1. Regression-Based Tests for Mediation and Moderation
Brian K. Miller, Ph.D.
Associate Professor of Management
Faculty Fellow for
Testing, Research-Support, and Evaluation Center

2. Presentation Objectives
Differentiate between mediation & moderation
Differentiate between hierarchical and stepwise regression
Run hierarchical regression in SPSS
Interpret regression output
Compute interaction terms
Mean center variables
Graphing interactions
Baron and Kenny’s (1986) steps for mediation
Run Sobel tests

3. 1. Differentiate between mediation and moderation

4. Moderation vs. Mediation
Moderator: third variable that affects strength of relationship between two other variables
Ex: relationship between performance and salary depends on gender
Ex: relationship between X1 and Y is strong, especially if X2 is also strong
Mediator: third variable that acts as generative mechanism between two other variables
Ex: performance mediates relationship between job knowledge and salary
Ex: X1 “causes” X2 which “causes” Y

5. Mediation Diagram
Job Knowledge
Job Performance
Salary

6. Moderation Diagram
Gender
Job Performance
Salary

7. 2. Differentiate between hierarchical regression & stepwise regression

8. Hierarchical Regression
Variables entered in equation at various stages
Important statistics and tests
Change in R2 (i.e. ΔR2)
Change in F-score (i.e. ΔF)
Overall equation F-score
Uses:
Control variables
Interaction terms
NOT same as stepwise regression

9. Stepwise Regression All variables entered at one stage
Software enters every combination of variables
Seeks to maximize R2
Capitalizes on chance characteristics of THIS sample 
Like throwing spaghetti against the wall to see what sticks 
Not recommended or even allowed by some journals

10. 3. Run hierarchical regression in SPSS

11. SPSS Example #1 Click “Analyze” / “Regression” / “Linear”
Choose “DVPerceptionsOfOrganizationalPolitics” as DV
Choose “Sex”, “Age” as IVs
Is there theoretical rationale for these control variables?
However, no need for stated hypotheses
Click “Next”
Choose “IVLocusofControl”, “IVLeaderMemberExchange” as IVs
Click “Statistics”
Put check in box for “R squared change”
Click “Continue”, click “OK”

12. 3. Interpreting Regression Output

13. Interpreting the SPSS Output
What is the R2 for Model 1?
What is the ΔR2 from Model 1 to Model 2?
What is the overall R2?
Is the change from Model 1 to Model 2 significant?
Is the overall F-score significant?

14. 5. Compute Interaction Terms

15. Review Diagram
Gender
Job Performance
Salary

16. Reconceptualizing Diagram as 2 x 2
Gender
Male Female
Highest Salary
Medium Salary
Low High
Job Performance
Medium Salary
Lowest Salary

17. Caution!
Never dichotomize continuous variables (e.g. cutting LMX in half at the median)
Results in loss of useful information
Treats scores close to each other as if far from each other
In ANOVA, no need to dichotomize since all IVs are categorical
Regression subsumes ANOVA
But, different method of creating interaction terms needed

18. Problem: Dichotomizing Continuous Variable
Observations 2 7 8
Job Performance Scores

19. Reconceptualizing Diagram as Graphic Plot
Males (r = .7)
Salary
Females (r = .4)
Job Performance

20. Moderation Effects
Strength of relationship between IV and DV depends upon Moderator
If one is female the correlation between PERFORMANCE and SALARY is different from same correlation for males
In regression
correlations manifest themselves as regression weights, or
slopes of lines
So…slope of lines is different based upon gender
Caution: Moderating variables is ALSO an IV

21. Moderation Effects (cont’d)
MUST include both main effects (i.e. both IVs) in regression before including Interaction Term
Model 1 has both main effects but NO interaction term
Model 2 has main effects AND interaction term

22. Example Hypothesis
There is an interaction effect between Locus of Control (LOC) and Leader-Member Exchange (LMX) in the prediction of Perceptions of Organizational Politics (POP), such that high levels of LOC combined with high levels of LMX will lead to higher levels of POP than will low levels of either or both of LOC and LMX.

23. Alternatively Written Hypotheses
The strength of the relationship between LOC and POP depends upon LMX, such that it is strongest when LMX is high and weakest when LMX is low.
There is a positive relationship between LOC and POP especially if LMX is also high.

