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Todd D. LittleUniversity of Kansas

Director, Quantitative Training ProgramDirector, Center for Research Methods and Data Analysis

Director, Undergraduate Social and Behavioral Sciences Methodology MinorMember, Developmental Psychology Training Program

crmda.KU.eduColloquium presented 04-05-2013 @ Purdue University

Special Thanks to Noel A. Card, James P. Selig, & Kristopher Preacher

Representing Time in Longitudinal Research: Assessment Lag as Moderator

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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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Design Independent

Variables Confounded Effect

Cohort-Sequential

Age & Cohort

Age x Cohort Interaction is confounded with Time

Time- Sequential

Age & Time Age x Time Interaction is confounded with Cohort

Cross-Sequential

Cohort & Time

Cohort x Time Interaction is confounded with Age

What’s Confounded?

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Transforming to Accelerated Longitudinal

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Accelerated Longitudinal Designs

Fall 6

Spr6

Fall7

Spr7

Fall8

Spr8

Fall9

Grade 6

Grade 7

Grade 8

Grade

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Accelerated Growth Curve Model(L13.1.GC.LevelCUBIC.Accelerated)

Fall 6

Spr.6

Fall7

Spr.7

Fall8

Spr.8

Fall9

==

==

= ==

===

==

==

-4*

5*0*

-3*-3* 0*

5* -1*1*1*0*-1*-1* 1*

-3*-2*-1* 0*1*2*3*

1* 1*1*1*

1*

1*1*

a1

Inter-cept

Linear

a2

Quad-ratic

a3

Cubic

a4

Grade8

8=11*

0*

Grade7

7=11*

0*

= = = = = = =

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Plot of Estimated Trends

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Fall 6 Spr 6 Fall 7 Spr 7 Fall 8 Spr 8 Fall 9

Positive Affect

Negative Affect

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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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• Use 2-time point data with variable time-lags to measure a growth trajectory + practice effects (McArdle & Woodcock, 1997)

Developmental time-lag model

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T1 T2Age

1student

2345678

0 1 2Time

3 4 5 65;65;34;94;64;115;75;25;4

5;75;84;115;05;45;105;35;8

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T0 T1 T2 T3 T4 T5 T6

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T0 T1

1

T2 T3 T4 T5 T6

Intercept

11

11

1t t tY I B G A P

111

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T0 T1

1

T2 T3 T4 T5 T6

1

Intercept

12 3 4

Growth

0

1t t tY I B G A P Linear growth

56

11

11

111

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T0 T1

1

T2 T3 T4 T5 T6

1 1

Intercept

11

1

Practice

1

Growth

1t t tY I B G A P Constant Practice Effect

11

11

111 1

011

2 3 40

56

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T0 T1

1

T2 T3 T4 T5 T6

1 1

Intercept

.45 .35

PracticeGrowth

1t t tY I B G A P Exponential Practice Decline

11

11

111 112 3 4

05

6

.55.67

.870

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T0Y I1T1Y I G P 2T2Y I G P

3T3Y I G P

4 4TY I G P

5 5TY I G P

6 6TY I G P

The Equations for Each Time Point

T0Y I1 1.0T1Y I G P 2 .82T2Y I G P

3 .67T3Y I G P

4 4 .55TY I G P

5 5 .45TY I G P

6 6 .37TY I G P

Constant Practice Effect Declining Practice Effect

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• 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

Developmental time-lag model

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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

• 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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Lag as Moderator (LAM) Models

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X1X2X3X4X5

Xn

Y1Y2

Y3Y4

Y5

Yn

T1 Tmin Tmax

•••

25

Variable Actual Assessments

T2

X6 Y6X7 Y7X8 Y8X9 Y9Xi YiXj Yj

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• 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

0 1 2 3i i i i iY b b X b Lag b X Lag

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Multiple Regression LAM model

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• 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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An Empirical Example

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Lag (Mean Centered)

Eff

ect o

f H

OM

ET

1 on

MD

I T2

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

2 0 1 T1 2 3 1TMDI b b HOME b Lag b HOME Lag

HOME predicting MDI

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• 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

Implications of LAM Models

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Randomly Distributed AssessmentX1X2X3X4X5

Xn

Y1

Yn

T1 Tbegin Tend

•••

Tmid

X6X7X8X9

Y1 Y1Y1 Y1Y2Y2 Y2Y2 Y2

Y3Y3 Y3Y3 Y3Y4Y4 Y4Y4 Y4Y5Y5 Y5Y5 Y5Y6Y6 Y6Y6 Y6

Y7Y7 Y7Y7 Y7Y8Y8 Y8Y8 Y8

Y9 Y9Y9Y9 Y9

Yn Yn Yn Yn

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Early Communication Indicators

MO6 MO9 MO12 MO15 MO18 MO21 MO24 MO27 MO30 MO33 MO360.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

GesturesVocalizationsSingle Word UtterancesMultiple Word Utterances

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

Ψ31 = -.003 (ns)

6 9 12 15 18 21 24

S1_GES I_GES15 S2_GES

-.031.69.07Ψ21 = .05

Ψ22 = .86 Ψ33 = .01Ψ11 = .01

Ψ32 = -.06

IFSPIFSP

Vocalizations

Ψ31 = -.006

6 9 12 15 18 21 24 27 30 33 36

S1_VOC I_VOC18 S2_VOC

-.133.70.18Ψ21 = .20

Ψ22 = 2.59 Ψ33 = .01Ψ11 = .02

Ψ32 = -.13

IFSPIFSPIFSP

Single Word Utterances

Ψ21 = .10

12 15 18 21 24 27 30 33 36

S_WRD I_WRD36

Ψ22 = 2.47Ψ11 = .004

IFSPIFSP

3.81.16

Multiple Word Utterances

Ψ21 = .43

18 21 24 27 30 33 36

S_MUL I_MUL36

Ψ22 = 7.79

4.30 .24Ψ11 = .02

IFSPIFSP

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Todd D. LittleUniversity of Kansas

Director, Quantitative Training ProgramDirector, Center for Research Methods and Data Analysis

Director, Undergraduate Social and Behavioral Sciences Methodology MinorMember, Developmental Psychology Training Program

crmda.KU.eduColloquium presented 04-06-2013 @

Purdue University

Thank You!

39www.Quant.KU.edu

Update

Dr. Todd Little is currently at

Texas Tech UniversityDirector, 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)