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. Cluster Analysis of Cases.
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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
Two Good Ways to Find Distinct Developmental Trajectories
Growth Curve Mixture Modeling with MPLUS
Free download from Carnegie Mellon.
Runs as a SAS PROC
Produces trajectorycharts automatically
Model published in Psychological Methods(1999, 2001)
Many studies injuvenile justice
Nagin & Jones’ PROC TRAJ
Two-group logistic model from CM WWW site.
Trajectories Based on Continuous Variables
Individual Growth Curves of Continuous Outcome
Distress of child i at time t
Distress(i,t) = βi0 + βi1t+ βi2t2 . . .
β0 the intercept, the score at time zero
β1 the linear slope, indicating a constant increase or decrease over time
β2 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
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)
Trajectories Based on an Indicator
Membership in a delinquent group at a given time (No, Yes)
Trajectories in Health Psychology
Well-done example of trajectory analysis with PROC TRAJ
N = 287 women surviving cancer
Parallel Trajectories Are Less Informative
Scott Holupka & Debra Rog
VU Washington DC
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
The Average Child? Try to Find Them!
The average child with ADHD gets somewhat better over 3 years
Bickman, Wolraich, Lambert & Simmons in prep
One size fits none
ADHD Trajectories (unsuccessful)Only one group resembles the mean
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.
Children with RAP, Recurrent Abdominal PainGrand Means Are Deceiving
Bayesian Information Criterion (BIC) values for CSI models
How do we find trajectories?Best Fit: 2, 3, or 4 Clusters
How do we find trajectories?Rejecting a 4-trajectory solution
Group Size Prob.
1 (21.2%) .
2 (67.1%) 0.0004
3 ( 1.7%) 0.0022Gp 3 tiny
4 (10.0%) 0.0874Gp 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 . . .
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
Three Common Trajectories for CSI
Long term risk
Short term risk
** p < .01
*** p < .001 one-way ANOVA
Comparison of Characteristics by CSI Symptom Trajectory Group
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