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Todd D. Little University of Kansas Director, Quantitative Training Program

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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Todd D. Little University of Kansas Director, Quantitative Training Program

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  1. 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. 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 www.crmda.ku.edu

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

  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 www.crmda.ku.edu

  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 www.crmda.ku.edu

  6. 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

  7. Multivariate Time-series(Multiple Variables x Multiple Occasions for 1 Person) www.crmda.ku.edu

  8. 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

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

  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 www.crmda.ku.edu

  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 www.crmda.ku.edu

  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 www.crmda.ku.edu

  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 www.crmda.ku.edu

  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 www.crmda.ku.edu

  15. (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

  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’ www.crmda.ku.edu

  17. 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

  18. 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

  19. 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

  20. (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

  21. (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

  22. (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

  23. (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

  24. (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

  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) www.crmda.ku.edu

  26. 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

  27. P-technique Data Transformation www.crmda.ku.edu

  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 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

  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 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

  30. Raw Means and Standard Deviations Energy ratings on a 5-point scale: N = 75 [15 weeks x 5 subjects] www.crmda.ku.edu

  31. 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

  32. 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

  33. 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

  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 www.crmda.ku.edu

  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 Email: yhat@ttu.edu IMMAP (immap.educ.ttu.edu) Stats Camp (Statscamp.org) www.Quant.KU.edu

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