1. Multilevel Mediation Zhen Zhang W. P. Carey School of Business
Arizona State University
CARMA Webcast
April 8, 2016

2. Overview Introduction 2-1-1 Mediation Extensions
Mediation and multilevel analysis
Two important notes for multilevel mediation
2-1-1 Mediation
Conflated, between-level, and within-level effects
Implications for theory and hypothesis development
Technical issues regarding random slopes, bootstrapping, and using latent means
Extensions
Other types of mediation models
Moderation analysis

3. Mediation Testing mediation effects Essential for theory development
Traditionally used Baron and Kenny’s (1986) approach
No longer need to examine path c (total effect X  Y) according to recent studies (Kenny & Judd, 2014; Rucker et al., 2011)
We will use a*b to quantify mediation (indirect) effects
We assume you already have a theory to support the implied causal directions
A mediator is a “variable, which represents the generative mechanism through which a focal independent variable is able to influence the dependent variable of interest” (Baron & Kenny, 1986, p.1173) M a b c X Y c' X Y

4. Multilevel Analysis Multilevel analysis
Accounts for nestedness in organizational data
Provides a richer understanding of level-related variances and relationships (Raudenbush & Bryk, 2002)
Adds complexity to mediation testing
We will start with a particular 2-level mediation model, 2-1-1, and then expand to other models X M Y
Level 2 Xj
Level 1
Mij
Yij

5. Note 1 Between- vs. within-level relationships
Are often different in magnitude and have different meanings
Simpson’s Paradox (e.g., Kievit et al., 2013)

6. Simpson’s Paradox Risk of heart attack Person A Person B Person C …
Amount of Daily Exercise

7. Note 1 Between- vs. within-level relationships
Are often different in magnitude and have different meanings
Simpson’s Paradox (e.g., Kievit et al., 2013)
Centering methods for level-1 variables are crucial; they can change the substantive meaning of the variable (Enders & Tofighi, 2007)
Group-mean centering the level-1 variables and adding back their means (either latent or observed means) at level 2 can help uncover the level-specific relationships

8. Note 2 Level of measurement ≠ the level where it is modeled
Researchers often use “level 1” and “level 2” in describing the models
Preacher et al. (2010) recommend using Between (B) level and Within (W) level to highlight where the variable is modeled
A level-1 (L1) variable may have both B- and W-level variances and effects
A level-2 (L2) variable has only B-level variance/effect

9. 2-1-1 Mediation 2, 1, and 1 refer to the levels of measurement
This model is conventionally depicted as: Xj c'
Level 2 a
Level 1
Mij
Yij b
Mediation effects were often calculated as a*b, where a and b came from multilevel modeling equations (e.g., HLM, SAS Proc Mixed, STATA Mixed)

10. 2-1-1 Mediation Why is this model important?
Prevalent in many disciplines and areas of research
Human Resources: Having a flexible-work system (L2 X) can increase employee satisfaction (L1 M) and therefore job performance (L1 Y)
Organizational Psychology/Behavior: Work team diversity (L2 X) may affect employee feedback-seeking behavior (L1 M) and therefore creativity (L1 Y)

11. 2-1-1 Mediation Why is this model important (cont’d) ?
Strategic Management: With firm panel data (years nested under a CEO, and assume only one CEO per firm), CEO personality (L2 X) can affect firm’s acquisitions in a given year (L1 M) and then next year’s financial performance (L1 Y)
Educational Psychology: With cluster-randomized trials (students nested within classrooms/teachers), a teacher training intervention (L2 X) may increase student motivation (L1 M) and then student achievement (L1 Y)

12. A Critical Look at 2-1-1 (Note 1)
The 1-1 relationship consists of B- and W-level effects
As shown in Zhang, Zyphur, and Preacher (2009), if grand-mean centering or no centering is used, the coefficient b is a conflated estimate of the true B- and W-level effects
Level 2
Level 1
Mij
Yij b

13. A Critical Look at 2-1-1 (Note 2)
Can L2 Xj influence the W-part of Mij?
Xj only has B-level variance and therefore path a is B-level
e.g., a teacher-level intervention is able to change the class average of student motivation (mean Mij), but not the within-class component of Mij X
Level 2 a
Level 1 M

14. B-level coefficient of Xj
Using HLM Equations
Within group variations of Mij
Group mean of Mij
B-level coefficient of Xj
If L2 Xj only affects the B-part of Mij
Then, should the W-level effect of M on Y in the 1-1 relationship be included in the mediation (causal) chain?
No, according to Preacher and colleagues (2009, 2010, in press)

15. A Clearer Graph xj mij yij mB yB mW yW X c' Level 2 a Level 1 M Y b
Between xj
mij
yij
Observed
Within mW yW

16. A Clearer Graph xj mij yij yB mB xB yW mW X c' Level 2 a Level 1 M Y b
Between
Within xj
mij
yij
Observed

17. What Does It Mean to Hypothesis Development?
We need to align hypothesis and analysis
Be sensitive to the separate B and W effects
“Decomposed-first” strategy vs. “conflated-first” strategy (Preacher et al., in press)
If you are only interested in a B-level phenomenon and the B parts of Mij and Yij have substantive meanings
Then the hypotheses need to focus on “collective” constructs such as team collective efficacy
If there is no a priori theoretical basis for the B parts of Mij and Yij
Then you may still hypothesize the effects of Xj on Mij or on Yij, but bear in mind that these effects only involve their B parts

