1. Past and Present Methods for testing Mediation

2. Outline Introduction Kenny’s basic model and procedure Sobel’s test
Bootstrap
Multiple Mediators

3. “Mediation models… are popular because they allow interesting associations to be decomposed into components that reveal possible causal mechanisms.” (Shrout & Bolger, 2002)
Examples:
(a) Parent unemployment Low parental quality  Child behavior problems
(b) Firefighters involvement in Sep 11 stress drinking problems.
(c) Job characteristics  critical psychological states , motivation

4. When does “simple” mediation occur?
A three variable system, simple mediation
(Can have also multiple mediators)
A mediator
A variable that fully, or partially transmits the effect of one or more independent
variables to one or more dependent variables

5. The Mediation Model:
initial variable
Outcome
The effect of X on Y may be mediated by a mediating variable M, and the variable X may still affect Y. 

6. Organizational Research Methods (ORM) Apr 2008
Kenny: “Reflections on Mediations”
“I would have never guessed that mediational analyses would become so common that a sizable percent of articles in the social sciences employ some form of mediational analysis…”

7. Reflections on Mediation Kenny ( 2008) , ORM
Good and bad side of the popularity of Baron and Kenny (1986):
Clear advice how to conduct the analysis
The causal assumptions are often ignored
If the presumed model is not correct, the results from the mediational analysis are of little value. 
Statistics is used to evaluate a presumed mediation model.

8. The Mediation Model:
initial variable
Outcome
The effect of X on Y may be mediated by a mediating variable M, and the variable X may still affect Y. 
Path c is called the total effect. 
The mediated model is :
C’ is the direct effect

9. My Comment: on notation
Most papers do not use “careful notations” to distinguish between estimates and parameters.
We adopt the common use and will clarify when the notation refers to parameters or their estimates
Regression equations:

10. Complete mediation Partial mediation
The mediator has also been called an intervening variable. 
Complete mediation
Variable X no longer affects Y, after M has been controlled for, so path c' is zero.
Partial mediation
The path from X to Y is reduced in absolute size, but is still different from zero when the mediator is controlled.    

11. Baron and Kenny Steps - Causal Step Approach
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,
The researcher draws upon evidence from several regressions

12. Step 1:  This step establishes that there is an effect that may be mediated.
(test path c).
Show that X significantly accounts for the variability in Y.
Y as the criterion variable in a regression equation and X as a predictor
MacKinnon et al , 2004:

13. Show that X significantly accounts for the variability in M
Step 2:
Show that X significantly accounts for the variability in M
Use M as the criterion variable in the regression equation ,and X as a predictor (estimate and test path a). 
This step essentially involves treating the mediator as if it were an outcome variable.
MacKinnon et al , 2004

14. Step 3: Show that the mediator affects the outcome variable.
Use Y as the criterion variable in a regression with X and M as predictors (estimate and test path b).
Show that M significantly accounts for the variability in Y when controlling for X and the effect of X on Y decreases substantially when M is entered simultaneously with X as a predictor for Y.
MacKinnon et al , 2004

15.   Most analysts believe that the essential steps in establishing mediation are Steps 2 and 3.
“To establish that M completely mediates the X-Y relationship, the effect of X on Y controlling for M (path c') should be zero (???)”….

16. Note that the steps are stated by KENNY in terms of zero and nonzero coefficients, not in terms of statistical significance.  
“Trivially small coefficients can be statistically significant with large sample sizes and very large coefficients can be non-significant with small sample sizes.
Statistical significance is informative, but other information should be part of statistical decision making”.

17. Measuring Mediation
The amount of mediation, (the indirect effect), is defined as the reduction of the effect of the initial variable on the outcome
= c - c' 
This difference in coefficients is theoretically exactly the same as the product of the effect of X on M times the effect of M on Y , ab;
thus it holds that
ab =c - c'. 

