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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

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

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:

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:

Spurious Relationship

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

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:

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

Moderated Causal Relationships

The variable that “moderates” the relationship is called a moderator variable.

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

Interactions, when translated into causal analysis, focus on moderated relationships

When I encounter an interaction effect, I think:

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

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.

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

A “Trick” We Will Use: Linear Transformations

Y = a + b1 X + e

Satisfaction = a + b1 Grade + e

Satisfaction = 12 + -.50 Grade + e

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

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

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

Categorical IV and Categorical MV

Y = Relationship satisfaction (0 to 10)

X = Gender (female = 1, male = 0)

Z = Grade (6th = 1, 7th = 0)

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?

Categorical IV and Categorical MV

Gender effect for 6th grade: 8 – 7 = 1

Categorical IV and Categorical MV

Gender effect for 6th grade: 8 – 7 = 1

Gender effect for 7th grade: 7 – 4 = 3

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

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)

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)

Categorical IV and Categorical MV

Extend to groups > 2 (add 8th grade)

Inclusion of covariates

How to generate means and tables

Continuous IV and Categorical MV

Y = Relationship satisfaction (0 to 10)

X = Time spent together (in hours)

Z = Gender (female = 1, male = 0)

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

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)

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)

Continuous IV and Categorical MV

Do not estimate slopes separately; use flipped reference group strategy

Extend to groups > 2 (use grade as example)

Categorical IV and Continuous MV

Study conducted in Miami with bi-lingual Latinos

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)

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

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

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….

Categorical IV and Continuous MV

Identity Mean English – Mean Spanish

- 1.50
- 1.00
- 0.50
- 0.00
- -0.50
- -1.00
- 7 -1.50

Categorical IV and Continuous MV

Identity Mean English – Mean Spanish

- 1.50
- 1.00
- 0.50
- 0.00
- -0.50
- -1.00
- 7 -1.50

Y = a + b1 Ad language + b2 Identity + b3 Ad X Identity

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 + b1 Ad language + b2 Identity + b3 Ad X Identity

Categorical IV and Continuous MV

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

Categorical IV and Continuous MV

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

Mean difference for Latino ID = 1 is 1.707 (p < 0.05)

Categorical IV and Continuous MV

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

Mean difference for Latino ID = 1 is 1.707 (p < 0.05)

Mean attitude for English ad for Latino ID = 1 is 4.922

Categorical IV and Continuous MV

Identity Mean English Mean Spanish Difference

- 4.922 3.215 1.707*
- 4.915 3.662 1.253*
- 7

Categorical IV and Continuous MV

Identity Mean English Mean Spanish Difference

- 4.922 3.215 1.707*
- 4.915 3.662 1.253*
- 4.908 4.108 0.800*
- 7

Categorical IV and Continuous MV

Identity Mean English Mean Spanish Difference

- 4.922 3.215 1.707*
- 4.915 3.662 1.253*
- 4.908 4.108 0.800*
- 4.901 4.555 0.346*
- 4.895 5.002 -0.107
- 4.888 5.449 -0.561*
- 7 4.882 5.896 -1.014*

(Common practice, Mean = 3, SD = 1.2; Show R program)

Continuous IV and Continuous MV

Y: Child anxiety (0 to 20)

X: Parent anxiety (0 to 20)

Z: Parenting behavior: Control (0 to 20)

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

7 .10

8 .20

9 .30

10 .40

11 .50

12 .60

13 .70

Continuous IV and Continuous MV

Control b for Y onto X

7 .10

8 .20

9 .30

10 .40

11 .50

12 .60

13 .70

Y = a + b1 Control + 0.10 PA + 0.10 (Control)(PA)

(Common practice versus regions of significance)

(Why we include component parts)

Identify focal independent variable

Identify first order moderator variable

Identify second order moderator variable

Three Way Interactions

European American

Latinos

IC = (6-5) – (6-4) = -1

IC = (6-6) – (6-6) = 0

Three Way Interactions

European American

Latinos

IC = (6-5) – (6-4) = -1

IC = (6-6) – (6-6) = 0

TW = [(6-5) – (6-4)] - [(6-6) – (6-6)] = -1

Three Way Interactions

European American (1)

