DB #3

On the Moderation of Mechanisms: A Conceptual Overview of Conditional Process Analysis

Andrew F. Hayes Professor of Quantitative Psychology

The Ohio State University Department of Psychology

www.afhayes.com These are slides at: www.afhayes.com/public/mobc.pdf

•

…

define the conditional indirect and direct effect.

•

…

do a brief review of the analysis of indirect and conditional effects.

•

…

do a conceptual introduction to “conditional process analysis.”

•

…

do one simple example, with relevant computer output.

•

…. talk about some interesting extensions of basic principles

•

…

turn you into a conditional process analysis expert.

•

…

teach you how to estimate such models in your chosen software.

•

…

get all your questions answered.

•

…

leave you with many new questions unanswered.

I will and will not

•

…

point you toward where you can learn more.

My objective is primarily to whet your appetite for learning more. Knowing what is possible analytically can influence how we think about problems theoretically.

•

…. speak mostly in abstractions. You can fill in the blanks concretely.

What is “Conditional Process Analysis”

“Conditional process analysis”

is a modeling strategy undertaken with the goal of describing the conditional

or

contingent

nature of the mechanism(s) by which a variable transmits its effect on another, and testing hypotheses about such contingent effects.

“Process analysis”, used to quantify and examine the direct and indirect pathways through which an antecedent variable X

transmits its effect on a consequent variable Y through an intermediary M. Better known as “mediation analysis”

these days.

M

X Y

M

X Y

A melding of two ideas conceptually and analytically:

“Moderation analysis” used to examine how the effect of an antecedent X

on an consequent Y

depends on a third moderator variable M (a.k.a. “interaction”)

Mechanisms are quantified with indirect effects. Indirect effects can be moderated, meaning mechanisms can be contingent. We can model such contingencies using rudimentary linear modeling principles. It is not difficult once you learn the fundamentals.

YX

M

The “simple mediation”

model

X→M→Y

is a causal chain of events. A mediator variable can be a psychological state, a cognitive process, an affective response, a biological change, or any other conceivable “mechanism”

variable through which X

exerts an effect on Y. But it must be causally between X and Y.

Mediation

A mediation model links a putative cause (X) to a presumed effect (Y) at least in part through an intermediary or “mediator”

variable (M).

“Indirect effect”

ab

= “indirect effect”

of X

on Y

through M c'

= “direct effect”

of X

on Y

c

c

= “total effect”

of X

on Y

Using OLS or ML, with Y

as continuous:

X YcXiY += 1ˆ

a b

c'

aXiM += 2ˆ

bMXciY +′+= 3ˆ X M Y

Using OLS or ML, with M as continuous:

The indirect effect quantifies the effect of X

on Y

through M. Evidence that ab

is different from zero is consistent with mediation. Evidence that path c

is different from zero is not a requirement of 21st

century mediation analysis. Correlation between X

and Y

is neither sufficient nor necessary

to claim that X

affects Y.

c

= c' + ab

Moderation. The effect of X

on Y

can be said to be moderated

if its size or direction is dependent on M. It tells us about the conditions that facilitate, enhance, or inhibit the effect, or for whom or what the effect is large vs. small, present versus absent, and so forth.

Moderation

M is depicted here to moderate

the size of the effect of X

on Y, meaning that the size of the effect of X on Y

depends on M. We say M is the moderator

of the X

→ Y

relationship, or X

and M interact

in their influence on Y.

YX

M

“Linear moderation”

and “Conditional effect”

YX

M Y

M

X

Conceptual diagram depicting X’s effect on Y

moderated by M. Linear moderation as a statistical model

XM

b1

b2

b3

“Simple linear moderation”

is typically estimated by allowing X’s effect on Y

to be a linear function of M (other forms of moderation are possible):

In this model, the conditional effect of X

on Y, θX→Y

,

is b1

+ b3

M:

XMbMbXbiMbXMbbiY 32112311 )(ˆ +++=+++=

MbXiY YX 21ˆ ++= →θ MbbYX 31 +=→θ

θX→Y

There is no effect of X

on Y

that one can reduce to a single estimate, for the effect of X

on Y

depends on M unless b3

is zero. An inference about the coefficient for XM

in the model is a widely used test of linear moderation.

where

Integrating mediation and moderation analysis

Combining moderation and mediation analysis, at least in principle, is not new at all. Many have talked about it in the distant past (e.g., Judd & Kenny, 1981; James & Brett, 1984; Baron and Kenny, 1986). It goes by various names that often confuse,

including “moderated mediation”

and “mediated moderation.”

