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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. 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

Cluster analysis of cases
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

Two good ways to find distinct developmental trajectories

Two Good Ways to Find Distinct Developmental Trajectories

  • Muthen2 Growth curve mixture modeling with MPLUS


  • Nagin & Jones PROC TRAJ

Nagin jones proc traj

Free download from Carnegie Mellon.

Easily installed.

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.

Basic readings
Basic Readings

  • 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.

  • 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.

  • Nagin, D.S., Analyzing developmental trajectories: A semiparametric, group-based approach. Psychological Methods, 1999. 4(2): p. 139-157.

Trajectories based on continuous variables

Trajectories Based on Continuous Variables

J. Child


& Psychiatry

Trajectory analysis vs longitudinal hlm
Trajectory Analysis vs. Longitudinal HLM


  • HLM, pre-existing groups

  • TRAJ, discover groups

  • HLM, confirmatory

  • TRAJ exploratory

  • HLM, powerful general purpose tool

  • TRAJ, interesting special purpose tool


  • Individual growth curves

  • Model based, within children & between children

  • Iterative software

  • Graphic results, not just p values

Individual growth curves of continuous outcome

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

Polynomial approximation just a description
Polynomial ApproximationJust a description

  • Polynomials can approximate almost any shape

  • Monte Carlo curves

    • Random betas

    • Time4

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)

Trajectories based on an indicator

Trajectories Based on an Indicator

Membership in a delinquent group at a given time (No, Yes)

Trajectories in health psychology

Trajectories in Health Psychology

Well-done example of trajectory analysis with PROC TRAJ

N = 287 women surviving cancer

Parallel trajectories are less informative

Parallel Trajectories Are Less Informative

Scott Holupka & Debra Rog

VU Washington DC

Two examples of trajectory analysis

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? 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

Adhd trajectories unsuccessful only one group resembles the mean

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

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.

Children with RAP, Recurrent Abdominal PainGrand Means Are Deceiving

How do we find trajectories trajplot macro shows each solution
How do we find trajectories? deviations)TRAJPLOT Macro Shows each solution

How do we find trajectories best fit 2 3 or 4 clusters

Bayesian Information Criterion (BIC) values for CSI models deviations)

How do we find trajectories?Best Fit: 2, 3, or 4 Clusters

How do we find trajectories rejecting a 4 trajectory solution

How do we find trajectories? deviations)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 . . .

Example of a 3 group model proc traj
Example of a 3 Group Model deviations)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

Three common trajectories for csi

Three Common Trajectories for CSI deviations)

Long term risk

Short term risk

Low risk

** p < .01

*** p < .001 one-way ANOVA

Try to Explain the Trajectories with Ordinary Analysis (oneway ANOVA, regression, chi2 etc)Theory based > Fishing

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