Download Craving - Bay Area SAS Users Group

Real Data, Real Headache? Using Proc Mixed
and Maximum Entropy Correlated Equilibria to
Longitudinally Analyze Small Sample Data
David Bell
State of California
Industrial Relations Information Services
Presentation Objectives
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Demonstrate the power of mixed longitudinal
hierarchical linear models (i.e., Proc Mixed) to
measure individual change within a
treatment program with small N and over
only 6 months time.
Demonstrate the use of Maximum Entropy
Correlated Equilibria to show latent behavioral
“strategies” employed by the individuals.
General Application
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Although this demonstration was applied to the study
of an outpatient forensic treatment program similar
applications have been used to look at the adaptation
sub-units within a larger environmental context such
as:
Sub county areas adapting to new socio-economic changes
happening to a large county context over time
How is a particular business company adapting to a changing
commercial environment
Longitudinal Mixed Models
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Also can be known as Hierarchical Linear
Models (HLMs)
SAS Proc Mixed or variants thereof are used
for this analysis
The modeling often is to measure individual
or subunit growth/change within a larger
group context that is also changing over time
(e.g., individual within a treatment group, or
census tract within a county in a GIS
application, injured subgroups within a larger
group of injured workers,etc.)
Application to a Forensic Outpatient
Substance Abuse Treatment Program
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N=9 adult women judicially supervised.
All had prior hx. Of substance abuse.
All had prior hx. Of incarceration.
Treatment program setting was within
an inner city.
Duration of measured program was 6
months (one psych assess/month)
Confidence Bands
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Confidence bands were estimated at each temporal point using the following
formulae (from Singer and Willett, 2003):
To estimate the intercept of the Dependent Variable:
Ywavei  Intercept   i  WAVEi
Where:
i = Sample time period (six time periods)
Ywave = Estimated Dependent Variable value
β = slope value
Wave = time period
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Craving: The Strength of Craving Substance
Craving
Confidence
Bands
SAS without
Proc Mixed
Output
The SAS System
Model A: Unconditional growth model
The Mixed Procedure
Covariance Parameter Estimates
Cov Parm Subject
UN(1,1)
UN(2,1)
UN(2,2)
Residual
ID
ID
ID
Standard
Estimate
Error
Z
Value
Pr Z
0.2871
0.1035
2.78
0.0028 * variability of initial status t00 or time0: significant initial differences
-0.03622
0.01083
-3.34 0.0008 * covariance of init status and growth t10, t01. Persons with most crave improve most
0
.
.
.
* variability in growth rates t11: no measurable individual differences in improvement rates.
0.3136
0.06807
4.61
<.0001
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Craving: The Strength of Craving Substance
Craving
Confidence
Bands
SAS without
Proc Mixed
Output
The SAS System
Solution for Fixed Effects
Effect
Intercept
wave
Estimate
1.7433
-0.1272
Standard
Error
DF
0.2519
0.04559
t Value
8
8
6.92
-2.79
Pr > |t|
0.0001
0.0236
Crave Graph Output
Confidence Bands
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To estimate upper and lower confidence
limits for the confidence band :
CIwavei  InterceptCI  (  CI WAVEi )
Where:
CIα = Confidence Interval for α significance level (.95, .99,…)
i = Sample time period (six time periods)
Intercept = Intercept estimate for confidence limit
β = Adjusted slope value
Wave = time period
Craving with Confidence Bands
Now for Razzle Dazzle!
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Proc Mixed gave us a lot of information
on the significance of change on the
group and individual levels.
Now let’s go a little deeper. What
forces shaped their strategies? What
was in their heads consciously or not so
consciously? Now let’s try a little game
theory on their crave…
Taking Entropy to the Max
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In 1949 Claude Shannon, while working at Bell Labs, developed
entropy as the central role of information theory sometimes
referred as the measure of uncertainty.
Decades later entropy has been applied to game theory in terms
of estimating correlated equilibria to neural networks and
dynamic multilayer perceptron (DMP) mechanics, neurolinguistic programming, economics, and genetics.
One of the most exhaustively written books on the application
of entropy to probability theory was written by E.T.Jaynes
entitled “Probability Theory: The Logic of Science.” Jaynes
does an excellent job of defining and applying the Maximum
Entropy principle or MaxEnt.
Applying MaxEnt
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Maximum entropy is the maximum
amount of disorder or random noise
contained in a collection of data.
Since the estimates randomness are not
mapped to specific external theoretical
distributions, inferences are also called
“data driven” or “case based”
inferences.
Applying MaxEnt to Game
Theory: Correlated Equilibria
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Luis Ortiz, et al used an extension of the
MaxEnt Markov Model (MEMM) to estimate
correlated equilibria vectors.
The general MEMM model is
General MaxEnt Markov Model
  f ( o ,s )
1

i i i
Ps( s | o) 

Z (o, s)
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Where:
Z= normalizing constant
i= individual/feature/unit
s= state or equilibrium state
λ= weight (MaxEnt derived)
o = observation,score, or mean
The MEMM Correlate Equilibria
Generate Vectors
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The vectors “gain strength” from
repulsion or attraction in terms of
borrowing or crossover. It is not
uncommon for the combination of
repulsion and attraction to determine
the Nash equilibrium estimate
Push, Pull and Crossover
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Push vectors
Push, Pull and Crossover
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Pull vectors
Push, Pull and Crossover:
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Crossover Vectors
Back to the Crave
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We recall the basic graphic
output:
Graphic Analysis: Major
Vectors Equilibria and Median
Graphic Analysis: The Whole
Shebang
The Output Analysis
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General Descriptive Statistics
Size = 54
std deviation = 0.544844303953988
Variance = 0.29685531555110567
SS= 16.030187039759706
Mean = 1.3263888888888886
Median = 1.1458333335000002
N = 54.0
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General Equilibria Parameter Estimates
Z= 0.4989759539887422
Vector Projections
Lambda(1)
Lambda(2)
h(1,1)= 1.219228823840153 ; h(1,2)= 4.894430791576568 ;
h(2,1)= 1.1781009797200759; h(2,2)= 5.043253215480595 ;
h(3,1)= 1.1383604876120992; h(3,2)= 5.1966008058033175 ;
h(4,1)= 1.099960548427998; h(4,2)= 5.35461115693797 ;
h(5,1)= 1.0628559417377677; h(5,2)= 5.517426047039291 ;
h(6,1)= 1.0270029725172667; h(6,2)= 5.6851915652369875 ;
Sub. Lamba(1) = -0.033732670451915935
logOdds 0.05055901095664123
OR= 1.0518589327254706 P = 0.5126370609349316
Sub. Lamda(2) = 0.030406482437172054
logOdds -0.0532519826555934
OR= 0.9481410672745295 P = 0.486690149497741
Exploratory Findings
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The odds of the participants selecting actions that decrease
craving for substance are 1.052 to one versus 0.95 in selecting
actions to increase craving. Note: in MEMM, even small
differences in OR values are meaningful.
The downward change in localized High value vector suggests a
downward shift in “centrist” values which were found to be
significant in the Mixed regression results.
The extremal high/low vectors show a push relationship
indicating that the decease in craving is resistive in nature in
this environment. However given the downward adjustment to
the localized High vector, even considering drugs is becoming
less likely.
Conclusion
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We explored real data with some real
problems
We used mixed regression to statistically
analyze group/individual growth
We demonstrated how game theory can be
used for exploratory analysis of strategies
used by the parties previously analyzed.
Questions?