More recently:

Muller, Judd, and Yzerbyt (2005): Describe analytical models and steps for assessing when “mediation is moderated”

and “moderation is mediated.” Edwards and Lambert (2007): Take a path analysis perspective and show how various effects in a simple mediation model can be conditioned on a third variable.

Preacher, Rucker, and Hayes (2007): Provide a formal definition of the conditional indirect effect

and give formulas, standard errors, and a bootstrap approach for estimating and testing hypotheses about moderated mediation in five different models. MacKinnon and colleagues (e.g., Fairchild & MacKinnon, 2009): Explicate various analytical approaches to testing hypotheses about mediated moderation and moderated mediation. Hayes (2013) and Hayes and Preacher (2013). Introduce the term “conditional process modeling”

and (in Hayes and Preacher, 2013) take a structural equation modeling approach to estimating the contingent nature of direct and indirect effects.

Examples in substance use research

As a result of these recent discussions and the analytical approaches described therein, models that combine moderation and mediation are seen in the literature with increasing frequency, including in alcoholism research.

Stein, L. A. R., Minugh, P. A. et al. (2009). Readiness to change as a mediator of the effect of a brief motivational intervention on posttreatment

alcohol-related consequences of injured emergency department hazardous drinkers. Psychology of Addictive Behaviors, 23, 185-195.

Houben, K., Wiers, R. W., & Jansen, A. (2011). Getting a grip on drinking behavior: Training working memory to reduce alcohol abuse. Psychological Science, 22, 968-975.

Malouf, E., Stuewig, J., & Tangney, J. (2012). Self-control and jail inmates’ substance misuse post-release: Mediation by friends’

substance use and moderation by age. Addictive Behaviors, 37, 1198-1204.

Witkiewitz, K., & Bowen, S. (2010). Depressing, craving, and substance use Following a randomized trial of mindfulness-based relapse prevention. Journal of Consulting and Clinical Psychology,

78, 362-374.

Integrating mediation and moderation



The indirect effect of X

on Y

through M is estimated as the product of the a

and b

paths 

But what if size of a

or b

(or both) depends on another variable (i.e., is moderated)? 

If so, then the magnitude of the indirect effect therefore depends on a third variable, meaning that “mediation is moderated”.



When a

or b

is moderated, it is sensible then to estimate “conditional indirect effects”—values of indirect effect conditioned on values of the moderator

variable that moderates a

and/or b. 

Direct effects can also be conditional. For instance, in the above, W moderates X’s direct effect on Y.

YX

M

c'

a b

W Z

MBRP : Randomly assigned to treatment as usual (0) or mindfulness-based relapse prevention therapy (1)

CRAVE2: Score on the Penn Alcohol Craving Scale at 2 month follow-up.

Example inspired by …

168 clients of a public service agency providing treatment for alcohol and substance use disorders.

BDIP: Beck Depression Inventory scores immediately following completion of therapy.

USE4: Alcohol and other substance use at 4-month follow-up measured with the Timeline Follow-Back.

Witkiewitz, K., & Bowen, S. (2010). Depression, craving, and substance use following a randomized trial of mindfulness- based relapse prevention. Journal of Consulting and Clinical Psychology, 78, 362-374.

Covariates in the model included depression at start of therapy (BDI0), craving at baseline (CRAVE0) and hours in treatment (TREATHRS).

The model

Question: Do the skills acquired through MBRP therapy moderate craving as the mechanism through which negative affect influences alcohol and other substance use? This is a “first stage”

moderated mediation model that also allows for the direct effect of X

to be moderated.

YX

M W

U

Craving

Substance use Depression

Treatment as usual (0) versus MBRP therapy (1).

Baseline craving (U1: CRAVE0) and depression (U2: BDI0) and hours in treatment (U3: TREATHRS)

CRAVE2

USE4 BDIP

MBRP

The model in equation form

M

X Y

W

A conditional process model with a common moderator of the first

stage path of the X→ M→Y

indirect effect (the mechanism) as well as the direct effect of

X

on Y.

