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A General Model for Testing Mediation and Moderation Effects

Amanda J. Fairchild and

Department of Psychology, University of South Carolina, Barnwell College, 1512 Pendleton St.,

Columbia, SC 29208, USA

David P. MacKinnon

Research in Prevention Lab, Department of Psychology, Arizona State University, P.O. Box

871104, Tempe, AZ 85287-1104, USA

Amanda J. Fairchild: afairchi@mailbox.sc.edu

Abstract

This paper describes methods for testing mediation and moderation effects in a dataset, both

together and separately. Investigations of this kind are especially valuable in prevention research

to obtain information on the process by which a program achieves its effects and whether the

program is effective for subgroups of individuals. A general model that simultaneously estimates

mediation and moderation effects is presented, and the utility of combining the effects into a single

model is described. Possible effects of interest in the model are explained, as are statistical

methods to assess these effects. The methods are further illustrated in a hypothetical prevention

program example.

Keywords

Mediation; Indirect effect; Moderation; Mediated moderation; Moderated mediation

Relations between variables are often more complex than simple bivariate relations between

a predictor and a criterion. Rather these relations may be modified by, or informed by, the

addition of a third variable in the research design. Examples of third variables include

suppressors, confounders, covariates, mediators, and moderators (MacKinnon et al. 2000).

Many of these third variable effects have been investigated in the research literature, and

more recent research has examined the influences of more than one third variable effect in

an analysis. The importance of investigating mediation and moderation effects together has

been recognized for some time in prevention science, but statistical methods to conduct

these analyses are only now being developed. Investigations of this kind are especially

valuable in prevention research where data may present several mediation and moderation

relations.

Previous research has described the differences between mediation and moderation and has

provided methods to analyze them separately (e.g., Dearing and Hamilton 2006; Frazier et

al. 2004; Gogineni et al. 1995; Rose et al. 2004). More recent research has presented models

to simultaneously estimate mediation and moderation to investigate how the effects work

together (e.g., Edwards and Lambert 2007; MacKinnon 2008; Muller et al. 2005; Preacher et

al. 2007). A review of the substantive literature illustrates that few applied research

examples have used these models, however. Although analyzing mediation and moderation

separately for the same data may be useful, as described later in this paper, simultaneous

Correspondence to: Amanda J. Fairchild, afairchi@mailbox.sc.edu.

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examination of the effects is often relevant and allows for the investigation of more varied,

complex research hypotheses.

What Type of Research Questions Can Be Addressed with the

Simultaneous Analysis of Mediation and Moderation Effects?

Is the Process By Which a Program Has an Effect the Same Across Different Types of

Participants?

In prevention and intervention research, the mediation model has been used to understand

the mechanism(s) by which program effects occur. To determine the generalizability of

these mechanisms or to explain an unexpectedly small mediated effect it may be of interest

to investigate whether the mediation relation, or the indirect effect, holds across different

subgroups (e.g., men vs. women or low-risk vs. high-risk). To investigate these hypotheses,

a researcher asks whether the indirect effect is moderated, or whether the mediated effect

depends on levels of another variable. For example, suppose that a business implements a

worksite-wellness program (the independent variable, X) to reduce obesity-related health

risks in its employees. Program developers hypothesize that by increasing employee

knowledge about the benefits of eating fruits and vegetables (the mediator variable, M),

employee consumption of fruits and vegetables will increase (the dependent variable, Y),

thus reducing health risk. An estimate of the indirect effect of the program on employee fruit

and vegetable consumption through employee knowledge of the benefits of eating fruits and

vegetables is unexpectedly low. Through talks with employees, it becomes apparent that

participants were more or less motivated to gain and use knowledge from the program to

improve their diet based on whether they had a family history of obesity-related illness such

as diabetes or cardiovascular disease. Program developers hypothesize that participants’

family history of obesity-related illness may moderate the mediation relation in the data,

affecting the influence of the program on employee knowledge of fruits and vegetables and

its subsequent impact on fruit and vegetable consumption (See Fig. 1).

Can a Mediation Relation Explain an Interaction Effect in My Data?

Suppose a similar worksite-wellness program was implemented in a larger sister company

and program effects had been dependent on whether the participant was a full or part-time

employee at the company. To investigate the underlying reasons for this unexpected

interaction, or moderation relation, program analysts could investigate a mediation

hypothesis where the interaction effect predicts a mediator variable which predicts the

outcome, defined here as the mediation of a moderator effect. For example, perhaps in

addition to increasing employee knowledge of fruit and vegetable benefits with the wellness

curriculum, the program (X) also introduced a work culture, or a social norm (M), of healthy

eating which contributed to employee fruit and vegetable consumption (Y; See Fig. 2).

