1. Structural Equation Modeling
Dynamic P-Technique
Structural Equation Modeling
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
Workshop presented
Society for Research in Adolescence Peer Preconference
Special Thanks to: Ihno Lee, Chapter co-author in Handbook.
crmda.KU.edu

2. 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

3. Cattell’s Data Box Occasions of Measurement Variables (or Tests)
Persons (or Entities)

4. 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

5. 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

6. P-Technique Data Setup
Lag 0
Observational Record O 1 2 3 4 n -1
Selected Variables V

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

8. 1st 15 days for Subject 4, Lag 0
The Obtained Correlations All Days
Positive Items Negative Items
1.000

9. Three Indicators of the Same Construct in a Time Series
Var 1
Var 2
Var 3
Time

10. L15.1.s1.Lag0.LS8
-.19
(-.64)
Positive
Negative
.19
.56
.88
.52
1.15
.99
.86
.81
1.27
.92
Active
Lively
Peppy
Sluggish
Tired
Weary
.09
.18
.18
.21
.08
.13 X
.21
.15
-.35
.03
.01
-.04
Model Fit: χ2(8, n=101) = 9.36, p = .31, RMSEA = .039(.000;.128), TLI/NNFI = .994, CFI=.997

11. L15.1.s2.Lag0.LS8
-.74
(-.65)
Positive
Negative
.93
1.43
1.09
.96
1.04
1.10
.86
.92
1.03
1.05
Active
Lively
Peppy
Sluggish
Tired
Weary
.41
.04
.19
.72
.22
.21 X
.27
-.06
-.21
.01
.01
-.02
Model Fit: χ2(8, n=101) = 8.36, p = .40, RMSEA = .014(.000;.119), TLI/NNFI = .999, CFI=.999

12. L15.1.s3.Lag0.LS8
-.21
(-.43)
Positive
Negative
.77
.32
1.26
.28
1.07
1.11
.83
.73
1.17
1.10
Active
Lively
Peppy
Sluggish
Tired
Weary
.40
.19
.33
.14
.10
.09 X
.31
-.11
-.20
.00
.01
-.01
Model Fit: χ2(8, n=101) = 9.70, p = .31, RMSEA = .050(.000;.134), TLI/NNFI = .992, CFI=.997

13. L15.1.s4.Lag0.LS8
-.82
(-.81)
Positive
Negative
.97
1.05
1.86
1.05
.91
1.01
1.08
.95
1.05
1.00
Active
Lively
Peppy
Sluggish
Tired
Weary
.20
.16
.15
.48
.28
.32 X
.19
.03
-.22
-.13
.11
.03
Model Fit: χ2(8, n=101) = 14.6, p = .07, RMSEA = .084(.000;.158), TLI/NNFI = .983, CFI=.991

14. L15.1.s5.Lag0.LS8
-.59
(-.60)
Positive
Negative
1.19
.81
1.15
1.03
1.03
.96
1.02
.08
1.67
1.25
Active
Lively
Peppy
Sluggish
Tired
Weary
.35
.52
.63
.17
.46
1.20 X
.09
.16
-.25
-.03
.21
-.18
Model Fit: χ2(8, n=101) = 5.11, p = .75, RMSEA = .000(.000;.073), TLI/NNFI = 1.02, CFI=1.0

15. Measurement Invariance by Participant
(L3.alternative null fit.xls)
Measurement Invariance by Participant
Model χ2 df p RMSEA % CI TLI/NNFI CFI Constraint Tenable
Null <
Configural
Invariance
Loading < No
Intercept < No
Partial < Yes
(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)

16. 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’

17. Dynamic P-Technique Setup
Lag 0
Non-matched record
Observational Record O 1 2 3 n -1
Selected Variables ( V )
Lag 1 4 5
Selected Variables ( * V, or
V+V*

18. A Lagged Covariance Matrix12 13 CL
21*
12*
1*3*
2*3*
1*2*
13*
23*
31*
32* AR
11*
22*
33* 23 2 1 3 1* 2* 3*
Variable 1
Variable 2
Variable 3
Variable 1*
Variable 2*
Variable 3*
Lag 0
Lag 1
AR = Autoregressive Correlation
CL = Cross-lagged Correlation
C = Within Lag Covariance

19. 1st 15 days for Subject 4, 3 Lags

20. L15.4.s4.3lags: Subject 4 Positive Lag 1 Positive Lag 2 Positive Lag 0
(Initial model: L15.3.s4.3lags)
L15.4.s4.3lags: Subject 4
Positive
Lag 1
.95
Positive
Lag 2
.95 1*
Positive
Lag 0
.23
.23
-.79
-.88
-.88
.36
.36
Negative
Lag 0
Negative
Lag 1
.84
Negative
Lag 2
.82
.65
.65 1*
Model Fit: χ2(142, n=101) = 154.3, p = .23; RMSEA = .02; TLI/NNFI = .99

21. L15.4.s1.3lags: Subject 1 Positive Lag 1 Positive Lag 2 Positive Lag 0
(Initial model: L15.3.s1.3lags)
L15.4.s1.3lags: Subject 1
Positive
Lag 1 1
Positive
Lag 2 1 1*
Positive
Lag 0
-.64
-.66
-.66
Negative
Lag 0
Negative
Lag 1
.94
Negative
Lag 2
.94
.24
.24 1*
Model Fit: χ2(144, n=101) = 159.9, p = .17; RMSEA = .05; TLI/NNFI = .99

