1. Mediation and Moderation:
What are they? Are they for me?

2. Overview Definitions of M&M and Differences between M&M
Methods for testing for mediation
Advanced topics in mediation
Methods for testing for moderation
Advanced topics in moderation
Advanced topics in mediation and moderation

3. Simple Mediation and Moderation: Similarities
Both involve 3 variables
The goal is to determine how a third variable affects the simple relationship between two variables
Both can be analyzed with regression or structural equation modeling (SEM) software

4. Simple Mediation and Moderation: Differences
Moderation involves product terms, not mediation
Mediation is best used with longitudinal data, but this is not required for moderation
Centering variables is usually required in moderation, but not in mediation
Graphing is essential in moderation, but only highly recommended in mediation

5. Definitions and Important Questions
Mediation
A mediating variable is the mechanism by which an effect occurs between a predictor and an outcome
The important question is whether a third variable (Med) partially or completely controls the relationship between a predictor (X) and an outcome (Y)

6. Definitions and Important Questions
Moderation
A moderating variable is one that affects the direction and/or strength of the relationship between a predictor variable and a criterion variable
The important question is whether the relationship between the predictor (X) and the outcome (Y) differs across the levels of the moderator (mod)

7. Frequency of Mediation Analyses
Number of mentions of “mediation” “mediator” or “mediates” in the title of psychology journal articles,

8. Introduction to Mediation
Simple Model c X Y

9. Introduction to Mediation
Mediation Model
Med b a c' Y X

10. Introduction to Mediation
Rephrasing the important question:
Does the simple relationship between X and Y [c] shrink or disappear when the mediator is present in the model (i.e., is c΄<c, or is c΄=0)?
A relationship that shrinks indicates partial mediation, whereas a relationship that disappears indicates full mediation
More on full and partial mediation to come

11. Decomposition of Effects
Total Effect = Indirect Effect + Direct Effect
c = c΄ + ab
Thus, ab = c - c΄
This holds exactly for least squares estimates (e.g., standard multiple regression), but is only an approximation with other estimation methods

12. Baron & Kenny … I wish I was cited like that
The most famous reference to mediation is Baron & Kenny (1986)
Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51,
This paper has been referenced more than times

13. Why was the Baron & Kenny article so popular?
It laid out mediation in a set of simple to implement steps:
Test X -> Y (path c)
Test X -> M (path a)
Test M -> Y, controlling for X (path b)
Test X -> Y, controlling for M (path c΄)
If steps 1 to 3 are significant, and c΄ is reduced then we have partial mediation (or if c΄ = 0 we have full mediation)

14. Criticisms of the Baron & Kenny Steps
The main criticisms of the Baron & Kenny approach relate to the necessity of the steps
A significant X to Y path is not required (e.g., the direct and indirect effect may be of different signs)
Other researchers have argued that the M to Y path may not be significant when X and M are highly correlated (multicollinearity)
Further, a statistical test of X -> Y in the last step is meaningless since it cannot prove partial or full mediation

15. How do we test for partial mediation?
There are two popular approaches for testing whether c΄ < c (i.e., whether partial mediation has occurred)
Sobel Test
OK test when sample sizes are large (>100)
Bootstrapping
Good test for small or large sample sizes

16. Sobel Test
The Sobel test assesses the statistical significance of the indirect effect (ab)
In other words, it assesses whether a significant part of the relationship between X and Y is controlled by the mediator
Let sa and sb represent the standard errors of a and b, and reject Ho: ab = 0 if z > z1-α, where:

17. Problem with the Sobel Test
The distribution of ab is highly positively skewed, and thus relying on a method that assumes a normally distributed sampling distribution (i.e., Sobel test) reduces power
In other words, the tail probability is going to be larger (larger p-value) if we assume a normal distribution than if we assume a skewed distribution

