Methods for AnalyzingMediation and Moderation

Jeffrey R. EdwardsUniversity of North Carolina

In honor of Lawrence R. JamesGeorgia Institute of Technology

Overview• Two perspectives on mediation– Structural equation modeling– Causal steps approach

• Mediation versus moderation• Integrating moderation and mediation– Structural models with contingent paths– Adaptation of the causal steps approach

• Key lessons and take-aways

Modest Beginnings• Some of the most profound scientific

insights have modest beginnings.• For example, a researcher might come

across a source of confusion in an area, see through the confusion with clarity and insight, and convey their views in ways that fundamentally reshape the thinking of other researchers.

• This describes the work of Larry James on mediation and moderation.

The Back Story• “I reviewed a manuscript for JAP in which I

commented to the editor, Robert Guion, that the paper appeared to confound mediation with moderation. Bob . . . asked if I would be interested in preparing a paper on this subject . . . I had no burning attraction to mediation per se . . . my interest in mediation is embedded in my overall attraction to structural equation modeling” (James, 2008, p. 359).

SEM Approach to Mediation• James and colleagues (James & Brett,

1984; James, Mulaik, & Brett, 1982) presented methods for conceptualizing and analyzing mediation based on SEM.

• By using SEM as a foundation, the meaning and analysis of mediation is straightforward, because mediated models are special cases of structural equation models.

James et al. (1982)• “if w is a mediating mechanism for the

function relating y to x, then the model has the form x → w → y. This model indicates two functional equations, namely, w = Bwxx + dw and y = Byww + dy. This suggests that explicit inclusion of w renders the (x,y) relationship indirect, where the effect of x on y must now pass through w” (p. 30).

James and Brett (1984)•“m is a mediator of the probabilistic

relation y = f(x) if m is a probabilistic function of x (i.e., m = f[x]) and y is a probabilistic function of m (i.e., y = f[m]), where x, m, and y have different ontological content (i.e., represent different hypothetical constructs or latent variables)” (p. 310).

^^

Model Specification• The equations for a complete mediation

model are as follows (for simplicity, we assume x, m, and y are standardized):

m = bmxx + em

y = bymm + ey

• This model is depicted as follows:

x m y

em eybmx bym

Model Testing• The complete mediation model is tested

using the following criteria:– bmx differs from zero; – bym differs from zero;– The correlation between x and y does not differ

from the product of bmx with bym, which represents the indirect effect of x on y transmitted through m.

• In SEM, the third criterion is a test of model fit with one degree of freedom.

Comparison With Other Methods

• The SEM approach to conceptualizing and testing mediation developed by James and colleagues is straightforward, incisive, and refreshingly clear.

• This approach stands in contrast to the original presentation of the causal steps approach (Baron & Kenny, 1986), which had an alluring simplicity that made it popular among applied researchers.

Baron and Kenny (1986)• “A variable functions as a mediator when

it meets the following conditions:– Variations in levels of the independent variable

significantly account for variation in the presumed mediator;

– Variations in the mediator significantly account for variations in the dependent variable;

– A previously significant relation between the independent and dependent variables is no longer significant” (p. 1176).

Model Specification• The equations for the causal steps

procedure are as follows:y = cyxx + ey

m = amx + em

y = bym + c’yx + e’y

• These equations represent two models, one that excludes the mediator, and another that includes the mediator.

Model Specification• These figures depict the models for

the causal steps approach:

x y

ey

cyx

x m y

em e’yamx bym

c’yx

Model Testing• With the causal steps procedure, mediation

is evidenced by the following criteria:– cyx differs from zero;– amx differs from zero;– bym differs from zero;– c’yx does not differ from zero.

• If the first three criteria are met, the fourth criteria gives the “strongest demonstration of mediation” (p. 1176).

Model Testing• The paths that should differ from

zero are now shown in red.

x y

ey

cyx

x m y

em e’yamx bym

c’yx

Comparing the Approaches• Model specification:– The SEM approach specifies the complete

mediation model as hypothesized.– The causal steps approach specifies two

models that include or exclude m, and the model with m adds a path from x to y.

• Under complete mediation, the SEM model is correctly specified, whereas both of the causal steps models are misspecified.

Another Look at the Models• The complete mediation model is:

x m y

em eybmx bym

• The first causal steps model excludes m:

x y

ey

cyx

Another Look at the Models• The complete mediation model is:

x m y

em eybmx bym

• The second causal steps model adds a path from x to y when none is predicted:

x m y

em e’yamx bym

c’yx

Comparing the Approaches• Model testing:– The SEM approach tests the two paths of the

mediated effect and the fit of the model, which addresses the omitted path.

– The causal steps approach tests a path from x to y in the model that excludes m and then tests the three paths of a partial mediation model.

• The SEM tests are aligned with the model, whereas the causal steps tests deviate from the model.

Revised Causal Steps Approach

• “One might ask whether all of the steps have to be met for there to be mediation. Certainly, step 4 does not have to be met unless the expectation is for complete mediation . . . step 1 is not required, but a path from the initial variable to the outcome variable is implied if steps 2 and 3 are met. So the essential steps in establishing mediation are steps 2 and 3” (Kenny et al., 1998, p. 260).

Reflections by Kenny (2008)• “Why is the Baron and Kenny (1986) article

so popular? After all, there were many good sources on the topic, most notably James and Brett (1984) . . . On the good side, it provided clear definitions of both mediation and moderation. It also provided clear and explicit advice on how to conduct a mediational analysis. On the bad side, Baron and Kenny (1986) are much too formulaic” (Kenny, 2008, p. 355).