22. L15.4.s5.3lags: Subject 5 Positive Lag 1 Positive Lag 2 Positive Lag 0
(Initial model: L15.3.s5.3lags)
L15.4.s5.3lags: Subject 5
Positive
Lag 1
.94
Positive
Lag 2
.94 1*
Positive
Lag 0
.24
.24
-.61
-.66
-.66
.24
Negative
Lag 0
Negative
Lag 1 1
Negative
Lag 2
.94 1*
Model Fit: χ2(143, n=101) = 93.9, p = .99; RMSEA = .00; TLI/NNFI = 1.05

23. L15.4.s3.3lags: Subject 3 Positive Lag 1 Positive Lag 2 Positive Lag 0
(Initial model: L15.3.s3.3lags)
L15.4.s3.3lags: Subject 3
Positive
Lag 1 1
Positive
Lag 2
.88 1*
.37
Positive
Lag 0
-.41
-.51
-.51
.31
.31
Negative
Lag 0
Negative
Lag 1
.94
Negative
Lag 2
.92
.24
.24 1*
Model Fit: χ2(142, n=101) = 139.5, p = 1.0; RMSEA = .0; TLI/NNFI = 1.0

24. L15.4.s2.3lags: Subject 2 Positive Lag 1 Positive Lag 2 Positive Lag 0
(Initial model: L15.3.s2.3lags)
L15.4.s2.3lags: Subject 2
Positive
Lag 1
.95
Positive
Lag 2
.94 1*
Positive
Lag 0
-.63
-.63
-.63
-.24
-.24
Negative
Lag 0
Negative
Lag 1
.95
-.17
Negative
Lag 2
.91
.24
.24 1*
Model Fit: χ2(142, n=101) = 115.2, p = .95; RMSEA = .0; TLI/NNFI = 1.0

25. 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)

26. Weekly Growth Trends Week 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,

27. P-technique Data Transformation
Traditional
P-technique
Dynamic P-tech, Arbitrary
Dynamic P-tech, Structured
Single
person
- 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
- Unstructured time points
- Structured time points
- Compare equivalent relationships across a sample

28. 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 Tu
1 We
1 Th
1 Fr
1 Sa
1 Su
1 Mo
1 Tu
1 We
Impute empty records
Create parcels by averaging 3 positive/negative items

29. Data Restructuring
Retain selected rows (with Monday as the beginning of the week)
Stack participant data sets
Subj Day PA_Mo PA_Tu PA_We PA_Th PA_Fr PA_Sa PA_Su
1 Mo
1 Mo
1 Mo
1 Mo
2 Mo
2 Mo
2 Mo
5 Mo
Note: meaning assigned to arbitrary time points

30. 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)
(.97)
1.24
(1.10)
1.32
(1.01)
1.18
(.94)
1.29
(1.02)
Negative /
Low Energy
0.97
(1.14)
0.92
(1.17)
0.90
0.81
0.96
0.84
(1.06)
1.05
(1.08)
N = 75
[15 weeks x 5 subjects]

31. Level and Shape model a1 a2 Pos Pos Intercept Slope a1 a2 Neg Neg
.13
1.08 1* S4 S3 S2 S1 a1 a2
.002
Pos
Intercept
Pos
Slope
.08
.04
.06
-.10
.06
.12
1.35 a1
-.30 a2
-.04
Neg
Intercept
Neg
Slope
.24 1*
.01 0* 1* 1* 1* 1* 1* 1* 1*
(L15.7lags.LevShape)
Mon
Tues
Wed
Thurs
Fri
Sat
Sun
Model fit: χ2 (116) = , p = .23, RMSEA = .000, CFI = .98, TLI/NNFI = .98

32. Positive Affect model a1 a2 a3 Pos Intercept Friday Sunday .01 1.23
(L15.7lags.pos) a1
1.23 a2
.09
.07 a3
.07
.002
Pos
Intercept
Friday
Sunday
.19
.09
.05 1* 1* 1* 1* 1* 1* 1* 1* 1*
Mon
Tues
Wed
Thurs
Fri
Sat
Sun
.79
Model fit: χ2 (25) = 25.96, p = .41, RMSEA = .021, CFI = .99, TLI/NNFI = .99

33. Negative Affect model a1 a4 Neg Intercept Neg Slope Friday Sunday a2
(L15.7lags.neg)
-.03
.21
.84
.10
.003 a1 a4
.02
.001
-.001
Neg
Intercept
Neg
Slope
Friday
Sunday
.01
.09
.12
.40 1* a2 a3 1* 1*
.05
.13 1* 1* 1* 1* 1* 3* 2* 1*
Mon
Tues
Wed
Thurs
Fri
Sat
Sun
.70
Model fit: χ2 (20) = 18.46, p = .56, RMSEA = .000, CFI = 1.00, TLI/NNFI = 1.01

34. 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

35. 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
IMMAP (immap.educ.ttu.edu)
Stats Camp (Statscamp.org)