Analysis of Interaction Effects

1. Analysis of Interaction Effects James Jaccard New York University

2. Overview Will cover the basics of interaction analysis, highlighting multiple regression based strategies Will discuss advanced issues and complications in interaction analysis. This treatment will be somewhat superficial but hopefully informative

3. Conceptual Foundations of Interaction Analysis

4. Causal Theories Most (but not all) theories rely heavily on the concept of causality, i.e., we seek to identify the determinants of a behavior or mental state and/or the consequences of a behavior or environmental/mental state I am going to ground interaction analysis in a causal framework

5. Causal Theories Causal theories can be complicated, but at their core, there are five types of causal relationships in causal theories Direct Causal Relationships A direct causal relationship is when a variable, X, has a direct causal influence on another variable, Y:

6. Direct Causal Relationships

7. Direct Causal Relationships

8. Indirect Causal Relationships Indirect Causal Relationships An indirect causal relationship is when a variable, X, has a causal influence on another variable, Y, through an intermediary variable, M:

9. Indirect Causal Relationships

10. Spurious Relationship Spurious Relationship A spurious relationship is one where two variables that are not causally related share a common cause:

11. Bidirectional Causal Relationships Bidirectional Causal Relationships A bidirectional causal relationship is when a variable, X, has a causal influence on another variable, Y, and that effect, Y, has a “simultaneous” impact on X:

12. Bidirectional Causal Relationships

13. Moderated Causal Relationships Moderated Causal Relationships A moderated causal relationship is when the impact of a variable, X, on another variable, Y, differs depending on the value of a third variable, Z

14. Moderated Causal Relationships

15. Moderated Causal Relationships

16. Moderated Causal Relationships The variable that “moderates” the relationship is called a moderator variable.

17. Causal Theories We put all these ideas together to build complex theories of phenomena. Here is one example:

18. Interaction Analysis Interactions, when translated into causal analysis, focus on moderated relationships When I encounter an interaction effect, I think:

19. Interaction Analysis Key step in interaction analysis is to identify the focal independent variable and the moderator variable. Sometimes it is obvious – such as with the analysis of a treatment for depression on depression as moderated by gender

20. Interaction Analysis Sometimes it is not obvious – such as an analysis of the effects of gender and ethnicity on the amount of time an adolescent spends with his or her mother Statistically, it matters not which variables take on which role. Conceptually, it does.

24. The Statistical Analysis of Interactions

25. Some Common Practices Omnibus tests – I do not use these Hierarchical regression – I use sparingly Focus on unstandardized coefficients - we tend to stay away from standardized coefficients in interaction analysis because they can be misleading and they do not have “clean” mathematical properties

26. A “Trick” We Will Use: Linear Transformations Y = a + b1 X + e Satisfaction = a + b1 Grade + e Satisfaction = 12 + -.50 Grade + e

27. A “Trick” We Will Use: Linear Transformations Y = a + b1 X + e Satisfaction = a + b1 Grade + e Satisfaction = 12 + -.50 Grade + e Satisfaction = 9 + -.50 (Grade – 6) + e

28. A “Trick” We Will Use: Linear Transformations Y = a + b1 X + e Satisfaction = a + b1 Grade + e Satisfaction = 12 + -.50 Grade + e Satisfaction = 9 + -.50 (Grade – 6) + e “Mean centering” is when we subtract the mean

29. Interaction Analysis Will focus on four cases: Categorical IV and Categorical MV Continuous IV and Categorical MV Categorical IV and Continuous MV Continuous IV and Continuous MV Assume you know the basics of multiple regression and dummy variables in multiple regression

30. Categorical IV and Categorical MV

31. Categorical IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Gender (female = 1, male = 0) Z = Grade (6th = 1, 7th = 0)

32. Categorical IV and Categorical MV Three questions: Is there a gender difference for 6th graders? Is there a gender difference for 7th graders? Are these gender effects different?

33. Categorical IV and Categorical MV Gender effect for 6th grade: 8 – 7 = 1

34. Categorical IV and Categorical MV Gender effect for 6th grade: 8 – 7 = 1 Gender effect for 7th grade: 7 – 4 = 3

35. Categorical IV and Categorical MV Gender effect for 6th grade: 8 – 7 = 1 Gender effect for 7th grade: 7 – 4 = 3 Interaction contrast: (8-7) – (7– 4) = -2

36. Categorical IV and Categorical MV Y = a + b1 Gender + b2 Grade + b3 (Gender)(Grade) Y = 4.0 + 3.0 Gender + b2 Grade + -2.0 (Gender)(Grade)

37. Categorical IV and Categorical MV Y = a + b1 Gender + b2 Grade + b3 (Gender)(Grade) Y = 4.0 + 3.0 Gender + b2 Grade + -2.0 (Gender)(Grade) Flipped: Y = 7.0 + 1.0 Gender + b2 Grade + 2.0 (Gender)(Grade)

38. Categorical IV and Categorical MV Extend to groups > 2 (add 8th grade) Inclusion of covariates How to generate means and tables

39. Continuous IV and Categorical MV

40. Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0)

41. Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0) Three questions: For females: b = 0.33 For males: b = 0.20 Are the effects different: 0.33 – 0.20

42. Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0) For females: b = 0.33 For males: b = 0.20 Y = a + b1 Gender + 0.20 Time + 0.13 (Gender)(Time)

43. Continuous IV and Categorical MV Y = Relationship satisfaction (0 to 10) X = Time spent together (in hours) Z = Gender (female = 1, male = 0) For females: b = 0.33 For males: b = 0.20 Y = a + b1 Gender + 0.20 Time + 0.13 (Gender)(Time) Flipped: Y = a + b1 Gender + 0.33 Time + -0.13 (Gender)(Time)

44. Continuous IV and Categorical MV Do not estimate slopes separately; use flipped reference group strategy Extend to groups > 2 (use grade as example)

45. Categorical IV and Continuous MV

46. Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos

47. Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos Ad language: Half shown ad in Spanish (0) and half in English (1)

48. Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos Ad language: Half shown ad in Spanish (0) and half in English (1) Latino identity: 1 = not at all, 7 = strong identify

49. Categorical IV and Continuous MV Study conducted in Miami with bi-lingual Latinos Ad language: Half shown ad in Spanish (0) and half in English (1) Latino identity: 1 = not at all, 7 = strong identify Outcome = Attitude toward product (1 = unfavorable, 7 = unfavorable) Hypothesized moderated relationship

50. Common Analysis Form: Median Split Many researchers not sure how to analyze this, so use median split for continuous moderator variable and conduct ANOVA Why this is bad practice….