18. Inconsistent mediation
MacKinnon, Fairchild, and Fritz (2007) used the term "inconsistent mediation” when c' is opposite in sign to ab
In this case the mediator acts like a suppressor variable.
Step 1 would not necessarily be met, but there is still mediation c = c‘+ab 
MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S.  (2007).
Mediation analysis. Annual Review of Psychology, 58,,

19. Measuring Mediation
For multilevel models, logistic analysis and structural equation model with latent variables  it is inadvisable to compute c from Step 1, but rather c should be inferred to be
c' + ab
and not directly computed

20. Sobel test Tests the hypothesis that
is the estimated variance of the estimated coefficient a
is the estimated variance of the estimated coefficient b

21. Sobel test In practice they give almost same result since
Other versions:
In practice they give almost same result since
Is very close to zero

22. Tests for the mediated effect based on normal theory can yield inaccurate confidence limits and significance tests as the product of two normally distributed variables is not itself normally distributed.

23. Preacher & Leonardelli
The Sobel test works well only in large samples.
We recommend using this test only if the user has no access to raw data.
If you have the raw data, bootstrapping offers a much better alternative that imposes no distributional assumptions.
Preacher, K. J., & Hayes, A. F. (2004).
Behavior Research Methods, Instruments & Computers, 36(4),

24. FROM WIKIPEDIA
Bootstrapping is becoming the most popular method of testing mediation because it does not require the normality assumption to be met, and because it can be effectively utilized with smaller sample sizes .
However, mediation continues to be (perhaps inappropriately) most frequently determined using the logic of Baron and Kenny or the Sobel test.

25. Bootstrap
A short tutorial

26. Computer Intensive Methods
Enable to do simulations

27. It is an applicable alternative to inference based on parametric assumptions when
those assumptions are in doubt
or where parametric inference is impossible or it requires very complicated formulas for the calculation of standard errors

28. The advantage of bootstrapping over analytical methods is its great simplicity.
It is straightforward to apply the bootstrap to derive estimates of
standard errors and confidence intervals
for complex estimators of complex parameters

29. Inference on Mean We know its sampling distribution
What do we mean by the “sampling distribution”?

30. Sampling Distribution
If we drew another sample under the SAME conditions and recalculate the mean for each sample, what would be the distribution of those

31. Sampling Distribution
If we drew another sample under the SAME conditions and recalculate the mean for each sample, we know that
And we use it for calculating confidence intervals, and doing inference on

32. Sampling Distribution
Suppose we are interested in a ratio,
( e.g. Sample of teachers):
Y number of absence hours
X number of absence days
Estimate required for

33. Sampling Distribution
We use as an estimate
What is its sampling distribution?
What is the sampling distribution of a correlation r?

34. Sampling Distribution
If we drew other samples under the SAME conditions and recalculate the statistic for each sample, what would be the distribution of those values?

35. Sampling Distribution
The distribution of the population is represented by the “sample”.
For univariate data it is the n observations , each with the same “weight” 1/n
For bivariate data, it is the n pairs , each pair with probability 1/n

36. An alternative approach Bootstrap
Bootstrap the sampling distribution of ab and derive a confidence interval with the empirically derived bootstrapped sampling distribution.
The bootstrapping is accomplished by taking a large number of samples of size n (where n is the original sample size) from the data, sampling with replacement, and computing the indirect effect, ab, in each sample.

37. BOOTSTRAP very small illustration
Maya=4
Guy=3
Ron=2
Dan=5
Gil=5
Tom=5
Boot Sample 1000
N=7; Mean=4.14
Maya=4
Ron=2
Dan=5
Gil=5
Boot Sample 1
N=7; Mean=3.86
Maya=4
Dan=5
Gil=5
Tom=5
Boot Sample 2
N=7; Mean=4.71
Maya=4
Ron=2
Dan=5
Gil=5
Tom=5
Boot Sample 3
N=7; Mean=4.29
,…,
Random selection with replacements: A given case can be selected as part
of a bootstrap sample not at all, once, twice, or even multiple times.
Maya=4
Guy=3
Ron=2
Dan=5
Gil=5
Tom=5
Ben=5
Original Sample
N=7; Mean=4.14

38. Bootstrap Preacher& Hayes(2004) Behavior Research Methods, Instruments, & Computers
Percentile Bootstrap
The sample values at the a/2 and 1-a/2 percentiles of the bootstrap sampling distribution are used as the lower and upper confidence limits.