Program developers hypothesize that the more hours an employee worked in a week

determined how much they were subjected to the social norm which ultimately influenced

their fruit and vegetable consumption.

Current Research

The purpose of this article is to provide a straightforward, methodological resource on

models to simultaneously test mediation and moderation effects for the substantive

researcher. To that end, we organize methods for simultaneously testing mediation and

moderation into a single framework that allows for point estimation and construction of

confidence intervals. Interpretation and effect computation are provided, and the model is

applied to a substantive dataset to illustrate the methods. To ensure common ground for this

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discussion, basic mediation and moderation effects from which the model is formed are first

reviewed.

Review of the Mediation Model

The mediation model offers an explanation for how, or why, two variables are related, where

an intervening or mediating variable, M, is hypothesized to be intermediate in the relation

between an independent variable, X, and an outcome, Y (See Fig. 3). Early presentations of

mediation in prevention research (e.g., Baron and Kenny 1986;Judd and Kenny 1981a;

1981b) illustrated causal step methods to test for mediation, but more recent research has

supported tests for statistical mediation based on coefficients from two or more of the

following regression equations (MacKinnon and Dwyer 1993):

(1)

(2)

(3)

Where c is the overall effect of the independent variable on Y; c′ is the effect of the

independent variable on Y controlling for M; b is the effect of the mediating variable on Y;

a is the effect of the independent variable on the mediator; i1, i2, and i3 are the intercepts for

each equation; and e1, e2, and e3 are the corresponding residuals in each equation (see Fig.

3).

Although there are alternative ways to estimate mediation, the product of coefficients is

most easily applied to complex models and is used in this paper. The product of coefficients

test computes the mediated effect as the product of the â and b̂ coefficients from Eq. 2 and

3. Sobel (1982,1986) derived the variance of âb̂ product based on the multivariate delta

method. This formula has been widely used to estimate the normal theory standard error of

âb̂:

(4)

Where is the variance of the â coefficient and is the variance of the b̂ coefficient.

MacKinnon et al. (1998) and MacKinnon and Lockwood (2001) showed that tests for the

mediated effect based on normal theory can yield inaccurate confidence limits and

significance tests, however, as the product of two normally distributed variables is not itself

normally distributed. Alternative tests based on the asymmetric distribution of the product of

two normally distributed variables are available and have been shown to outperform

traditional methods (MacKinnon et al. 2002; MacKinnon et al. 2004). A new program called

“PRODCLIN” (MacKinnon et al. 2007) has automated computation of the distribution of

the product test for mediation so that it is widely accessible. The researcher need only

specify values of â, b̂, the standard error of â, the standard error of b̂, and the statistical

significance level desired.

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Assumptions of the mediation model include the usual OLS estimation assumptions (e.g.,

correct specification of the model’s functional form, no omitted variables, no measurement

error; Cohen et al. 2003). Mediation analysis also assumes correct causal ordering of the

variables, no reverse causality effects, and no XM interaction.

Review of the Moderation Model

The moderation model tests whether the prediction of a dependent variable, Y, from an

independent variable, X, differs across levels of a third variable, Z (See Fig. 4). Moderator

variables affect the strength and/or direction of the relation between a predictor and an

outcome: enhancing, reducing, or changing the influence of the predictor. Moderation

effects are typically discussed as an interaction between factors or variables, where the

effects of one variable depend on levels of the other variable in analysis. Detailed

descriptions of moderator effects and a framework for their estimation and interpretation

were presented in Aiken and West (1991).

Moderation effects are tested with multiple regression analysis, where all predictor variables

and their interaction term are centered prior to model estimation to improve interpretation of

regression coefficients. A single regression equation forms the basic moderation model:

(5)

Where β1 is the coefficient relating the independent variable, X, to the outcome, Y, when Z

= 0, β2 is the coefficient relating the moderator variable, Z, to the outcome when X = 0, i5

the intercept in the equation, and e5 is the residual in the equation.

The regression coefficient for the interaction term, β3, provides an estimate of the

moderation effect. If β3 is statistically different from zero, there is significant moderation of

the X-Y relation in the data. Plotting interaction effects aids in the interpretation of

moderation to show how the slope of Y on X is dependent on the value of the moderator

variable. Regression slopes that correspond to the prediction of Y from X at a single value of

Z are termed simple slopes.

Assumptions of the moderation model include OLS regression assumptions, as described

earlier, and homogeneity of error variance. The latter assumption requires that the residual

variance in the outcome that remains after predicting Y from X is equivalent across values

of the moderator variable.

