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.

8

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)

10

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 ф1

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:

β1

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’

15

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:

θ2

θ1

16

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 = β1θ2 = ф1 - θ1

Direct effect= θ1 Total effect= β1θ2 + θ1 = ф 1

17

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

18

Epidemiology Equations with Covariate C.

E[Y|A=a, C=c] = ф0+ ф1 A + ф2 C

E[Y|A=a, M=m, C=c] = θ0+ θ1 A + θ2 M + θ4 C

E[M|A=a, C=c] = β0+ β1 A + β2 C

With AM interaction

E[Y|A=a, M=m, C=c] = θ0+ θ1 A + θ2 M + θ3 AM + θ4 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 unmeasured—that 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).

23

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

mediational and outcome constructs assessed at a point in time prior

to the delivery of the treatment. … Here again we are assuming a

randomized experimental research design, so that treatment is not

related to any of the pretreatment measures. … we are reducing

bias in the estimation of the mediational process by controlling

for pretreatment differences on all mediating and outcome

variables. … The success of this strategy depends on meeting two

assumptions besides the usual assumptions of ANCOVA …

constructs must be assessed without error in order to adequately

control for them. Second, assuming that the effects of all omitted

variables that cause … Time 2 variables are mediated through

the Time 1 variables

27

Mediation is a Longitudinal Model

A mediator is a variable in a chain whereby an

independent variable causes the mediating variable

which in turn causes the outcome variable—these

are longitudinal relations. X, M, and Y in single

mediator model imply longitudinal relations even

if measured at the same time.

For a single mediator model, temporal order for X

is clear when it represents random assignment, but

the temporal order of M and Y must be based on

prior research or theory.

28

Timing of Relations

When does X affect M or M affect Y?

Triggering, cascading, and other timing processes (Tang & DeRubeis, 1999; Howe et al., 2002)

Tang & DeRubeis (1999) found evidence that change in therapy occurs within the first few sessions.

How are decisions made about timing? Not often considered in research projects except with respect to when a manipulation is made and the easiest time for data collection.

Timing is crucial for deciding when to collect longitudinal measures (Collins & Graham, 2003).

29

Cross-sectional models

Cross-section is a snapshot of relations.

Models assume that a system has reached an equilibrium so observed relations are not just due to the particular point of observation. But systems may be dynamic and change over time in complicated ways.

Meaning of cross-sectional relations (relation of rank order of level) is different from longitudinal relations (relation of rank order of change).

Cross-sectional mediation may differ in many ways from longitudinal mediation.

May take time for effects to occur. Size of effect depends on time lag-effect 1 day apart is likely different from an effect 1 year apart.

(Cole & Maxwell, 2003; Gollob & Reichardt, 1991; MacKinnon, 2008; Maxwell & Cole 2007; Maxwell et al., 2012 and Commentaries in Multivariate Behavioral Research)

30

Benefits of Longitudinal Data

• Time-ordering of X to M to Y is investigated. Can shed light on whether changes in M precede changes in Y.

• Both cross-sectional and longitudinal relations can be examined.

• Removes some alternative explanations of effects, e.g., effects of static variables can be removed.

31

What if repeated measures of X, M, and Y are available?

• Measures of X, M, and Y at two time points allow

for several options; difference score, ANCOVA,

residualized change score, relative change…

• Measures of X, M, and Y at three or more time

points allow for many alternative longitudinal

models.

• For many examples, X is measured once and

represents random assignment of participants to one

of two groups. Other variables often do not

represent random assignment.

32

Stability, Stationarity, and Equilibrium

• Stability-the extent to which the mean of a

measure is the same across time.

• Stationarity-the extent to which relations among

variables are the same across time.

• Equilibrium-the extent to which a system has

stabilized so that the relations examined are the

same over time.

Cole & Maxwell, 2003; Dwyer, 1983; Kenny, 1979;

MacKinnon, 2008; Wohlwill, 1973

33

Models for Two Waves

Difference Scores for X, M, and Y in the mediation regression equations.

Analysis of Covariance where the baseline value of X, M, and Y is included as a predictor of the follow-up value of X, M, and Y.

Residual Change. Predict time 2 with time 1 and use the difference between the time 2 score and predicted time 2 score as the dependent variable.

