Multilevel Mediation Overview. -Mediation -Multilevel data as a nuisance and an opportunity -Mediation in Multilevel Models -http://www.public.asu.edu/~davidpm/ -Research Funded by National Institute on Drug Abuse and Prevention Science Methodology Group. Mediation Statements.
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-Multilevel data as a nuisance and an opportunity
-Mediation in Multilevel Models
-Research Funded by National Institute on Drug Abuse and Prevention Science Methodology Group
A variable that is intermediate in the causal process relating an independent to a dependent variable.
Antecedent to Mediating to Consequent (James & Brett, 1984)
Initial to Mediator to Outcome (Kenny, Kashy & Bolger, 1998)
Program to surrogate endpoint to ultimate endpoint (Prentice, 1989)
Independent to Mediating to Dependent used in this presentation.
2. The independent variable is related to the potential mediator:
M = i2 + aX + e2
3. The mediator is related to the dependent variable controlling for exposure to the independent variable:
Y = i3+ c’X + bM + e3
Mediated effect=ab Standard error=
Mediated effect=ab=c-c’ (MacKinnon et al., 1995)
Direct effect=c’ Total effect=ab+c’=c
Test for significant mediation:
z’= Compare to empirical distribution
of the mediated effect
Observations in groups may lead to dependency among respondents in the same group.
The dependency could be due to communication among persons in the same group, similar backgrounds, or similar response biases.
Violation of independent observations an assumption of many statistical analyses.
ICC provides a measure of extent to which observations in a group tend to respond in the same way compared to other groups.
ICC ranges from 1 to –1/(k-1) where k is the number of subjects in each group.
ICC =τoo / (τoo + σ2)
where τoo is variance among groups and σ2 is the variance among individuals.
Many different ICCs depending on additional predictors in the model.
where ntotal is the total sample size and ncluster is the number of persons in each cluster.
Individual, Level 1: Yij = β0j + eij
Group, Level 2: β0j = γ00 + cjXj + u0j
Individual, Level 1 : Yij = β0j + bi Mij + eij
Group, Level 2: β0j = γ00 + c’jXj + u0j
Individual, Level 1 : Mij = β0j + eij
Group, Level 2: β0j = γ00 + ajXj + u0j
Level of X, M, and Y can be used to describe different types of multilevel models. Assume X, M, and Y are all measured at the individual level.
1 1 1; X, M, and Y measured at the individual level.
2 1 1; X at level 2, M and Y at the individual level.
2 2 1; X and M at level 2, Y at the individual level.
2 2 2; X, M, and Y level 2.
Models with more than two levels.
The ab and c-c’ estimators of the mediated effect, algebraically equivalent in single-level models, are not exactly equivalent in the multilevel models (Krull & MacKinnon, 1999). This is because the weighting matrix used to estimate the model properly in the multilevel equations is typically not identical for each of the three equations. The non-equivalence between ab and c-c’, however, is typically small and tends to vanish at larger sample sizes (Krull & MacKinnon, 1999).
The standard error of the mediated effect is calculated using the same formulas described above, except that the estimates and standard errors of a and b may come from equations at different levels of analysis and if both coefficients are random they may require the covariance between a and b.
The random coefficients a and b may be correlated so the covariance between a and b must be included in the standard error (Kenny, Bolger, & Korchmaros, 2003).
abrandom =ab + covariance(ab)
Var(abrandom) = a2sb2 +b2sa2 + sa2sb2 + 2absbsarab + sa2sb2 rab2
rab is the correlation between the a and b random coefficients.
Three variable longitudinal growth model where the relation of X to Y varies across individuals and the relation of M to Y varies across individuals.
Kenny et al. (2003) describe an example with daily measures of stressors, coping, and mood. The stressor to coping and coping to mood relations were random, i.e., varied across individuals.
Kenny et al. (2003) used a data driven approach where the values for a and b in each cluster were correlated.
Bauer et al. (2006) use a method so that all coefficients are estimated simultaneously so that the covariance between a and b is given.
New version of Mplus will estimate the correlation/covariance between random coefficients such as a and b.
Two views of multilevel data: (1) a nuisance in the statistical analysis and (2) an opportunity to investigate effects at different levels.
New Mplus version allows for estimation of models for random a and b effects. Bauer et al., (2006) describe a SAS approach to finding this covariance.
Can have very complicated models with many levels and potential mediation across and between levels.