1. Estimating and Testing Mediation
David A. Kenny
University of Connecticut

2. Overview Introduction (click to go there)
The Four-Step Approach (click)
The Modern Approach (click)
Test the Indirect Effect (click)
Power (click)
Assumptions (click)
DataToText (click)

3. Introduction

4. Interest in Mediation
Mentions of “mediation” or “mediator” in psychology abstracts:
1980: 36
1990: 122
2000: 339
2010: 1,198 4

5. Why the Interest in Mediation?
Understand the mechanism
theoretical concerns
cost and efficiency concerns
Find more proximal endpoints
Understand why the intervention did not work
finding the missing link
compensatory processes

6. The Beginning Model: Basic ResearchY X

7. The Mediational Model M b a c' Y X

8. The Beginning Model: Applied ResearchY

9. (standardized or unstandardized)
The Four Paths
X  Y: path c
X  M: path a
M  Y (controlling for X): path b
X  Y (controlling for M): path c′
(standardized or unstandardized)

10. The Four-Step Approach

11. In the 1980s Different Pairs of Researchers Proposed a Series of Steps to Test Mediation
Judd & Kenny (1981)
James & Brett (1984)
Baron & Kenny (1986)

12. The Four Steps
Step 1: X  Y (test path c)
Step 2: X  M (test path a)
Step 3: M (and X)  Y (test path b)
Step 4: X (and M)  Y (test path c′)
Note that Steps 3 and 4 use the same regression equation.

13. Total and Partial Mediation
Total Mediation
Meet steps 1, 2, and 3 and find that c′ equal zero.
Partial Mediation
Meet steps 1, 2, and 3 and find that c′ is smaller in absolute value than c.

14. (davidakenny.net/papers/twmr/example.txt)
Example Dataset
Morse et al.
J. of Community Psychology, 1994
treatment  housing contacts  days of stable housing
persons randomly assigned to treatment groups.
109 people
(davidakenny.net/papers/twmr/example.txt)

15. Variables in the Example
Treatment
1 = treated (intensive case management)
0 = treatment as usual
Housing Contacts: total number of contacts per during the 9 months after the intervention began
Stable Housing
days per month with adequate housing (0 to 30)
Averaged over 7 months from month 10 to month 16, after the intervention began

16. Morse et al. Example Step 1: X  Y c = 6.558, p = .009 Step 2: X  M
a = 5.502, p = .013
Step 3: M (and X)  Y
b = 0.466, p < .001
Step 4: X (and M)  Y
c′ = 3.992, p = .090

17. When Can We Conclude Complete Mediation?
When c′ is not statistically different from zero, but c is? NOT REALLY
Ideally when a and b are substantial and c′ is small, not just non-significant.

18. Standardized vs. Unstandardized
For most everything in this presentation, the variables can be standardized making the coefficients Betas.
The key thing is to be consistent: If M is standardized in Step 2, it is also standardized in Step 3.

19. Presenting Mediation 19

20. The Modern Approach

21. Dissatisfaction with the Steps Approach
Low power in the test of Step 1
No single measure and test of mediation
Work by MacKinnon, Hayes, Preacher and others has led to the “Modern Approach.”

22. Decomposition of Effects
Total Effect = Direct Effect + Indirect Effect
c = c′ + ab
(This equality exactly holds for multiple regression, but not necessarily for other estimation methods.)
The Indirect Effect or ab provides one number that summarizes the amount of mediation.
Example:
6.558 = (5.502)(0.466)
And 100(2.56/6.56) = 39% of the total effect is explained (ab/c or equivalently 1 - c′/c).

23. Two Sides of the Same Coin
Note that
ab = c - c′
That is, the indirect effect exactly equals the amount of the reduction in the total effect (c) after the mediator is introduced.
Some papers (e.g., Baron & Kenny) emphasize one side and others emphasize the other. Current work in mediation is now focused on the “ab” side.

24. Estimating the Total Effect (c)
The total effect or c can be inferred from direct and indirect effect as c′ + ab.
We need not perform the Step 1 regression to estimate c.
This can be useful in situations when c does not exactly equally c′ + ab.

25. Inconsistent Mediation
ab and c′ have a different sign
X as a “suppressor” variable
Example: Stress and Mood with Coping as a Mediator
Consequences
The Total Effect or c may not be significant
Percent mediated greater than 100%
Do we have mediation?
Yes. There is an indirect effect (ab > 0).
No. There is no effect that is “mediated.”