18. Bootstrapping
Bootstrapping allows researchers the opportunity to resample from the data in order to generate an empirical sampling distribution of ab
Researchers can then use the empirical (bootstrap) estimate of ab, along with the standard error of the bootstrap estimates, to compute a z statistic or a confidence interval (which will generally be asymmetric)

19. Example
Let’s say we are interested in determining if anxiety (anx) mediates the relationship between hindrances to doing well in a stats course (hindr) and the exam average (exavg)
r(anx,hindr)=.452*
r(anx,exavg)=-.367*
r(hindr,exavg)=-.303*
Let’s use α = .05

20. Mediational Model
Anxiety
Hindrances
Exam
Average

21. Regression Equations Predicting exavg from hindr:
Exavg = (hindr) (p < .001)
Predicting anx from hindr:
Anx = .552 (hindr) (p < .001)
Predicting exavg from both hindr and anx:
Exavg = (hindr) (anx)
p-value for hindr = .062; p-value for anx = .002
Therefore, since p(hindr)> α, many would say (but we won’t) that anx fully mediates the prediction of exavg from hindr

22. Added Variable Plots
The relationship between examavg and hindrances is weaker after controlling for anxiety
b=-3.87
b=-2.20

23. Sobel Test
We can evaluate whether or not anx is a significant mediator by using the Sobel test
More specifically, we can evaluate the significance of the indirect effect using the Sobel test
We will need the regression coefficients (and standard errors) for: 1) predicting anx from hindr and; 2) predicting exavg from anx, with hindr in the equation:
Anx = (hindr)
[SE (hindr) = .097]
Exavg = (hindr) (anx)
[SE (anx) = .959]

24. Sobel Test Calculation
Example
Sobel Test Calculation
Therefore, with Sobel z = (p = .0058), we can say that anxiety significantly mediates the relationship between stats related hindrances and the exam average
A Sobel calculator is available at:

25. Bootstrap Analyses Method 1: AMOS Indirect Effect: -1.671
or any other SEM software that computes bootstrapped indirect effects and standard errors
Indirect Effect:
Standard Error: .668
z = /.668 = (p = .0062)

26. Bootstrap Analyses Method 2: SPSS/SAS/MPlus Macros
Indirect Effect:
95% CI: ,
Standard Error: .678
z = /.678 = (p = .0066)

27. Effect Size Measures Completely Standardized Indirect Effect
Product of the Beta coefficients for a and b
βa = .452
βb = -.289
CSIE = .452 * = -.131
According to Cohen, is small, is medium, and is large
Thus we have a medium sized indirect effect

28. Effect Size Measures Proportion of the Total Effect that is Mediated
An alternative to the CSIE, that is very intuitive
% Mediated = indirect effect / total effect = ab / c
For our example:
% Mediated = / = 43%
Therefore, 43% of the relationship between hindrances to doing well in stats and the exam average is mediated by stats-related anxiety

29. Full Mediation
There is a lot of debate over how to determine full mediation, or whether full mediation could ever exist in psychology
Initially, it was suggested by Baron & Kenny that a nonsignificant predictor effect, after controlling for the mediator (path c΄), would indicate full mediation
However, statistical significance on its own is not good, and with a small sample size, or a weak X-Y relationship, we would find full mediation too often

30. Full Mediation, cont’d
It has been suggested that a significant Sobel test, and a % Mediated > .80, is indicative of full mediation (davidakenny.net/cm/mediate.htm)
Constance Mara derived an equivalence- and bootstrap-based test of full mediation that involves demonstrating that the raw correlation (rxy) is equivalent to the correlation implied by the indirect effect (rxy*) = βaβb

31. Multiple Mediators
Many models evaluate the effects of more than one mediator
It is best to investigate all potential mediators in one model so that you control for the effects of the different mediators
It is also advantageous to use statistical software to conduct the separate mediational analyses simultaneously

32. Mediational Model
Anxiety
Hindrances
Exam
Average
Exam
Ability

33. Multiple Mediator Model in AMOS
A disadvantage of AMOS is that generally it only provides a total indirect effect, not the separate indirect effects for each mediator
We can use “ghost variables” to be able to obtain the separate indirect effect estimates

