1crmda.KU.edu

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.eduWorkshop presented 05-24-2012 @

University of Turku, Finland

Special Thanks to: Ihno Lee, Chapter co-author in Handbook.

Dynamic P-Technique Structural Equation Modeling

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Cattell’s Data Box

• Cattell invented the Box to help us think ‘outside the box’

• Given the three primary dimensions of variables, persons, and occasions, at least 6 different structural relationships can be utilized to address specific research questions

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Cattell’s Data Box

Occasions of Measurement

Variables

(or T

ests)

Per

son

s (o

r E

nti

ties

)

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Cattell’s Data Box• R-Technique: Variables by Persons

• Most common Factor Analysis approach• Q-Technique: Persons by Variables

• Cluster analysis – subgroups of people• P-Technique: Variables by Occasions

• Intra-individual time series analyses• O-Technique: Occasions by Variables

• Time-dependent (historical) clusters• S-Technique: People by Occasions

• People clustering based on growth patterns• T-Technique: Occasions by People

• Time-dependent clusters based on people

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Michael Lebo’s Example Data

• Lebo asked 5 people to rate their energy for 103 straight days

• The 5 folks rated their energy on 6 items using a 4 point scale:• Active, Lively, Peppy• Sluggish, Tired, Weary

• A priori, we would expect two constructs, positive energy and negative energy

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Lag 0

Observational RecordO1

Observational RecordO2

Observational RecordO3

Observational RecordO4

Observational RecordOnObservational RecordOn-1On-1

On

Selected VariablesV

P-Technique Data Setup

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Multivariate Time-series(Multiple Variables x Multiple Occasions for 1 Person)

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1st 15 days for Subject 4, Lag 0 1 111 212 2 333 011 3 111 333 4 333 011 5 233 111 6 333 111 7 344 000 8 222 111 9 222 111 10 333 001 11 434 011 12 101 443 13 343 111 14 334 111 15 110 343

The Obtained Correlations All Days

Positive Items Negative Items

1.000 0.849 1.000 0.837 0.864 1.000 -0.568 -0.602 -0.660 1.000 -0.575 -0.650 -0.687 0.746 1.000 -0.579 -0.679 -0.724 0.687 0.786 1.000

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Var 1

Var 2

Three Indicators of the Same Construct in a Time Series

Var 3

Time

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L15.1.s1.Lag0.LS8

Positive Negative

1.15 .99 .86 .81 1.27 .92

-.19(-.64)

.09 .18 .18 .21 .08 .13

Active WearyTiredSluggishPeppyLively

.19 .56

Model Fit: χ2(8, n=101) = 9.36, p = .31, RMSEA = .039(.000;.128), TLI/NNFI = .994, CFI=.997

X.21 .15 -.35 .03 .01 -.04

.88 .52

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L15.1.s2.Lag0.LS8

Positive Negative

1.04 1.10 .86 .92 1.03 1.05

-.74(-.65)

.41 .04 .19 .72 .22 .21

Active WearyTiredSluggishPeppyLively

.93 1.43

Model Fit: χ2(8, n=101) = 8.36, p = .40, RMSEA = .014(.000;.119), TLI/NNFI = .999, CFI=.999

X.27 -.06 -.21 .01 .01 -.02

1.09 .96

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L15.1.s3.Lag0.LS8

Positive Negative

1.07 1.11 .83 .73 1.17 1.10

-.21(-.43)

.40 .19 .33 .14 .10 .09

Active WearyTiredSluggishPeppyLively

.77 .32

Model Fit: χ2(8, n=101) = 9.70, p = .31, RMSEA = .050(.000;.134), TLI/NNFI = .992, CFI=.997

X.31 -.11 -.20 .00 .01 -.01

1.26 .28

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L15.1.s4.Lag0.LS8

Positive Negative

.91 1.01 1.08 .95 1.05 1.00

-.82(-.81)

.20 .16 .15 .48 .28 .32

Active WearyTiredSluggishPeppyLively

.97 1.05

Model Fit: χ2(8, n=101) = 14.6, p = .07, RMSEA = .084(.000;.158), TLI/NNFI = .983, CFI=.991

X.19 .03 -.22 -.13 .11 .03

1.86 1.05

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L15.1.s5.Lag0.LS8

Positive Negative

1.03 .96 1.02 .08 1.67 1.25

-.59(-.60)

.35 .52 .63 .17 .46 1.20

Active WearyTiredSluggishPeppyLively

1.19 .81

Model Fit: χ2(8, n=101) = 5.11, p = .75, RMSEA = .000(.000;.073), TLI/NNFI = 1.02, CFI=1.0

X.09 .16 -.25 -.03 .21 -.18

1.15 1.03

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Measurement Invariance by Participant

Model χ2 df p RMSEA 90% CI TLI/NNFI CFI Constraint Tenable

Null 3351.349 123 <.001 --- --- - --- --- --- ---

Configural 47.161 40 .203 .038 .000-.082 0.993 0. 998 ---Invariance

Loading 166.392 56 <.001 .137 .113-.162 0.925 0.966 NoInvariance

Intercept 373.738 72 <.001 .192 .172-.213 0.843 0.907 NoInvariance

Partial 90.255 63 <.014 .063 .025-.092 0.984 0.982 YesInvariance

(L3.alternative null fit.xls)

