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1

A Longitudinal Look at Longitudinal

Mediation Models David P. MacKinnon, Arizona State University

Causal Mediation Analysis

Ghent, Belgium

University of Ghent

January 28-29, 2013

Introduction

Assumptions

Unique Issues with Longitudinal Relations

Two-wave Mediation Models

Three or more wave Mediation Models

Application to a Health Promotion Study *Thanks to National Institute on Drug Abuse and Yasemin Kisbu-Sakarya and

Matt Valente.

2

Mediator Definitions

A mediator is a variable in a chain whereby an independent variable causes the mediator which in turn causes the outcome variable (Sobel, 1990)

The generative mechanism through which the focal independent variable is able to influence the dependent variable (Baron & Kenny, 1986)

A variable that occurs in a causal pathway from an independent variable to a dependent variable. It causes variation in the dependent variable and itself is caused to vary by the independent variable (Last, 1988)

3

Single Mediator Model

MEDIATOR

M

INDEPENDENT

VARIABLE

X Y

DEPENDENT

VARIABLE

a b

c

Directed Acyclic Graph

4

X Y

M

5

Mediation is important

because

Central questions in many fields are about

mediating processes

Important for basic research on mechanisms of

effects

Critical for applied research, especially prevention

and treatment

Many interesting statistical and mathematical issues

6

Applications

Two overlapping applications of mediation analysis:

(1) Mediation for Explanation

(2) Mediation by Design

7

Mediation by Design

Select mediating variables that are causally related to an outcome variable.

Intervention is designed to change these mediators.

If mediators are causally related to the outcome, then an intervention that changes the mediator will change the outcome.

Common in applied research like prevention and treatment.

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Intervention Mediation Model

MEDIATORS

M1, M2, M3,

INTEREVENTION

PROGRAM

X Y

OUTCOMES

Action

theory

If the mediators selected are causally related to Y, then changing the

mediators will change Y. Test of each theory is important when

total effect is nonsignificant.

Conceptual

Theory

9

Mediation Regression Equations

Tests of mediation for a single mediator use information from some or all of three equations.

The coefficients in the equations may be obtained using methods such as ordinary least squares regression, covariance structure analysis, or logistic regression. The following equations are in terms of linear regression and expectations.

(Hyman, 1955; Judd & Kenny, 1981; Baron & Kenny, 1986)

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Equation 1 Social Science

MEDIATOR

M

INDEPENDENT

VARIABLE

X Y

DEPENDENT

VARIABLE c

1. The independent variable is related to the dependent variable:

11

Equation 1 Epidemiology

MEDIATOR

M

INDEPENDENT

VARIABLE

A Y

DEPENDENT

VARIABLE 1

1. The independent variable is related to the dependent variable:

12

Equation 2 Social Science

MEDIATOR

M

INDEPENDENT

VARIABLE

X Y

DEPENDENT

VARIABLE

2. The independent variable is related to the potential mediator:

a

13

Equation 2 Epidemiology

MEDIATOR

M

INDEPENDENT

VARIABLE

A Y

DEPENDENT

VARIABLE

2. The independent variable is related to the potential mediator:

1

14

Equation 3 Social Science

MEDIATOR

M

INDEPENDENT

VARIABLE

X Y

DEPENDENT

VARIABLE

a

3. The mediator is related to the dependent variable controlling for

exposure to the independent variable:

b

c

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Equation 3 Epidemiology

MEDIATOR

M

INDEPENDENT

VARIABLE

A Y

DEPENDENT

VARIABLE

3. The mediator is related to the dependent variable controlling for

exposure to the independent variable:

2

1

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Effect Measures

Natural Indirect Effect = ab = c-c

ab = c-c for ordinary least squares regression not

nonlinear models like logistic regression.

Direct effect= c Total effect= ab+c=c

Natural Indirect Effect = 12 = 1 - 1

Direct effect= 1 Total effect= 12 + 1 = 1

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Social Science Equations with Covariate C.

