1crm da. KU. edu 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 University of Turku, Finland Special Thanks to: Ihno Lee, Chapter co-author in Handbook. Dynamic P-Technique Structural Equation Modeling Dynamic P-Technique Structural Equation Modeling

Cattells 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

Cattells Data Box Occasions of Measurement Variables (or Tests) Persons (or Entities)

Cattells 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

Michael Lebos 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

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

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

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

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

L15.1.s1.Lag0.LS8 PositiveNegative (-.64) ActiveWearyTiredSluggishPeppyLively Model Fit: χ 2 (8, n=101) = 9.36, p =.31, RMSEA =.039 (.000;.128), TLI/NNFI =.994, CFI=.997 X.88.52

L15.1.s2.Lag0.LS8 PositiveNegative (-.65) ActiveWearyTiredSluggishPeppyLively Model Fit: χ 2 (8, n=101) = 8.36, p =.40, RMSEA =.014 (.000;.119), TLI/NNFI =.999, CFI=.999 X

L15.1.s3.Lag0.LS8 PositiveNegative (-.43) ActiveWearyTiredSluggishPeppyLively Model Fit: χ 2 (8, n=101) = 9.70, p =.31, RMSEA =.050 (.000;.134), TLI/NNFI =.992, CFI=.997 X

L15.1.s4.Lag0.LS8 PositiveNegative (-.81) ActiveWearyTiredSluggishPeppyLively Model Fit: χ 2 (8, n=101) = 14.6, p =.07, RMSEA =.084 (.000;.158), TLI/NNFI =.983, CFI=.991 X

L15.1.s5.Lag0.LS8 PositiveNegative (-.60) ActiveWearyTiredSluggishPeppyLively Model Fit: χ 2 (8, n=101) = 5.11, p =.75, RMSEA =.000 (.000;.073), TLI/NNFI = 1.02, CFI=1.0 X

Measurement Invariance by Participant Model χ2 df p RMSEA 90% CI TLI/NNFI CFI Constraint Tenable Null < Configural Invariance Loading < No Invariance Intercept < No Invariance Partial < Yes Invariance (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)

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

Dynamic P-Technique Setup

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 * 2 2* 2 3* Variable 1 Variable 2 Variable 3 Variable 1* Variable 2* Variable 3* Variable 1Variable 2Variable 3Variable 1*Variable 2*Variable 3* Lag 0Lag 1 A Lagged Covariance Matrix AR = Autoregressive Correlation CL = Cross-lagged Correlation C = Within Lag Covariance

1 st 15 days for Subject 4, 3 Lags

L15.4.s4.3lags: Subject 4 Negative Lag 0 Positive 1* Negative Lag 1.84 Positive Lag 1.95 Negative Lag 2.82 Positive Lag Model Fit: χ 2 (142, n=101) = 154.3, p =.23; RMSEA =.02; TLI/NNFI =.99 (Initial model: L15.3.s4.3lags)

L15.4.s1.3lags: Subject 1 Negative Lag 0 Positive 1* Negative Lag 1.94 Positive Lag 1 1 Negative Lag 2.94 Positive Lag Model Fit: χ 2 (144, n=101) = 159.9, p =.17; RMSEA =.05; TLI/NNFI =.99 (Initial model: L15.3.s1.3lags)

L15.4.s5.3lags: Subject 5 Negative Lag 0 Positive 1* Negative Lag 1 1 Positive.94 Negative Lag 2.94 Positive Lag Model Fit: χ 2 (143, n=101) = 93.9, p =.99; RMSEA =.00; TLI/NNFI = (Initial model: L15.3.s5.3lags)

L15.4.s3.3lags: Subject 3 Negative Lag 0 Positive 1* Negative Lag 1.94 Positive Lag 1 1 Negative Lag 2.92 Positive Lag Model Fit: χ 2 (142, n=101) = 139.5, p = 1.0; RMSEA =.0; TLI/NNFI = 1.0 (Initial model: L15.3.s3.3lags)

L15.4.s2.3lags: Subject 2 Negative Lag 0 Positive 1* Negative Lag 1.95 Positive Lag 1.95 Negative Lag 2.91 Positive Lag Model Fit: χ 2 (142, n=101) = 115.2, p =.95; RMSEA =.0; TLI/NNFI = 1.0 (Initial model: L15.3.s2.3lags)

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)

Weekly Growth Trends Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Carrig, M., Wirth, R.J., & Curran, P.J. (2004)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,

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 - Time dependency - Unstructured time points - Time dependency - Structured time points - Compare equivalent relationships across a sample

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

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 Mo Mo Mo Mo Mo Mo Mo Note: meaning assigned to arbitrary time points

Raw Means and Standard Deviations Energy ratings on a 5-point scale: MonTuesWedThursFriSatSun 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]

Level and Shape model 0* Mon NegIntercept NegSlope 1* * TuesWed Thurs FriSat Sun PosSlope * S4 S3 S2 S1.04 PosIntercept Model fit: χ2 (116) = , p =.23, RMSEA =.000, CFI =.98, TLI/NNFI = (L15.7lags.LevShape)

Positive Affect model 1* 1* SundayFriday Model fit: χ2 (25) = 25.96, p =.41, RMSEA =.021, CFI =.99, TLI/NNFI =.99 Mon TuesWed Thurs FriSat Sun 1*.79 (L15.7lags.pos) PosIntercept

Negative Affect model Model fit: χ2 (20) = 18.46, p =.56, RMSEA =.000, CFI = 1.00, TLI/NNFI = 1.01 Mon TuesWed Thurs FriSat Sun.70 1* FridaySunday 2* 3* (L15.7lags.neg) NegInterceptNegSlope

Cost-benefit analysis Extrapolates the average within-person change from pooled time series data But obscures unique information about each individuals variability and growth patterns Does not utilize the strengths of P-technique data Add subject covariates to detect individual differences at the mean level

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) 35ww w.Q uan t.K U.e du