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- September 13, 2020
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Question

- Please free grammar errors and free plagiarism
- Read the requirements of the question deeply and answer each one
- Attached four resources, two books indicated the chapter of each to be read, and two articles
- Your answer must be from the materials of the four resources in addition to your resources

The question

Examine the literature in your topic area and identify five articles published within the past five years that investigate mediating, moderating, or independent variables in an attempt to contribute to theory in the topic area. Write a paper in which for each article, you:

1.Describes the theory the researchers explore. What are the key constructs in the theory? How are they related? Identify which ones are cause, effect, mediating, or moderating constructs. How are the constructs operationalized?

2.Briefly describe the study, including the number of participants and research methods.

3.Briefly describe the statistical analyses used

4.Briefly described the findings and how the researchers interpreted them and their contribution to theory.

Using some or all of the five articles, argue for a gap in the knowledge in the topic area and briefly describe a study involving mediator and or moderator variables that can contribute to theory.

Support your paper with a minimum of 5 resources. In addition to these specified resources, other appropriate scholarly resources, including older articles, may be included.

Length: 5-7 pages not including title and reference pages

References: Minimum of 5 scholarly resources.

References

Creswell, J. W. (2013) Research design: Qualitative, quantitative, and mixed methods approaches. |
Read Pages 49-61 |

Trochim, W. M. K., & Donnelly, J. P. (2008) The research methods knowledge base. |
Read Pages 61-62 |

Prev Sci (2009) 10:87?99

DOI 10.1007/s11121-008-0109-6 A General Model for Testing Mediation

and Moderation Effects

Amanda J. Fairchild & David P. MacKinnon Published online: 12 November 2008

# Society for Prevention Research 2008 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

A. J. Fairchild (*)

Department of Psychology, University of South Carolina,

Barnwell College,

1512 Pendleton St.,

Columbia, SC 29208, USA

e-mail: [email protected]

D. P. MacKinnon

Research in Prevention Lab, Department of Psychology,

Arizona State University,

P.O. Box 871104, Tempe, AZ 85287-1104, USA 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 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 88 Prev Sci (2009) 10:87?99 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 obesityrelated 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 Fig. 1 Conceptual diagram for

the moderation of an indirect

effect example Family

History of

Obesityrelated

Illness WorksiteWellness

Program Hours

worked

(full time

vs. part

time) Norm for

consuming

fruit &

vegetables WorksiteWellness

Program Fruits &

Vegetables

Consumed Fig. 2 Conceptual diagram for the mediation of a moderator effect

example 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 discussion, basic mediation and

moderation effects from which the model is formed are

first reviewed. Knowledge

of Fruit &

Vegetable

Benefits Family

History of

Obesityrelated

Illness Fruits &

Vegetables

Consumed Prev Sci (2009) 10:87?99 89 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):

Y ¼ i1 þ cX þ e1 ð1Þ Y ¼ i2 þ c'X þ bM þ e2 ð2Þ M ¼ i3 þ aX þ e3 ð3Þ 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). X Y

c 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 a and b coefficients from Eq. 2 and 3. Sobel (1982,

^^

1986) derived the variance of ab product based on the

multivariate delta method. This formula has been widely

^^

used to estimate the normal theory standard error of ab:

r???????????????????????

s^ ^ ¼

ab ^ 2

2a

s^ b2 þ s^ ^ 2

a b ð4Þ 2

2

^

Where sa is the variance of the a coefficient and S ^ is the

^

b

^

variance of the b 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 a, b, the

^

^ the standard error of b, and the

standard error of a,

statistical significance level desired.

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. (Total Effect of X on Y) Review of the Moderation Model M

a b X Y

c' Fig. 3 Path diagram for the single-mediator model. Note. X= the

independent variable, Y= the dependent variable, and M= the

mediating variable. The mediation model decomposes the total effect

of X on Y (c), into two parts: the indirect effect of X on Y, quantified

by ab (the product of a and b), and the direct effect of X on Y with the

effect of the mediator removed, quantified by c?. c=ab+c? 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 90 Prev Sci (2009) 10:87?99 the residual variance in the outcome that remains after

predicting Y from X is equivalent across values of the

moderator variable. Z Combining Mediation and Moderation Analyses

Analyzing the Models Separately

X X Y ?1 ?2

Z Y ?3

XZ Fig. 4 Alternate path diagram representations of the moderation

model. Note. X= the independent variable, Y= the dependent variable,

Z= the moderator variable, XZ= the product of X and the moderator

variable, ?1 = the effect of X on Y, ?2 = the effect of Z on Y, and ?3 =

the effect of XZ on Y term are centered prior to model estimation to improve

interpretation of regression coefficients. A single regression

equation forms the basic moderation model:

Y ¼ i5 þ b1 X þ b2 Z þ b3 XZ þ e5 ð5Þ Where ?1 is the coefficient relating the independent

variable, X, to the outcome, Y, when Z = 0, ?2 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, ?3,

provides an estimate of the moderation effect. If ?3 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 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. 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 Construct

Program

Theory Conduct

Pilot Work

and Collect

Empirical

Feedback Refine

Program

Theory (Repeat link until satisfied) Implement

& Evaluate

Full

Program (Use Program Implementation and Evaluation to Inform Original Theory) Fig. 5 Refining program theory: an empirical-theoretical exchange.

Note. Figure is based on Donaldson (2001) diagram that occurs on

p. 472 of that text Prev Sci (2009) 10:87?99 91 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:

Y ¼ i6 þ c1 X þ c2 Z þ c3 XZ þ e6 ð6Þ M ¼ i7 þ a1 X þ a2 Z þ a3 XZ þ e7

Y ¼ i8 þ c01 X þ c02 Z þ c03 XZ þ b1 M þ b2 MZ

þ hXM þ jXMZ þ e8 ð7Þ ð8Þ 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 92 Prev Sci (2009) 10:87?99 Fig. 6 MacKinnon (2008) General Joint Analysis Mode. Note.

X= the independent variable, Y=

the dependent variable, Z= the

moderator variable, M= the mediating variable, XZ= the interaction of X and Z, MZ=the

interaction of M and Z, XM=

the interaction of X and M, and

XMZ= the three-way interaction

between X, M, and Z X c1

c2 Z Y c3

XZ XMZ j

XM h

MZ b2 c'1 X a1 c'2 Y Z a2

XZ c'3 a3 b1

M 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). 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 heterogene- Prev Sci (2009) 10:87?99 93 ity 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: Case 1:

Case 2: Mediated Effect in Group 1

(a = -2)(b = -2)

(a = 1)(b = 2) Mediated Effect in Group 2

(a = 2)(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 identi…