David Bell State of California Industrial Relations Information Services

1. 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

2. 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.

3. 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

4. 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.)

5. 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)

6. 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: Where: i = Sample time period (six time periods) Ywave = Estimated Dependent Variable value β = slope value Wave = time period

7. 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

8. 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

9. Crave Graph Output

10. 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

11. Craving with Confidence Bands

12. 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…

13. 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.

14. 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.

15. 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

16. 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

17. 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

18. Push, Pull and Crossover • Push vectors

19. Push, Pull and Crossover • Pull vectors

20. Push, Pull and Crossover: • Crossover Vectors

21. Back to the Crave • We recall the basic graphic output:

22. Graphic Analysis: Major Vectors Equilibria and Median

23. Graphic Analysis: The Whole Shebang

24. 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

25. 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.

26. 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?