Latinos (0)

IC = (6-5) – (6-4) = -1

IC = (6-6) – (6-6) = 0

TW = [(6-5) – (6-4)] - [(6-6) – (6-6)] = -1

Y = 6.0 + 0 Gender + b2 Grade + b3 Ethnic + 0 (Gender)(Grade)

+ b5 (Gender)(Ethnic) + b6 (Grade)(Ethnic) + -1 (Gender)(Grade)(Ethnic)

Modeling Non-Linear Interactions

Y = α + β1 X + β2 Z + ε

β1 = α’ + β3 Z + β4 Z2

Substitute right hand side for β1:

Y = α + (α’ + β3 Z + β4 Z2) X + β2 Z + ε

Modeling Non-Linear Interactions

Y = α + β1 X + β2 Z + ε

β1 = α’ + β3 Z + β4 Z2

Substitute right hand side for β1:

Y = α + (α’ + β3 Z + β4 Z2) X + β2 Z + ε

Expand:

Y = α + α’X + β3 XZ + β4 XZ2 + β2 Z + ε

Modeling Non-Linear Interactions

Y = α + α’X + β3 XZ + β4 XZ2 + β2 Z + ε

Re-arrange terms:

Y = α + α’X + β2 Z + β3 XZ + β4 XZ2 + ε

Modeling Non-Linear Interactions

Y = α + α’X + β3 XZ + β4 XZ2 + β2 Z + ε

Re-arrange terms:

Y = α + α’X + β2 Z + β3 XZ + β4 XZ2 + ε

Re-label and you have your model:

Y = α + β1 X + β2 Z + β3 XZ + β4 XZ2 + ε

Modeling Non-Linear Interactions

Y = α + α’X + β3 XZ + β4 XZ2 + β2 Z + ε

Re-arrange terms:

Y = α + α’X + β2 Z + β3 XZ + β4 XZ2 + ε

Re-label and you have your model:

Y = α + β1 X + β2 Z + β3 XZ + β4 XZ2 + ε

Use centering strategy to isolate effect of X on Y (β1 ) at any given value of Z; also consider modeling intercept

Exploratory Interaction Analysis

Use program in R

Y = Tenured or not (using MLPM)

X = Number of articles published

Z = Number of years since hired

Y = α + β1 X + ε

N M Value X Slope

478 1.000 .000

475 2.000 .002

457 3.000 .007

408 4.000 .007

330 5.000 .009

246 6.000 .008

166 7.000 .005

115 8.000 .009

74 9.000 .011

48 10.000 .001

BI = α + β1 Aact + β2 PN + β3 PBC + ε

When we regress Y onto a set of predictors, we assume that people are drawn from a single population with common linear coefficients

But, in reality, we probably are mixing heterogeneous population segments with different coefficients characterizing the segments

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

Mixture Model for Heavy Episodic Drinking

A four class model fits data best (entries are linear coefficients)

Aact SNDNPBC

Segment 1 (42%): .33 .02 .01 -.01

Segment 2 (17%): .10 .29 .30 .01

Segment 3 (21%): .30 .29 .05 .04

Segment 4 (20%): .48 .09 .25 -.03

It is common for people to conclude that an effect “generalizes” in the absence of a statistically significant interaction effect

Example with RCT of obesity treatment and gender

Problem is that we can never accept the null hypothesis of a zero interaction contrast

Solution: Adopt the framework of equivalence testing

Step 1: Specify a threshold value that will be used to define functional equivalence

Step 2: Specify the range of functional equivalence

Step 3: Calculate the 95% CI for the interaction contrast

Step 4: Determine if the CI is completely within the range of functional equivalence

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

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 using multiple group analysis

If assumptions of normality or variance homogeneity are suspect

Use approaches with robust standard errors

Bootstrapping

Huber-White sandwich estimators

Be careful of outlier resistant robust methods

Rand Wilcox work with smoothers

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