39. Percentile Bootstrap Explanation:
Assume for illustration that 1,000 bootstrap samples have been taken.
The point estimate of ab is simply the mean ab
computed over the 1,000 samples. The sd of the values is the SE.
To derive the 1-α confidence interval:
SORT the elements of the vector of 1,000 estimates of ab from low to high.
The lower limit of the confidence interval is the (α/2)1000 value, and the upper limit is the
(1-α/2)1000 ‘th value in the sorted sample.

40. MeShrout & Bolger (2002) Simulation Results
It is not always the case that inferences based on bootstrap procedures will differ from the standard normal-theory method. For example, the results for c’ would lead to similar conclusions.
Estimates and SE based on one sample N=80 from a population in which 𝑎=0.4;𝑏=0.3;𝑎𝑛𝑑 𝑐^′=0
Mean and SD’s of 1,000 bootstrap estimates

41. Percentile Bootstrap
The confidence interval can and often is asymmetric ( in accordance with the skewness of the sampling distribution of ab)
If 0 is outside the CI , the null hypothesis of no mediation is rejected at the α level of significance.

42. Bootstrap-McKinnon et al.(2004)
Bias Corrected Bootstrap
In the paper of McKinnon et al. several re- sampling methods are described and assessed. The bias corrected was found best.
The bias is expressed by the Z score of the value obtained from the proportion of the bootstrap samples below the original estimate

43. Bootstrap-McKinnon et al.(2004)
Bias Corrected Bootstrap
“The single best method overall was the bias- corrected bootstrap which had Type1 error rates closest to the nominal level along with more power than the other methods”

44. EXAMPLE:
Estimate of ab was with SE=0.147
With B=1000
The average of the a*b*'s was
with SD=0.047
95% CI Using Bootstrap Percentile ( 0.01,0.20)
Conclusion:………..

45. Minor Drawbacks of Bootstrap
More time consuming for the computer
Bootstrap yields slightly different CI’s each time you apply this method to the same data.

46. Distal and Proximal Mediation
             Hoyle and Kenny (1999) define a proximal mediator as a being greater than b (all variables standardized) and a distal mediator as b being greater than a.
           

47. Multicolinerity
Multicollinearity is to be expected in a mediational analysis and it cannot be avoided.
Effective sample size is approximately N(1 - r2) where N is the total sample size and r is the correlation between X and M. 
So, if M is a strong mediator (path a is large), to achieve equivalent power the sample size would have to be larger than what it would be if M were a weak mediator.  

48. MacKinnon et al. ( 2000) Mediation , Confounding, Suppression
In these three cases, three variables.
The interest is in the role of a third variable in the relationship between X and Y.
In Mediation: X causes M which causes Y

49. Confounding The concept of a confounding variable has been
developed primarily in the context of the health sciences and epidemiological research
A confounder is a variable related to two factors of interest that falsely obscures or accentuates the relationship between them
Adjustment for the confounder provides an undistorted estimate of the relationship between the independent and dependent variables

50. MacKinnon et al. ( 2000) Confounding
The third variable, relates to X and Y and therefore explains the relationship between X and Y.
The direction of the arrow a is reversed.
Example:
X income, Y cancer incidence , age is a confounder
BUT not mediator, income does not cause age…

51. MacKinnon et al. ( 2000) Mediation, Confounding
The statistical adjustment for the third variable, in both cases, reduces the magnitude of the relationship between X and Y.

52. MacKinnon et al. ( 2000) Suppression
When the statistical adjustment for the third variable, increases the magnitude of the relationship between X and Y.

53. Suppressor Help-seeking Stressor strain
Stressor
Help-seeking
strain
A theoretically interesting association can become empirically weak in magnitude, not only because it diminishes over time. Even proximal effects can be substantially diminished if
they are suppressed by a competing process.
Why we expect path c’ to be positive: It is possible that stressors does not lead to strain for people who seek help, so if we control for help-seeking we expect to see a positive relationship between stressor and strain.
In a population of persons who possess good coping skills, the magnitude of c’ may be similar to that of a× b but opposite in sign (when people are high of cpoing skills, the seek help in response to stressors so we will see a relatively large magnitude of indirect effect through help-seeking). In this case, the total effect may be close to zero (MacKinnon et al., 2000).
Clearly, the bivariate effect X → Y obscures the complexity of the causal relations between these variables.

