1. Trajectory 1. Physics. The path of any body moving under the action of given forces... especially the curve described by a projectile in its flight through the air. [O.E.D.] Multimedia Physics Studios

2. 2 Cluster Analysis of Cases Cluster analysis of cases “dissects” a sample into distinct groups of individuals. It takes a sample and makes a pie chart. Clusters are a categorical variable. Everyone becomes a 1, 2 or 3. –Generally not discovering “God-given” categories –Constructing a data-based typology

3. Two Good Ways to Find Distinct Developmental Trajectories Muthen 2 Growth curve mixture modeling with MPLUS http://www.statmodel.com/index2.html Nagin & Jones PROC TRAJ http://www.ncovr.org/docs/Special_Project/Tr ajectory/index.htm

4. Growth Curve Mixture Modeling with MPLUS

5. Nagin & Jones’ PROC TRAJ Free download from Carnegie Mellon. Easily installed. Runs as a SAS PROC Produces trajectory charts automatically Parsimonious Model published in Psychological Methods (1999, 2001) Many studies in juvenile justice Two-group logistic model from CM WWW site.

6. 6 Basic Readings 1.Helgeson, V.S., et al., Psychological and physical adjustment to breast cancer over 4 years: identifying distinct trajectories of change. Health psychology : official journal of the Division of Health Psychology, American Psychological Association., 2004. 23(1): p. 3-15. 2.Jones, B.L., D.S. Nagin, and K. Roeder, A SAS Procedure based on mixture models for estimating developmental trajectories. Sociological Methods and Research, 2001. 29: p. 374-393. 3.Nagin, D.S., Analyzing developmental trajectories: A semiparametric, group- based approach. Psychological Methods, 1999. 4(2): p. 139-157.

7. Trajectories Based on Continuous Variables J. Child Psychology & Psychiatry

8. 8 Trajectory Analysis vs. Longitudinal HLM Different HLM, pre-existing groups TRAJ, discover groups HLM, confirmatory TRAJ exploratory HLM, powerful general purpose tool TRAJ, interesting special purpose tool Similar Individual growth curves Model based, within children & between children Iterative software Graphic results, not just p values

9. Individual Growth Curves of Continuous Outcome Distress(i,t) = β i 0 + β i 1 t+ β i 2 t 2... β 0 the intercept, the score at time zero β 1 the linear slope, indicating a constant increase or decrease over time β 2 is the quadratic slope indicating a curve of acceleration or deceleration Individuals with missing observations can be included Proc TRAJ goes up to degree 5 Distress of child i at time t

10. 10 Polynomial Approximation Just a description Polynomials can approximate almost any shape Monte Carlo curves Random betas Time 4

11. 11 How PROC TRAJ Works Each child has an observed trajectory Each trajectory has an approximate model description Each child is described by several numbers Cases can be sorted into clusters by the several numbers Nagin developed the statistical theory Jones wrote PROC TRAJ software (free download)

12. Trajectories Based on an Indicator Membership in a delinquent group at a given time (No, Yes)

13. Trajectories in Health Psychology Well-done example of trajectory analysis with PROC TRAJ N = 287 women surviving cancer

14. Parallel Trajectories Are Less Informative Scott Holupka & Debra Rog VU Washington DC

15. Two Examples of Trajectory Analysis Inattentive ADHD Symptoms in a High Risk Group N = 267 school children Recurrent Abdominal Pain (RAP) in children with no medical diagnosis

16. The Average Child? Try to Find Them! Children, K to 4 243 school children “at risk” 68% boys At risk for ADHD MD diagnosis Teacher screen or Followed 3 years Teacher ratings Inattentive symptoms Six or more is positive Bickman, Wolraich, Lambert & Simmons in prep One size fits none The average child with ADHD gets somewhat better over 3 years

17. ADHD Trajectories (unsuccessful) Only one group resembles the mean Clinical chronic 61% Starts clinical (6 or more) Slight improvement Clinical improving 30% Starts clinical Remission by 1 year Subclinical Starts normal Stays normal

18. Children with RAP, Recurrent Abdominal Pain Grand Means Are Deceiving CSI trajectories for entire sample (means and standard deviations) “Children with RAP improve briefly then stay the same at a moderate level for 4-5 years.” Is that what this chart really says? (Note Std Dev) Let’s try PROC TRAJ.

19. 19 How do we find trajectories? TRAJPLOT Macro Shows each solution

20. How do we find trajectories? Best Fit: 2, 3, or 4 Clusters Bayesian Information Criterion (BIC) values for CSI models

21. How do we find trajectories? Rejecting a 4-trajectory solution Group Size Prob. 1 (21.2%). 2 (67.1%) 0.0004 3 ( 1.7%) 0.0022  Gp 3 tiny 4 (10.0%) 0.0874  Gp 1-Gp 4 nonsig Four clusters not acceptable because there are not 4 distinct clusters. In addition, cluster 3 is too small. Knowing when to stop...

22. 22 Example of a 3 Group Model PROC TRAJ Group Parameter Estimate Prob > |T| 1 Intercept 15.44068 0.0000 Linear -0.68666 0.0002 Quadratic 0.01047 0.0002 2 Intercept 42.87414 0.0000 Linear -2.42765 0.0000 Quadratic 0.03429 0.0000 3 Intercept 33.74592 0.0000 Linear 0.65764 0.1630 Quadratic -0.00597 0.4160

23. Three Common Trajectories for CSI Long term risk Short term risk Low risk ** p <.01 *** p <.001 one-way ANOVA

24. 24 Try to Explain the Trajectories with Ordinary Analysis (oneway ANOVA, regression, chi 2 etc) Theory based > Fishing Comparison of Characteristics by CSI Symptom Trajectory Group Variable Low Risk N=94 M (SD) Short Term Risk N=17 M (SD) Long Term Risk N=21 M (SD) F Ratio Child Depression Inventory 7.72 (5.68)11.88 (7.33)14.57 (8.01)11.65*** Life Events5.14 (3.66)6.88 (4.78)9.76 (4.42)12.07***

25. OK, Officer, we’re fishing for a hypothesis. We’ll test it with fresh data. Like cluster analysis or exploratory factor analysis, TRAJ needs cross validation