Combining Mediation and Moderation Analyses

Analyzing the Models Separately

Much of the work combining mediation and moderation analyses has been presented in the

context of prevention program design and development, where examining mediation and

moderation effects together aims to improve program implementation by combining theory-

driven ideas and empirical evidence. For example, Donaldson (2001) indicates that

multivariate relations between variables in a treatment program tend to be one of three

types: (a) direct effects, (b) mediated effects, and (c) moderated effects. By combining the

examination of these effects in a single analysis, the researcher may not only identify

mediating processes through which the program achieves its effects but may also identify

effective program components and/or particular characteristics of the participants or the

environment that moderate the effectiveness of the program. If the theoretical underpinnings

of a treatment or prevention program serve as a starting point for its curriculum, separate

analyses of mediation and moderation may be used to iteratively refine program theory.

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These analyses may be used to collect empirical feedback and to conduct pilot work of the

program before large-scale implementation of the curriculum (See Fig. 5). Specifically, by

examining mediation one is able to investigate how effective a program curriculum was in

changing target behaviors, and whether the program aimed to alter appropriate mediators of

desired outcomes. Analyzing moderation effects in this context allows the researcher to

identify variables that may improve or reduce the program’s ability to alter mediating

variables, as well as to examine the external validity, or generalizability, of the model across

different groups or settings (Hoyle and Robinson 2003). Hypothesized moderator variables

may be more or less amenable to program tailoring, however. Although program subgroups

may be formed on moderators such as age or gender with little difficulty, forming program

subgroups based on other moderator variables such as ethnicity or family risk may be

impractical and/or unethical. Nonetheless, the identification of subgroups for which a

program is most effective is useful, and the examination of moderation and mediation

effects in this context increases the scientific understanding of behaviors and improves

program efficacy. West and Aiken (1997) have argued that these analyses are especially

useful after the successful implementation and evaluation of a treatment program. This

allows for the continual development and improvement of a program, but after an effective

first evaluation.

Analyzing the Models Simultaneously

By simultaneously investigating mediation and moderation, the effects may not only be

disentangled and analyzed separately but can also be evaluated together. There have been

two primary effects analyzed in the literature: (a) the mediation of a moderator effect, and

(b) the moderation of an indirect effect. The mediation of a moderator effect involves

exploring mediating mechanisms to explain an overall interaction of XZ in predicting Y,

whereas the moderation of an indirect effect involves investigating whether a mediated

relation holds across levels of a fourth, moderating variable. These effects have previously

been referred to as mediated-moderation and moderated-mediation in the literature,

respectively. These alternative descriptions may enhance the distinction between the two.

Previous models to simultaneously test mediation and moderation effects have been

presented with varying notation (e.g., Edwards and Lambert 2007; James and Brett 1984;

Muller et al. 2005; Preacher et al. 2007) or without testable equations (e.g., Baron and

Kenny 1986; Wegener and Fabrigar 2000), making it difficult to understand similarities and

differences among the methods. Moreover the criteria for testing the effects have varied

across sources, making it hard to extrapolate recommendations for use. It is possible to

create a general model to test these effects, however, that subsumes several previous

frameworks by including all possible interactions between variables in the mediation and

moderation models (MacKinnon 2008). Such a model unifies the methods into a single

presentation where different models are represented as special cases of the larger

framework. Three regression equations form the model:

(6)

(7)

(8)

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where all predictors in the model are centered at zero to improve interpretation of the lower

order coefficients. In Eq. 6, c1 is the effect of the independent variable on the outcome when

Z = 0 (also the average effect of X on Y because the mean of Z = 0), c2 is the effect of the

moderator variable on the outcome when X = 0 (also the average effect of Z on Y because

the mean of X = 0), c3 is the effect of the interaction between the independent variable and

the moderator on the outcome, and i6 and e6 are the intercept and the residual in the

equation, respectively. In Eq. 7, a1 is the effect of the independent variable on the mediator

when Z = 0 (also the average effect of X on M because the mean of Z = 0), a2 is the effect of

the moderator variable on the mediator (also the average effect of Z on M because the mean

of X = 0), a3 is the effect of the interaction between the independent and moderator variables

on the mediator, and i7 and e7 are the intercept and the residual in the equation, respectively.

In Eq. 8, c′1 is the effect of the independent variable on the outcome when M = 0 and Z = 0

(the average effect of X on Y), c′2 is the effect of the moderator on the outcome when X = 0

and M = 0 (the average effect of Z on Y), c′3 is the effect of the interaction between the

independent and moderator variables on the outcome when M = 0 (the average effect of XZ

on Y), b1 is the effect of the mediator on the outcome when X = 0 and Z = 0 (the average

effect of M on Y), b2 is the effect of the interaction between the moderator and mediator

variables on the outcome when X = 0 (the average effect of MZ on Y), h is the effect of the

interaction between the independent and mediator variables on the outcome when Z = 0 (the

average effect of XM on Y), and j is the effect of the three-way interaction of the mediating,

moderating, and independent variables on the outcome. The intercept and residual in Eq. 8

are coded i8 and e8, respectively. A path diagram for the model is presented in Fig. 6.

