Get documents about "PowerPoint Presentation
Document Sample
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
•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
2
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/Tr
ajectory/index.htm
Growth Curve Mixture Modeling
with MPLUS
Nagin & Jones’ PROC TRAJ
•Free download from Carnegie Mellon.
•Easily installed.
•Runs as a SAS PROC Two-group logistic model from CM WWW site.
•Produces trajectory
charts automatically
•Parsimonious
•Model published
in Psychological
Methods
(1999, 2001)
•Many studies in
juvenile justice
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.
6
Trajectories
Based on
Continuous
Variables
J. Child
Psychology
& Psychiatry
Trajectory Analysis
vs.
Longitudinal HLM
Similar Different
• HLM, pre-existing groups
•Individual growth curves
• TRAJ, discover groups
•Model based, within children • HLM, confirmatory
& between children
• TRAJ exploratory
•Iterative software • HLM, powerful general
•Graphic results, not just p purpose tool
values • TRAJ, interesting special
purpose tool
8
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
•Polynomials can
approximate almost
any shape
•Monte Carlo curves
•Random betas
•Time4
10
How PROC TRAJ Works
•Each child has an observed
trajectory 75
Hypothetical example:
•Each trajectory has an 2
CBCL = 65 - 3 * Time + 0.4 * Time
approximate model 70
description Child has 3 numbers
•Each child is described by
65
several numbers CBCL
•Cases can be sorted into
clusters by the several 60
numbers
•Nagin developed the 55
statistical theory
•Jones wrote PROC TRAJ
50
software (free download) 0 1 2 3 4 5 6 7 8 11 9
Time
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!
•Children, K to 4
The average child with ADHD
•243 school children “at risk”
•68% boys gets somewhat better over 3 years
•At risk for ADHD
MD diagnosis
Teacher screen or
•Followed 3 years
•Teacher ratings
•Inattentive symptoms
•Six or more is positive
Bickman, Wolraich,
One size fits none
Lambert & Simmons in
prep
ADHD Trajectories (unsuccessful)
Only one group resembles the mean
9
Inattentive Symptoms
Observed means
•Clinical chronic 61%
•Starts clinical (6 or more)
Clinical chronic •Slight improvement
6
61%
•Clinical improving 30%
•Starts clinical
Model
3 (line) Subclinical •Remission by 1 year
9% •Subclinical
Clinical
improving 30% •Starts normal
0 •Stays normal
0 1 2 3
Years
Children with RAP, Recurrent Abdominal Pain
Grand Means Are Deceiving
50
worse
1 standard deviation
40
CSI trajectories for entire
sample (means and standard
deviations)
CSI Score
30
“Children with RAP improve
briefly then stay the same at a
20 moderate level for 4-5 years.”
Is that what this chart really
10
says? (Note Std Dev)
better
0 Let’s try PROC TRAJ.
0 12 60
1.5
Months
How do we find trajectories?
TRAJPLOT Macro Shows each solution
19
How do we find trajectories?
Best Fit: 2, 3, or 4 Clusters
BIC Bayesian Information Criterion
-2000 Better fit
Bayesian
Information
Criterion
3 (BIC)
-2025 values for
CSI models
-2050
CSI
Worse Fit
-2075
1 2 3 4 5 6 7 8
Number of Trajectories
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 . . .
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
22
Three Common Trajectories for CSI
50
Observed • Long term risk
40 • Short term risk
Model
Long Term Risk: 14%
• Low risk
CSI Score
30
Short Term Risk: 15%
20
10
Low Risk: 71%
0
0 1.5 12 60 ** p < .01
Months *** 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
Short Term Long Term
Low Risk
Risk Risk
Variable N=94 F Ratio
N=17 N=21
M (SD)
M (SD) M (SD)
Child
Depression 7.72 (5.68) 11.88 (7.33) 14.57 (8.01) 11.65***
Inventory
Life Events 5.14 (3.66) 6.88 (4.78) 9.76 (4.42) 12.07***
24
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