(L15.s1-s5.0.Lag0.null)(L15.s1-s5.1.Lag0.config)(L15.s1-s5.2.Lag0.weak)

(L15.s1-s5.3.Lag0.partial)(L15.s1-s5.4.Lag0.strong)

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Some Thoughts

• The partial invariance across persons highlights the ideographic appeal of p-technique

• Nomothetic comparisons of the constructs is doable, but the composition of the constructs is allowed to vary for some persons (e.g., person 5 did not endorse ‘sluggish’).

• In fact, Nesselroade has an idea that turns the concept of invariance ‘on its head’

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Lag 0

Non-matched recordObservational RecordO1

Observational RecordO2

Observational RecordO3

Observational RecordOnObservational RecordOn-1On-1

On

Selected Variables(V )

Lag 1

Observational RecordO1

Observational RecordO2

Observational RecordO3

Observational RecordO4

Observational RecordO4 Observational RecordO5

Non-matched recordObservational RecordOn

Selected Variables (V*)2V,or V+V*

Dynamic P-Technique Setup

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C 12

C 13

CL 21*

CL 12*

C 1*3* C 2*3*

C 1*2*

CL 13* CL 23*

CL 31*

CL 32*

AR 11*

AR 22*

AR 33*

C 23

21

22

23

21*

22*

23*

Variable 1

Variable 2

Variable 3

Variable 1*

Variable 2*

Variable 3*

Variable 1 Variable 2 Variable 3 Variable 1* Variable 2* Variable 3*

Lag 0 Lag 1

A Lagged Covariance Matrix

AR = Autoregressive Correlation

CL = Cross-lagged Correlation

C = Within Lag Covariance

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1st 15 days for Subject 4, 3 Lags 1 111 212 333 011 111 333 2 333 011 111 333 333 011 3 111 333 333 011 233 111 4 333 011 233 111 333 111 5 233 111 333 111 344 000 6 333 111 344 000 222 111 7 344 000 222 111 222 111 8 222 111 222 111 333 001 9 222 111 333 001 434 011 10 333 001 434 011 101 443 11 434 011 101 443 343 111 12 101 443 343 111 334 111 13 343 111 334 111 110 343 14 334 111 110 343 444 000 15 110 343 444 000 333 120

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L15.4.s4.3lags: Subject 4

NegativeLag 0

PositiveLag 0

1*

1*

NegativeLag 1

.84

PositiveLag 1

.95

NegativeLag 2

.82

PositiveLag 2

.95

-.79 -.88 -.88

.65

.23

.65

.23

.36 .36

Model Fit: χ2(142, n=101) = 154.3, p = .23; RMSEA = .02; TLI/NNFI = .99

(Initial model: L15.3.s4.3lags)

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L15.4.s1.3lags: Subject 1

NegativeLag 0

PositiveLag 0

1*

1*

NegativeLag 1

.94

PositiveLag 1

1

NegativeLag 2

.94

PositiveLag 2

1

-.64 -.66 -.66

.24 .24

Model Fit: χ2(144, n=101) = 159.9, p = .17; RMSEA = .05; TLI/NNFI = .99

(Initial model: L15.3.s1.3lags)

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L15.4.s5.3lags: Subject 5

NegativeLag 0

PositiveLag 0

1*

1*

NegativeLag 1

1

PositiveLag 1

.94

NegativeLag 2

.94

PositiveLag 2

.94

-.61 -.66 -.66

.24 .24

Model Fit: χ2(143, n=101) = 93.9, p = .99; RMSEA = .00; TLI/NNFI = 1.05

.24

(Initial model: L15.3.s5.3lags)

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L15.4.s3.3lags: Subject 3

NegativeLag 0

PositiveLag 0

1*

1*

NegativeLag 1

.94

PositiveLag 1

1

NegativeLag 2

.92

PositiveLag 2

.88

-.41 -.51 -.51

.24 .24

.37

.31 .31

Model Fit: χ2(142, n=101) = 139.5, p = 1.0; RMSEA = .0; TLI/NNFI = 1.0

(Initial model: L15.3.s3.3lags)

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L15.4.s2.3lags: Subject 2

NegativeLag 0

PositiveLag 0

1*

1*

NegativeLag 1

.95

PositiveLag 1

.95

NegativeLag 2

.91

PositiveLag 2

.94

-.63 -.63 -.63

.24 .24

-.17

-.24 -.24

Model Fit: χ2(142, n=101) = 115.2, p = .95; RMSEA = .0; TLI/NNFI = 1.0

(Initial model: L15.3.s2.3lags)

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As Represented in Growth Curve Models

• How does mood fluctuate during the course of a week?

