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Copyright 2003, D. P. MacKinnon 1
Mediator and Moderator MethodsDavid P. MacKinnon
Arizona State University
AABT November 22, 2003
PAlternative Methods to Assess Mediation and Moderation
PNational Institute on Drug Abuse GrantPhttp://www.public.asu.edu/~davidpm/
P“Everyone talks about the weather but nobody does anything about it.” (Mark Twain)
Copyright 2003, D. P. MacKinnon 2
Mediators and Moderators
Nursing “.. Should consider hypotheses about mediators and moderators that could provide additional information about why an observed phenomenon occurs” (Bennett, 2000).
Children’s programs “.. Including even one mediator and one moderator in a program theory and testing it with the evaluation .. will yield more fruit….” (Petrosino, 2000)
Child mental health “rapid progress … depends on efforts to identify moderators and mediators of treatment outcome. We recommend randomized clinical trials routinely include and report such analyses” (Kraemer et al., 2002).
Copyright 2003, D. P. MacKinnon 3
2, 3, 4, or more variable effects. Two variables: X Y, Y X , X Y are reciprocally related. Measures of effect include the correlation, covariance, regression coefficient, odds ratio, mean difference.
Three variables: X M Y, XY M, YXM, and all combinations of reciprocal relations. Special names for third variable effects, confounder, mediator, moderator/interaction.
Four variables: many possible relations among variables, e.g., XZMY
Copyright 2003, D. P. MacKinnon 4
MediatorA variable that is intermediate in the causal process relating an
independent to a dependent variable.
PChild Psychotherapy induces catharsis, insight and other mediators which lead to a better
outcome (Freedheim & Russ, 1981)
PPsychotherapy changes attributional style which reduces depression (Hollon, Evans, & DeRubies,
1991)
PParenting programs reduce parents negative discipline which reduces symptoms among
children with ADHD (Hinshaw, 2002).
Copyright 2003, D. P. MacKinnon 5
Single Mediator Model
MEDIATOR
M
INDEPENDENT VARIABLE
X Y
DEPENDENT VARIABLE
a b
c’
Copyright 2003, D. P. MacKinnon 6
More Mediator Definitions A mediator is a variable in a chain whereby an
independent variable causes the mediator which in turn causes the outcome variable (Sobel, 1990)
The generative mechanism through which the focal independent variable is able to influence the dependent variable (Baron & Kenny, 1986)
A variable that occurs in a causal pathway from an independent variable to a dependent variable. It causes variation in the dependent variable and itself is caused to vary by the independent variable (Last, 1988)
Copyright 2003, D. P. MacKinnon 7
Other names for Mediators and the Mediated Effect
Intervening variable is a variable that comes in between two others.
Process variable because it represents the process by which X affects Y.
Intermediate or surrogate endpoint is a variable that can used in place of an ultimate endpoint.
Indirect Effect to indicate that there is a direct effect of X on Y and there is an indirect effect of X on Y through M.
Proximal to distal variables
Copyright 2003, D. P. MacKinnon 8
Mediation in Psychotherapy (Freedheim & Russ, 1992)
Program outcome
Catharsis
Correction of
emotional experienc
e
Insight and
emotional resolution
Problem Solving
and Coping
Copyright 2003, D. P. MacKinnon 9
Reasons for Mediation Analysis in Psychotherapy Research
• Mediation is important for clinical science. Practical implications include reduced cost and more effective treatments.
• Mediation analysis is based on theory for the processes underlying treatments. Action theory corresponds to how the treatment will affect mediators—the X to M relation. Conceptual Theory focuses on how the mediators are related to the outcome variables—the M to Y relation (Chen, 1990, Lipsey, 1993).
Copyright 2003, D. P. MacKinnon 10
Questions about mediators selected for therapy
• Are these the right mediators? Are they causally related to the outcome? Is self-esteem causally related to symptoms? Conceptual Theory
• Can these mediators be changed? Can personality be changed? Action Theory
• Will the change in these mediators that we can muster with our treatment be sufficient to lead to desired change in the outcome? Do we have the resources to change self-esteem in four sessions? Both Action and Conceptual Theory.
Copyright 2003, D. P. MacKinnon 11
Mediation Causal Steps Test
Series of steps described in Judd & Kenny (1981) and Baron & Kenny (1986).
One of the most widely used methods to assess mediation in psychology.
Consists of a series of tests required for mediation as shown in the next slides.
Copyright 2003, D. P. MacKinnon 12
Step 1
MEDIATOR
M
INDEPENDENT VARIABLE
X Y
DEPENDENT VARIABLE
c
1. The independent variable causes the dependent variable:
Y = i1 + cX +
Copyright 2003, D. P. MacKinnon 13
Step 2
MEDIATOR
M
INDEPENDENT VARIABLE
X Y
DEPENDENT VARIABLE
2. The independent variable causes the potential mediator:
M = i2 + aX +
a
Copyright 2003, D. P. MacKinnon 14
Step 3
MEDIATOR
M
INDEPENDENT VARIABLE
X Y
DEPENDENT VARIABLE
a
3. The mediator must cause the dependent variable controlling for the independent variable:
Y = i3+ c’X + bM +
b
c’
Copyright 2003, D. P. MacKinnon 15
Mediated Effect Measures
Mediated effect=ab Standard error=
Mediated effect=ab=c-c’ (see MacKinnon et al., 1995 for a
proof)
Direct effect= c’ Total effect= ab+c’=c
Test for significant mediation:
z’= or compute Confidence Limits
2 22 2aba bs s
ab
2 22 2aba bs s
Copyright 2003, D. P. MacKinnon 16
Results of Statistical Simulation Study (MacKinnon et al., 2002)
Substantial differences in Type I error rates and power across causal steps, difference in
coefficients (c-c’), and product of coefficients
(ab) methods. Causal steps described in Baron and Kenny (1986) have low power for small
effects.A product of coefficient test has good balance of
power and Type I error rates, can be extended to longitudinal and multiple mediators, and has clear links with action and conceptual theory.