Assumptions of the general model include assumptions of the mediation and moderation

models as described earlier. Issues of causal inference in non-additive models may also

require additional stipulations for estimation. Note that the presence of any significant two-

way interactions in the model implies that the main effects of X and M do not provide a

complete interpretation of effects. The presence of a significant three-way interaction in the

model also implies that lower order two-way interactions do not provide a complete

interpretation of effects. If there are significant interactions, point estimates can be probed

with plots and tests of simple effects to probe the interaction effects. Edwards and Lambert

(2007), Preacher et al. (2007), and Tein et al. (2004) provide methods to perform these

analyses.

Testing effects: Criteria for the moderation of an indirect effect—To examine

whether an indirect effect is moderated, it is of interest to investigate whether the mediated

effect (ab) differs across levels of a fourth, moderating variable. Previous sources have

argued that this effect can be defined by either a moderated a path, a moderated b path, or

both moderated a and b paths in the mediation model (James and Brett 1984; Muller et al.

2005; Preacher et al. 2007; Wegener and Fabrigar 2000), such that if there is moderation in

either path of the indirect effect then the mediated relation depends on the level of a

moderator variable. There are circumstances, however, in which a heterogeneous a or b path

does not imply a heterogeneous ab product term.

Although significant heterogeneity in either the a or b path may imply significant

heterogeneity in the ab product term in some cases, examining moderation of the product

term or moderation of both paths versus examining moderation of single paths in the

mediation model are not conceptually identical. Consider the following example where a

moderated a path in the mediation model means something different from both moderated a

and b paths in the model. Presume that X is calcium intake, M is bone density, Y is the

number of broken bones, and Z is gender. Calcium intake is known to have an effect on the

bone density of women, and the relation between calcium intake and bone density is

stronger in women than it is in men (i.e., heterogeneity in the a path in the model).

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Specifically, men have greater bone density in general and thus yield fewer gains from

supplemental calcium intake. However, bone density affects the fragility of bones in a

constant way across males and females, such that low bone density leads to more broken

bones (i.e., no heterogeneity of the b path in the model). Previous models would deem this

scenario as moderation of the indirect effect, arguing that moderation of the a path suffices

as a test for the effect. There are two problems with this argument. First, testing the

heterogeneity of only the a or b path in the mediation model is not a test of mediation

because only a single link in the mediated effect is tested in each case. Second, a

heterogeneous a path in this model suggests something different from both heterogeneous a

and b paths or a heterogeneous ab product. Heterogeneity in both paths of the mediated

effect would suggest that gender not only moderates the effect of calcium intake on bone

density, but that gender also moderates the effect of bone density on broken bones.

Heterogeneity of the product estimate of the mediated effect would suggest that gender

moderates the mechanism by which calcium intake affects bone loss; this may or may not be

true based on the research literature. Although the moderation of a single path may imply

moderation of the product term in some cases, it is critical to differentiate the scenarios as

they correspond to different research hypotheses.

There are also numerical examples that show instances when heterogeneity in individual

paths of the mediation model does not imply heterogeneity of the product term. Consider the

following mediated effect scenarios in two moderator-based subgroups:

Mediated Effect in Group 1Mediated Effect in Group 2

Case 1: (a = −2)(b = −2)(a = 2)(b = 2)

Case 2: (a = 1)(b = 2)(a = 2)(b = 1)

In both scenarios the a and b paths are heterogeneous across groups thus satisfying criteria

for the moderation of a mediated effect as defined by Edward and Lambert (2007), James

and Brett (1984), Morgan-Lopez and MacKinnon (2006), Preacher et al. (2007), and

Wegener and Fabrigar (2000). However the ab product is identical across groups, indicating

that there is no moderation of the indirect effect. Although tests for the moderation of a

mediated effect based on the heterogeneity of individual path coefficients in the mediation

model will be more powerful than a test based on the heterogeneity of the product term

(given the usual low power to detect interactions), these tests may also have elevated Type 1

error rates.

