Trajectory

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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
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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

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Two Good Ways to Find Distinct Developmental
Trajectories

• Muthen2 Growth curve mixture modeling with MPLUS
• http//www.statmodel.com/index2.html
• Nagin Jones PROC TRAJ
• http//www.ncovr.org/docs/Special_Project/Trajecto
  ry/index.htm

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Growth Curve Mixture Modeling with MPLUS
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Nagin Jones PROC TRAJ

• Free download from Carnegie Mellon.
• Easily installed.
• Runs as a SAS PROC
• Produces trajectorycharts automatically
• Parsimonious
• Model published in Psychological Methods
  (1999, 2001)
• Many studies injuvenile justice

Two-group logistic model from CM WWW site.
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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.

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Trajectories Based on Continuous Variables
J. Child Psychology Psychiatry
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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

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Individual Growth Curves of Continuous Outcome
Distress of child i at time t
Distress(i,t) ßi0 ßi1t ßi2t2 . . .
ß0 the intercept, the score at time zero ß1 the
linear slope, indicating a constant increase or
decrease over time ß2 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
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Polynomial ApproximationJust a description

• Polynomials can approximate almost any shape
• Monte Carlo curves
• Random betas
• Time4

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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)

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Trajectories Based on an Indicator
Membership in a delinquent group at a given time
(No, Yes)
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Trajectories in Health Psychology
Well-done example of trajectory analysis with
PROC TRAJ N 287 women surviving cancer
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Parallel Trajectories Are Less Informative
Scott Holupka Debra Rog VU Washington DC
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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

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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

The average child with ADHD gets somewhat better
over 3 years
Bickman, Wolraich, Lambert Simmons in prep
One size fits none
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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

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Children with RAP, Recurrent Abdominal PainGrand
Means Are Deceiving
 
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How do we find trajectories?TRAJPLOT Macro Shows
each solution
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How do we find trajectories?Best Fit 2, 3, or 4
Clusters
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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 . . .
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Example of a 3 Group ModelPROC TRAJ

• Group Parameter Estimate Prob gt 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

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Three Common Trajectories for CSI
 

• Long term risk
• Short term risk
• Low risk

 
p lt .01 p lt .001 one-way ANOVA
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Try to Explain the Trajectories with Ordinary
Analysis (oneway ANOVA, regression, chi2
etc)Theory based gt Fishing
Comparison of Characteristics by CSI Symptom
Trajectory Group
Variable Low Risk N94 M (SD) Short Term Risk N17 M (SD) Long Term Risk N21 M (SD) F Ratio
Child Depression Inventory 7.72 (5.68) 11.88 (7.33) 14.57 (8.01) 11.65
Life Events 5.14 (3.66) 6.88 (4.78) 9.76 (4.42) 12.07
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OK, Officer, were fishing for a hypothesis.
Well test it with fresh data.
Like cluster analysis or exploratory factor
analysis, TRAJ needs cross validation