Representing Time in Longitudinal
Research: Assessment Lag as Moderator
Todd D. Little
University of Kansas
Director, Quantitative Training Program
Director, Center for Research Methods and Data Analysis
Director, Undergraduate Social and Behavioral Sciences Methodology Minor
Member, Developmental Psychology Training Program
crmda.KU.edu
Colloquium presented 04-05-2013 @ Purdue University
Special Thanks to Noel A. Card, James P. Selig, & Kristopher Preacher
crmda.KU.edu
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Overview
• Conceptualizing and Representing Time in
Longitudinal Research
• B = ƒ(age) vs. Δ = ƒ(time)
• The Accelerated Longitudinal Design
• Developmental-Lag Model
• The Lag as Moderator Model
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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/practice 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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Transforming to Accelerated Longitudinal
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Accelerated Longitudinal Designs
Grade
Fall
6
Spr
6
Fall
7
Spr
7
Fall
8
Spr
8
Fall
9
Grade 6
Grade 7
Grade 8
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Accelerated Growth Curve Model
(L13.1.GC.LevelCUBIC.Accelerated)
a2
a1
3*
2*
-3*
-2* -1* 0*1*
1*
1*
1*
1*
1* 1* 1*
Fall
6
=
Spr.
6
Fall
7
=
=
=
=
=
=
a4
Quadratic
Linear
Intercept
=
a3
5*
0*
-3*-4*
5*
0*
-3*
Spr.
7
Fall
8
=
=
=
=
Grade ==
7 1*
=
=
1*
7=1 0*
0*
Cubic
-1*
1*
1*
0*
-1*
-1* 1*
=
Spr.
8
=
=
=
Fall
9
=
Grade
8
8=1
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Plot of Estimated Trends
4.0
3.5
3.0
Positive Affect
2.5
Negative Affect
2.0
1.5
1.0
Fall 6
S pr 6
Fall 7
S pr 7
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Fall 8
S pr 8
Fall 9
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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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Developmental time-lag model
• Use 2-time point data with variable time-lags to
measure a growth trajectory + practice effects
(McArdle & Woodcock, 1997)
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Time
Age
student
T1
T2
1
5;6
5;7
2
5;3
5;8
3
4;9
4;11
4
4;6
5;0
5
4;11
5;4
6
5;7
5;10
7
5;2
5;3
8
5;4
5;8
0
1
2
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3
4
5
6
14
T0
T1
T2
T3
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T4
T5
T6
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Yt  1 I  Bt G  At P
1
Intercept
1
T0
1
1
T1
1
1
1
1
T2
T3
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T4
T5
T6
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Yt  1 I  Bt G  At P
Linear growth
1
1
Intercept
1
T0
1
Growth
1
T1
1
1
1
1
0 1
T2
2
3
4
T3
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5
6
T4
T5
T6
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Yt  1 I  Bt G  At P
Constant Practice Effect
1
1
Intercept
1
T0
1
1
Growth
1
T1
1
1
1
1
0 1
T2
2
3
4
T3
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Practice
5
6
T4
0
11
1
1
T5
1
1
T6
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Yt  1 I  Bt G  At P
Exponential Practice Decline
1
1
Intercept
1
T0
1
1
Growth
1
T1
1
1
1
1
0 1
T2
2
3
4
T3
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Practice
5
6
T4
0
1 .87
.67
.55
T5
.45
.35
T6
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The Equations for Each Time Point
Constant Practice Effect
Declining Practice Effect
YT0  I
YT1  I  1G  P
YT0  I
YT1  I  1G  1.0 P
YT2  I  2G  P
YT2  I  2G  .82 P
YT3  I  3G  P
