Longitudinal Structural Equation Modeling
SOME PRACTICES TO IMPROVE DEVELOPMENTAL SCIENCE
Todd D. Little, Director
Institute for Measurement, Methodology, Analysis and Policy
(iMMAP)
Workshop presented March 19, 2014 @
Society for Research in Adolescence
Pre-conference in Austin, TX
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General Roadmap
Overview of the three primary approaches to
longitudinal analysis:
 Panel models
 Growth curve models
 Dynamic P techniques
Understand common issues in longitudinal research:
 Thinking about Context
 Thinking about Change
 Conceptualizing Time and Growth
 Validity Threats and Measurement Issues
 Missing data (planned and unplanned)
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Three techniques of longitudinal analysis
Panel models
 Evaluates association between X and Y, with Y measured at a

later point in time than X
Control for earlier levels of Y, so prediction is of (residual)
change
Time 1
X
Time 1
Y
Time 2
Y
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Three techniques of longitudinal analysis
Panel models
 Answer the question…
• Are individuals who are relatively high on X at Time 1 relatively high (positive
association) or low (negative) on changes in Y later (Time 2)?
Time 1
X
Time 1
Y
Time 2
Y
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Three techniques of longitudinal analysis
Panel models
 Simplest analysis with multiple regression
•
•
•
•
Assumes error-free measurement
YT2 = b0 + b1 YT1 + b2 XT1 + e
Cross-lag path evaluated as b2
Interindividual (rank order) stability evaluated as b1
Time 1
X
Time 1
Y
b2
b1
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Time 2
Y
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Three techniques of longitudinal analysis
Panel models
 Evaluating both directions of prediction, in regression:
•
•
•
•
YT2 = b0 + b1 YT1 + b2 XT1 + e & XT2 = b0 + b1 XT1 + b2 YT1 + e
One cross-lag prediction is significant, the other is not.
– Typically taken as evidence for direction of effect. (must be cautious about
differential reliabilities)
Neither cross-lag significant.
– Either no longitudinal relation, or problems with timing, power, or reliability
(or combination of the three).
Both cross-lags significant.
– Either reciprocal influence or a third variable is ‘true’ cause.
Time 1
X
Time 2
X
Time 1
Y
Time 2
Y
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Three techniques of longitudinal analysis
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A Panel Model with Strong
Directionality Implicated
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
1. Use Latent Variables
 Find or develop multiple indicators of all constructs
 Requires only weak measurement assumptions
• Congeneric indicators
 Measurement model informs content and criterion validity
• Constructs are corrected for measurement error
• Can test and establish factorial invariance across groups and time
“Indicators are our worldly
window into the latent space”
- John R. Nesselroade
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
2. Use MACS Modeling
 Mean and Covariance Structures (MACS) modeling
 Tailor the statistical analysis model to fit the research question
• Avoid trendy analyses if they don’t fit the question!
 Advances in SEM allow almost any statistical model.
• Multilevel, Mixture, Multiple-group, Longitudinal, Time Series, State Space etc.
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Comparing constructs across time
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Three techniques of longitudinal analysis
Growth curve models
 Fits a growth function to how each person increases or decreases
on a variable over time
(within-person, or intraindividual, change)
Score (YT)
•
Time
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Three techniques of longitudinal analysis
Growth curve models
•
•
 Intraindividual change
 Over multiple people, can examine…
Average patterns of growth
Variability in growth (interindividual variability in intraindividual change)
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Three techniques of longitudinal analysis
Growth curve models
 Multilevel modeling for growth curves:
• Combines conceptual steps of regression analyses
• Multilevel equations:
– Level 1: Within-person
–
Yit = π0i + π1i(Time) + π2i(Time-varying predictor) + εit
Level 2: Across-person
π0i = γ00 + γ01(Time-invariant predictor) + ζ0i
π1i = γ10 + γ11(Time-invariant predictor) + ζ1i
π2i = γ20 + γ21(Time-invariant predictor) + ζ2i
Note: I = individual; t = time point
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Three techniques of longitudinal analysis
Growth curve models
 Multilevel modeling for growth curves
 Some Limits of MLM for growth curves
• Assumes (but can not test) measurement equivalence across time
• Typically specify shape of growth without evaluating
• Can only model one DV
– Other variables must be included as either time-varying or time-invariant
predictors
• Typical usage does not involve time lag – therefore, can not make
conclusions regarding causality / temporal primacy
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Three techniques of longitudinal analysis
P-techniques
 Treating each time point as a ‘case’
Dynamic P-techniques
 Involves introducing a lag into the data set / analysis
 Then evaluate whether score at time t is predicted by previous
scores at time t-1
• Multiple regression
• Path analysis / SEM
Time
Xt
Xt-1
Yt
Yt-1
1
xt1
--
yt1
--
2
xt2
xt1
yt2
yt1
3
xt3
xt2
yt3
yt2
4
xt4
xt3
yt4
yt3
…
…
…
…
…
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Context
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Context
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Social Ecology
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Physical Ecology
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Personal Ecology
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Contexts as Statistical Relationships
Direct effects