Relative Change. The change divided by the baseline measure or the natural logarithm of time 2 divided by time 1 (Tornqvist et al., 1985).

Controversy over difference score versus ANCOVA models.

34

Regression to the Mean

Galton’s regression to mediocrity. Tall parents tend to

have shorter children. Short parents tend to have

taller children.

Occurs when two variables are imperfectly related.

Examples are the sophomore jinx and spontaneous

remission.

Galton squeeze diagrams to investigate regression to

the mean.

Lord’s (1967) paradox

35

Two-wave Longitudinal Model

BASELINE

OUTCOME

BASELINE

MEDIATOR

POST-TEST

OUTCOME

POST-TEST

MEDIATOR

PROGRAM

Mediated effect=a4b5

Direct effect = c’3

b1

c’3

a4

b5

b2

DAG with Confounder U

X

M1

Y1 Y2

M2

U

37

Summary of Two-Wave Models

Difference score versus ANCOVA models. For randomized X, ANCOVA has more statistical power. If there is a difference in the results between the two models, check for baseline differences.

Difference score and residualized change measures are useful because they transform two measures to one measure, i.e., the difference score combines the time 1 mediator and time 2 mediator so all the models can be applied.

Meaning of mediation with the different models differ: Correlated change scores, correlated adjusted time 2 scores. Note issue of Lord’s paradox for the M to Y relation because M is not randomized.

More options with more waves of data. More complexity too though.

38

Models for Three or More Waves

Autoregressive Models

Latent Growth Curve Models (LGM)

Latent Change Score Models (LCS)

Autoregressive and Latent Growth Curve Models

(ALT)

Differential Equation Models (DEM) Others: Area Under the Curve, Multilevel Structural Equation Models, Survival

Analysis, fractional polynomial (Royston & Altman, 1994), spline (Borghi et

al., 2006), functional data analysis (Ramsay, 2005)

39

1Χ 3Χ2Χ1b 2b

2 3

21

2

3

2

2

2

1

3223

2112

,,,, bb

XbX

XbX

x

Autoregressive (Jöreskog, 1974)

40

Autoregressive Model with Time-Ordered Mediation,

Cole & Maxwell, (2003); MacKinnon (1994, 2008)

1

1

1X

2

2

3

3

2b

2c

1a

2s1c

2s

3s 3s

1b

Note: Residuals at the same time are

correlated

41

Autoregressive Models

• Many mediated effects. Standard error of the sum of

(or any function) the indirect effects can be derived

with the multivariate delta method or resampling

methods.

• Model does not allow for random effects for

individual change and does not include modeling of

means. Change in growth of means is an important

aspect of longitudinal data.

42

Latent Growth Model (LGM)

I S

X1 X2 X3

1 1 1

1 2

ε1 ε2 ε3

Meredith & Tisak (1990)

Means

SIXSIXSIX

ISSI ,,,,,,

210

222

3

2

2

2

1

33

22

11

43

Χ

1Μ 2Μ 3Μ

1Υ 2Υ 3Υ

ms

yi ys

mi

Latent Growth Curve (Model Cheong et al., 2003)

b'c

a

44

Latent Growth Models (LGM)

• LGM models change over time by estimating an intercept

and slope for change in variables. These models can be used

to investigate mediation by estimating change over time for

the mediator and change over time for the outcome. The

relation between the change in the mediator and change in

the outcome represents the b path (Cheong et al. 2003).

• The causal direction of correlated change is ambiguous.

Another LGM estimates change in the mediator at earlier

time points and relates to change in the outcome at later

time points providing more evidence for temporal

precedence of the mediator.

D.P. MacKinnon

Latent Change Score (LCS) McArdle (2001)

ε2 ε3

β1

D2

β2

D1

1

1

X1 X2 X3 1

1

Means

XBDXBD

DXXDXX

x ,,,,, 212

3

2

2

2

1

222

111

3223

2112

D.P. MacKinnon

Latent Change Score Mediation Model

3Y 2Y 3Y

1M

2M

3M

1Y 2Y 3Y

1M

2M

3M

M

M

Y Y

1

1

1

1

1

1

1 1

1

1

1

1a

2a

1b2b

3b

1c2c

1X

47

Latent Change Score Models (LCS)

• LCS parameterizes models so that change between adjacent

waves is analyzed.

