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 05-24-2012 @
University of Turku, Finland
Special Thanks to: Ihno Lee, Chapter co-author in Handbook.
1
crm
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
www.crmda.ku.edu
2
Cattell’s Data Box
Persons (or Entities)
Occasions of Measurement
www.crmda.ku.edu
3
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
•
S-Technique: People by Occasions
•
• Time-dependent (historical) clusters
•
People clustering based on growth patterns
T-Technique: Occasions by People
•
Time-dependent clusters based on people
www.crmda.ku.edu
4
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
www.crmda.ku.edu
5
P-Technique Data Setup
Lag 0
Selected Variables
Observational Record O1 V
Observational Record O2
Observational Record O3
Observational Record O4
On-1 Observational Record On-1
On Observational Record On
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6
Multivariate Time-series
(Multiple Variables x Multiple Occasions for 1 Person)
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7
st
1
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
www.crmda.ku.edu
8
Three Indicators of the Same Construct in a
Time Series
Var 1
Var 2
Var 3
Time
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9
L15.1.s1.Lag0.LS8
-.19
.19
1.15
(-.64)
Positive
.99
Active
.09
Lively
.18
.21
.88
Negative
.52
.86
.81
Peppy
1.27
Sluggish
.18
.15
-.35
.03
.92
Tired
.21
.08
.01
.56
Weary
.13
-.04
X
Model Fit: χ2(8, n=101) = 9.36, p = .31, RMSEA = .039(.000;.128), TLI/NNFI = .994, CFI=.997
www.crmda.ku.edu
10
L15.1.s2.Lag0.LS8
-.74
.93
(-.65)
Positive
1.04 1.10
Active
.41
Lively
.04
.27
1.09
Negative
.96
.86
Peppy
.19
-.06
-.21
.92
1.03 1.05
Sluggish
Tired
.72
.01
.22
.01
1.43
Weary
.21
-.02
X
Model Fit: χ2(8, n=101) = 8.36, p = .40, RMSEA = .014(.000;.119), TLI/NNFI = .999, CFI=.999
www.crmda.ku.edu
11
L15.1.s3.Lag0.LS8
-.21
.77
(-.43)
Positive
1.07 1.11
Active
.40
Lively
.19
.31
1.26
Negative
.28
.83
Peppy
.33
-.11
-.20
.73
1.17 1.10
Sluggish
Tired
.14
.00
.10
.01
.32
Weary
.09
-.01
X
Model Fit: χ2(8, n=101) = 9.70, p = .31, RMSEA = .050(.000;.134), TLI/NNFI = .992, CFI=.997
www.crmda.ku.edu
12
L15.1.s4.Lag0.LS8
-.82
.97
(-.81)
Positive
1.86
1.05
.91 1.01 1.08
Active
.20
Lively
.16
.19
Negative
Peppy
.15
.03
-.22
.95
1.05 1.00
Sluggish
Tired
.48
-.13
1.05
Weary
.28
.11
.32
.03
X
Model Fit: χ2(8, n=101) = 14.6, p = .07, RMSEA = .084(.000;.158), TLI/NNFI = .983, CFI=.991
www.crmda.ku.edu
13
L15.1.s5.Lag0.LS8
-.59
1.19
1.03
(-.60)
Positive
Lively
.52
.09
1.03
.96 1.02
Active
.35
1.15
Negative
Peppy
.63
.16
-.25
.08
1.67 1.25
Sluggish
Tired
.17
-.03
.46
.21
.81
Weary
1.20
-.18
X
Model Fit: χ2(8, n=101) = 5.11, p = .75, RMSEA = .000(.000;.073), TLI/NNFI = 1.02, CFI=1.0
www.crmda.ku.edu
14
(L3.alternative null fit.xls)
Measurement Invariance by Participant
Model
Null
χ2
df
p
3351.349 123 <.001
RMSEA
90% CI TLI/NNFI
CFI
Constraint Tenable
---
--- - ---
---
---
---
Configural 47.161 40
Invariance
.203
.038
.000-.082
0.993
0. 998