Coming Full Circle on Mediation

• From the outset, James and colleagues framed mediation in terms of SEM with a model that directly incorporated the paths involved in the mediated effect.

• The causal steps approach complicated matters but eventually returned to the perspective advanced by James.

• The moral of the story is that we should specify and test models that represent our theory.

Moderation• James and Brett (1984) presented a

clear description of moderation and distinguished it from mediation.

• The James and Brett (1984) discussion of moderation is consistent with other informed treatments and has stood the test of time as a valid and useful point of reference.

Moderation According to James and Brett (1984)

• “a variable z is a moderator if the relationship between two . . . variables, say x and y, is a function of the level of z. This definition indicates an x by z interaction, or a nonadditive relation, where y is regarded as a probabilistic function of x and z . . . the function being y = b1x + b2z + b3xz + e for deviation scores and a model linear in the parameters” (p. 310).

Moderation According to Baron and Kenny (1986)

• “a moderator is a . . . variable that affects the direction and/or strength of the relation between an independent or predictor variable and a dependent or criterion variable . . . if the independent variable is denoted as X, the moderator as Z, and the dependent variable as Y, Y is regressed on X, Z, and XZ” (p. 1176).

Moderation According toAiken and West (1991)

• “The XZ interaction signifies that the regression of Y on X depends on the specific value of Z at which the slope of Y on X is measured. There is a different line for the regression of Y on X at each and every value of Z” (p. 10).

Moderated Mediation• James and Brett (1984) incorporated

moderation into mediated models by adding product terms to the equations used to analyze mediation, e.g.:

m = bmxx + bmzz + bm(xz)xz + em

y = bymx + byzz + by(mz)mz + ey

• James and Brett (1984) mapped these equations onto plots of simple slopes and diagrams of moderated path models.

Moderated Mediation• James and Brett (1984) gave a concrete

example of moderated mediation:– The effect of performance feedback on

attributions depends on self-esteem:• People with high self-esteem will attribute poor

performance to their effort, not their ability.• People with low self-esteem will attribute poor

performance to their ability, not their effort.– Attributions influence intended persistence on

subsequent tasks, with effort attributions leading to increased persistence.

Moderated Mediation• James and Brett (1984) depict their

moderated mediation model as follows:PerformanceFeedback

EffortAttribution

IntendedPersistence

HighSelf-Esteem

PerformanceFeedback

AbilityAttribution

IntendedPersistence

LowSelf-Esteem

• “The salient point here is that moderation may be functionally involved in the first-stage of a mediation relation” (p. 313).

An Alternative Based on the Causal Steps Approach

• Baron and Kenny (1986) present a causal steps approach to moderated mediation.

• This approach involves three equations: y = cyxx + cyzz + cy(xz)xz + ey

m = amxx + amzz + am(xz)xz + em

y = bymm + c’yxx + c’yzz + c’y(xz)xz + e’y

• A fourth equation adds xm, such that the effect of m is moderated by x rather than z. We will disregard this equation.

The Steps Used to Establish Mediated Moderation

• The conditions for the moderated causal steps procedure are as follows:– cyx, cyz, and cy(xz) differ from zero;– amx, amz, and am(xz) differ from zero;– bymm differs from zero;– c’yxx and c’yzz do not differ from zero;– c’y(xz)xz is less than cy(xz)xz.

• Mediated moderation is said to depend primarily on cy(xz), am(xz), bymm, and c’y(xz)xz.

Model Without the Mediator• The first step is based on a model

that excludes the mediator:

z y

ey

x

xz• The paths that should differ from

zero are shown in red.

Model With the Mediator• The remaining steps are based on a

model that includes the mediator:

• The pink path from xz to y should be smaller when m is added to the model.

zy

ey

x

xz

m

em

Model With the Mediator• The remaining steps are based on a

model that includes the mediator:

zy

ey

x

xz• The pink path from xz to y should be

smaller when m is added to the model.

m

emThe flashing paths are deemed critical to moderated mediation.

Comparing the Approaches• The approach presented by James and

Brett (1984) indicates that the paths of a mediated model depend on the level of the moderator variable.

• This approach targets the essence of moderated mediation, provides a clear analytical framework, and foreshadowed subsequent developments (Edwards & Lambert, 2007).

Comparing the Approaches• The causal steps approach to moderated

mediation has several drawbacks:– The path on x is tested despite the fact that

this path have no unique value, given that it depends on the level of z.

– The xz term is treated as a causal variable, even though it has no causal potential of its own. Rather, xz is simply a mathematical device to capture a contingent effect of x on y that depends on the level of z.

Comparing the Approaches• The causal steps approach to moderated

mediation has several drawbacks:– The change in the coefficient on xz when m

is added to the model does not itself show how the form of the interaction is altered.

–When moderated mediation is predicted, the initial model that excludes m is misspecified.

– The model that includes m also incorporates a moderated effect of x on y, regardless of whether it is hypothesized.

What We Have Learned From James and Colleagues• Specify the model that represents your

theory, and test that model.• Mediated models are structural equation

models, and we should view them as such.• SEM handles complex mediated models

that tax the causal steps approach.• Moderated mediation represents a model

with paths that vary across levels of the moderator.

Thank You, Larry James• We owe a debt of gratitude to Larry

James for deciphering the complexities of mediation, moderation, and how they can be combined and integrated.

• The clarity with which he has addressed these issues provides a model for those of us who aspire to follow in his steps.