YT3  I  3G  .67 P
YT 4  I  4G  P
YT 4  I  4G  .55P
YT 5  I  5G  P
YT 6  I  6G  P
YT 5  I  5G  .45P
YT 6  I  6G  .37 P
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Developmental time-lag model
• Summary
– 2 measured time points are formatted according to
time-lag
– This formatting allows a growth-curve to be fit,
measuring growth and practice effects
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Temporal Design
• Changes (and causes) take time to Unfold
• The ability to detect an 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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Types of Change Effects
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Lag as Moderator (LAM) Models
• One possible way to address the issue of lag choice
is to treat lag as a moderator
• Following this approach lag is treated as a
continuous variable that can vary across
individuals
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Variable Actual Assessments
X1
Y1
X2
Y2
X3
Y3
X4
Y4
X5
Y5
X6
X7
X8
X9
Xi
Xj
••
•
Xn
T1
Y6
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
function of Lagi
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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
MDI T 2  b0  b1HOMET1  b2 Lag  b3 HOME1  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 of lag and hypothesized effects
• Selecting/Sampling 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
Y1
Y2
X2
X3
X6
X7
X8
X9
Y1
Y1
Y2
Y2
Y2
Y3
Y3
Y4
X4
X5
Y1
Y3
Y4
Y5
Y5
Y6
Y6
Y7
Y3
Y8
Y4
Y5
Y6
Y7
Y9
Y4
Y5
Y6
Y5
Y6
Y7
Y8
Y9
Y3
Y4
Y7
Y8
Y1
Y2
Y8
Y7
Y8
Y9
Y9
Y9
••
•
Xn
T1
Yn
Tbegin
Yn
Yn
Tmid
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Yn
Yn
Tend
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Early Communication Indicators
5.00
4.50
4.00
3.50
3.00
Gestures
Vocalizations
2.50
Single Word Utterances
2.00
Multiple Word Utterances
1.50
1.00
0.50
0.00
MO6
MO9
MO12
MO15
MO18
MO21
MO24
MO27
MO30
MO33
MO36
T-Scores
• Individual-likelihood Based Estimation
– Allows individually varying values of time
yit = αi + βiλit + εit
– Ages in months ((days/365)*12) were calculated
and centered around locations of latent
intercepts
T-Scores
Gestures
IFSP
IFSP
Ψ31 = -.003 (ns)
Ψ21 = .05
Ψ32 = -.06
.07
1.69
Ψ11 = .01
S1_GES
6
-.03
Ψ22 = .86
9
Ψ33 = .01
I_GES15
12
15
S2_GES
18
21
24
Vocalizations
IFSP
IFSP
IFSP
Ψ31 = -.006
.18
Ψ11 = .02
Ψ21 = .20
Ψ22 = 2.59
S1_VOC
6
9
Ψ32 = -.13
3.70
12
I_VOC18
15
-.13
Ψ33 = .01
18
S2_VOC
21
24
27
30
33
36
Single Word Utterances
IFSP
IFSP
Ψ21 = .10
.16
3.81
Ψ11 = .004
Ψ22 = 2.47
S_WRD
12
15
18
I_WRD36
21
24
27
30
33
36
Multiple Word Utterances
IFSP
IFSP
Ψ21 = .43
.24
4.30
Ψ22 = 7.79
Ψ11 = .02
S_MUL
18
21
I_MUL36
24
27
30
33
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Thank You!
Todd D. Little
University of Kansas
Director, Quantitative Training Program
Director, Center for Research Methods and Data Analysis
Director, Undergraduate Social and Behavioral Sciences Methodology Minor
Member, Developmental Psychology Training Program
crmda.KU.edu
Colloquium presented 04-06-2013 @
Purdue University
crmda.KU.edu
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Update
Dr. Todd Little is currently at
Texas Tech University
Director, Institute for Measurement, Methodology, Analysis and Policy (IMMAP)
Director, “Stats Camp”
Professor, Educational Psychology and Leadership
Email: yhat@ttu.edu
IMMAP (immap.educ.ttu.edu)
Stats Camp (Statscamp.org)
www.Quant.KU.edu
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