todd d. little university of kansas director, quantitative training program director, center for...
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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 3-7-2012 @
Society for Research in Adolescence Peer Preconference
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)
Pers
ons (
or E
ntiti
es)
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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 1Var 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 ParticipantModel 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 CorrelationCL = Cross-lagged CorrelationC = 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 TransformationTraditional 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.LevShape)
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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
www.Quant.KU.edu 35
UpdateDr. 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.eduIMMAP (immap.educ.ttu.edu)Stats Camp (Statscamp.org)
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