ONE-WAY BETWEEN SUBJECTS ANOVA

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. Group 1 Group 2 Group 3 4 6 8 4 6 8 4 6 8 Example of Between Groups variance only:

11. Example of Within Groups variance only: Group 1 Group 2 Group 3 4 6 4 8 4 8 6 8 6

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 noise Low noise High noise 15 15 12 17 19 10 18 14 10 14 12 12 x=16 x=15 x=11 grand mean = 14

19. ANOVA Summary Table Source SS df MS F p Between Within Total

20. STEP 1: SS Total = S(x-xG)2 grand mean x x-x (x-x)2 15 1 1 17 3 9 18 4 16 14 0 0 15 1 1 19 5 25 14 0 0 12 -2 4 12 -2 4 10 -4 16 10 -4 16 12 -2 4 S = SS Total = 96

21. x x-x (x-x)2 16 2 4 16 2 4 16 2 4 16 2 4 15 1 1 15 1 1 15 1 1 15 1 1 11 -3 9 11 -3 9 11 -3 9 11 -3 9 S = SS Between = 56 STEP 2: SS Between = S(xg-xG)2 group mean

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

23. ANOVA Summary Table Source SS df MS F p Between 56 Within 40 Total 96

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

25. ANOVA Summary Table Source SS df MS F p Between 56 2 Within 40 9 Total 96 11

26. STEP 5: Calculate Mean Squares

27. ANOVA Summary Table Source SS df MS F p Between 56 2 28.00 Within 40 9 4.44 Total 96 11

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 Source SS df MS F p Between 56 2 28.00 6.31 <.05 Within 40 9 4.44 Total 96 11

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. Calculating the Kruskal-Wallis ANOVA 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)

35. No noise Low noise High noise 17 19 9 18 16 8 14 12 12 16 11 8 13 10 7

36. No noise Low noise High noise 17 13 19 15 9 4 18 14 16 11.5 8 2.5 14 10 12 7.5 12 7.5 16 11.5 11 6 8 2.5 13 9 10 5 7 1 STEP 1: Rank scores. STEP 2: Sum ranks for each group. SR1 = 57.5 SR2 = 45 SR3 = 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