1. ANOVA II (Part 2) Class 16

2. Implications of Interaction 1. Main effects, alone, will not fully describe the results. 2. Each factor (or IV) must be interpreted in terms of the factor(s) with which it interacts. 3. Analysis of findings, when an interaction is present, will focus on individual treatment means rather than on overall factor (IV) means. 4. Interaction indicates moderation.

3. Interactions are Non-Additive Relationships Between Factors 1. Additive : When presence of one factor changes the expression of another factor consistently, across all levels. 2. Non-Additive : When the presence of one factor changes the expression of another factor differently, at different levels.

4. Ordinal and Disordinal Interactions XXXX Interaction YYY Interaction

5. Ordinal and Disordinal Interactions Ordinal Interaction Disordinal Interaction

7. Birth Order Main Effect:NO Gender Main Effect:NO Interaction:NO

8. Birth Order Main Effect:YES Gender Main Effect:NO Interaction:NO

9. Birth Order Main Effect:NO Gender Main Effect:YES Interaction:N0

10. Birth Order Main Effect:YES Gender Main Effect:YES Interaction:NO

11. Birth Order Main Effect:NO Gender Main Effect:NO Interaction:YES

12. Birth Order Main Effect:YES Gender Main Effect:NO Interaction:YES

13. Birth Order Main Effect:NO Gender Main Effect:YES Interaction:YES

14. Birth Order Main Effect:YES Gender Main Effect:YES Interaction:YES

15. Birth Order Means

17. Development of ANOVA Analytic Components 1. Individual scores  Condition (cell) sums 2. Condition sums  Condition means 3. Cond. means – ind. scores  Deviations  Deviations 2 4. Deviations 2  Sums of squares (SS between, SS within ) 5. Sum Sqrs / df  Mean squares (Between and Within) 6. MS Between  F Ratio MS Within F (X, Y df)  Probability of null ( p ) p  Accept null, or accept alt.

19. Birth Order and Ratings of “Activity” Deviation Scores AS Total Between Within (AS – T) = (A – T) +(AS –A) 1.33 (-2.97)= (-1.17) +(-1.80) 2.00(-2.30)=(-1.17) +(-1.13) 3.33(-0.97)=(-1.17) + ( 0.20) 4.33(0.03)=(-1.17) +( 1.20) 4.67(0.37)=(-1.17) + ( 1.54) Level a 1: Oldest Child Level a 2: Youngest Child 4.33 (0.03)= (1.17) +(-1.14) 5.00(0.07)= (1.17) +(-0.47) 5.33(1.03)= (1.17) + (-0.14) 5.67(1.37)= (1.17) +( 0.20) 7.00(2.70)= (1.17) + ( 1.53) Sum: (0) = (0) + (0) Mean scores: Oldest = 3.13Youngest = 5.47 Total = 4.30

20. Sum of Squared Deviations Total Sum of Squares = Sum of Squared between-group deviations + Sum of Squared within-group deviations SS Total = SS Between + SS Within

21. Computing Sums of Squares from Deviation Scores Birth Order and Activity Ratings (continued) SS = Sum of squared diffs, AKA “sum of squares” SS T =Sum of squares., total (all subjects) SS A = Sum of squares, between groups (treatment) SS s/A =Sum of squares, within groups (error) SS T = (2.97) 2 + (2.30) 2 + … + (1.37) 2 + (2.70) 2 = 25.88 SS A = (-1.17) 2 + (-1.17) 2 + … + (1.17) 2 + (1.17) 2 = 13.61 SS s/A = (-1.80) 2 + (-1.13) 2 + … + (0.20) 2 + (1.53) 2 = 12.27 Total (SS A + SS s/A ) = 25.88

22. Variance CodeCalculationMeaning Mean Square Between Groups MS A SS A df A Between groups variance Mean Square Within Groups MS S/A SS S/A df S/A Within groups variance Variance CodeCalculationDataResult Mean Square Between Groups MS A SS A df A 13.61 1 13.61 Mean Square Within Groups MS S/A SS S/A df S/A 12.27 8 1.53 Mean Squares Calculations

23. F Ratio Computation F =13.61 1.51 = 8.78 F = MS A = Between Group Variance MS S/A Within Group Variance

25. Conceptual Approach to Two Way ANOVA SS total = SS between groups + SS within groups Oneway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) XXXX YYYY