Copyright 2003, D. P. MacKinnon 17
Mediation Methods
Mediated effect=ab Standard error=Confidence intervals based on the distribution of the
product of two random variables are more accurate than existing methods and methods in common use
have low power (MacKinnon et al., 2002). Confidence intervals based on the bias-corrected bootstrap are most accurate overall (MacKinnon,
Lockwood, & Williams, in press).
2 22 2aba bs s
Copyright 2003, D. P. MacKinnon 18
Mediator versus Moderator
Moderator is a variable that affects the strength or form of the relation between two variables. The variable is not intermediate in the causal sequence, so it is not a mediator.
Moderator is also an interaction, the relation between X and Y depends on a third variable. There are other more detailed definitions of a moderator.
Tested by including interaction effects in addition to main effects of X.
Copyright 2003, D. P. MacKinnon 19
Moderators
Moderators determine for whom the program is most effective. Could be used to match treatments.
Moderators may also include mediation, e.g., the relation between X and M differs across groups, a, M and Y differs across groups, b, or whether the mediated effect ab, differs across groups.
Copyright 2003, D. P. MacKinnon 20
MacArthur Five Types of Risk Factors (see Kraemer et al. 2001)
1. Proxy- correlated risk factors that represent only part of the true risk factor, e.g., physical symptom attributions are a proxy for attributional style, the critical risk factor.
2. Overlapping-correlated risk factors but no temporal precedence.
3. Independent- uncorrelated risk factors.
4. Mediators-correlated risk factors measured after treatment.
5. Moderators-must be measured at the beginning of the study
Copyright 2003, D. P. MacKinnon 21
MacArthur Model
Often exploratory because of our limited knowledge about moderators and mediators in psychotherapy.
The type of risk factor is based on observed measures such as Spearman correlations and regression coefficients.
Enter variables and when they were measured. The method determines each variable as one of the five possible risk factors based on temporal position, correlation between risk factors, and relation to the outcome.
Copyright 2003, D. P. MacKinnon 22
MacArthur Steps for the Single Mediator Model
• X precedes M which precedes Y in a sequence.• Step 1: Test whether X is correlated with Y. Test whether
M is correlated with Y. If the Spearman correlation is greater than a certain value proceed. Sometimes reasonable to proceed if X is not correlated with Y.
• Step 2: Show that the effect of X on Y can be explained in part by M. The analysis method can be linear regressoin or some other method. If the interaction of X and M significantly predicts Y then there is also evidence of mediation.
• Test of mediation consists of steps.
Copyright 2003, D. P. MacKinnon 23
MacArthur Mediator and Moderator Model
• MacArthur model is similar as the BK tests in that it tests the relation between X and Y. Reduced power. MM does not require this association and recent BK articles do not make this requirement (MacKinnon et al., 2002).
• MM differs from BK as the correlation between M and Y may be compared to a criterion. BK does not have this requirement but it emerges later in BK when the relation between M and Y must be significant, adjusted for X.
• MM and BK tests reduce to the same thing, a separate test of the a path and a test of the b path.
• XM interaction in BK is tested based on theory or assumption. XM interaction of a primary part of MM.
Copyright 2003, D. P. MacKinnon 24
Similarities and Differences• Agree that investigating mediators and moderators is a good idea!• MM and BK can be used for exploratory and confirmatory analysis.• MM highlights the importance of temporal precedence and the
possibility of differential relations between M and Y across X. BK includes XM interaction as an assumption that is tested based on theory. Temporal precedence is important for both BK and MM for both moderators and mediators.
• MM a type of mediation test based on steps that is very similar to BK and other causal step tests.
• MM does not assume linearity or normality. Although not explicitly stated, most applications of BK make these assumptions.
Copyright 2003, D. P. MacKinnon 25
Mediation Assumptions I Temporal precedence X before M before Y:
Collect longitudinal experimental data. No measurement error: use reliable measures or
estimate measurement models (Hoyle & Kenny, 1999), timing and spacing of measurements.
No omitted variables: Program of study to examine candidate explanatory variables.
No missing data: Use existing adjustments and investigate sources of missing data.
Normally distributed scores: Apply methods for categorical variables or use resampling methods, MM does not make this assumption
Copyright 2003, D. P. MacKinnon 26
Assumptions II• Mediation chain is correct: Any mediation model is
part of a longer mediation chain. The researcher decides what part of the micromediational chain to examine. Similar decisions must be made about outcomes and timing. Niles (1922) and Wright (1923) agreed on this.
• Homogeneous effects across subgroups: It assumed that the relation from X to M and from M to Y are homogeneous across subgroups or other characteristics of participants in the study. This means there are not moderator effects. Central part of MM
Copyright 2003, D. P. MacKinnon 27
Future Directions
• More applications of mediation and moderation analysis.
• Programs of research to solve the limitations of single studies including the best experiments to follow a mediation and moderation analysis.
• Continue work extending mediation methods in survival analysis, longitudinal growth curve modeling, and multilevel models.
• Address action theory in addition to conceptual theory. Measure mediators and moderators in every study.
Copyright 2003, D. P. MacKinnon 28
Symposium on Mediators and Moderators: Hypothesized Effects
Mediator and Moderator
Symposium
# Studies with Med. and Mod.
Analysis
Interest in Med. and Mod.
Analysis
Norms Regarding Reporting Results of Studies
Comprehension of Reasons for Med. and Mod.
Analysis
Beliefs About the Importance
of Theory Testing