Despite potential problems for making inferences on moderation of the ab product from

information on the moderation of individual paths, initial simulation work suggests that

extending a test of joint significance (where the test for mediation is based on the

significance of component paths in the model such that if both â and b̂ are significant then

the mediated effect âb̂ is deemed significant) to models for mediation and moderation may

be acceptable. Specifically, if conclusions about the moderation of âb̂ are based on whether

both â and b̂ are significantly affected by the moderator variable, Z, Type 1 error rates never

exceed .0550 (Fairchild 2008). Effects of the moderator variable on component paths of the

mediation model are examined using Eq. 7 and 8, where â3 quantifies the effect of Z on the

a path and b̂2 quantifies the effect of Z on the b path, respectively (See Fig. 6). If both

coefficients are significant, it may be claimed that there is significant moderation of the

indirect effect. To obtain either a point estimate or confidence limits for the effect, a product

of coefficients test can be used.

To estimate a product of coefficients test for moderation of the indirect effect in the case of a

dichotomous moderator variable, separate mediation models can be estimated for each group

and equivalence of the âb̂ point estimates can be compared across moderator-based

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subgroups. An example of a dichotomous moderator variable might be gender or clinical

diagnosis. The null hypothesis associated with the test is that the difference between the two

mediated effect point estimates in each group is zero:

(9)

If the point estimates in each group are statistically different from one another, there is

significant moderation of the indirect effect (i.e., heterogeneity in the ab product), such that

the mediated effect is moderated by group membership. To test the estimate in Eq. 9 for

statistical significance, the difference is divided by a standard error for the estimate to form

a z statistic. If the groups are independent, the standard error of the difference between the

two coefficients is:

(10)

Where

the mediated effect in group 2.To test heterogeneity of the indirect effect in the case of a

continuous moderator variable, variance in the estimates of the ab product across levels of

the moderator variable is examined. An example of a continuous moderator variable might

be individual motivation to improve. Because a z test as shown above can only

accommodate moderator variables with two levels (or a small number of levels with

contrasts between two variables) and because levels of continuous moderators often do not

represent distinct groups, tests with continuous moderators are more complicated. The

question becomes how to assess differences in the ab product across a large number of

levels of the moderator variable, and the answer to that question is incomplete at this time.

Random coefficient models assess variance in regression coefficients across multiple levels

such as multiple levels of a moderator. If the moderator is thought of as a higher order

variable across which lower order effects (such as ab) may vary, the random coefficient

modeling framework may be suitable to assess variance in the indirect effect across levels of

a continuous moderator. Kenny, Korchmaros, and Bolger (2003) describe an estimate of the

variance of ab when a and b are correlated for the case of random effects based on Aroian

(1947):

is the variance of the mediated effect in group 1 and is the variance of

(11)

Bauer, Preacher, and Gil (2006) make an important distinction between the variance of ab

and the variance of the average value of ab in the multilevel model. It is the variance of ab

that is relevant to the question of whether an indirect effect is moderated as we would like to

know if there is substantial variability in ab across levels of the moderator variable.

It is also possible to test whether individual paths in the mediation model differ across levels

of a moderator variable. These tests can investigate moderation of the direct effect of X on Y

(c′) or evaluate the generalizability of action and conceptual theory for a program. The null

hypothesis to test homogeneity of a program’s direct effect is:

(12)

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This hypothesis is tested by examining the significance of in Eq. 8; if the regression

coefficient is significant, there is significant moderation of the program’s direct effect.

Recall that conceptual theory for a program corresponds to the b path of the mediation

model, which defines the theory that links variables or psychological constructs (e.g., M) to

behavioral outcomes. The relationships examined in this piece of the model are driven by

previous models or theories presented in the literature that explain motivations for behavior.

The null hypothesis to test homogeneity of a program’s conceptual theory across levels of

the moderator variable is:

(13)

This hypothesis is tested by examining the significance of b̂2 in Eq. 8; if the regression

coefficient is significant there is significant moderation of the program conceptual theory.

Action theory for a program corresponds to the a path of the mediation model, which

defines what components of the program are designed to manipulate the mechanisms of

change. This piece of the model illustrates how the program intervenes to modify

hypothesized mediators. The null hypothesis to test homogeneity of a program’s action

theory across levels of the moderator variable is:

(14)

This hypothesis is tested by examining the significance of â3 in Eq. 7; if the regression

coefficient is significant there is significant moderation of the program action theory. See

Chen (1990) for more details on program action and conceptual theory.

Testing effects: Criteria for the mediation of a moderator effect—A test for the

mediation of a moderator effect examines whether the magnitude of an overall interaction

effect of the independent variable (X) and the moderator variable (Z) on the dependent

variable (Y) is reduced once the mediator is accounted for in the model (Muller et al. 2005).

Thus, the examination of the mediation of a moderator effect considers the mediation model

as a means to explain why a treatment effect of X on Y is moderated by a third variable, Z.

In this way, the mediation of a moderator effect hypothesis probes mediation as a possible

process that accounts for the interaction of the treatment and the outcome. There is no need

to differentiate methods for categorical and continuous moderator variables here.

