Chapter 9 and 10

1. Chapter 9 and 10 Simple ANOVA with Post Hoc and Repeated Measures ANOVA

2. Analysis of Variance (ANOVA) • Used to determine differences between 2+ groups. • General: ANOVA tests the likelihood that the samples came from the same population. Are the mean scores different. • Between Groups Variance – how groups vary about the Grand mean. • Within Groups Variance – how scores in each group vary about the group mean.

3. Multiple t-tests? • Multiple tests inflate alpha • Chance of error increases • Product of alpha and # of tests • Bonferroni adjustment - divide alpha by # of tests (decrease type I error but increase type II error) • t-test only uses data from 2 samples • ANOVA uses all groups data

4. ANOVA • If there are no differences between groups (come from same population), then the between and within groups variance = one. • Equation for F = between groups variance/within groups variance • If HO is true, and there is no sampling error, then F=1.

5. Single Factor ANOVA (one-way, simple) • One independent variable and one dependent variable • Example: compare reaction time between three groups, each given a different drug prior to the test (caffeine, beta blocker, placebo). • Subjects randomly assigned to the three groups. • IV = drug, DV = reaction time

6. Single Factor ANOVA Example • F = MSB/MSW, where B = between, W = within, and MS = mean square = SS/df. • MSB = SSB/dfB • MSW = SSW/dfW • SSTotal = SST = (dev)2 of all scores about the grand mean (51.9). • SST=SSB+SSW

7. SS total Difference from the grand mean

8. SS total Difference from the grand mean

9. SS total Difference from the grand mean

10. Single Factor ANOVA Example • SST = (dev)2 of all scores about the grand mean. • SST = 52.05c + 93.25b + 15.65p = 160.95 • SST=SSb+SSw

11. SSWithin = SSW = (dev)2 of scores about their respective group mean; df = n1 + n2 + n3 - 3

14. Single Factor ANOVA Example • SSW = (dev)2 of all scores about their respective group mean. • SSW = 10c + 38.8b + 15.2p = 64 • dfW = 5 + 5 + 5 – 3 = 12

15. SSBetween = SSB = (dev)2 of group means about the grand mean; dfB = # groups – 1. SSB =42.05c+54.45b+0.45p= 96.95 dfB = 3-1 = 2

16. F = MSB/MSW ratio (expected &observed) MSB = SSB/dfBMSW = SSW/dfW Summary Table • p<.05 (Fcrit = 3.89 with 2,12 degrees of freedom) • Reject HO (page 266 for .05/page 267 for .01)

17. Equation for t Equation for chi-square

18. ANOVA • If calculated F > Fcrit reject HO. • Fcrit depends on , dfNUM, and dfDENOM • Follow-up Procedures • “significant” F only tells us there are differences, not where specific differences lie (familywisep). Post hoc tests, e.g., (pairwise p) comparisons • C vs. BB • C vs. P • BB vs. P

19. Post Hoc Procedures •  inflation • If compare C vs. BB (type I risk = .05), C vs. P (.05) and BB vs. P (.05), then the probability of making one Type I error across all comparisons (.05 x 3 = 0.15) • Approaches to control  inflation. • Bonferroni adjustment: PC = FW/c, where c = # of comparisons, PC = “per comparison”, and FW = “familywise”. (.05/3 = .016) • Other post hoc procedures: Tukey, Scheffe, Newman-Keuhls, Duncan, etc.

20. Relationship Between t and F • F = t2 • F and t are based on the same mathematical model and t is just a special case of ANOVA. • It is ok to use F test when comparing 2 means. • When only two values – t-test and ANOVA are identical.

21. Assumptions of ANOVA • Homogeneity of Variance – violations result in F values that are too big and Type I error risk is > . • The assumption is that the within groups variances are equal. • Violations are serious if: • Sample sizes are unequal. • Violation is mainly due to one deviant group. • Rarely checked. • Solution: evaluate F at lower .

22. Repeated Measures ANOVA • “within-subjects” design - subjects are measured on DV more than once as in training studies or learning studies • Advantages • increased power - due to decreased variance. • can use smaller sample sizes • allows for study over time

23. Repeated Measures ANOVA • Disadvantages • carryover effects - early treatments affect later ones. • practice effects - subjects’ experience with test can influence their score on the DV. • fatigue • sensitization - subjects’ awareness of treatment is heightened due to repeated exposure to test.

24. Repeated Measures ANOVA • Sum the rows and columns • Square each row • Compute SS between columns (treatment variability) • Compute SS between rows (subject variability) • Compute SS total (C+R+residual) • Compute SS error (total-C&R) • Calculate DF (-1) for each (C,R) & E (RxC) • Calculate MS for each C,R&E (SS/DF) • F=MS of column/MS error • MS of rows is subject variability and is used in ICC

25. Repeated Measures ANOVA • Sphericity – equal variance and equal covariance • violations inflate F • common corrections - Huynh-Feldt, Greenhouse-Geisser.

26. Repeated Measures ANOVA • aka “within-subjects” design • Reaction Time Study - could be modified into RM design by having one group of subjects perform under all conditions • Main effects – the effect of 1 IV on 1 DV (whole group collapsed) • Interactions – the effect of multiple IV’s on 1 DV (groups reacted differently)

27. Lab Exercise • One way ANOVA (single factor) • Post hoc • Mean plots • Repeated measures ANOVA • Main effect • Interactions • Post hoc • Mean plots