E[Y|X=x, C=c] = i1+ c X + c2 C

E[Y|X=x, M=m, C=c] = i2+ c X + b M + c3 C

E[M|X=x, C=c] = i3+ a X + a2 C

With XM interaction

E[Y|X=x, M=m, C=c] = i4+ c X + b M + h XM + c4 C

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Epidemiology Equations with Covariate C.

E[Y|A=a, C=c] = 0+ 1 A + 2 C

E[Y|A=a, M=m, C=c] = 0+ 1 A + 2 M + 4 C

E[M|A=a, C=c] = 0+ 1 A + 2 C

With AM interaction

E[Y|A=a, M=m, C=c] = 0+ 1 A + 2 M + 3 AM + 4 C

VanderWeele (2010)

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Identification Assumptions

1. No unmeasured X to Y confounders given

covariates.

2. No unmeasured M to Y confounders given

covariates.

3. No unmeasured X to M confounders given

covariates.

4. There is no effect of X that confounds the M to Y

relation.

VanderWeele & VanSteelandt (2009)

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Omitted Variables/Confounders

(Judd & Kenny, 1981 p. 607): a mediational analysis may also

yield biased estimates because of omitted variables that cause both

the outcome and one or more of the mediating variables. If

variables that affect the outcome and .mediating variables are not

controlled in the analysis, biased estimates of the mediation process

will result, even .. a randomized experimental research design ...

(James & Brett, 1984 p. 317-318): misspecification due to a

"serious" unmeasured variables problem. By a serious unmeasured

variables problem is meant that a stable variable exists that (a) has a

unique, nonminor, direct influence on an effect (either m or y, or

both); (b) is related at least moderately to a measured cause of the

effect (e.g., is related to x in the functional equation for m); and (c)

is unmeasuredthat is, is not included explicitly in the causal

model and the confirmatory analysis (James, 1980; James et al.,

1982).

21

Assumptions

Reliable and valid measures.

Data are a random sample from the population of

interest.

Coefficients, a, b, c reflect true causal relations

and the correct functional form.

Mediation chain is correct. Temporal ordering is

correct: X before M before Y.

No moderator effects. The relation from X to M

and from M to Y are homogeneous across

subgroups or other participant characteristics.

22

Significance Testing and Confidence

Limits

Product of coefficients estimation, ab, of the mediated effect and standard error is the most general approach with best statistical properties for the linear model given assumptions. Best tests are the Joint Significance, Distribution of the Product, and Bootstrap for confidence limit estimation and significance testing again under model assumptions.

For nonlinear models and/or violation of model assumptions, the usual estimators are not necessarily accurate. New developments based on potential outcome approaches provide more accurate estimators (Robins & Greenland, 1992; Pearl, 2001).

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Testing Mediation When the Total Effect is Not Statistically Significant

Test of ab can be more powerful than test of c, i.e.,

mediation more precisely explains how X affects Y.

Lack of statistically significant c is very important for

mediation analysis because failure of treatment, relapse,

or both theories is critical for future studies.

Inconsistent mediation relations are possible because

adding a mediator may reveal a mediation relation.

Note the test of c is important in its own right but is a

different test than the test for mediation. It is also a causal

estimator.

24

More on Temporal Order Assumption

Assume temporal ordering is correct: X before M before Y.

Assume that relations among X, M, and Y are at equilibrium so the observed relations are not solely due to when they are measured, i.e., if measured 1 hour later a different model would apply.

Assume correct timing and spacing of measures to detect effects.

But manipulations target specific times with many patterns of change over time.

25

Judd & Kenny (1981)

(Judd & Kenny, p. 613): While the estimation of

longitudinal multiple indicator process models is

complex, it is also likely to be quite rewarding,

since only through such an analysis can we

glimpse the process whereby treatment effects are

produced. Without knowledge of this process,

generalizing treatment effects may be difficult.

26

Judd & Kenny (1981)

(Judd & Kenny, 1981 p. 611): Specifically we might include the

me