26. What to Present Decomposition of Effects Total Effect
Indirect Effect(s)
Direct Effects
Less of an emphasis on complete versus partial mediation.

27. Test of the Indirect Effect

28. Work on Determining How to Test ab = 0
The key piece of information in the modern approach is the indirect effect.
Many researchers developed approaches to this problem. 28

29. Strategies to Test ab = 0 Test a and b separately Sobel test
Bootstrapping
Monte Carlo Method

30. Test a and b Separately Easy to do Works fairly well
Does not provide a method for a confidence interval for ab.
Seem too much like the old-fashioned Four-Step Approach.

31. Sobel Test of Mediation
Compute the square root of a2sb2 + b2sa2 which is denoted as sab
Note that sa and sb are the standard errors of a and b, respectively; ta = a/sa and tb = b/sb.
Divide ab by sab and treat that value as a Z.
So if ab/sab greater than 1.96 in absolute value, reject the null hypothesis that the indirect effect is zero.

32. Website: http://www.people.ku.edu/~preacher/sobel/sobel.htm
Example
a = and b = 0.466
sa = and sb = 0.100
ab = 2.56; sab = 1.157
Sobel test Z is 2.218, p = .027
We conclude that the indirect effect is statistically different from zero.
Website:

33. ab Large values of “ab” are more variable than small values (i.e., 0).ab
The distribution of ab is highly skewed which lowers the power of the test.

34. Bootstrapping
“Nonparametric” way of computing a sampling distribution.
Re-sampling (with replacement)
Many trials (computationally intensive)
Current thinking is NOT to correct for bias (i.e., the mean of the bootstrap estimate differs slightly from the estimate).
Compute a confidence interval which is asymmetric.
Slight changes because empirically derived.

35. Results of Bootstrapping
95% Percentile Confidence Interval:
Lower Upper
Note that the CI is asymmetric for an estimate of Also values differ to sampling error.
(Done using the Hayes & Preacher macro from

36. Monte Carlo Method
Save a, b, sa (the standard error of a) and sb from the mediation analysis.
For some methods you have the covariance of ab or sab to save.
Use them and the assumption of normality to generate estimates of a and b and so ab.
Use this distribution of ab’s to get a confidence interval or a p value.
Less computationally intensive than bootstrapping and many trials are feasible.

37. Results of the Monte Carlo Method
95% Monte Carlo Confidence Interval (20,000 trials):
Lower Upper
Done using the Selig & Preacher web program at if correlated use Tofighi’s Rmediation at

38. Power

39. Power of the Test of c If c′ is zero, then c equals ab.
If both a and b have a moderate effect size, then c has a “smaller than small” effect size (assuming c′ is zero).
Thus, it is very possible to have tests of significance for a and b be statistically significant, but c is not.
Note that for N = 100, if a and b have moderate effect sizes (r = .3) and c′ = 0, the power of the test of c is only .14 whereas the power for a is .87 and b is .83.

40. Relative Power of c and ab
Several authors, as early as Cox (1960), have noted that the indirect effect has much more power than the test of c even when the two equal the same value as when c′ is zero.
Kenny and Judd (2014) show that sometimes the test of c need 75 times the number of cases to have the same power as the test of ab!

41. Relative Power of the Test of c and c′ when b = 0
When b = 0, c = c′ and in this case c might be statistically significant, yet c′ might not be significant even though. Why?
There is multicollinearity between X and M due to path a.
Note that as M becomes a more successful mediator and path a gets larger, multicollinearity becomes more of a issue in the testing of c′ (and b).

42. Conclusion: Power of the test of b declines as a increases.
Power and the Test of b
Let b = .3 (standardized). What happens to N need to have 80% power as path a gets larger:
a N
(a is standardized; sample size needed for 80% power to reject the null hypothesis that b = 0)
Conclusion: Power of the test of b declines as a increases.

43. Tool MedPowR R based program to help in power analyses.
Can either give the power for a given value of N and a, b, and c’ or give the N needed to achieve a desired level of power.
Program can be downloaded at

45. Power calculations have begun...
Effect Size Power N c a b c' ab
Alpha for all power calculations set to .050.
Power calculations complete.