Varies at the level of the individual and influences the individual directly
Indirect (mediated) effects

X varies at the level of the individual and influences the individual through its effect
on an intervening variable, M
Mediating effects

Distal context influences proximal context which influences the individual
Moderating effects


Interactive influences that change the strength of any of the above effects
Discrete vs. Continuous
Reciprocal effects and feedback loops

Cross-time associations that can express as indirect, mediated/mediating, or
moderated/moderating.
Hierarchically nested effects

Larger units of context that can have direct, indirect, mediating, or moderating effects
or be mediated and or moderated.
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Types of Variables
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Types of Variables
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
3. Focus on Measurement!
 Rethink Likert scales
• Our great-great-great-great academic progenitors used them!
 Develop measures that are sensitive to change
 Take advantage of touch screen technology and software
• Using “rulers” is now easy and efficient
"Whatever exists at all exists in some amount. To know it
thoroughly involves knowing its quantity as well as its quality"
- E. L. Thorndike (1918)
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Context and Measurement
We should measure persons and contexts well

Measures should be appropriate for the construct
•
•

Contexts should be quantified (borrow from sociology, for example)
Developmental measures should address change
– The tragic legacy of test-retest reliability
Measures and analyses should not be haphazard
•
•
•
•
•
Avoid: “Hey, this new method is cool, let’s try it on this data?”
Question -> Measurement -> Statistical Model
Avoid short forms of existing scales (use intentionally missing design)
– (‘allure of the bloated specific’ idea)
Develop or modify to make sure the measurement tool is right
Take time to refine and pilot measures (even well-established ones).
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Context of Measurement
Homotypic vs. heterotypic expressions across ages
 e.g., Aggression
Surface (proximal) structure vs. deep (ultimate)
structure of behavior
 e.g., helping as resource-directed behavior
Typological (subgroups) differences
 Identification issues and procedures
• Muthen’s m-Plus, Nagin’s Proc Traj, Bergman’s Sleipner
n-adic (dyadic, triadic, etc.) overlay on all of the various
modeling approaches
 e.g., SRM, APIM, Siena
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Measurement Calibration
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Context of Change
Interindividual differences vs. Intraindividal differences
 Ergodicity conundrum
Associations (within and between time)
 Covariances and Correlations vs. Regressions
 Direct and Indirect effects
•

Auto-regressive vs. Cross-lagged
– 1st-order vs. higher-order
Linear vs. non-linear
Means and Variances
Mediation vs. Moderation vs. Additive Effects
B = ƒ(age) vs. Δ = ƒ(time)
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Validity Threats in Longitudinal Work
Threats to Validity
 Maturation




• In pre-post experiment effects may be due to maturation not the treatment
• Most longitudinal studies, maturation is the focus.
Regression to the mean
• Only applicable with measurement error
Instrumentation effects (factorial invariance)
Test-retest effects (ugh)
Selection Effects
• Sample Selectivity vs. Selective Attrition
Age, Cohort, and Time of Measurement are confounded
 Sequential designs attempt to unconfound these.
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The Sequential Designs
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What’s Confounded?
Design
Independent
Variables
Confounded Effect
CohortSequential
Age &
Cohort
Age x Cohort Interaction is
confounded with Time
TimeSequential
Age & Time
Age x Time Interaction is
confounded with Cohort
CrossSequential
Cohort &
Time
Cohort x Time Interaction is
confounded with Age
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Appropriate Time and Intervals
Age in years, months, days.
Experiential time: Amount of time something is experienced
 Years of schooling, length of relationship, amount of practice
 Calibrate on beginning of event, measure time experienced
Episodic time: Time of onset of a life event
 Toilet trained, driver license, puberty, birth of child, retirement
 Early onset, on-time, late onset: used to classify or calibrate.
 Time since onset or time from normative or expected occurrence.
Measurement Intervals (rate and span)
 How fast is the developmental process?
 Intervals must be equal to or less than expected processes of change
 Measurement occasions must span the expected period of change
 Cyclical processes
•
E.g., schooling studies at yearly intervals vs. half-year intervals
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Transforming to Episodic Time
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Transforming to Accelerated
Longitudinal
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Transforming to Accelerated
Longitudinal
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Temporal Design
Changes (and causes) take time to Unfold
The ability to detect the effect depends on the
measurement interval
The ability to model the shape of the effect requires
adequate sampling of time intervals.
The ability to model the optimal effect requires
knowing the shape in order to pick the optimal
(peak) interval.
Lag within Occasion: the Lag as Moderator Model
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Variable Actual Assessments
X1
Y1
Y2
X2
Y3
X3
Y4
X4
Y5
X5
Y6
X6
X7
X8
X9
Xi
Xj
••
•
Xn
T1
Y7
Y8
Y9
Yi
Yj
Yn
Tmin
T2
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Tmax
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Multiple Regression LAM model
Yˆi  b0  b1 X i  b2 Lagi  b3 X i  Lagi
Xi is the focal predictor of outcome Yi
Lagi can vary across persons
b1 describes the effect of Xi on Yi when Lagi is zero
b2 describes the effect of Lagi on Yi when Xi is zero
b3 describes change in the Xi → Yi relationship as a
of Lagi
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function
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An Empirical Example
Data are from the Early Head Start (EHS) Research and Evaluation
study (N = 1,823)
Data were collected at Time 1 when the focal children were
approximately 14 months of age and again at Time 2 when the children
were approximately 24 months of age
The average lag between Time 1 and Time 2 observations was 10.3
months with values ranging from 3.0 to 17.3 months
Measures:

The Home Observation for the Measurement of the Environment (HOME)
assessed the quality of stimulation in the home at Time 1.

The Mental Development Index (MDI) from the Bayley Scales of Infant
Development measured developmental status of children at Time 2.
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HOME predicting MDI
𝑀𝐷𝐼𝑇2 = 𝑏𝑜 + 𝑏1 𝐻𝑂𝑀𝐸𝑇1 + 𝑏2 𝐿𝑎𝑔 + 𝑏3 (𝐻𝑂𝑀𝐸𝑇1 𝑥 𝐿𝑎𝑔)
Effect of HOMET1 on MDIT2
3
2.5
2
1.5
1
0.5
0
-7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
Lag (Mean Centered)
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Implications of LAM Models
Lag is embraced
 LAM models allow us to model, not ignore, interactions
Selecting Lag is critical
 Sampling only a single lag may limit generalizability
Theory Building
 LAM models may yield a better understanding of relationships
and richer theory regarding those relationships
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Randomly Distributed Assessment
X1
X2
X3
Y1
Y2
X6
X7
X8
X9
Y1
Y1
Y2
Y2
Y2
Y3
Y3
Y4
X4
X5
Y1
Y4
Y5
Y6
Y7
Y3
Y5
Y6
Y8
Y4
Y7
Y8
Y8
Y9
Y3
Y5
Y6
Y7
Y1
Y2
Y9
Y3
Y4
Y4
Y5
Y6
Y5
Y6
Y7
Y8
Y7
Y8
Y9
Y9
Y9
••
•
Xn
T1
Yn
Tbegin
Yn
Yn
Tmid
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Yn
Yn
Tend
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
4. Use Modern Missing Data Treatments!
 Multiple Imputation (MI)
 Full Information Maximum Likelihood (FIML)
• Use inclusive auxiliary variables
 Only MI and FIML can avoid bias and retain power
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
5. Use Planned Missing Data Designs
 Increase validity, save money, and reduce unplanned missing,
• Multiform questionnaire protocols to reduce burden, fatigue, and test reactivity
• Two method measurement strategies to increase power with minimal costs
• Wave missing to reduce costs and minimize dropout
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
6. Model Your Data
 The rules of hypothesis testing and the principals of staunch
empiricism are road blocks to understanding and discovery
• Nuance in and refinement of models should be embraced
 Have the dialog between model and data
• Like the influence of deviant peers, make good choices and you will avoid
trouble!
• Be wise and be transparent
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Characteristics of Good Models
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PRACTICES TO IMPROVE
DEVELOPMENTAL SCIENCE
7. The Randomized Clinical Trail (RCT) is not Gold
 It’s not fools gold either – it’s a good design
• It is not the only design that we can use to make compelling statements of “causality”
 Regression discontinuity design, for example, is just as valid
• Often is a better, and more just, design to test the efficacy of an intervention
• Tom says: Include a comparison regression function to increase power and broaden
generalizability.
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INSTITUTE FOR MEASUREMENT,
METHODOLOGY, ANALYSIS AND POLICY
TODD D. LITTLE, DIRECTOR
Recommended readings
Little, T.D. (2013). Longitudinal Structural Equation Modeling. New York, NY:
Guilford Press

Duh.
Card, N. A., & Little, T. D. (2007). Longitudinal modeling of developmental
processes. International Journal of Behavioral Development, 31, 297-302

Introduction to special issue, but first part identifies basic issues in longitudinal
modeling and then points you to the innovations covered in the special issue.
Little, T. D., Card, N. A., Preacher, K. J., & McConnell, E. (in press). Modeling
longitudinal data from research on adolescence. In R. Lerner & L. Steinberg
(Eds.)., Handbook of Adolescent Psychology (4th Ed.). Wiley.

Provides a broad summary of the three classes of techniques
Card, N. A., Little, T. Ds., & Bovaird, J. A. (2007). Modeling ecological and
contextual effects in longitudinal studies of human development. In T. D. Little,
J. A., Bovaird, & N. A. Card (Eds.), Modeling contextual effects in longitudinal
studies (pp. 1-11). Mahwah, NJ: LEA

Introduction to our book that points you to a lot of really great chapters covering
many of these issues in detail
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