• Really a special case of latent growth curve modeling but

with growth between adjacent waves.

• More complicated change over time can be made by picking

different coefficients and second order factors. Second order

factors can be change in change or second derivatives.

• Promising model not often used for mediation analysis.

Promising in that often theory predicts effects at different

change points.

48

Longitudinal models for a steroid prevention project (ATLAS)

• Adolescents Teaching and Learning to Avoid Steroids (ATLAS). Randomized high school football teams in Oregon and Washington to receive the steroid prevention program or an information only group. Just individual data here.

• Measured the same persons over repeated occasions. Here we will look at preliminary models for four repeated measures. The dependent variable is intentions to use steroids.

• Linn Goldberg (OHSU) principal investigator. For more on the program see Goldberg et al. (1996) and for mediation see MacKinnon et al., (2001). LGM Cheong et al., (2003).

• Program delivered after baseline measurement. In general, timing of the mediators should be relatively quick for knowledge and beliefs measures. It may take longer for norms measures. Four waves of measurement for the models studied.

49

Analysis decisions

• LGM model, slope coded as 0 1 * 1 where * indicates a

free parameter. Note that there was a booster after the

3rd measurement. If the model was not identified, then

loadings were 0 2.5 * 14.5 to represent the months from

baseline. All LGM models had RMSEA lower than .041

(lowest .019).

• Autoregressive model. Tested for stationarity in the a

and b paths. Stationarity observed more often for b paths

and less often for a paths, as expected. All RMSEAs

lower than .088 (lowest was .068).

50

LGM and Autoregressive mediation effects

Mediator LGM Autoregressive

ab(se) z a1b2(se) z Knowledge -.28(.12) -4.88 -.08(.02) -4.90

Coach Tol -.11(.05) -2.27 -.02(.01) -3.24

Team as info -.21(.06) -3.42 -.04(.02) -3.30

Peer as info -.12(.05) -2.43 -.04(.01) -2.30

Reasons not use -.12(.04) -2.98 -.02(.01) -3.01

Normative bel -.12(.07) -1.64 -.00(.00) -0.14

51

Modern Causal Inference for Longitudinal Data 1

Time varying effects lead to complexities when

interpreting causal effects.

Changes at earlier waves could cause subsequent

variables that complicate model interpretation.

For example, the relation of M to Y at each wave

can lead to complications. Should earlier

measures of M or Y be included in the

prediction of later waves of data? Problem of

collider bias.

52

Modern Causal Inference for Longitudinal Data 2

Specify longitudinal models in a potential outcome

and causal framework.

G-computation ~ standardization where predictions

are made for factual and counterfactual data.

G-estimation to obtain a parameter value that

removes effect of interest.

Marginal Structural Model with inverse probability

weighting to weight observations by amount of

confounding.

(Robins 1986, 1989, 1999 and colleagues)

53

Marginal Structural Model

IPW Example Obtain predictors of M that will render M unaffected by

confounders L. Note that this assumes that all

confounders are in the statistical model-the no

unmeasured confounders assumption.

The method uses inverse probability weighting to

reweight participants according to exposure to

treatment and values of confounders. (Coffman,

2011; Robins, Hernan, & Brumbeck, 2000).

54

ATLAS IPW Analysis

• Confounders may explain the relation of M to Y in

these data. It would be useful to apply a method that

adjusts for this discrepancy.

• A large number of measures were used in the

propensity model.

• The effects of the intervention were not large so it is

possible that these effects would be attenuated after

adjustment.

• Team social norm mediator and nutrition behavior

outcome.

55

Predictors of M

Ethnicity, grade in school, school has a gym

Baseline grade point average, peers as an

information source, body image, intent to use

steroids, communication skills, perceived

susceptibility to steroid use, knowledge of anabolic

steroid effects, positive aspects of steroids, self

esteem, depression, win-at-all costs attitude,

perceived severity of steroid use, coach tolerance

for steroids, attitudes about steroid users.