One way to test whether an XZ interaction in the data is accounted for, at least in part, by a

mediating relation is to examine whether the magnitude of the regression coefficient

corresponding to the overall interaction effect, c3, is reduced once the mediator is added to

the model. Using coefficients from Equations 6 and 8, where ĉ′3 represents the direct effect

of the interaction effect on Y once the mediator is included in the model, a point estimate

and standard error of this difference can be computed:

(15)

(16)

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Dividing an estimate of the difference in Eq. 15 by its standard error in Eq. 16 provides a

test of significance for the estimate. If there is a significant difference in the coefficients

then the overall moderation effect of XZ on Y is significantly explained, at least in part, by a

mediated relation. The null hypothesis corresponding to this effect is that there is no

reduction in the overall interaction once accounting for the mediator:

(17)

A test of the mediation of a moderator effect with a product of coefficients estimator

circumvents the need for testing the overall interaction of XZ on Y as shown above.

Morgan-Lopez and MacKinnon (2006) presented a point estimator and standard error for the

product of coefficients method:

(18)

(19)

The product of coefficients estimator in Eq. 18 illustrates that the effect of the interaction of

the independent and moderator variables on the outcome (represented by â3) is transmitted

through a mediating variable (represented by b̂1). Dividing the estimate of the product by an

estimate of its standard error in Eq. 19 provides a test of significance for the estimate based

on normal theory. However, as described for the product of coefficients estimator in the

single mediator model, asymmetric confidence intervals for the distribution of the product

are more accurate than tests based on the normal distribution and should be implemented

(note also that a test of joint significance of the coefficients could be conducted). If the

product is significantly different from zero, then the moderator effect is explained, at least in

part, by a mediating mechanism. The null hypothesis corresponding to this test is that the

product of the coefficients is equal to zero:

(20)

Numerical Examples

Returning now to the research scenarios presented at the beginning of this paper, numerical

examples can be provided to show how these analyses may appear in practice. Two

simulated datasets were created to explore the questions. Scenario #1 asked “Is the process

by which a program has an effect the same across different types of participants?” The

researcher investigates whether the indirect effect is moderated to answer this question. Note

that although it is necessary to estimate the XM interaction in Equation 8 to avoid bias in the

XZ term (Yzerbyt, Muller, and Judd 2004), the XMZ interaction need not be estimated if

there is no supporting hypothesis for its existence. Likewise, Equation 6 need not be

estimated if overall effects are not part of the research question. Recall that for this example

the independent variable, X, was a worksite-wellness program to reduce obesity-related

health risk. The mediating variable, M, was knowledge of the benefits of eating fruits and

vegetables, and the outcome, Y, was employee consumption of fruits and vegetables. The

moderator variable in analysis, Z, was family history of obesity-related illness with 107

employees having no history of disease (i.e., ngroup1 = 107) and 93 employees having a

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history of disease (i.e., ngroup2 = 93). The null hypothesis for the research question in

Scenario #1 is that there is no difference in the indirect effect of the program on employee

fruit and vegetable consumption through knowledge of fruit and vegetable benefits across

employees who have a family history of disease versus those who have no family history of

disease. To test this hypothesis, an estimate of the mediated effect can be computed in each

group and compared for differences. Estimating Equations 2 and 3, the mediated effect, âb̂

(sâb̂), for employees with no family history of obesity-related disease was computed as .

0004(.0186), with an asymmetric confidence interval of −.0400, .0413, where â(sâ) = .

0041(.2018)and b̂(sb̂) = .0923(.0969) were used as inputs into the PRODCLIN program

(MacKinnon et al. 2007). The mediated effect for employees with a family history of

obesity-related disease was .2158(.0821), with an asymmetric confidence interval of .0693, .

3909, where â(sâ) = .5479(.1854) and b̂(sb̂) = .3938(.0687) were used as inputs into the

PRODCLIN program (MacKinnon et al. 2007). Initial inspection of the indirect effects

shows that the mediation relation in the data is significant for individuals with a family

history of obesity-related illness (the simple mediated effect for group 1) and nonsignificant

for individuals with no family history of disease (the simple mediated effect for group 2).