54. Indirect effects and mediation
It is possible for M to be causally between X and Y even if X and Y aren’t associated.
In this case, some prefer to avoid the term mediator when describing M and instead refer simply to X’s indirect effect on Y through M
Mathieu & Taylor (Journal of Organizational Behavior,2006)

55. Wood et al. ORM(2008)
For all approaches, the assumption of causality is implicit in the definition of mediation. A mediator is defined as an explanatory mechanism through which one variable affects another. However, including a mediator in a study does not guarantee that the commonly accepted conditions for inferring causality are met.

56. Addictive behaviors and addiction-prone personality
traits:
Associations with a dopamine multilocus genetic profile
Davis &Loxton
Addictive Behaviors 38 (2013) 2306–2312

57. Path A was tested by regressing the addictive personality
scale on MLGP (controlling for age, gender, and ethnicity).
Results indicated that a higher MLGP score – reflective of a stronger
dopamine signal in the striatal region of the brain – was positively
and significantly associated with greater addictive personality traits
(B = 0.85, BSE = 0.32, p = 0.009; RSQUARE=0.04
Path B was tested by regressing addictive behaviors
on the addictive personality scale (controlling for age, gender,
and ethnicity). Results confirmed a highly significant relationship
between these two variables (B = 4.03, BSE = 0.54, p<0.0001;
R2 = 0.24)

58. In the final step, the results indicated
that the measure of addictive personality traits was still a highly significant predictor in the model, but that MLGP no longer contributed significantly to the variance in addictive behaviors (B = 1.90, BSE = 2.46, p = 0.44).

59. In accordance with the recommendations of Hayes (2009), the
mediated (i.e. indirect) effect was tested for significance using the
bootstrapping procedures outlined by Preacher and Hayes (2004).
Bootstrapping uses the original sample as the population from
which random samples with replacement are used to provide the
Best (????) estimate of the true indirect effect. Employing the Preacher and
Hayes (2008) SPSS INDIRECT macro, 5000 bootstrap samples were
created to estimate bias-corrected standard errors and 95% percentile
confidence intervals for the indirect effect of MLGP scores on addictions
via an addictive personality (controlling for age, gender, and
ethnicity).
MY COMMENT/QUESTION:
WAS BOOTSTRAP REQUIRED??????

60. Results of the bias-corrected bootstrapped analyses supported the findings of the earlier mediation analyses by showing that the MLGP score had a significant indirect effect on the frequency and strength of addictive behaviors via addictive personality traits (B = 3.42, BSE = 1.29), with a 95% BC confidence interval ranging from 1.15 to The absence of zero within the confidence interval range supports the hypothesis that addictive personality scores significantly mediated the relationship between a genotypic risk profile and the proneness to addictive behaviors.

61. Multiple Mediator Models
c - the unstandardized coefficient
represents the total effect of X on Y
The path c’ is the direct effect of X on Y
The indirect effect of X on Y is via
the j mediators

62. Multiple Mediator Models
The specific indirect effect of X on Y via mediator i is defined as the product of the two unstandardized paths linking X to Y via mediator Mi
=aibi.
Advantages: are analogous to using one multiple regression model instead of several simple regressions.

63. Multiple Mediator Models
We then can determine to which extent specific M variables transmit the X→Y effect, conditional on the presence of the other mediators in the model.
It allows to compare the effects of the mediators.

64. Multiple Mediator Models
In the past was not often used, probably due to lack of “tools”.
Nowadays, is getting more popular.
Applicable easily by using BOOTSTRAP

65. No Changes to the bootstrapping are required to
implement it in the multiple mediator case.
We generate the bootstrap sampling distributions
of the TOTAL and specific indirect effects
by taking samples with replacement
from the data k times.

66. Multilevel-Mediation
Mediation with Hierarchical Data

67. Conditional Indirect Effects:
(Moderated Mediation, Mediated Moderation)