56

ATLAS Data No Adjustment

Standard 95% Confidence Parameter Estimate Error Limits Z Pr > |Z|

Intercept 0.0081 0.0389 -0.0682 0.0844 0.21 0.8351

X 0.4262 0.0668 0.2954 0.5571 6.38

57

ATLAS IPW Results

Standard 95% Confidence

Parameter Estimate Error Limits Z Pr > |Z|

Intercept -0.0035 0.0395 -0.0810 0.0739 -0.09 0.9290

X 0.4237 0.0681 0.2903 0.5572 6.22

58

More Complexities of Longitudinal Modeling

• Does the measure have the same meaning at each

wave? X, M, and Y at an earlier developmental

stage may differ from X, M, and Y at a later stage.

Relations between change in X on M and change in

M on Y may differ.

• Timing of measurement should match theoretical

change. Transitions are important, e.g., home to

elementary school, elementary to high school, high

school to workforce/college. Sleeper Effects.

Measure at appropriate times to capture effects.

59

Types of change over time

• Cumulative: There may be cumulative effects such that more M yields more Y.

• Threshold: Once a mediator gets to a certain level, then it will change Y. Once a level of a mediator is reached, the individual changes to a new level, e.g., learning a concept in algebra.

• Saltation and stasis: Change occurs rapidly after no previous change, e.g., human growth (Lampl et al., 1992)

• The types of changes may differ over time. And change for X to M to may differ for M to Y—nonlinear relations.

60

Person-oriented Methods

• Focus on patterns of responses by individuals.

• Classifies individuals based on their responses, such

as whether their responses are consistent with

mediation or not.

• Configural Frequency Analysis, Latent Class

Analysis, Markov Models, are examples…

• Complement for variable-oriented methods, may

provide different information.

• Combination of both approaches for mediation in

mixture models is an active area of research.

61

Summary

• Mediation is a Longitudinal model.

• Many alternative models that provide different

information about mediation effects.

• Often requires an iterative process to model

longitudinal data.

• Lots of methods work needs to be done to

understand these models: causal inference, model

equivalence, validity of assumptions.

• Need examples of applying the models to real data.

62

Thank You

References available by sending me an e-mail at David.MacKinnon@asu.edu

Most topics are covered in MacKinnon (2008). Introduction to Statistical Mediation Analysis, Erlbaum; Mahwah, NJ. e.g., Causal Inference circa 2008 Chapter 13, Longitudinal Mediation models in Chapter 8, and background for mediation in Chapters 1 and 2.

See website for Research In Prevention Laboratory

http://www.public.asu.edu/~davidpm/

mailto:David.MacKinnon@asu.eduhttp://www.public.asu.edu/~davidpm/

63

Hypothesized Effects of Ghent

Mediation Analysis Presentation

Ghent

Presentation

# Accurate

Mediation

Analyses

Mediation

Awareness

Knowledge of

Longitudinal

Models

Models for different

number of waves

Importance of

Causal Inference

Methods

64

General Autoregressive Model

1

1

1X

2

2

2X

3

3

3X

Note: All residuals are correlated

at the same time are correlated

65

Autoregressive Model with Time-Ordered Mediation,

Cole & Maxwell (2003)

1

1

1X

2

2

2X

3

3

3X

2b

2c

1s

2a1a

2s

1s

1c

2s

3s 3s

1b

Note: All residuals at the same time are

correlated

66

1Χ 2Χ 3Χ

1Μ 2Μ 3Μ

1Υ 2Υ 3Υ

xi

ms

xs

yi ys

mi

Latent Growth Curve Model

a

b

1c

Nutrit1 Nutrit2 Nutrit3 Nutrit5

11

1

111

Program

PerIn1 PerIn2 PerIn3 PerIn5

1 1

1

0

11

PerIn6PerIn4

1

1

1

1

Nutrit6Nutrit4

1

1

10

2

3

4

1

1

1

a

c’

b

68 Nutrit3 Nutrit5

1

111

Program

PerIn1 PerIn2 PerIn3

1

1

0

1 1

Nutrit6Nutrit4

1

1

10

2

3

4

1

1

1

a

c’

b

2/1/2013 D.P. MacKinnon

Latent Change Score Mediation Model

3X1X 2X

1X 3X2X

3Y 2Y 3Y

1M

2M

3M

x

1Y 2Y 3Y

x 1M

2M

3M

M

M

Y Y

1 1 1

11

11

1

1

1

1

1

1

1 1

1

1

1

1a

2a

1b2b

3a

3b

1c2c 3c