46. Assumptions

47. Taking Assumptions Seriously
Older Mediation Analysis ignored the strong assumptions required for the analysis.
Current work, especially that within the Causal Inference tradition, focuses much more on them.
Researchers need to conduct “Sensitivity Analyses” or “What If?”analyses. 47

48. Assumptions: Multiple Regression
Linearity
Is a problem in the example; housing contacts has a quadratic effect on stable housing (p = .034).
Normal Distribution of Errors
Interval level of measurement of M and Y
Equal Error Variance
Independence
No clustering
X and M Do Not Interact to Cause Y 48

49. No XM Interaction: Linear Mediation
Called “Moderation” in Baron & Kenny
Add XM (and possibly other interaction terms, e.g., X2M) when explaining Y.
Many contemporary analysts now see XM interaction as part of mediation.
Not significant for the example (p = .476)

50. (Guaranteed if X is manipulated.)
Causal Assumptions
Perfect Reliability
for M and X
No Omitted Variables
all common causes of M and Y, X and M, and X and Y measured and controlled
No Reverse Causal Effects
M and Y not cause X
Y may not cause M
(Guaranteed if X is manipulated.) 50

51. Basic Mediational Causal Model
Note that U1 and U2 are theoretical variables and not “errors” from a regression equation.

52. Unreliability Usually safe to assume that X is perfectly reliable.
Measurement error in Y does not bias unstandardized regression coefficients.
Measurement error in M is problematic.

53. Unreliability in M

54. Effect of Unreliability in M
b is attenuated (closer to zero)
c′ is inflated (given consistent mediation)
more as a increases
more as b increases
Note that the bigger the indirect, the greater the bias in c′.

55. What to Do about Unreliability in M?
Improve the reliability
Adjust estimates using Structural Equation Modeling
Conduct Sensitivity Analyses assuming different values of reliability.

56. Omitted Variables

57. Other Terms for an Omitted Variable
Third variable.
Confounder
Term used in epidemiology
Becoming increasingly popular

58. What is the Effect of Omitted Variables?
Usually, but not always, the sign of ef is the same as b.
Inflating the estimate of b
Deflates the estimate of c′ (could produce inconsistent mediation).

59. Effect of Vitamin A Supplements in Northern Sumatra59

60. "Standard" Analysis 60
Compliance should totally mediate this effect. Why is c' negative?

61. Results with an Omitted Variable61
Path c' fixed to zero. Omitted variables that causes M and Y.

62. What to Do about Omitted Variables?
Do not omit them: Include them in the analysis as covariates.
If there is good reason to believe that c′ = 0, they can be allowed for.
Sometimes the omitted variable is “shared method effects.” If an issue, measure M and Y by different methods.
Conduct sensitivity analyses. 62

63. A Comforting Fact Unreliability in M deflates ab and inflates c′.
An omitted variable usually deflates ab and inflates c′.
As Fritz, Kenny, & MacKinnon have shown these two biases can almost exactly offset each other. 63

64. Reverse Causation

65. Effect of Reverse Causation
Typically, b and g have the same sign, which likely makes the value of b inflated and the value of c′ deflated.

66. What to Do about Reverse Causation?
Longitudinal designs
If c′ = 0, then the model can be estimated.
Instrumental variable method.

67. Timing of the Measurement of the Mediator
Mediator should be measured after X but before Y.
X might be measured at the same time as Y (e.g., number of treatment sessions), but it must be assumed that X has not changed since when it affected Y.

68. Controlling for Prior Values
Obtain baseline measures of M and Y.
Control for baseline M and Y in the analysis.

69. DataToText

70. DataToText Project
Have the researcher tell DataToText what is the research question.
DataToText performs the requisite analyses.
DataToText gives the results from those analyses:
computer output
a written description 70 70

71. DataToText Macros
Macro developed to provide text, tables, and figures of a simple mediational analysis.
SPSS version: MedText
R version: MedTextR 71

72. Advantages of DataToText
Does the analyses that should be done, but often are not, e.g., tests for outliers and nonlinearity.
MedTextR issues up to 20 different warnings.
Produces a 3 page text describing the results.
Surprisingly “intelligent”
Graphics 72

74. Links davidakenny.net/dtt/datatotext.htm davidakenny.net/dtt/mediate.htm davidakenny.net/dtt/mediateR.htm davidakenny.net/dtt/MedTextR.pdf 74 74

75. Topics Not Discussed Use of SEM simultaneous estimation
latent variables (reflective and formative)
instrumental variable estimation
Causal Inference Approach
Mediators or outcomes that are categorical or counted
Clustering and multilevel mediation
Mediated moderation and moderated mediation 75 75

76. Conclusion Mediational Analyses Are Very Simple
Mediational Analyses Are Very Difficult
Difficulties are more in conceptualization and measurement than in the statistical analysis.
davidakenny.net/cm/mediationN.ppt