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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

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slide1

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 3-7-2012 @

Society for Research in Adolescence Peer Preconference

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

crmda.KU.edu

cattell s data box
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

slide3

Cattell’s Data Box

Occasions of Measurement

Variables (or Tests)

Persons (or Entities)

www.crmda.ku.edu

cattell s data box1
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

www.crmda.ku.edu

michael lebo s example data
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

slide6

Lag 0

Selected Variables

V

Observational Record

O

1

Observational Record

O

2

Observational Record

O

3

Observational Record

O

4

O

Observational Record

O

n

-1

n

-1

O

Observational Record

O

n

n

P-Technique Data Setup

www.crmda.ku.edu

slide7

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

www.crmda.ku.edu

1 st 15 days for subject 4 lag 0
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

www.crmda.ku.edu

slide10

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

www.crmda.ku.edu

slide11

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

www.crmda.ku.edu

slide12

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

www.crmda.ku.edu

slide13

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

www.crmda.ku.edu

slide14

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

www.crmda.ku.edu

measurement invariance by participant

(L3.alternative null fit.xls)

Measurement Invariance by Participant

Model χ2dfp RMSEA90% 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 No

Invariance

Intercept 373.738 72 <.001 .192 .172-.213 0.843 0.907 No

Invariance

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

Invariance

(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

some thoughts
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’

www.crmda.ku.edu

slide17

Lag 0

Lag 1

Selected Variables

(

V

)

Selected Variables (

V

*

)

2

V,

Non-matched record

Observational Record

O

1

or

V+V*

Observational Record

O

Observational Record

O

1

2

Observational Record

O

Observational Record

O

2

3

Observational Record

O

Observational Record

O

3

4

Observational Record

O

Observational Record

O

4

5

O

Observational Record

O

Observational Record

O

n

-1

n

n

-1

O

Observational Record

O

Non-matched record

n

n

Dynamic P-Technique Setup

www.crmda.ku.edu

slide18

Lag 0

Lag 1

Variable 1

Variable 2

Variable 3

Variable 1*

Variable 2*

Variable 3*

2

Variable 1

1

C

2

12

Variable 2

2

C

2

C

Variable 3

13

23

3

AR

CL

CL

2

Variable 1*

11*

21*

31*

1*

CL

C

CL

AR

2

Variable 2*

32*

1*2*

12*

22*

2*

CL

CL

AR

C

C

2

Variable 3*

13*

23*

33*

1*3*

2*3*

3*

A Lagged Covariance Matrix

AR = Autoregressive Correlation

CL = Cross-lagged Correlation

C = Within Lag Covariance

www.crmda.ku.edu

1 st 15 days for subject 4 3 lags
1st 15 days for Subject 4, 3 Lags

1 111 212 333 011111 333

2 333 011111 333333 011

3 111 333333 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

slide20

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

.95

.95

Positive

Lag 1

Positive

Lag 2

Negative

Lag 1

Negative

Lag 2

.84

.82

L15.4.s4.3lags: Subject 4

1*

Positive

Lag 0

.23

.23

-.79

-.88

-.88

.36

.36

Negative

Lag 0

.65

.65

1*

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

www.crmda.ku.edu

slide21

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

1

1

Positive

Lag 2

Positive

Lag 1

Negative

Lag 1

Negative

Lag 2

.94

.94

L15.4.s1.3lags: Subject 1

1*

Positive

Lag 0

-.64

-.66

-.66

Negative

Lag 0

.24

.24

1*

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

www.crmda.ku.edu

slide22

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

.94

.94

Positive

Lag 2

Positive

Lag 1

Negative

Lag 1

Negative

Lag 2

1

.94

L15.4.s5.3lags: Subject 5

1*

Positive

Lag 0

.24

.24

-.61

-.66

-.66

.24

Negative

Lag 0

1*

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

www.crmda.ku.edu

slide23

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

.88

1

Positive

Lag 2

Positive

Lag 1

Negative

Lag 2

Negative

Lag 1

.92

.94

L15.4.s3.3lags: Subject 3

1*

.37

Positive

Lag 0

-.41

-.51

-.51

.31

.31

Negative

Lag 0

.24

.24

1*

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

www.crmda.ku.edu

slide24

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

.94

.95

Positive

Lag 2

Positive

Lag 1

Negative

Lag 2

Negative

Lag 1

.91

.95

L15.4.s2.3lags: Subject 2

1*

Positive

Lag 0

-.63

-.63

-.63

-.24

-.24

Negative

Lag 0

-.17

.24

.24

1*

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

www.crmda.ku.edu

as represented in growth curve models
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

weekly growth trends

Week 1

Week 2

Week 3

Week 6

Week 5

Week 4

Weekly Growth Trends

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

data restructuring
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

www.crmda.ku.edu

data restructuring1
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 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

www.crmda.ku.edu

raw means and standard deviations
Raw Means and Standard Deviations

Energy ratings on a 5-point scale:

N = 75

[15 weeks x 5 subjects]

www.crmda.ku.edu

level and shape model

S1

S4

1*

1*

S3

1*

S2

1*

Level and Shape model

.13

1.08

a1

a2

.002

Pos

Intercept

Pos

Slope

.08

.04

.06

-.10

.06

.12

1.35

a1

-.30

a2

-.04

Neg

Intercept

Neg

Slope

.24

.01

1*

0*

1*

1*

1*

1*

1*

1*

1*

(L15.7lags.LevShape)

Mon

Tues

Wed

Thurs

Sun

Fri

Sat

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

www.crmda.ku.edu

positive affect model
Positive Affect model

(L15.7lags.pos)

.01

a2

1.23

a1

.07

a3

.09

.07

.002

Pos

Intercept

Friday

Sunday

.19

.09

.05

1*

1*

1*

1*

1*

1*

1*

1*

1*

Mon

Tues

Wed

Thurs

Sun

Fri

Sat

.79

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

www.crmda.ku.edu

negative affect model
Negative Affect model

(L15.7lags.neg)

-.03

.21

.003

.10

.84

a1

a4

.001

.02

-.001

Neg

Intercept

Neg

Slope

Friday

Sunday

.01

.09

.12

.40

1*

a2

a3

1*

.13

.05

1*

1*

1*

1*

1*

1*

3*

2*

1*

Mon

Tues

Wed

Thurs

Sun

Fri

Sat

.70

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

www.crmda.ku.edu

cost benefit analysis
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

update
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)

www.Quant.KU.edu