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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/-Research 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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MediatorA 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

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

a b

c’

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Relation of X to Y

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

c

1. The independent variable is related to the dependent variable:

Y = i1 + cX +

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Relation of X to M

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

2. The independent variable is related to the potential mediator:

M = i2 + aX +

a

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Relation of X and M to Y

MEDIATOR

M

INDEPENDENT VARIABLE

X Y

DEPENDENT VARIABLE

a

3. The mediator is related to the dependent variable controlling for exposure to the independent variable:

Y = i3+ c’X + bM +

b

c’

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Mediated Effect Measures

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

2 22 2aba bs s

ab

2 22 2aba bs s

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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 =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.

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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/[(1+ncluster-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 = β0j + eij

Group, Level 2: β0j = γ00 + cjXj + u0j

ith individual in the jth group. The group level intercept, β0j, 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 = β0j + bi Mij + eij

Group, Level 2: β0j = γ00 + c’jXj + u0j

The bi parameter is at the individual level because the mediator is assumed to work through

individuals and the c’j 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 = β0j + eij

Group, Level 2: β0j = γ00 + 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 +b2sa

2 + sa2sb

2 + 2absbsarab + sa2sb

2 rab

2

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.