1. Moderation, Mediation, and Other Issues in Regression

2. Regression Assumptions
Dependent variable a linear function of the IVs
Each observation drawn independently (No relationships between cases such that errors for each case are independent from others)(HLM)
Variance of errors around line should be fairly constant: Homoscedascticity
Errors are normally distributed
Critical
Robust

3. To Interpret Regression as Effects
The DV does not influence any of the IVs (cause and effect)
The IVs are measured without error
Must include all common causes of the presumed cause and presumed effect

4. Common Cause
A common cause is a variable that affects both a presumed cause and a presumed effect. Here, variable a and variable b are correlated because they are both affected by the common cause. See chapter 11 for more detail

5. No Multicollinearity
Tolerance = ranges from 0 (no independence from other variables) to 1 (complete independence)
Multicollinearity - when Tolerance is low (closer to 0) and VIF is high (around 6, 7)

6. Multicollinearity Present?

7. Moderation = Interaction
Interaction effects = Moderation, or when the magnitude of the effect of one variable depends on another
There is also a handout on this topic in the files containing answers to exercises

8. TV viewing by Ability Level
The effect of TV viewing on achievement is moderated by ability level

9. Pretest - Posttest
The effect of pre-test scores is moderated by condition

10. Intervening Variables
Think of a relationship between two variables (one IV1 and one DV)
Can you think of an intervening variable (IV2) that might help to explain the relationship between IV1 and your DV?
Write down example and discuss with your neighbor

11. Mediation
Mediation = indirect effect. Mediators are intervening variables that partially account for the relationship between an influence and outcome

12. Mediation can be total…
…or partial. Here, variable 1 continues to have a direct effect on variable 2, in addition to its indirect effect through the mediator

13. Conditions for Mediation
Step 1: Is influence a significant predictor of outcome?
Regress DV on IV1
Influence
Outcome

14. Conditions for Mediation
Step 2: Is the mediator a significant predictor of outcome?
Regress DV on Mediator
Outcome
Mediator

15. Conditions for Mediation
Step 3: Is there a relationship between the influence variable and the mediator? Regress Mediator on Influence
Influence
Mediator

16. Conditions for Mediation
Step 4: Does the magnitude of the direct path from influence
to outcome decrease in the presence of the mediator?
Regress Outcome on both Mediator and IV1.
Influence
Outcome
Mediator

17. Conditions for Mediation
Step 5: Is the mediation effect meaningful and significant?
(more later)
Influence
Outcome
Mediator
Websites:

18. Example of Mediation

19. In SPSS Regress Achievement on Ability (Path B)
Regress Achievement on Motivation (Path C)
Regress Motivation on Ability (Path A)
Regress Achievement on both Ability (Path B’) AND Motivation (Path C’)
Compare Path B to Path B’

20. SATMath
TotalCoursePts

21. TotalCoursePts
Self-Efficacy

22. SATMath
Self-efficacy

23. SAT Math
TotalCoursePts
Self-efficacy

24. Sobel Statistic a = .004 (Sa) = .001 b = 33.143 (Sb) = 4.551
a = .004 (Sa) = .001
b = (Sb) = 4.551

25. Jargon Structural Equation Modeling (SEM) AKA:
Analysis of Covariance Structures; Causal Analysis; LISREL
Includes:
Latent variable SEM; CFA; Path Analysis
Path Analysis
Simplest form of SEM
Uses measured variables
Can be solved via MR

26. Correlations Among 3 Variables

27. A Causal Model

28. The Tracing Rule
The correlation between two variables X and Z is equal to the sum of the product of all paths from each possible tracing between X and Z [in Figure 2]. EXCEPT:
the same variable is not entered twice per tracing and
a variable is not both entered and exited through an arrowhead.

29. The Solved Model

30. Path Analysis with MR Regress Achievement on Ability and Motivation
Regress Motivation on Ability
See SPSS output

31. Regression Results
Paths to Achievement:
Path to Ability:

32. A More Complete Model
Disturbances = error = residuals = SQRT(1-R2)

33. HW 9
The purpose of this homework is to test a mediational model. Based on theory, test a mediational model using your HSLS dataset.
Write up your results. In doing so, make sure you:
Briefly justify your model using theory and past research
Provide a brief data analytic plan (in other words, describe what analyses you will be running to test your mediational model)
Report and interpret statistics related to your mediational style using APA style
Interpret your findings