24. Plotting Interactions
High LMX (r = .6)
Perceptions of Politics
Low LMX (r = -.2)
Locus of Control

25. Creating Multiplicative Terms
Most (but not all!) IVs in regression are continuous
For 2 IVs, create third term that serves as interaction
Simply multiply both IVs to create new term
Heads up: sometimes new term is collinear with component terms

26. Collinearity Issues
Product of two variables is almost always collinear with its constituent parts
When two variables are so strongly correlated with each other that they affect interpretation of regression
Can make Beta weights exceed ± 
Tolerance / Variance Inflation Factor (VIF)
Tolerance is reciprocal of VIF
Collinearity indicated if:
Tolerance < .10, or…
…VIF > 10

27. Computing Interaction Terms in SPSS
To create a multiplicative terms, click: Transform/Compute
In “Target Variable” box type new variable name:
lmxXloc
Double click on “IVLocusOfControl”
Click the asterisk sign (for multiplication) from list of operators to the right
OR use the keyboard to type the asterisk sign, i.e. hold shift key, type “*” (above the 8 key)
Double click on “IVLeaderMemberExchange”
Click OK

28. SPSS Example #2: Moderated Multiple Regression
Click Analyze/Regression/Linear or Dialog Recall button
Click “Reset” to start with all new variables
Choose “DVPerceptionsOfOrganizationalPolitics” as DV
Choose “IVLocusOfControl” and “IVLeaderMemberExchange” as IVs
Click “Next”
Choose new term “lmxXloc” (interaction term just created from product of two IVs above) as IV
Click “Statistics”
Put check in boxes for “R square change” and “Collinearity Diagnostics”
Click “Continue”, click “OK”

29. 6. Mean Centering Variables

30. Mean Centering Variables
Mean centering of variables reduces impact of collinearity
This is NOT same as standardizing variables!
Allows for better interpretation of regression weights
Requires:
Calculation of mean of the variable
Creation of new variable that…
…is difference between measured variable and mean of variable

31. SPSS Example #3: Mean Centering Variables
Find mean of variable by:
Analyze>Frequencies
Double click “IVLocusOfControl” and “IVLeaderMemberExchange” to move to box on right
Click “Statistics” button
Put check mark in box for “Means” under “Central Tendency”
Write mean of “IVLocusOfControl” and mean of “IVLeaderMemberExchange”

32. SPSS Example #3: Continued
Click “Transform”, “Compute”, then “Reset”
Type new variable name: meancenteredloc
Double click on “IVLocusOfControl” to move it to Numeric Expression box
Next type a minus sign followed by the mean of LOC
Mean =
Click OK
Do same for calculating new “meancenteredlmx”
Mean =

33. SPSS Example #3: Continued
Create new multiplicative interaction term from newly computed mean centered variables
Then, rerun previous regression model using new mean centered IVs
What’s different about the key statistics and tests?

34. 7. Graphing Interactions

35. Graphing Interactions
DV on vertical (Y) axis
IV on horizontal (X) axis
Calculate values of Moderator (other IV) 1 sd above and below mean
Calculate values of IV 1 sd above and below mean
Insert value of Moderator at lower sd in regression equation
Insert value of Moderator at upper sd in regression equation
See

36. Using Regression Formula to Plot Interactions
Use the regression equation:
Y = X X X1X2
Where:
X1 = Locus of Control
X2 = Leader Member Exchange
Find 4 different values (points) for Y

37. Using Regression Formula (cont’d)
Calculate one equation for:
Insert value of LMX at 1sd above mean
Insert value of LOC at 1sd above mean
Calculate another equation for:
Insert value of LMX at 1sd below mean
Insert value of LOC at 1sd below mean
Connect the dots (i.e. draw the line segments)

38. Sample Graph

39. 8. Baron and Kenny’s (1986) Steps for Mediation

40. Diagram of Mediation EffectIV
Mediator DV

41. Mediation Refresher
Is the effect of one variable transmitted to another via a third variable?
Does some variable serve as a generative mechanism by which the effects of an IV are transmitted to a DV?
Q: General Mental Ability predicts Job Performance, but how or why?
A: GMA leads to greater Job Knowledge and greater Job Knowledge leads to greater Job Performance (Schmidt & Hunter, 1998)
Mediator explains HOW something occurs or WHY it occurs
vs.
Moderator explains WHEN something occurs