... ...ˆ

3212

3211

++′+′+′+= ++++= bMXWcWcXciY

XWaWaXaiM

This model is estimated (using OLS, for example) as:

...)(ˆ ...)(ˆ

2312

2311

++′+′+′+=

++++=

bMWcXWcciY

WaXWaaiM or

equivalently

U

CRAVE2

USE4BDIP

MBRP CRAVE0 BDI0 TREATHRS

...ˆ ...ˆ

22

21

++′++=

+++=

→

→

bMWcXiY

WaXiM

YX

MX

θ

θ

Using my “theta notation”:

where WaaMX 31 +=→θ WccYX 31 ′+′=→θand

The conditional indirect effect

M

X Y

W

... ...ˆ

3212

3211

++′+′+′+= ++++= bMXWcWcXciY

XWaWaXaiM

This model is estimated as:

or

equivalently

U

...ˆ ...ˆ

22

21

++′++=

+++=

→

→

bMWcXiY

WaXiM

YX

MX

θ

θ

WaaMX 31 +=→θ WccYX 31 ′+′=→θand

θX→M

θX→Y

b

The indirect effect of X

on Y

through M is the product of the effect of X

on M and the effect of M on Y: ωM

= θX→M

b = (a1 + a3

W)b

= a1

b + a3

bW. This is a function

of W. Plug in a value of W and you get the “conditional indirect effect”

of X

on Y

through M, conditioned on that value of W. An inference about that conditional indirect effect is an inference about “conditional”

mediation. In this example, W is dichotomous, but it doesn’t have to be.

CRAVE2

USE4BDIP

MBRP CRAVE0 BDI0 TREATHRS

The conditional direct effect

M

X Y

W

... ...ˆ

3212

3211

++′+′+′+= ++++= bMXWcWcXciY

XWaWaXaiM

This model is estimated as:

or

equivalently

U

...ˆ ...ˆ

22

21

++′++=

+++=

→

→

bMWcXiY

WaXiM

YX

MX

θ

θ

WaaMX 31 +=→θ WccYX 31 ′+′=→θand

θX→M

θX→Y

b

The direct effect of X

on Y

through M is θX→M = c'1

+ c'3

W. This is a function

of W. Plug in a value of W and you get the “conditional direct effect”

of X

on Y. An inference about that conditional direct effect is an inference about whether X

affects Y

independent of the mechanism through M, conditioned on that value of W.

CRAVE2

USE4BDIP

MBRP CRAVE0 BDI0 TREATHRS

Easy to do with software you are (probably) already using

The PROCESS macro for SPSS and SAS is turn-key and easy to use but less flexible because the user is constrained to models PROCESS is programmed to estimate. PROCESS is freely available at www.afhayes.com

SPSS: process vars=crave2 use4 bdip mbrp crave0 bdi0 treathrs/y=use4/m=crave2 /x=bdip/w=mbrp/model=8/boot=10000.

SAS: %process (data=meditate,vars=crave2 use4 bdip mbrp crave0 bdi0 treathrs, y=use4,m=crave2,x=bdip,w=mbrp, model=8,boot=10000);

Mplus can also be used. It requires more programming skill but is more versatile with more benefits and fewer limitations.

Model 8

PROCESS output

PROCESS Output PROCESS output

(a1 + a3

W)bW Bootstrap confidence intervals

W (c’1 + c’3

W) NHSTs and confidence intervals

Difference between conditional indirect effects (with bootstrap confidence interval)

Direct and indirect effects of depression on substance use are positive and statistically different from zero among those given therapy as usual. No direct or indirect effects of depression on substance use among those given MBRP therapy. The indirect effect through craving differs between the two groups---”moderated mediation”

Some other examples

Stein, L. A. R., Minugh, P. A. et al. (2009). Readiness to change as a mediator of the effect of a brief motivational intervention on post-treatment alcohol-related consequences of injured emergency department hazardous drinkers. Psychology of Addictive Behaviors, 23, 185-195.

Houben, K., Wiers, R. W., & Jansen, A. (2011). Getting a grip on drinking behavior: Training working memory to reduce alcohol abuse. Psychological Science, 22, 968-975.

Malouf, E., Stuewig, J., & Tangney, J. (2012). Self-control and jail inmates’ substance misuse post-release: Mediation by friends’

substance use and moderation by age. Addictive Behaviors, 37, 1198-1204.

We just examined a “first stage”

model. But moderation can occur in the “second stage” of the mechanism as well:

….or a variable can moderate both stages of the mechanism.

There are many possibilities. The math is different, but the principles are the same.

An intriguing possibility

A causal agent modifying the operation of its own mechanism by which it affects an outcome.

M

X Y

XMbMbXciY

aXiM

212

11

ˆ

ˆ

++′+=

+=

This model is estimated as:

or

MXbbXciY

aXiM

)(ˆ

ˆ

212

11

++′+=

+=

equivalently

The effect of X

on M is just “a”, but the the effect of M on Y

depends on X: b1

+ b2

X. The indirect effect of X

is the product of these effects: a(b1 + b2

X) = ab1 + ab2

X and so depends on X. This makes sense to do only if X

is not dichotomous.