Statistical inspection of the difference between the two indirect effects shows a difference

of .2154. Using Eq. 11, the pooled standard error for this difference was computed as .0421,

yielding a z-statistic of 5.1143. Because the obtained z-value is greater than the critical value

(1.96) associated with α = .05, there is significant moderation in the indirect effect such that

ab is heterogeneous across moderator-based subgroups. Alternatively estimating Equations

7 and 8 and examining a test of joint significance of the â3(sâ3) and b̂2(sb̂2) coefficients to

examine whether the indirect effect is moderated yields the same conclusion. Both

parameter estimates are significant: â3(sâ3) = .2760(.1386) and b̂2(sb̂2) = .1919(.0697),

respectively. Because the moderator variable in this example only had two levels, there is no

need to follow up the finding with simple effects; the difference in the mediated effects

occurs between the simple mediated effect in group 1 and the simple mediated effect in

group 2. Had there been three or more levels of the moderator variable in the example, the

researcher could conduct follow up analyses to investigate simple mediated effects at

different levels of the moderator variable to see where specific differences occur. Had there

also been a significant XM interaction in the data, the researcher would need to estimate

simple mediated effects across levels of X as well.

Scenario #2 asked “Can a mediation relation explain an interaction effect in my data?” To

answer this question, the researcher investigates whether the interaction effect can predict a

mediating variable which in turn predicts the outcome. Recall that X was the worksite-

wellness program described above, M was a social norm for eating fruits and vegetables, Z

was part-time or full-time work status (191 part-time employees, 209 full-time employees),

and Y was fruit and vegetable consumption. The null hypothesis for the research question is

that a mediating variable cannot explain any part of the interaction. To test this hypothesis,

the â3b̂1 point estimate is computed for the data and tested for significance. As indicated in

the numerical example for Research Scenario #1, although it is necessary to estimate the

XM interaction to avoid bias in the XZ term (Yzerbyt et al. 2004), there is no need to model

the XMZ interaction if there is no hypothesis to support its estimation. Estimating Eq. 7 and

8, â3b̂1 was computed as .1781 (.0814), with an asymmetric confidence interval of .0192, .

3383, where â3(sâ3) = .1550(.0706) and b̂1(sb̂1) = 1.1495(.0405) were used as inputs into the

PRODCLIN program (MacKinnon et al. 2007). Failure to include zero in the confidence

interval indicates significance of the point estimate, such that the interaction of the program

effect and part-time versus full-time work status is explained at least in part by the mediating

mechanism of social norms for eating fruits and vegetables. Example SAS datasets, program

code and output, and complete worked examples for the general model to simultaneously

test mediation and moderation effects are available from the first author.

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Issues of Power in Models to Test Mediation and Moderation Effects

Power to detect interaction effects is often low because of the small effect sizes observed in

social science (Aiken and West 1991). Models that simultaneously examine mediation and

moderation effects are at an even greater disadvantage as they involve several interaction

terms as well as estimation of indirect effects. Fairchild (2008) investigated power for the

general model using previous arguments made in the literature that interaction effects in real

data typically range from explaining 1% to 3% of the variance in the dependent variable

(Champoux and Peters 1987; Evans 1985; McClelland and Judd 1993). Results showed that

when the effect size of the â3 and b̂2 parameters both explained 1% of the variance, a test of

joint significance for moderation of the indirect effect required N ≥ 1000 for .8 power. If the

effect size of the parameters both explained 3% of the variance, the sample size requirement

for .8 power reduced to N ≥ 500. Morgan-Lopez and MacKinnon (2006) showed similar

results for tests to examine the mediation of an interaction effect.

The large sample size requirement of these tests makes investigation of mediation and

moderation effects in small samples challenging. Effect size analysis usually offers a guide

to the practical utility of effects that is independent of significance and sample size, but

effect size measures for models that simultaneously analyze mediation and moderation

effects are not yet fully researched. Although increasing sample size has historically been

most utilized to increase the power of a test, there are several alternatives that may increase

power when accessing additional data is not possible (Beck 1994).

Statistical significance of a test is based on four parameters: sample size, effect size, power,

and the alpha-level set for the experiment. Thus to achieve .8 power when sample size is

fixed either the effect size of the treatment and/or the alpha level of the experiment may be

manipulated to increase sensitivity of the research design. For example, rather than

implementing the nominal .05 criterion that allows for errors in null-hypothesis-

significance-testing 5 times out of every 100, the researcher may choose to evaluate results

against a more liberal .10 criterion that allows for errors 10% of the time. Increasing the

Type 1 error rate for the study decreases the possibility of making a Type 2 error, or the

failure to find a significant effect when one truly exists in the data. This option may be

appropriate when an effect of interest is especially important and avoiding Type 2 errors is

critical. A second option for increasing power when sample size is limited in a study is to

increase the effect size of the manipulation. This option involves increasing the between-

group variance on the independent variable in the design and decreasing the within-group

variance, or error variance, of the variables (Beck 1994). To increase the between-group

variance on the independent variable the researcher may impose an extreme-groups design

(see Borich and Godbout 1974) or deliver a stronger manipulation so that differences

between the treatment and control groups are exaggerated. When the strength of the

intervention is not easily increased or formation of extreme groups is not feasible,

researchers may opt to focus on decreasing within-group variance, or error variance, in the

design. Statistical methods to reduce error variance include improving reliability of the

measurement tools used in analysis and implementing analysis of covariance models that

partition covariates out of treatment effects. Research design tools that are able to reduce

error variance in a model include blocking and matching designs that attempt to equate

treatment and control groups on non-intervention dimensions in which they may differ

(Cook and Campbell 1979).