34. Multiple Mediational Model in AMOS
Anxiety b1 a1
Hindrances
Exam
Average a1 a2 a2 b2
Exam
Ability b1 b2

35. Bootstrap Analyses Method 1: AMOS Anxiety as Mediator:
Indirect Effect:
Standard Error: .670
z = /.689 = (p = .0036)
Exam Ability as Mediator
Indirect Effect: .265
Standard Error: .244
z = .265/.244 = (p = .1387)

36. Bootstrap Analyses Method 2: SPSS Macro from Preacher/Hayes
Anxiety as Mediator:
Indirect Effect:
95% CI: ,
Standard Error: .689
z = /.689 = (p = .0036)
Exam Ability as Mediator
Indirect Effect: .275
95% CI: , .9573
Standard Error: .247
z = .275/.247 = (p = .1330)

37. Further Topics in Mediation
Latent Variable Mediation Models
Categorical Outcomes
Dichotomous or Multicategorical Predictors
Nonlinear relationships
4 variable mediation chains
X  M1  M2  Y

38. Introduction to Moderation
What is moderation? …. INTERACTION!
The effect of X on Y differs at different levels of the moderator
Ex: The effects of provoked anger on aggression differs at different levels of trait aggressiveness
We could deal with interactions in ANOVA (all categorical predictors), but ‘moderation’ often implies interaction with at least one continuous variable (and we would never categorize a continuous variable in order to use ANOVA!)

39. Visual Representations of ModerationX Y X Y
Mod
X*Mod

40. Understanding Moderation
Interaction terms represent a relationship between X and Y, and Mod and Y, that is more than just additive
In other words, the combined effect of X and Mod is more substantial than what would be expected by just summing the effects of X and Mod

41. Two-way Model with Additive Predictor Effects (i.e., no interaction)X b0
Low Mod
High Mod
Medium Mod Y
Y΄ = β0 + β1 X + β2 Mod

42. Two-way Model with Interaction
High Mod X b0
Low Mod
Med Mod
Y΄ = β0 + β1X + β2Mod + β3X*Mod

43. Creating the Interaction Term
The interaction term is created as the product of the X and Mod variables
Typically, continuous variables are mean centered (i.e., X – M) and categorical variables are dummy coded (e.g., 0,1) before creating the interaction term
Note that the main effect terms must be included in the model with the interaction term, or the interaction term is confounded

44. Centering Centering does not affect the interaction term (β3)
In fact, centering is not necessary for simply screening for the presence of an interaction
However, if you are going to interpret the main effects with the interaction term in the model (e.g., for conducting simple effects), then the interaction term should be created using appropriate centering

45. More on Centering
When an interaction term is present in a model, the effect of X is interpreted as the effect of X when Mod = 0 (or the effect of Mod when X = 0), which is often uninformative
Thus, we want to create an interaction term which allows us to interpret the main effects in a meaningful way, for example the effect of X at the mean of Mod (which will be important for understanding the nature of interactions)

46. Example: Two Continuous Predictors
We are interested in determining if Classroom Independence (ci), Student Independence (si), or the interaction between them are important predictors of grades in a class
cic, sic are the mean-centred versions of si and ci, and cic:sic (cic*sic) is the interaction term
We start by exploring the interaction

47. Statistical Software Output
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Intercept e < 2e-16
cic e e-16
sic e e-14
cic:sic e e-07
- Since the interaction is significant, we want to explore the nature of the interaction (and avoid trying to interpret the main effects)

48. R Plot of the Interaction

49. Interpreting the Nature of the Interaction
As si increases, the relationship between ci and grades gets stronger
Although this R plot is very informative, often researchers follow traditions such as creating a plot of the effect of ci on grades at the mean and +/- one standard deviation from the mean of si, and obtaining the significance of the simple slopes
These plots/analyses are done using the calculators at: and excel files are available at:

