Copyright 2003, D. P. MacKinnon 1 Mediator and Moderator Methods David P. MacKinnon Arizona State University AABT November 22, 2003 PAlternative Methods to Assess Mediation and Moderation PNational Institute on Drug Abuse Grant Phttp:// 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, X  Y  M, Y  X  M, 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., X  Z  M  Y

Copyright 2003, D. P. MacKinnon 4 Mediator A variable that is intermediate in the causal process relating an independent to a dependent variable. P Child Psychotherapy induces catharsis, insight and other mediators which lead to a better outcome (Freedheim & Russ, 1981) P Psychotherapy changes attributional style which reduces depression (Hollon, Evans, & DeRubies, 1991) P Parenting 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 XY DEPENDENT VARIABLE ab 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 experience 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 XY DEPENDENT VARIABLE c 1.The independent variable causes the dependent variable: Y = i 1 + c X +  

Copyright 2003, D. P. MacKinnon 13 Step 2 MEDIATOR M INDEPENDENT VARIABLE XY DEPENDENT VARIABLE 2. The independent variable causes the potential mediator: M = i 2 + a X +   a

Copyright 2003, D. P. MacKinnon 14 Step 3 MEDIATOR M INDEPENDENT VARIABLE XY DEPENDENT VARIABLE a 3. The mediator must cause the dependent variable controlling for the independent variable: Y = i 3 + c’ X + b M +   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 abab

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).

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