Recommendations

To best implement the simultaneous analysis of mediation and moderator effects, the

researcher should first identify which third variable model characterizes his or her primary

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research question. This a priori model identification acts as the starting point of

interpretation and can be guided by theory or previous research. If the primary research

question is determined to be mediation, the researcher may examine all possible interaction

effects in the model or a subset of theoretically-relevant interactions and discuss which

moderated effects exist. Methods for examining the moderation of the indirect effect have

been discussed in this manuscript, as well as methods for examining moderation in

individual links of the mediation model such as the direct effect. Exploring the moderation

of either the total or direct effect in the mediation model is simply a means of examining the

significance of the individual regression coefficients corresponding to those effects (i.e., c3

and c′3, respectively), and any significant interaction effect in the model will be interpreted

with reference to mediation. Interactions may be further probed with simple slopes analyses

to examine variable relations across levels of the moderator variable (Edwards and Lambert

2007; Preacher et al. 2007; Tein et al. 2004).

If the statistical model appropriate to answer the primary research question is determined to

be moderation, mediation may be investigated as a means to explain an overall moderated

treatment effect. To that end, the interpretation of interaction effects in the model focus on

whether a mediating mechanism is responsible, at least in part, for overall moderation in the

data. As presented in the manuscript, this assessment may be made with either a difference

) or product of coefficients (â3b̂1) estimator. Both tests examine

whether there is a decrease in the overall XZ interaction once a mediating variable is

modeled. However, given the availability of more accurate significance testing for the

product of coefficients estimator, it is recommended that researchers analyze â3b̂1.

Preferably, the exploration of possible interactions in either model are well-thought-out and

delineated by theory to avoid confusion and unwieldy models, as well as to facilitate

interpretation of effects in the model. Regardless of which third variable model is identified

with the primary research question, it is necessary to model the XM interaction to avoid bias

in the XZ product term (Yzerbyt, Muller, and Judd 2004).

in coefficients (

Limitations and Future Directions

The cost of the generalizability of the general model to test mediation and moderation

effects is possible inflation of Type I error, lack of power, and difficulty with interpretation

of model parameters if several effects are present. The model may be simplified, however, to

represent more specialized cases of mediation and moderation joint effects such as baseline

by treatment interactions by constraining paths in the model to be zero.

An additional limitation of models with moderation and mediation is the extensive

assumptions required for accurate assessment of relations among variables (Holland 1988).

The sensitivity of conclusions to violations of assumptions is not yet known and correct

conclusions will likely require repeated applications in any substantive research area. In

particular, often the X variable is the only variable that represents random assignment,

making interpretation of causal relations between other variables in the model susceptible to

omitted variable bias. In many applications, the model results may represent descriptive

information about how variables are related rather than elucidating true causal relations

among variables. Information on true causal relations will require programs of research to

replicate and extend results as well as information from other sources such as qualitative

information and replication studies in different substantive research areas.

Estimation of the methods and their performance can be investigated for categorical and

longitudinal data. Previous work has explored the needs of categorical data in the single

mediator model (MacKinnon and Dwyer 1993) and this research may serve as a resource for

extending the general model to test mediation and moderation effects into that domain. With

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regard to longitudinal data, the current model can be applied to two-wave data by using

difference scores or residualized change scores, but data with three or more waves cannot

yet be accommodated. The addition of cross-lagged effects in longitudinal data frameworks

such as autoregressive models or latent growth models increases the possible number of

interactions among variables and whether the investigation of all possible cross-lagged

effects is valuable or overly cumbersome will need to be determined.

In summary, many questions in prevention science involve how and for whom a program

achieves its effects. Mediation and moderation models are ideal for examining these

questions. By investigating both mediation and moderation effects in data from prevention

programs information about the mechanisms underlying program effects as well as the

generalizability of program effects and curricula can be evaluated. Models that

simultaneously estimate mediation and moderation effects not only allow for the

examination of these questions, but also permit the evaluation of more complex research

hypotheses such as whether a moderator effect in the data can be explained by a mediating

mechanism, or whether a mediating mechanism depends on the level of another variable. A

broad integration of these models into the substantive research literature will enhance the

information drawn from prevention work and will inform our knowledge on prevention

models.

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