Multilevel Mediation Overview

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Multilevel Mediation Overview
-Mediation -Multilevel data as a nuisance and an
opportunity -Mediation in Multilevel
Models -http//www.public.asu.edu/davidpm/ -Rese
arch Funded by National Institute on Drug Abuse
and Prevention Science Methodology Group
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Mediation Statements

• If norms become less tolerant about smoking then
  smoking will decrease.
• If at-risk children are taught in classrooms with
  appropriate management, they will have more
  educational success.
• If parents learn effective discipline, the
  negative effects of divorce will be reduced.
• If parental monitoring is increased then
  adolescents will be less likely to use drugs.

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Mediator

• 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.

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Single Mediator Model
MEDIATOR
M
a
b
INDEPENDENT VARIABLE
DEPENDENT VARIABLE
c
X
Y
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Relation of X to Y
MEDIATOR
M
INDEPENDENT VARIABLE
DEPENDENT VARIABLE
c
X
Y

• The independent variable is related to the
  dependent variable
• Y i1 cX e1

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Relation of X to M
MEDIATOR
M
a
INDEPENDENT VARIABLE
DEPENDENT VARIABLE
X
Y
2. The independent variable is related to the
potential mediator M i2 aX e2
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Relation of X and M to Y
MEDIATOR
M
a
b
INDEPENDENT VARIABLE
DEPENDENT VARIABLE
c
X
Y
3. The mediator is related to the dependent
variable controlling for exposure to the
independent variable Y i3 cX bM e3
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Mediated Effect Measures
Mediated effectab Standard error Mediated
effectabc-c (MacKinnon et al., 1995) Direct
effect c Total effect abcc Test
for significant mediation z Compare to
empirical distribution of the mediated effect
ab
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Mediation Assumptions I

• For each method of estimating the mediated effect
  based on Equations 2 and 3 (ab) or Equations 1
  and 3 (c c)
• Predictor variables are uncorrelated with the
  error in each equation.
• Errors are uncorrelated across equations.
• Predictor variables in one equation are
  uncorrelated with the error in other equations.
• Correctly specified model.
• Independent Observations Violations are the
  subject of this presentation

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Importance of Mediation in Prevention and
Treatment Research

• Mediation is important because it provides
  information about how a program works or fails to
  work. Practical implications include reduced cost
  and more effective interventions.
• Mediation analysis is an ideal way to test
  theory. A theory based approach focuses on the
  processes underlying programs. Action theory
  corresponds to how the program will affect
  mediators. Conceptual Theory focuses on how the
  mediators are related to the dependent variables
  (Chen, 1990, Lipsey, 1993).

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Grouping/Clustering Variables in Prevention
Research

• Schools, Clinics, Classrooms, Therapy Groups
• Families, Siblings, Dyads
• Cities, Counties, Courts, Zipcodes, Countries
• Also observations from Individuals observations
  from different times.

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Clustering and Independent Observations

• 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.

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Intraclass Correlation (ICC)

• 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 too / (too s2)
• where too is variance among groups and s2 is the
  variance among individuals.
• Many different ICCs depending on additional
  predictors in the model.

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Example ICC values

• .01 number of cigarettes smoked (Murray et al.,
  1994) and clustering by schools.
• .02 for physical activity among girls (Murray et
  al., 2004).
• .001 to .12 for mediators of social norms,
  attitudes, knowledge for football players in high
  schools (Krull MacKinnon, 1999).

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Why is a nonzero ICC a problem?

• Increases Type I error rates if it is ignored
  (Barcikowski, 1981).
• Actual sample size is smaller than observed
  sample size because of violation of independence
  (Hox, 2002).
• Effective sample size is
  Neffective ntotal/(1ncluster-1)ICC,
• where ntotal is the total sample size and
  ncluster is the number of persons in each
  cluster.

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Multilevel Mediation Examples

• Residential instability reduced collective
  efficacy which increased violence (neighborhoods,
  Sampson et al., 1997)
• Anabolic prevention program affects norms
  regarding healthy behavior which reduced
  intentions to use steroids (high school football
  teams, Krull MacKinnon, 1999 2001).
• Alcohol prevention program affected norms which
  reduced alcohol use, (schools, Komro et al., 2001)

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Symposium Multilevel Mediation Examples

• Stressors to coping to distress. Cluster is
  observations within individuals ( Dan Feaster et
  al., )
• Stress to communication to marital quality.
  Cluster is dyads of husband/wife (Getachew Dagne
  et al.,).
• Longitudinal relations between stress and
  depression. Cluster is observations within
  individuals (George Howe et al.).

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Model for the X to Y relation

• Individual, Level 1 Yij ß0j eij
• Group, Level 2 ß0j ?00 cjXj u0j
• ith individual in the jth group. The group level
  intercept, ß0j, is the dependent variable in the
  Level 2 equation. Note that cj is at the group
  level because assignment is at the group level
  for this example. It is possible to have
  individual ci, and/or group level cj coefficients.

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Model for Y Predicted by X and M

• Individual, Level 1 Yij ß0j bi Mij eij
• Group, Level 2 ß0j ?00 cjXj u0j
• The bi parameter is at the individual level
  because the mediator is assumed to work through
  individuals and the cj parameter is at the group
  level because of assignment by group. Other
  analyses may have b and c coefficients at
  different or all levels. Note the slopes, bj,
  may be the dependent variable in another equation
  so slopes are a random coefficient that differs
  across groups.

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Model for the X to M relation

• Individual, Level 1 Mij ß0j eij
• Group, Level 2 ß0j ?00 ajXj u0j
• X predicts the dependent variable M. The aj
  parameter is estimated at the group level because
  assignment to conditions is at the group level
  for this example. Again it is possible to have
  individual, ai, and/or group level, aj
  coefficients.

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Multilevel Mediation Opportunities

• Example with X at the group level and M and Y at
  the individual level is common.
• There are many other opportunities. Slopes may be
  random coefficients that differ across groups.
  The slope relating M to Y may differ across
  groups. If X codes assignment then the X to M
  relation is not random. But if X differs across
  individuals, then the X to M and M to Y slopes
  may both be random.

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Multilevel mediation effects at for two-level
models

• 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.

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The ab and c-c estimators

• 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).

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The ab standard error estimators

• 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.

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What if a and b coefficients represent random
effects?

• 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.

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When will a and b coefficients represent random
effects?

• 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.

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But how do you get the correlation between a and
b when they are random?

• 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.

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Summary and Future Directions

• 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.