• Restructure chained, dynamic p-technique data into latent growth curve models of daily mood fluctuation

• Examine the average pattern of growth • Variability in growth (interindividual

variability in intraindividual change)

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Weekly Growth TrendsWeek 1 Week 2 Week 3

Week 4 Week 5 Week 6

Carrig, M., Wirth, R.J., & Curran, P.J. (2004). A SAS Macro for Estimating and Visualizing Individual Growth Curves. Structural Equation Modeling: An Interdisciplinary Journal, 11, 132-149.

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P-technique Data Transformation

Traditional P-technique

Dynamic P-tech, Arbitrary

Dynamic P-tech, Structured

Singleperson

- Identical variable relationships (same r at every time point)- Independent observations

- With time lags, how do scores at T1 affect those at T2- Time points are unstructured(Time 1, Time 2)

- Time dependency- Time points are non-arbitrary (Mon, Tues, Wed)- Compare equivalent relationships

Chained / 2+ people

- Stacked subject data, pools intra-individual info- Assume identical relationships

- With time lags- Time dependency- Unstructured time points

- Time dependency- Structured time points- Compare equivalent relationships across a sample

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Data Restructuring• Add 7 lags – autoregressive effects of energy/mood within a

one-week period• Ex:

Subj Day Lag0 Lag1 Lag2 Lag3 Lag4 Lag5 Lag6 1 Mo . . . . . . 1 1 Tu . . . . . 1 2 1 We . . . . 1 2 1 1 Th . . . 1 2 1 0 1 Fr . . 1 2 1 0 1 1 Sa . 1 2 1 0 1 0 1 Su 1 2 1 0 1 0 1 1 Mo 2 1 0 1 0 1 2 1 Tu 1 0 1 0 1 2 2 1 We 0 1 0 1 2 2 1

• Impute empty records• Create parcels by averaging 3 positive/negative items

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Data Restructuring• Retain selected rows (with Monday as the

beginning of the week)

• Stack participant data setsSubj Day PA_Mo PA_Tu PA_We PA_Th PA_Fr PA_Sa PA_Su 1 Mo1 1.00 0.67 0.67 1.33 1.00 1.33 0.67 1 Mo2 0.67 0.67 1.00 1.00 1.33 0.67 1.00 1 Mo3 0.33 1.00 1.00 1.67 1.67 0.00 1.00 1 . . . . . . . . 1 Mo15 1.00 0.67 0.67 1.33 1.00 1.33 0.67 2 Mo1 1.00 0.33 0.67 0.33 0.67 2.33 0.00 2 Mo2 0.00 0.00 1.00 0.67 1.33 1.33 2.67 2 Mo3 1.33 3.00 1.33 3.00 1.67 0.00 2.67 . . . . . . . . . . . . . . . . . . 5 Mo15 0.00 1.67 0.00 1.33 0.67 1.00 0.33

• Note: meaning assigned to arbitrary time points

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Raw Means and Standard Deviations

Energy ratings on a 5-point scale:

Mon Tues Wed Thurs Fri Sat Sun

Positive /High Energy

1.23(1.05)

1.23(.97)

1.24(1.10)

1.24(.97)

1.32(1.01)

1.18(.94)

1.29(1.02)

Negative /Low Energy

0.97(1.14)

0.92(1.17)

0.90(1.05)

0.81(.97)

0.96(1.17)

0.84(1.06)

1.05(1.08)

N = 75[15 weeks x 5 subjects]

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Level and Shape model

0*

a1 a2

Mon

NegIntercept

NegSlope

1*1* 1* 1*

1.35 -.30

.01.24

-.04

1*

Tues Wed Thurs Fri Sat Sun

1*1*1*

a1 a2

PosSlope

1.08 .13

.08

.0021*S4

1*S3

1*S2

1*S1

.04Pos

Intercept

Model fit: χ2 (116) = 126.79, p = .23, RMSEA = .000, CFI = .98, TLI/NNFI = .98

.06 .12.06 -.10

(L15.7lags.L

evShape)

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Positive Affect model

1*

a1 a3

1*

1.23 .07

.05.19 .09 SundayFriday

.01

a2.07

.09.002

Model fit: χ2 (25) = 25.96, p = .41, RMSEA = .021, CFI = .99, TLI/NNFI = .99

Mon Tues Wed Thurs Fri Sat Sun

1* 1* 1* 1* 1*1*

1*

.79

(L15.7lags.pos)

Pos Intercept

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Negative Affect model

Model fit: χ2 (20) = 18.46, p = .56, RMSEA = .000, CFI = 1.00, TLI/NNFI = 1.01

Mon Tues Wed Thurs Fri Sat Sun.70

1*

1*

1* 1* 1* 1*1*

1*

Friday Sunday

1*2*3*

.40.01 .09 .12

.02

.10

-.03

.001

.003

-.001a1

.84a4

.21

a2 a3.05 .13

(L15.7lags.neg)

NegIntercept

NegSlope

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Cost-benefit analysis

• Extrapolates the average within-person change from pooled time series data

• But obscures unique information about each individual’s variability and growth patterns

• Does not utilize the strengths of P-technique data

• Add subject covariates to detect individual differences at the mean level

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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)