Presentation is loading. Please wait.

Presentation is loading. Please wait.

Analysis of Interaction Effects James Jaccard New York University.

Similar presentations


Presentation on theme: "Analysis of Interaction Effects James Jaccard New York University."— Presentation transcript:

1 Analysis of Interaction Effects James Jaccard New York University

2 Will cover the basics of interaction analysis, highlighting multiple regression based strategies Overview 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

8 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 A spurious relationship is one where two variables that are not causally related share a common cause:

11 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 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

16 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 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 Interaction Analysis

20 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 Interaction Analysis Statistically, it matters not which variables take on which role. Conceptually, it does.

21

22

23

24 The Statistical Analysis of Interactions

25 Omnibus tests – I do not use these Some Common Practices 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 Y = a + b 1 X + e A “Trick” We Will Use: Linear Transformations Satisfaction = a + b 1 Grade + e Satisfaction = Grade + e

27 Y = a + b 1 X + e A “Trick” We Will Use: Linear Transformations Satisfaction = a + b 1 Grade + e Satisfaction = Grade + e Satisfaction = (Grade – 6) + e

28 Y = a + b 1 X + e A “Trick” We Will Use: Linear Transformations Satisfaction = a + b 1 Grade + e Satisfaction = Grade + e Satisfaction = (Grade – 6) + e “Mean centering” is when we subtract the mean

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

30 Categorical IV and Categorical MV

31 Y = Relationship satisfaction (0 to 10) X = Gender (female = 1, male = 0) Z = Grade (6 th = 1, 7 th = 0) 6th7th Female Male7.04.0

32 Categorical IV and Categorical MV 6th7th Female Male Three questions: Is there a gender difference for 6 th graders? Is there a gender difference for 7 th graders? Are these gender effects different?

33 Categorical IV and Categorical MV 6th7th Female Male Gender effect for 6 th grade: 8 – 7 = 1

34 Categorical IV and Categorical MV 6th7th Female Male Gender effect for 6 th grade: 8 – 7 = 1 Gender effect for 7 th grade: 7 – 4 = 3

35 Categorical IV and Categorical MV 6th7th Female Male Gender effect for 6 th grade: 8 – 7 = 1 Gender effect for 7 th grade: 7 – 4 = 3 Interaction contrast: (8-7) – (7– 4) = -2

36 Y = a + b 1 Gender + b 2 Grade + b 3 (Gender)(Grade) Categorical IV and Categorical MV 6th7th Female Male Y = Gender + b 2 Grade (Gender)(Grade)

37 Y = a + b 1 Gender + b 2 Grade + b 3 (Gender)(Grade) Categorical IV and Categorical MV 6th7th Female Male Y = Gender + b 2 Grade (Gender)(Grade) Flipped: Y = Gender + b 2 Grade (Gender)(Grade)

38 Extend to groups > 2 (add 8 th grade) Categorical IV and Categorical MV 6th7th Female Male Inclusion of covariates How to generate means and tables

39 Continuous IV and Categorical MV

40 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) For females: b = 0.33 For males: b = 0.20 Three questions: 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 + b 1 Gender Time (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 + b 1 Gender Time (Gender)(Time) Flipped: Y = a + b 1 Gender Time (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 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….

51 Categorical IV and Continuous MV Identity Mean English – Mean Spanish

52 Categorical IV and Continuous MV Identity Mean English – Mean Spanish Y = a + b 1 Ad language + b 2 Identity + b 3 Ad X Identity

53 Categorical IV and Continuous MV In order to make intercept meaningful, subtracted 1 from Latino Identity measure, so ranged from 0 to 6 Y = a + b 1 Ad language + b 2 Identity + b 3 Ad X Identity

54 Categorical IV and Continuous MV

55 Mean attitude for Spanish ad for Latino ID = 1 is 3.215

56 Categorical IV and Continuous MV Mean attitude for Spanish ad for Latino ID = 1 is Mean difference for Latino ID = 1 is (p < 0.05)

57 Categorical IV and Continuous MV Mean attitude for Spanish ad for Latino ID = 1 is Mean attitude for English ad for Latino ID = 1 is Mean difference for Latino ID = 1 is (p < 0.05)

58 Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference *

59 Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference * *

60 Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference * * *

61 Categorical IV and Continuous MV Identity Mean English Mean Spanish Difference * * * * * * (Common practice, Mean = 3, SD = 1.2; Show R program)

62 Continuous IV and Continuous MV Y: Child anxiety (0 to 20) X: Parent anxiety (0 to 20) Z: Parenting behavior: Control (0 to 20)

63 Continuous IV and Continuous MV Y: Child anxiety (0 to 20) X: Parent anxiety (0 to 20) Z: Parenting behavior: Control (0 to 20) Control b for Y onto X

64 Continuous IV and Continuous MV Control b for Y onto X Y = a + b 1 Control PA (Control)(PA) (Common practice versus regions of significance) (Why we include component parts)

65 Advanced Topics

66 Three Way Interactions

67 Identify focal independent variable Identify first order moderator variable Identify second order moderator variable

68 Three Way Interactions Grade 7Grade 8 Female6.0 Male IC 1 = (6-5) - (6-4) = -1 IC = (6-5) – (6-4) = -1 European American

