1. ONE-WAY BETWEEN SUBJECTS ANOVA Also called One-Way Randomized ANOVA Purpose: Determine whether there is a difference among two or more groups – used mostly with three or more groups – does not show which groups differ (unless there are only two)

2. Design and Assumptions Design: – One way means one independent variable – Between subjects means different people in each group. Assumptions: same as for independent samples t-test

3. Why not t-tests? Multiple t-tests inflate the experimentwise alpha level. experimentwise alpha level is the total probability of Type I error for all tests of significance in the study. ANOVA controls the experimentwise alpha level.

4. If I am doing six t-tests, each with a.05 alpha level, what is the experimentwise alpha?

5. So, the probability of making one or more errors is 1 -.7351 =.2649.

6. Concept of ANOVA Analysis of Variance Variance is a measure of variability Two step process: – divides the variance into parts – compares the parts

7. About Variance Numerator is the Sum of Squares Denominator is the Degrees of Freedom

8. Mean Square Variance is also called Mean Square Formula for variance in ANOVA terms:

9. Part I: Dividing the Variance Total Variance is divided into two parts: – Between Groups Variance - only differences between groups. – Within Groups Variance - only differences within groups. Between Groups + Within Groups = Total

10. Example of Between Groups variance only: Group 1Group 2Group 3 468

11. Example of Within Groups variance only: Group 1Group 2Group 3 464 848 686

12. What Influences Between Groups Variance? effect of the i.v. (systematic) individual differences (non-systematic) measurement error (non-systematic)

13. What Influences Within Groups Variance? individual differences (non-systematic) measurement error (non-systematic)

14. Part II: Comparing the Variance

15. About the F-ratio Larger with a bigger effect of the IV Expected to be 1.0 if Ho is true Never significant below 1.0 Can’t be negative

16. Sampling Distribution of F 1.0

17. Computation of One-Way BS ANOVA EXAMPLE: Twelve participants were randomly assigned to one of three noise conditions and given a spelling test. Determine whether noise had a significant effect on spelling performance. (See data on next page)

18. No noiseLow noiseHigh noise 151512 171910 181410 141212 x=16x=15x=11 grand mean = 14

19. ANOVA Summary Table SourceSSdfMSFp Between Within Total

20. STEP 1: SS Total =  (x-x G ) 2 grand mean xx-x(x-x) 2 1511 1739 18416 1400 1511 19525 1400 12-24 10-416 12-24  = SS Total = 96

21. STEP 2: SS Between =  (x g -x G ) 2 group mean xx-x(x-x) 2 1624 1511 11-39 11-39  = SS Between = 56

22. STEP 3: SS Within = SS Total - SS Between SS Within = 96 - 56 = 40

23. ANOVA Summary Table SourceSSdfMSFp Between56 Within40 Total96

24. STEP 4: Calculate degrees of freedom. df Total = N-1 df Total = 12-1 = 11 df Between = k-1k=#groups df Between = 3-1 = 2 df Within = N-k df Within = 12-3 = 9

25. ANOVA Summary Table SourceSSdfMSFp Between562 Within409 Total9611

26. STEP 5: Calculate Mean Squares

27. ANOVA Summary Table SourceSSdfMSFp Between56228.00 Within4094.44 Total9611

28. STEP 6: Calculate F-ratio.

29. STEP 7: Look up critical value of F. df numerator = df Between df denominator = df Within F-crit (2,9) = 4.26

30. APA Format Sentence A One-Way Between Subjects ANOVA showed a significant effect of noise, F (2,9) = 6.31, p <.05.

31. ANOVA Summary Table SourceSSdfMSF p Between56228.006.31 <.05 Within4094.44 Total9611

32. Computing Effect Size Eta-squared is the proportion of variance in the DV that can be explained by the IV.

33. KRUSKAL-WALLIS ANOVA Non-parametric replacement for One-Way BS ANOVA Assumptions: – independent observations – at least ordinal level data – minimum 5 scores per group

34. EXAMPLE: Fifteen participants were randomly assigned to one of three noise conditions and given a spelling test. Determine whether noise had a significant effect on spelling performance. (See data on next page) Calculating the Kruskal-Wallis ANOVA

35. No noiseLow noiseHigh noise 17199 18168 141212 16118 13107

36. STEP 1: Rank scores. No noiseLow noiseHigh noise 1713191594 18141611.582.5 1410127.5127.5 1611.511682.5 13910571 STEP 2: Sum ranks for each group.  R 1 = 57.5  R 2 = 45  R 3 = 17.5

37. STEP 3: Compute H.

38. STEP 4: Compare to critical value from  2 table. df = 2, critical value = 5.99 A Kruskal-Wallis ANOVA showed a significant difference among the three noise conditions, H(2) =8.38, p <.05.

39. ANOVA for Within Subjects Designs When the IV is within subjects (or matched groups), a Repeated Measures ANOVA should be used The logic of the ANOVA is the same Calculation differs to take advantage of the design

40. ANOVA for Within Subjects Designs The Friedman ANOVA is the non- parametric replacement for One-Way Repeated Measures ANOVA