50. Plot of the Interaction
1 sd below mean, mean, 1 sd above mean

51. Simple Slopes Analysis
Simple Slopes at Conditional Values of SIC
At 1 sd below the mean on SIC
cic simple slope = , t=3.379, p=0.0011
At the mean on SIC
cic simple slope = , t=9.851, p<.0001
At 1 sd above the mean on SIC
cic simple slope = , t=9.7616, p<.0001

52. Simple Slopes Analyses
It is very easy to conduct simple slopes analyses without available macros
Since our interaction terms were created from centered variables, the effect of cic on grades is at the mean of sic (i.e., sic = 0) so that is our first simple slope
We can obtain the simple slope of cic on grades at one sd below the mean on sic by shifting our data so that sic = 0 is at one sd below the mean
We do that by ADDING one sd to the scores on the centred si variable before creating the interaction term
We reverse this procedure for obtaining the simple slope of ci on grades at one sd above the mean of si

53. More on Simple Slopes Analyses
Simple slopes analyses are often over-used
A significant interaction indicates that the effect of X on Y differs across the levels of mod
So, if you focus on the significance of the simple slopes, you might be missing the important point which is that the slopes differ significantly (regardless of the statistical significance of any particular simple slope)
This is the same as focusing on simple effects when following up an interaction in ANOVA

54. What if our Moderator is Dichotomous?
If our moderator is dichotomous, nothing changes except that we dummy code (0,1) the variable instead of mean centering it
If the interaction is significant, we are now interested in exploring the simple slopes of X on Y at mod = 0 and mod = 1
We do that by reverse coding the variable in order to obtain both simple slopes
As before we can graph and explain the nature of the interaction

55. What if our moderator is multicategorical?
If our moderator has more than two categories (c), then we need c-1 dummy variables
3 Category Example (first category as referent):
Dummy Variable 1 (DV1): c1=0, c2=1, c3=0
Dummy Variable 2 (DV2): c1=0, c2=0, c3=1
We now create 2 interaction variables:
Int 1 = DV1 * X (note that X is mean centered)
Int 2 = DV2 * X

56. What if our moderator is multicategorical?
Y΄ = β0 + β1X + β2DV1+ β3DV2 + β4X*DV1+ β5X*DV2
We include X, both dummy variables, and the two interaction variables in our regression equation
The significance of the interaction is determined using the R2 change from a hierarchical regression, where the interaction terms are added to the model after the main effects

57. What if our moderator is multicategorical?
To follow-up a significant interaction, it is again important to plot the interaction
The simple slopes from the plots are useful for interpretation (i.e., understanding how the relationship between X and Y differs for different levels of mod), but the significance of the simple slopes is usually uninformative
Useful tests for interpreting the interaction are the DV*X interactions, as each tests whether the slope differs for the referent and the comparison groups (and the referent group can be changed)

58. Higher-Order Interactions
If you are interested in testing 3-way interactions in regression (and I don’t recommend testing 4-way or higher interactions), all of the same centering, coding, etc. rules apply
The question being asked is whether the 2-way interaction between X and Z differs across the values of mod

59. Higher-Order Interactions
Y΄ = β0 + β1X + β2Z+ β3Mod + β4X*Z+ β5X*Mod + β6Mod*Z+ β7X*Z*Mod
Interpreting 3-way interactions REQUIRES plots
More specifically, start by plotting the X*Z interaction at different levels of mod, and then following up each as if it were a two-way interaction
There are different ways to plot the interaction, and the method you choose will depend on how the variables are coded and the theory being evaluated
As always, try to focus less on the significance of the simple slopes, or simple, simple slopes, and more on interpreting the nature of the interaction

60. Further Topics in Moderation and Mediation
Categorical outcome variables
Moderated Mediation
Mediated Moderation
Nonlinear interactions
Comparing Software for Analyzing Mediation and Moderation
Etc., Etc., Etc. ……