18. What Does It Mean to Theory Building?
Very often, our theory building ignores a “tailored” step in the relationship
L2 Xj’s effect on the mediator is via a “tailored” Xij
Implementing flexible-work is often tailored to each employee
Each team member experiences and perceives team diversity in his/her own way
CEO personality can have meaningful year-to-year variations
After a training intervention, each teacher still customizes his/her improved instructions to a given student
Bringing the implicit “tailored Xij” back to the theoretical arguments
This is not “analytical tools lag behind theory development”
Instead, theory development could have explicitly included the “tailored” predictor in the causal chain (i.e., become a model)

19. 1-1-1 Mediation 1, 1, and 1 refer to the levels of measurement
This model is conventionally depicted as:
Level 2
Paths a, b, and c’ can be fixed or random in multilevel models
Level 1 a b
Xij
Mij
Yij c'

20. A Clearer Graph xij mij yij xB yB xW mW yW Level 2 a Level 1 b Xij MijmB yB
Between
xij
mij
yij
Within xW mW yW

21. Fixed vs. Random Slopes in 1-1-1 Models
When paths aw and bw are both fixed slopes
W-level mediation effect is aw * bw
According to Bauer et al. (2006), when both paths are random slopes
W-level mediation effect is aw* bw + Cov(aw, bw)
xij
mij
yij
Within xW mW yW

22. Bootstrapping for Confidence Intervals
The product a*b is not normally distributed
Case-based bootstrapping
Best for single level mediation testing
Difficult to implement for multilevel data
Monte Carlo bootstrapping
Recommended for multilevel mediation models (Preacher & Selig, 2012)
Can be implemented using programs such as R and SPSS after obtaining the asymptotic covariance matrix of the parameter estimates
Online utility at quantpsy.org/medn.htm

23. Why Can’t We Simply Use Regressions?
Regressions using un-centered variables and cluster-robust standard errors
Adjust for nestedness and is better than OLS
Cannot differentiate the B- and W-level specific effects, which are theoretically meaningful
The estimated coefficients are still conflated; in the simple case of a univariate regression: Slope Reg(yij, xij) = ICCx * SlopeB + (1 ‒ ICCx) * SlopeW

24. Using Latent vs. Observed Means
As compared to latent means, using observed means introduces biases
Bias in B-level main effect of 1-1 relationship (see Lüdtke et al, 2008, pp )
Bias in B-level mediation effect in model (Preacher et al., 2010, pp )
The Mplus program uses latent means and the B-level estimates do not suffer from such biases

25. Other Multilevel Mediation Models
1-1-2, 1-2-2, and 1-2-1
Longer chains such as
Recall the numbers represent levels of measurement, rather than the actual levels being analyzed
When there is at least one L2 variable involved in the mediation chain, the mediation effect needs to be calculated at the B-level

26. Extensions to Moderation
Preacher, Zhang, and Zyphur (in press) took the “decomposed-first” strategy to examine multilevel moderation models
1 ×(1→1)
2 ×(1→1)
2 ×(2→1)
1 ×(2→1)
The same logic can be extended to multilevel moderated mediation analysis

27. Summary Align theory and analysis
Decompose the B- and W-level variances and relationships to avoid conflated estimates
Using latent means is preferred
Monte Carlo bootstrapped confidence intervals are preferred as compared with significance testing of a*b

28. References
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, 1173–1182.
Bauer, D. J., Preacher, K. J., & Gil, K. M. (2006). Conceptualizing and testing random indirect effects and moderated Mediation in multilevel models: New procedures and recommendations. Psychological Methods, 11,
Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12, 121–138.
Kenny, D.A., & Judd, C.M. (2014) Power anomalies in testing mediation. Psychological Science, 25, 334–339.
Kievit, R. A., Frankenhuis, W. E., Waldorp, L. J., & Borsboom, D. (2013). Simpson’s paradox in psychological science: A practical guide. Frontiers in Psychology, 4, 513.
Lüdtke, O., Marsh, H. W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies. Psychological Methods, 13, 203–229.

29. References
Preacher, K. J., & Selig, J. P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6,
Preacher, K. J., Zhang, Z., & Zyphur, M. J. (In press). Multilevel structural equation models for assessing moderation within and across levels of analysis. Psychological Methods.
Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Psychological Methods, 15, 209–233.
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Newbury Park, CA: Sage.
Rucker, D. D., Preacher, K. J., Tormala, Z. L., & Petty, R. E. (2011). Mediation analysis in social psychology: Current practices and new recommendations. Social and Personality Psychology Compass, 5/6,
Zhang, Z., Zyphur, M. J., & Preacher, K. J. (2009). Testing multilevel mediation using hierarchical linear models: Problems and solutions. Organizational Research Methods, 12, 695–719.

30. Thank you!
Zhen Zhang