26. Conceptual Approach to Two Way ANOVA SS total = SS between groups + SS within groups Oneway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA SS between groups = Factor A and its levels (e.g., birth order; older/younger) Factor B and its levels (e.g., gender; male / female) The interaction between Factors A and B (e.g., how ratings of help seeker are jointly affected by birth order and gender)

27. Total Mean (4.32) Distributions of All Four Conditions

28. Total Mean (4.32) Gender Effect (collapsing across birth order)

29. Total Mean (4.32) Birth Order Effect (collapsing across gender)

31. Understanding Effects of Individual Treatment Groups How much can the variance of any particular treatment group be explained by: Factor A Factor B The interaction of Factors A and B Quantification of AB Effects AB - T = (A effect) + (B effect) + ????? AB - T = (A - T) + (B - T) + (AB - A - B + T) (AB - A - B + T) = ??? AKA "????" (AB - T) - ( ? - T) - ( ? - T) = Interaction Error Term in Two-Way ANOVA Error = (ABS - AB)

32. Understanding Effects of Individual Treatment Groups How much can the variance of any particular treatment group be explained by: Factor A Factor B The interaction of Factors A and B Quantification of AB Effects AB - T = (A effect) + (B effect) + (A x B Interaction) AB - T = (A - T) + (B - T) + (AB - A - B + T) (AB - A - B + T) = Interaction AKA "residual" (AB - T) - (A - T) - (B - T) = Interaction Error Term in Two-Way ANOVA Error = (ABS - AB)

33. Deviation of an Individual Score in Two Way ANOVA ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij) Ind. score Total Mean ??? Effect ??? (w’n Effect)

34. Deviation of an Individual Score in Two Way ANOVA ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij) Ind. score Total Mean Factor A Effect Factor B Effect Interaction AXB Effect Error (w’n Effect)

35. Degrees of Freedom in 2-Way ANOVA Between Groups Factor A df A = a - 12 – 1 = 1 Factor B df B = b – 12 – 1 = 1 Interaction Effect Factor A X Factor B df A X B = (a –1) (b – 1) (2-1) x (2-1) = 1 Error Effect Subject Variance df s/AB = ab(s – 1) df s/AB = n - ab 20 – (2 x 2) = 16 Total Effect Variance for All Factors df Total = abs – 1 df Total = n – 1 20 – 1 = 19

36. Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Factor A Factor Ba 1 a 2 Sum b 1 # X B 1 b 2 X X X SumA 1 X T Factor A = Birth Order Factor B = Gender# = Known quantity

37. Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Birth Order GenderYoungest Oldest Sum Males Sum Females 4.50 5.50 9.00 11.00 4.50 5.50 20.0010.00 NOTE: “Fictional sums” for demonstration.

38. Conceptualizing Degrees of Freedom (df) in Factorial ANOVA Factor A Factor Ba 1 a 2 a 3 Sum b 1 # # X B 1 b 2 # # X B 2 b 3 X X X X Sum A 1 A 2 X T A, B, T, # = free to vary; X = determined by #s Once # are established, Xs are known

39. Analysis of Variance Summary Table: Two Factor (Two Way) ANOVA ASS A a - 1SS A df A MS A MS S/AB BSS B b - 1SS b df b MS B MS S/AB A X BSS A X B (a - 1)(b - 1)SS AB df A X B MS A X B MS S/AB Within (S/AB) SS S/A ab (s- 1)SS S/AB df S/AB TotalSS T abs - 1 Source of Variation Sum of Squares df Mean Square F Ratio (SS)(MS)

41. Effect of Multi-Factorial Design on Significance Levels Mean Men Mean Women Sum of Sqrs. Betw'n dt Betw'n MS Betw'n Sum of Sqrs. Within df Within MS Within Fp One Way 4.783.583.421 22.4582.811.22.30 Two Way 4.783.583.421 5.096.854.03.09

42. ONEWAY ANOVA AND GENDER MAIN EFFECT SourceSum of Squares dfMean Square FSig. Gender3.421 1.22.34 Error 22.45 8 2.81 SourceSum of Squares dfMean Square FSig. Gender3.421 4.03.09 Birth Order16.021 18.87.005 Interaction3.751 4.42.08 Error 5.09 6 0.85 Total 9 TWOWAY ANOVA AND GENDER MAIN EFFECT Oneway F: 3.42 =1.22Twoway F : 3.42 = 4.42 2.81.85