The effect of X

on M

The effect of M on Y

Multiple mechanisms modeled simultaneously

Kong, G., Bergman, A. (2010). A motivational model of alcohol misuse in emerging adulthood. Addictive Behaviors, 35, 855-860.

Osberg, T. M., Billingsley,K., Eggert,M.,& Insana, M. (2012). From Animal House

to Old School: A multiple mediation analysis of the association between college drinking movie exposure and freshman drinking and its consequences. Addictive Behaviors, 37, 922-930.

Litt, D. M., & Stock, M. L. (2011). Adolescent alcohol-related risk cognitions: The roles of social norms and social networking sites. Psychology of Addictive Behaviors, 25, 708-713.

Webb,J. R., Robinson, E. A. R., & Brower, K. J. (2013). Mental health, not social support, mediates the forgiveness-alcohol outcome relationships. Psychology of Addictive Behaviors, 25, 462-473.

Why estimate such a model?



Many causal effects probably operate through multiple mechanisms simultaneously. Better to estimate a model consistent with such real- world complexities.



If your proposed mediator is correlated with the “real”

mediator but not caused by the independent variable, a model with only your

proposed mediator in it will be a misspecification and will potentially misattribute the process to your proposed mediator rather than the real mediator— “epiphenomenality.”



Different theories may postulate different mediators as mechanisms. Including them all in a model simultaneously allows for a formal statistical comparison of indirect effects representing different theoretical mechanisms.



When combined with moderation, allows for the modeling of different mechanisms for different people defined by different values of a moderator.

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate. For example:

M1

M2

X Y

W

A conditional process model with a common moderator of both of the first stage paths of the mechanism.

22113

23222122

13121111

ˆ

ˆ

ˆ

MbMbXciY

XWaWaXaiM

XWaWaXaiM

++′+=

+++=

+++=

This model is estimated as:

22113

22232122

12131111

ˆ )(ˆ

)(ˆ

MbMbXciY

WaXWaaiM

WaXWaaiM

++′+=

+++=

+++= or

equivalently

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

M1

M2

X Y

W

A conditional process model with a common moderator of both of the first stage paths of the mechanism.

Indirect effect of X

on Y

through M1 depends on W:

Indirect effect of X

on Y

through M2 depends on W:

22113

22232122

12131111

ˆ )(ˆ

)(ˆ

MbMbXciY

WaXWaaiM

WaXWaaiM

++′+=

+++=

+++=

An interesting extension

Waa 1311 +

Waa 2321 +

1b

2b

Wbaba bWaa

223221

223212 )( +=

+=ω

Wbaba bWaa

113111

113111 )(

+=

+=ω

M1

M2

X Y

W

A conditional process model with a common moderator of both of the first stage paths of the mechanism.

Indirect effect of X

on Y

through M1 depends on W:

Wbaba bWaa

223221

223212 )( +=

+=ω

Indirect effect of X

on Y

through M2 depends on W:

Wbaba bWaa

113111

113111 )(

+=

+=ω

22113

22232122

12131111

ˆ )(ˆ

)(ˆ

MbMbXciY

WaXWaaiM

WaXWaaiM

++′+=

+++=

+++=a11

= 0.40 a13

= -0.40 a21

= 0.00 a23

= 0.60 b1

= 0.50 b2

= 0.30

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

Waa 1311 +

Waa 2321 +

1b

2b

M1

M2

X Y

W

A conditional process model with a common moderator of both of the first stage paths of the mechanism.

Indirect effect of X

on Y

through M1 depends on W:

Indirect effect of X

on Y

through M2 depends on W:

a11

= 0.40 a13

= -0.40 a21

= 0.00 a23

= 0.60 b1

= 0.50 b2

= 0.30

W W bWaa

20.020.0 50.0)40.040.0(

)(

1

1

113111

−= −=

+=

ω ω ω

W W bWaa

18.000.0 30.0)60.000.0(

)(

2

2

223212

+= +=

+=

ω ω ω

213

2222

1211

30.050.0ˆ )60.000.0(ˆ

)40.040.0(ˆ

MMXciY

WaXWiM

WaXWiM

++′+=

+++=

+−+=

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

M1

M2

X Y

W

Indirect effect of X

on Y

through M1 when W = 0:

Indirect effect of X

on Y

through M2 when W = 0:

For people of type A

(e.g., W = 0) X

affects Y

through M1

but not through M2

.

