Real Data, Real Headache? Using Proc Mixed and Maximum Entropy Correlated Equilibria to Longitudinally Analyze Small Sample Data

David BellState of California

Industrial Relations Information Services

Presentation Objectives 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 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 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 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 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:

iii WAVEInterceptYwave Where:

i = Sample time period (six time periods)

Ywave = Estimated Dependent Variable value

β = slope value

Wave = time period

Craving without Confidence Bands Craving: The Strength of Craving Substance SAS Proc Mixed Output

The SAS System Model A: Unconditional growth model

The Mixed Procedure

Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z

UN(1,1) ID 0.2871 0.1035 2.78 0.0028 * variability of initial status t00 or time0: significant initial differences UN(2,1) ID -0.03622 0.01083 -3.34 0.0008 * covariance of init status and growth t10, t01. Persons with most crave improve most UN(2,2) ID 0 . . . * variability in growth rates t11: no measurable individual differences in improvement rates. Residual 0.3136 0.06807 4.61 <.0001

Craving without Confidence Bands Craving: The Strength of Craving Substance SAS Proc Mixed Output

The SAS System

Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t|

Intercept 1.7433 0.2519 8 6.92 0.0001 wave -0.1272 0.04559 8 -2.79 0.0236

Crave Graph Output

Confidence Bands To estimate upper and lower

confidence limits for the confidence band :

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

)( iCICIi WAVEInterceptCIwave

Craving with Confidence Bands

Now for Razzle Dazzle! 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 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, neuro-linguistic 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 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 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

Where: Z= normalizing constant i= individual/feature/unit s= state or equilibrium state λ= weight (MaxEnt derived) o = observation,score, or mean

i ii sof

soZosPs ),(

),(1)|(

The MEMM Correlate Equilibria Generate Vectors 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 Push vectors

Push, Pull and Crossover Pull vectors

Push, Pull and Crossover: Crossover Vectors

Back to the Crave We recall the basic graphic

output:

Graphic Analysis: Major Vectors Equilibria and Median

Graphic Analysis: The Whole Shebang

The Output Analysis General Descriptive Statistics

Size = 54 std deviation = 0.544844303953988 Variance = 0.29685531555110567 SS= 16.030187039759706 Mean = 1.3263888888888886 Median = 1.1458333335000002 N = 54.0 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 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 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?