69 Three Way Interactions Grade 7Grade 8 Female6.0 Male IC 1 = (6-5) - (6-4) = -1 Grade 7Grade 8 Female6.0 Male6.0 IC = (6-5) – (6-4) = -1IC = (6-6) – (6-6) = 0 European AmericanLatinos

70 Three Way Interactions Grade 7Grade 8 Female6.0 Male IC 1 = (6-5) - (6-4) = -1 Grade 7Grade 8 Female6.0 Male6.0 IC = (6-5) – (6-4) = -1IC = (6-6) – (6-6) = 0 European AmericanLatinos TW = [(6-5) – (6-4)] - [(6-6) – (6-6)] = -1

71 Three Way Interactions G7 (1)G8 (0) Female (1)6.0 Male (0) IC 1 = (6-5) - (6-4) = -1 G7 (1)G8 (0) Female (1)6.0 Male (0)6.0 IC = (6-5) – (6-4) = -1IC = (6-6) – (6-6) = 0 European American (1)Latinos (0) Y = Gender + b2 Grade + b3 Ethnic + 0 (Gender)(Grade) + b5 (Gender)(Ethnic) + b6 (Grade)(Ethnic) + -1 (Gender)(Grade)(Ethnic) TW = [(6-5) – (6-4)] - [(6-6) – (6-6)] = -1

72 Modeling Non-Linear Interactions

73 Y = α + β 1 X + β 2 Z + ε Modeling Non-Linear Interactions β 1 = α’ + β 3 Z + β 4 Z 2

74 Y = α + β 1 X + β 2 Z + ε Substitute right hand side for β 1 : Modeling Non-Linear Interactions β 1 = α’ + β 3 Z + β 4 Z 2 Y = α + (α’ + β 3 Z + β 4 Z 2 ) X + β 2 Z + ε

75 Y = α + β 1 X + β 2 Z + ε Substitute right hand side for β 1 : Modeling Non-Linear Interactions β 1 = α’ + β 3 Z + β 4 Z 2 Y = α + (α’ + β 3 Z + β 4 Z 2 ) X + β 2 Z + ε Expand: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε

76 Modeling Non-Linear Interactions Re-arrange terms: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε Y = α + α’X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε

77 Modeling Non-Linear Interactions Re-arrange terms: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε Y = α + α’X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε Re-label and you have your model: Y = α + β 1 X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε

78 Modeling Non-Linear Interactions Re-arrange terms: Y = α + α’X + β 3 XZ + β 4 XZ 2 + β 2 Z + ε Y = α + α’X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε Re-label and you have your model: Y = α + β 1 X + β 2 Z + β 3 XZ + β 4 XZ 2 + ε Use centering strategy to isolate effect of X on Y (β 1 ) at any given value of Z; also consider modeling intercept

79 Exploratory Interaction Analysis

80 Use program in R Y = Tenured or not (using MLPM) X = Number of articles published Y = α + β 1 X + ε Z = Number of years since hired

81 X COEFFICENT AND M VALUES N M Value X Slope

82

83

84 Regression Mixture Modeling

85 BI = α + β 1 Aact + β 2 PN + β 3 PBC + ε Mixture Regression But, in reality, we probably are mixing heterogeneous population segments with different coefficients characterizing the segments When we regress Y onto a set of predictors, we assume that people are drawn from a single population with common linear coefficients

86 With “mixed” populations, the overall regression analysis can characterize neither segment very well and lead to sub-optimal inferences and intervention strategies Mixture Regression

87 Another Example of Aggregation Bias

88 Mixture Regression

89 A four class model fits data best (entries are linear coefficients) Segment 1 (42%): Mixture Model for Heavy Episodic Drinking Segment 2 (17%): Segment 3 (21%): Segment 4 (20%): Aact SN DN PBC

90 Interaction Analysis and Establishing Generalizability

91 It is common for people to conclude that an effect “generalizes” in the absence of a statistically significant interaction effect Problem is that we can never accept the null hypothesis of a zero interaction contrast Generalizability Example with RCT of obesity treatment and gender Solution: Adopt the framework of equivalence testing

92 Step 1: Specify a threshold value that will be used to define functional equivalence Step 2: Specify the range of functional equivalence Generalizability Step 3: Calculate the 95% CI for the interaction contrast Step 4: Determine if the CI is completely within the range of functional equivalence

93 Measurement Error

94 It is well known that measurement error can bias parameter estimates in multiple regression. This holds with vigor for interaction analysis One approach to dealing with measurement error in general is to use latent variable modeling Measurement Error

95

96

97 Latent Variable Regression

98 There are a about a half a dozen approaches to how best to model latent variable interactions (e.g., quasi-maximum likelihood; Bayesian). I recommend the approach developed by Herbert Marsh as a good balance between utility and complexity, coupled with Huber-White sandwich estimators for robustness Latent Variable Regression Latent variable regression using multiple group analysis

99 Multi-Group Modeling in SEM

100 Assumption Violations

101 If assumptions of normality or variance homogeneity are suspect Huber-White sandwich estimators Assumption Violations Use approaches with robust standard errors Be careful of outlier resistant robust methods Bootstrapping Rand Wilcox work with smoothers

102 Thank God It Has Ended!


Download ppt "Analysis of Interaction Effects James Jaccard New York University."

Similar presentations


Ads by Google