the variance Total variance Stage 1 Within cond. variance Between cond. variance

18. Partitioning the variance Total variance Stage 1 Within cond. variance Between cond. variance Between subjects variance Error variance Stage 2

19. Because we use the same people in each condition, none of this variability comes from having different people in different conditions Partitioning the variance Total variance Stage 1 Within cond. variance Between cond. variance • Treatment effect • Error or chance (without individual differences) • Individual differences • Other error Between subjects variance Error variance Stage 2 • Other error (without individual differences) • Individual differences

20. Observed variance F-ratio = Variance from chance Step 4: Computing the F-ratio • The F ratio • Ratio of the between-conditions variance estimate to the population error variance estimate

21. Partitioning the variance Total variance Stage 1 Within cond. variance Between cond. variance • Treatment effect • Error or chance (without individual differences) • Individual differences • Other error Between subjects variance Error variance Stage 2 • Other error (without individual differences) • Individual differences

22. Partitioning the variance Total variance Stage 1 Between cond. variance Within cond. variance

23. Partitioning the variance

24. Partitioning the variance Total variance Stage 1 Between cond. variance Within groups variance Between subjects variance Error variance Stage 2

25. What is ? Between subjects variance Partitioning the variance The average score for each person

26. What is ? Partitioning the variance The average score for each person Between subjects variance

27. Partitioning the variance Total variance Stage 1 Between groups variance Within groups variance Stage 2 Error variance Between subjects variance

28. Partitioning the variance Error variance

29. Partitioning the variance Total variance Stage 1 Between cond. variance Within cond. variance Stage 2 Error variance Between subjects variance

30. Recall: Partitioning the variance Now we return to variance. But, we call it Means Square (MS) Mean Squares (Variance) Between conditions variance Error variance

31. Partitioning the variance Total variance Stage 1 Between cond. variance Within cond. variance Stage 2 Error variance Between subjects variance

32. Within-subjects ANOVA • The F table • Need two df’s • dfbetween (numerator) • dferror (denominator) • Values in the table correspond to critical F’s • Reject the H0 if your computed value is greater than or equal to the critical F • Separate row in each cell for 0.05 & 0.01 Do we reject or fail to reject the H0? • From the table (assuming 0.05) with 3 and 12 degrees of freedom the critical F = 3.49. • So we reject H0 and conclude that not all groups are the same

33. Computational Formulas T = Group Total G = Grand Total P = Person Total n = number of participants k = number of conditions N = number of scores

34. Computational Formulas

35. From Before

36. Computational Formulas

37. From Before

38. Computational Formulas

39. From Before Between subjects variance

40. Computational Formulas

41. From Before Total variance Stage 1 Between cond. variance Within cond. variance Stage 2 Error variance Between subjects variance

42. Within-subjects ANOVA in SPSS • Setting up the file • Running the analysis • Looking at the output