2. TWO-WAY BETWEEN SUBJECTS ANOVA Also called: Two-Way Randomized ANOVA Also called: Two-Way Randomized ANOVA Purpose: Measure main effects and interaction of two independent variables Purpose: Measure main effects and interaction of two independent variables Design: factorial Design: factorial Assumptions: same as one-way BS ANOVA Assumptions: same as one-way BS ANOVA

3. Dividing the Variance Total = A + B + AxB + Within Groups Total = A + B + AxB + Within Groups A: differences between levels of A A: differences between levels of A B: differences between levels of B B: differences between levels of B AxB: other between group differences AxB: other between group differences Within Groups: differences within groups Within Groups: differences within groups

4. A Variance affected by: A Variance affected by: – effect of Factor A (systematic) – individual differences (non-systematic) – measurement error (non-systematic) B Variance affected by: B Variance affected by: – effect of Factor B (systematic) – individual differences (non-systematic) – measurement error (non-systematic)

5. AxB Variance affected by: AxB Variance affected by: – AxB interaction (systematic) – individual differences (non-systematic) – measurement error (non-systematic) Within Groups variance affected by: Within Groups variance affected by: – individual differences (non-systematic) – measurement error (non-systematic)

6. Comparing the Variance

7. ANOVA Summary Table SourceSSdfMSFp Factor A Factor B AxBWithinTotal

8. EXAMPLE: An oral or written spelling test was given in one of three noise levels. Determine whether there were significant effects of test type, noise level, and the interaction of test type with noise level. (See data on next page) Computation of Two-Way BS ANOVA

9. Noise Level No Low High oral 151512 Test171910 written181410 141212

10. Noise Level No Low High oral161711 14.67 Test written161311 13.33 16 15 11 Overall mean = 14 Means

11. SourceSSdfMSFp Test Type Noise Level Test x Noise Within Total ANOVA Summary Table

12. STEP 1: SS Between =  (x c -x) 2 condition mean x c x c -x(x c -x) 2 1624 1739 11-39 1624 13-11 11-39 11-39 SS Between = 72

13. STEP 2: SS A =  (x a -x) 2 (A is Test Type) mean for level of A x a x a -x(x a -x) 2 14.67.67.45 13.33 -.67.45 13.33 -.67.45 SS Test Type = 5.40

14. STEP 3: SS B =  (x b -x) 2 (B is Noise Level) mean for level of B x b x b -x(x b -x) 2 16 24 15 11 11 -39 11 -39 SS Noise Level = 56

15. STEP 4: SS AxB = SS Between - SS A - SS B SS Test Type x Noise Level = 72 - 5.40 - 56 = 10.60

16. STEP 5: SS Within =  (x- x c ) 2 xx-x c (x-x c ) 2 15-11 1711 1824 14-24 15-24 1924 1411 12-11 1211 10-11 1211 SS Within = 24

17. SourceSS df MS F p Test Type 5.40 Noise Level56.00 Test x Noise10.60 Within 24.00 Total96.00 ANOVA Summary Table

18. STEP 6: Calculate degrees of freedom. df A = a-1a= # levels of A df Test Type = 2-1 = 1 df B = b-1b= # levels of B df Noise Level = 3-1 = 2 df AxB = (a-1)(b-1) df Test x Noise = (1)(2) = 2 df Within = (a)(b)(n-1)n = # per group df Within = (2)(3)(1) = 6

19. Source SS df MS F p Test Type 5.40 1 Noise Level56.00 2 Test x Noise10.60 2 Within 24.00 6 Total96.00 11 ANOVA Summary Table

20. STEP 7: Calculate Mean Squares. MS Test Type = 5.40/1 = 5.40 MS Noise Level = 56/2 = 28.00 MS Test x Noise = 10.60/2 = 5.30 MS Within = 24/6 = 4.00

21. Source SS df MS F p Test Type 5.40 1 5.40 Noise Level56.00 2 28.00 Test x Noise10.60 2 5.30 Within 24.00 6 4.00 Total96.00 11 ANOVA Summary Table

22. STEP 8: Calculate F-ratios. F(Test Type) = 5.40 /4.00 = 1.35 F(Noise Level) = 28/4.00 = 7.00 F(Test x Noise) = 5.30/4.00 = 1.32

23. STEP 9: Look up critical values of F. Test Type F-crit (1,6) = 5.99 Noise Level F-crit (2,6) = 5.14 Test x Noise F-crit (2,6) = 5.14

24. STEP 10: Compare F to F-crit. If F is equal to or greater than F-crit, reject the Null Hypothesis. Test Type1.35 < 5.99 Not sig. Noise Level7.00 > 5.14 Sig. Test x Noise1.32 < 5.14 Not sig.

25. Source SS df MS F p Test Type 5.40 1 5.40 1.35 >.05 Noise Level56.00 2 28.00 7.00 <.05 Test x Noise10.60 2 5.30 1.32 >.05 Within 24.00 6 4.00 Total96.00 11 ANOVA Summary Table

26. APA Format Sentence A Two-Way Between Subjects ANOVA showed a significant main effect of Noise Level, F (2,6) = 7.00, p.05, and a nonsignificant interaction, F (2,6) = 1.32, p >.05. A Two-Way Between Subjects ANOVA showed a significant main effect of Noise Level, F (2,6) = 7.00, p.05, and a nonsignificant interaction, F (2,6) = 1.32, p >.05.

27. Computing Effect Size Compute  2 for each effect Compute  2 for each effect