42. Diagram of Mediation Effect
General Mental Ability
Job Knowledge
Job Performance

43. Diagram of Mediation EffectX Y

44. Baron & Kenny’s (1986) Steps
Establish that IV is related to DV
Basically, is there an effect to be mediated? If not, then stop further tests. Some have scrutinized this assertion, though
Establish that IV is related to Mediator
If not stop further tests
Establish that Mediator is related to DV
Examine if IV is reduced in magnitude when controlling for (i.e. entering in equation) Mediator
Note: Steps 3 and 4 accomplished simultaneously

45. Partial vs. Full Mediation
Uses hierarchical regression
If statistically significant IV decreases in magnitude but remains statistically significant, there is evidence of PARTIAL mediation
If statistically significant IV decreases in magnitude to a level of non-significance, there is evidence of FULL mediation

46. SPSS Example #4 Locus of Control Perceptions of Politics
Distributive Justice
Leader-Member Exchange

47. SPSS Example #4: Step 1 in Tests for Mediation
Click “Analyze”, “Regression”, “Linear” or use the Dialog Recall button
Click “Reset” to start with new variables
Choose “DVDistributiveJustice” as DV
Choose “IVLocusOfControl” and “IVLeaderMemberExchange” as IVs
Click “OK”

48. SPSS Example #4: Step 2 in Tests for Mediation
Click “Analyze”, “Regression”, “Linear”
Remove “DVDistributiveJustice” as DV
Choose “DVPerceptionsOfOrganizationalPolitics” as DV
Choose “IVLocusOfControl” and “IVLeaderMemberExchange” as IVs
Click “OK”

49. SPSS Example #4: Steps 3 & 4 in Tests for Mediation
Click “Analyze”, “Regression”, “Linear”
Remove “DVPerceptionsOfOrganizationalPolitics” as DV
Choose “DVDistributiveJustice” as DV
Choose “IVLocusOfControl” and “IVLeaderMemberExchange” as IVs
Click “Next” to add Mediator (DVPerceptionsOfOrganizationalPolitics) in hierarchical regression
Choose “DVPerceptionsOfOrganizationalPolitics” as IV
Click on “Statistics” button
Put check mark in “R squared change” box
Click “Continue” button
Click “OK”

50. Interpreting the SPSS Output
Were both IVs related to DV?
Were both IVs related to Mediator?
What happened to IVs when Mediator added to regression equation in prediction of DV?
Is there partial, full, or no mediation?

51. 9. Running Sobel Tests

52. Problems with Baron and Kenny Tests
Baron and Kenny’s (1986) mediation tests give a rather holistic assessment of presence of mediation
Full mediation
Partial mediation
No mediation
Not a precise test of statistical significance

53. Sobel Tests Precise test of significance of mediator!
Useful for large samples only and when raw data not available
Bootstrapping preferred for small samples or for raw data
Several variations on this test with different adjustments for different assumptions
Aroian test/adjustment
Goodman test/adjustment

54. Mediation Model (for notation purposes)

55. Sobel Formulae
a = raw (unstandardized) regression coefficient for association between IV and mediator
sa = standard error of a
b = raw coefficient for the association between the mediator and the DV (when the IV is also a predictor of the DV)
sb = standard error of b.
Q: What is the raw coefficient divided by its standard error?
A: t-value

56. Sobel Formulae (cont’d)
Sobel test equation z-value = a * b/SQRT(b2*sa2 + a2*sb2)
Aroian adjustment z-value = a * b/SQRT(b2*sa2 + a2*sb2 + sa2*sb2)
Goodman adjustment
z-value = a * b/SQRT(b2*sa2 + a2*sb2 - sa2*sb2)

57. Sobel Tests for This Data Set
Sobel test assumes very small standard errors
Aroian and Goodman adjustments do not!
From the SPSS output we see that for LMX:
a = .344
b = -.640
sa = .073
sb = .120
Thus…
Z = .344 * -.640/SQRT(-.642* *.122)
Z = (p < .001)
Yeah! We have a precise significance level for our mediator

58. Citations for Further Info
Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: conceptual, strategic, ans statistical considerations. Journal of Personality and Social Psychology, 51(6),
Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Thousand Oaks, CA: Sage Publications.
Aguinis, H. (2004). Regression analysis of categorical moderators. New York: Guilford Press.

59. That’s all folks!!!