---
Loading 166.392 56
Invariance
<.001
.137
.113-.162
0.925
0.966
No
Intercept 373.738 72
Invariance
<.001
.192
.172-.213
0.843
0.907
No
Partial
90.255 63
Invariance
<.014
.063
.025-.092
0.984
0.982
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)
www.crmda.ku.edu
15
Some Thoughts
• The partial invariance across persons
•
•
highlights the ideographic appeal of ptechnique
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’
www.crmda.ku.edu
16
Dynamic P-Technique Setup
Lag 0
Lag 1
Selected Variables (V )
Non-matched record
Observational Record O1
Observational Record O2
Observational Record O3
Observational Record O4
Selected Variables (V*)
Observational Record O1 2V,
or V+V*
Observational Record O2
Observational Record O3
Observational Record O4
Observational Record O5
On-1 Observational Record On-1 Observational Record On
On Observational Record On Non-matched record
www.crmda.ku.edu
17
A Lagged Covariance Matrix
Lag 0
Variable 1
Variable 1
Variable 2
Lag 1
Variable 3
2
1
Variable 1*
Variable 2*
Variable 3*
AR = Autoregressive Correlation
CL = Cross-lagged Correlation
Variable 2
C 12
2
2
Variable 3
C 13
C 23
Variable 1*
AR 11*
CL 21*
CL 31*
2
1*
Variable 2*
CL 12*
AR 22*
CL 32*
C 1*2*
Variable 3*
CL 13*
CL 23*
AR 33*
C 1*3*
C = Within Lag Covariance
2
3
www.crmda.ku.edu
2
2*
C 2*3*
2
3*
18
st
1
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
www.crmda.ku.edu
19
(Initial model: L15.3.s4.3lags)
L15.4.s4.3lags: Subject 4
.95
1*
Positive
Lag 0
-.79
.95
Positive
Lag 1
.23
-.88
-.88
.36
Negative
Lag 0
1*
Positive
Lag 2
.23
.65
.36
Negative
Lag 1
.84
.65
Negative
Lag 2
.82
Model Fit: χ2(142, n=101) = 154.3, p = .23; RMSEA = .02; TLI/NNFI = .99
www.crmda.ku.edu
20
(Initial model: L15.3.s1.3lags)
L15.4.s1.3lags: Subject 1
1*
1
1
Positive
Lag 0
Positive
Lag 1
Positive
Lag 2
-.64
-.66
Negative
Lag 0
1*
-.66
.24
Negative
Lag 1
.94
.24
Negative
Lag 2
.94
Model Fit: χ2(144, n=101) = 159.9, p = .17; RMSEA = .05; TLI/NNFI = .99
www.crmda.ku.edu
21
(Initial model: L15.3.s5.3lags)
L15.4.s5.3lags: Subject 5
.94
1*
Positive
Lag 0
-.61
.24
.94
Positive
Lag 1
-.66
.24
Positive
Lag 2
-.66
.24
Negative
Lag 0
Negative
Lag 1
Negative
Lag 2
1*
1
.94
Model Fit: χ2(143, n=101) = 93.9, p = .99; RMSEA = .00; TLI/NNFI = 1.05
www.crmda.ku.edu
22
(Initial model: L15.3.s3.3lags)
L15.4.s3.3lags: Subject 3
1*
1
Positive
Lag 0
Positive
Lag 1
-.41
-.51
Positive
Lag 2
-.51
.31
Negative
Lag 0
1*
.88
.37
.24
.31
Negative
Lag 1
.94
.24
Negative
Lag 2
.92
Model Fit: χ2(142, n=101) = 139.5, p = 1.0; RMSEA = .0; TLI/NNFI = 1.0
www.crmda.ku.edu
23
(Initial model: L15.3.s2.3lags)
L15.4.s2.3lags: Subject 2
1*
.95
.94
Positive
Lag 0
Positive
Lag 1
Positive
Lag 2
-.63
-.63
-.63
-.24
-.24
-.17
Negative
Lag 0
1*
.24
Negative
Lag 1
.95
.24
Negative
Lag 2
.91
Model Fit: χ2(142, n=101) = 115.2, p = .95; RMSEA = .0; TLI/NNFI = 1.0
www.crmda.ku.edu
24
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)
www.crmda.ku.edu
25
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, 132-149.
