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.
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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.
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.
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)
19
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)
20
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
3s3s
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 2Μ3Μ
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 ,,,,, 21
2
3
2
2
2
1
222
111
3223
2112
D.P. MacKinnon
Latent Change Score Mediation Model
3Y2Y 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
1c 2c
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
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.
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.