a11

= 0.40 a13

= -0.40 a21

= 0.00 a23

= 0.60 b1

= 0.50 b2

= 0.30

20.0020.020.0 50.0)040.040.0(

)(

1

1

113111

=×−= ×−=

+=

ω ω ω bWaa

W W bWaa

18.000.0 30.0)60.000.0(

)(

2

2

223212

+= +=

+=

ω ω ω

W 0

0

213

2222

1211

30.050.0ˆ )60.000.0(ˆ

)40.040.0(ˆ

MMXciY

WaXWiM

WaXWiM

++′+=

+++=

+−+=

Indirect effect = 0.20

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

M1

M2

X Y

W

Indirect effect of X

on Y

through M1 when W = 0:

Indirect effect of X

on Y

through M2 when W = 0:

For people of type A

(e.g., W = 0) X

affects Y

through M1

but not through M2

.

20.0020.020.0 50.0)040.040.0(

)(

1

1

113111

=×−= ×−=

+=

ω ω ω bWaa

a11

= 0.40 a13

= -0.40 a21

= 0.00 a23

= 0.60 b1

= 0.50 b2

= 0.30

00.0018.000.0 30.0)060.000.0(

)(

2

2

223212

=×+= ×+=

+=

ω ω ω bWaa W

0 0

213

2222

1211

30.050.0ˆ )60.000.0(ˆ

)40.040.0(ˆ

MMXciY

WaXWiM

WaXWiM

++′+=

+++=

+−+=

Indirect effect = 0.00

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

M1

M2

X Y

W

For people of type B

(e.g., W = 1) X

affects Y

through M2

but not through M1

.

a11

= 0.40 a13

= -0.40 a21

= 0.00 a23

= 0.60 b1

= 0.50 b2

= 0.30

Y Indirect effect of X

on Y

through M1 when W = 1:

Indirect effect of X

on Y

through M2 when W = 1:

00.0120.020.0 50.0)140.040.0(

)(

1

1

113111

=×−= ×−=

+=

ω ω ω bWaa

W W bWaa

18.000.0 30.0)60.000.0(

)(

2

2

223212

+= +=

+=

ω ω ω

213

2222

1211

30.050.0ˆ )60.000.0(ˆ

)40.040.0(ˆ

MMXciY

WaXWiM

WaXWiM

++′+=

+++=

+−+=

W 1

1

Indirect effect = 0.00

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

M1

M2

X Y

W

For people of type B

(e.g., W = 1) X

affects Y

through M2

but not through M1

.

a11

= 0.40 a13

= -0.40 a21

= 0.00 a23

= 0.60 b1

= 0.50 b2

= 0.30

Y Indirect effect of X

on Y

through M1 when W = 1:

Indirect effect of X

on Y

through M2 when W = 1:

00.0120.020.0 50.0)140.040.0(

)(

1

1

113111

=×−= ×−=

+=

ω ω ω bWaa

18.0118.000.0 30.0)160.000.0(

)(

2

2

223212

=×+= ×+=

+=

ω ω ω bWaa

213

2222

1211

30.050.0ˆ )60.000.0(ˆ

)40.040.0(ˆ

MMXciY

WaXWiM

WaXWiM

++′+=

+++=

+−+=

W 1

1

Indirect effect = 0.18

An interesting extension

Mechanisms might be different for different types of people. For some types, mechanism 1 may be dominant, whereas for other types, mechanism 2 may dominate.

In closing…

•

All causal effects operate through some kind of mechanism---a causal chain of events. But all effects are contingent on something.

•

Mechanisms that are contingent can be modeled if we understand or can at least hypothesize something about those contingencies.

•

Simple combinations of moderation and mediation can be put together to yield complex models that are yet fairly simple to estimate and interpret.

•

Quantifications of mechanisms (indirect effects) can be modeled as functions of other variables (moderators).

•

Statistical tools exist to make the modeling easy, and people are beginning to do this in earnest in many areas of research, including substance use.

•

Learning resources are scattered throughout the methodology journals. The advice they offer is often inconsistent, sometimes dated.

These are slides at www.afhayes.com/public/mobc.pdf

Some places to go for help

www.afhayes.com

Some places to go for help

www.statisticalhorizons.com

Some places to go for help

Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression based approach. New York: The Guilford Press.

Hayes, A. F., & Preacher, K. J. (2013). Conditional process modeling: Using SEM to examine contingent causal processes. In G. R. Hancock and R. O. Mueller (Eds.) Structural equation modeling: A second course

(2nd

Ed). Information Age Publishing.