www.crmda.ku.edu
26
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 nonarbitrary (Mon, Tues,
Wed)
- Compare equivalent
relationships
Chained /
2+ people
- Stacked subject
data, pools intraindividual info
- Assume identical
relationships
- With time lags
- Time dependency
- Unstructured time
points
- Time dependency
- Structured time
points
- Compare equivalent
relationships across a
sample
www.crmda.ku.edu
27
Data Restructuring
•
•
Add 7 lags – autoregressive effects of energy/mood within a
one-week period
Ex:
Subj
1
1
1
1
1
1
1
1
1
1
Day
Mo
Tu
We
Th
Fr
Sa
Su
Mo
Tu
We
Lag0
.
.
.
.
.
.
1
2
1
0
Lag1
.
.
.
.
.
1
2
1
0
1
Lag2
.
.
.
.
1
2
1
0
1
0
Lag3
.
.
.
1
2
1
0
1
0
1
Lag4
.
.
1
2
1
0
1
0
1
2
Lag5
.
1
2
1
0
1
0
1
2
2
Lag6
1
2
1
0
1
0
1
2
2
1
• Impute empty records
• Create parcels by averaging 3 positive/negative items
www.crmda.ku.edu
28
Data Restructuring
•
Retain selected rows (with Monday as the
beginning of the week)
•
Stack participant data sets
Subj
1
1
1
1
1
2
2
2
.
.
5
•
Day
Mo1
Mo2
Mo3
.
Mo15
Mo1
Mo2
Mo3
.
.
Mo15
PA_Mo
1.00
0.67
0.33
.
1.00
1.00
0.00
1.33
.
.
0.00
PA_Tu
0.67
0.67
1.00
.
0.67
0.33
0.00
3.00
.
.
1.67
PA_We
0.67
1.00
1.00
.
0.67
0.67
1.00
1.33
.
.
0.00
PA_Th
1.33
1.00
1.67
.
1.33
0.33
0.67
3.00
.
.
1.33
PA_Fr
1.00
1.33
1.67
.
1.00
0.67
1.33
1.67
.
.
0.67
PA_Sa
1.33
0.67
0.00
.
1.33
2.33
1.33
0.00
.
.
1.00
PA_Su
0.67
1.00
1.00
.
0.67
0.00
2.67
2.67
.
.
0.33
Note: meaning assigned to arbitrary time points
www.crmda.ku.edu
29
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]
www.crmda.ku.edu
30
Level and Shape model
1.08
.13
a1
.002
Pos
.08
Intercept
Pos
Slope
.06
1.35 a
1
.24
1*
S3
1*
S2
1*
1* 1*
Tues
a2
-.04
1*
1*
1*
Wed
0*
Neg
Slope
-.30
S1
1*
.01
1*
Thurs
Fri
Sat
Sun
Model fit: χ2 (116) = 126.79, p = .23, RMSEA = .000, CFI = .98, TLI/NNFI = .98
www.crmda.ku.edu
(L15.7lags.LevShape)
Mon
.04
.12
Neg
Intercept
1*
a2
-.10
.06
1*
S4
31
Positive Affect model
(L15.7lags.pos)
.01
a1 1.23
.19
Pos
Intercept 1*
1* 1*
1* 1*
.79
Mon
.07
Tues
.09 a2
.09
Friday
1*
1*
Wed
.002
.07 a
3
Sunday
1*
Thurs
Fri
.05
1*
Sat
Sun
Model fit: χ2 (25) = 25.96, p = .41, RMSEA = .021, CFI = .99, TLI/NNFI = .99
www.crmda.ku.edu
32
Negative Affect model
(L15.7lags.neg)
-.03
a1
.40
.001
.02
.01
Neg
Intercept 1*
Neg
Slope
1*
1* 1* 1* 1*
3*
.70
Mon
2*
Tues
.21
.003
.10
.84
.09
a4
-.001
Friday
a2
.12
Sunday
a3
.13
.05
1*
1*
1*
Wed
Thurs
Fri
Sat
Sun
Model fit: χ2 (20) = 18.46, p = .56, RMSEA = .000, CFI = 1.00, TLI/NNFI = 1.01
www.crmda.ku.edu
33
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
www.crmda.ku.edu
34
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)
35
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