Variable Selection for Optimal Decision Making Susan Murphy & Lacey Gunter University of Michigan Statistics Department Artificial Intelligence Seminar Joint work with Ji Zhu

Simple Motivating Example Nefazodone - CBASP Trial R Nefazodone Nefazodone + Cognative Behavioral-analysis System of Psychotherapy (CBASP) 50+ baseline covariates, both categorical and continuous

Complex Motivating Example

Outline Framework and notation for decision making Need for variable selection Variables that are important to decision making Introduce a new technique Simulated and real data results Future work

Optimal Decision Making 3 components: observations X = (X 1, X 2,…, X p ), action, A, and reward, R A policy, π, maps observations, X, to actions, A Policies compared via expected mean reward, V π = E π [R], called the Value of π (Sutton & Barto,1998) Long Term Goal: find a policy, π *, for which

Some Working Assumptions Data collection is difficult and expensive limited number of trajectories (<1000) training set with randomized actions many observations Finite horizon (only 1-4 time points) we will initially work with just one time point Noisy data with little knowledge about underlying system dynamics Little knowledge about which variables are most important for decision making

Simple Example A clinical trial to test two alternative drug treatments The goal: to discover which treatment is optimal for any given future patient Components X baseline variables such as patient's background, medical history, current symptoms, etc. A assigned treatment R patient's condition and symptoms post treatment

Variable Selection Multiple reasons for variable selection in decision making, for example Better performance: avoid inclusion of spurious variables that lead to bad policies Limited resources: only small number of variables can be collected when enacting policies in a real world setting Interpretability: policies with fewer variables are easier to understand

What are people currently using? Variable selection for reinforcement learning in medical settings predominantly guided by expert opinion Predictive selection techniques, such as Lasso (Loth et al., 2006) and decision trees (Ernst et al., 2005) have been proposed Good predictive variables are useful in decision making, but are only a small part of the puzzle Need variables that help determine optimal actions, variables that qualitatively interact with the action

Qualitative Interactions What is a qualitative interaction? X qualitatively interacts with A if at least two distinct, non-empty sets exist within the space of X for which the optimal action is different (Peto, 1982) No Interaction Non-qualitative Interaction Qualitative interaction Qualitative interactions tell us which actions are optimal

Qualitative Interactions We focus on two important factors The magnitude of the interaction between the variable and the action The proportion of patients whose optimal choice of action changes given knowledge of the variable big interaction small interaction big interaction big proportion big proportion small proportion

Variable Ranking for Qualitative Interactions We propose ranking the variables in X based on potential for a qualitative interaction with A We give a score for ranking the variables Given data on i = 1,.., n subjects with j = 1,…,p variables in X, along with an action, A, and a reward, R, for each subject For Ê[R| A=a] an estimator of E[R| A=a], define

Variable Ranking Components Ranking score based on 2 usefulness factors Interaction Factor: max = 1 – 0 = 1 min = 0.3 – 0.7 = D j = 1 – ( -.4) = 1.4

Variable Ranking Components Proportion Factor: 2 out of 7 subjects would change choice of optimal action given X j

Ranking Score Ranking Score: Score, U j, j=1,…,p can be used to rank the p variables in X based on their potential for a qualitative interaction with A

Variable Selection Algorithm 1. Select important main effects of X on R using some predictive variable selection method a. Choose tuning parameter value that gives best predictive model 2. Rank variables in X using score U j ; select top k in rank 3. Again use a predictive variable selection method, this time selecting among main effects of X from step 1, main effect of A, and ranked interactions from step 2 a. Choose tuning parameter value such that the total subset of variables selected leads to a policy with the highest estimated Value

Simulation Data simulated under wide variety of scenarios (with and without qualitative interactions) Used observation matrix, X, and actions, A, from a real data set Generated new rewards, R, based on several different realistic models Compared new ranking method U j versus a standard method 1000 repetitions: recorded percentage of time each interaction was selected for each method

Methods Used in Simulation Standard Method: Lasso on (X, A, X  A) (Tibshirani, 1996) The standard Lasso minimization criterion is where Z i is the vector of predictors for observation i and λ is a penalty parameter Coefficient for A, β p+1, not included in penalty term Value of λ chosen by cross-validation on the prediction error

Methods Used in Simulation New Method: 1. Select important main effects of X on R using Lasso a. Choose λ value by cross-validation on prediction error 2. Rank variables in X using score U j ; select top k in rank 3. Use Lasso to select among main effects of X chosen in step 1, main effect of A, and interactions chosen in step 2 a. Choose λ value such that the total subset of variables selected leads to a policy with the highest estimated Value

Simulation Results × Continuous Qualitative Interaction  Spurious Interaction × Binary Qualitative Interaction  Spurious Interaction

Simulation Results × Binary Qualitative Interaction  Non-qualitative Interaction  Spurious Interaction × Continuous Qualitative Interaction  Non-qualitative Interaction  Spurious Interaction

Depression Study Analysis Data from a randomized controlled trial comparing treatments for chronic depression (Keller et al., 2000) n = 440 patients, p = 64 observation variables in X, actions, A = Nefazodone or A = Nefazodone + Cognitive psychotherapy (CBASP), Reward, R = Hamilton Rating Scale for Depression score

Depression Study Results Ran both methods on 1000 bootstrap samples Resulting selection percentages: ALC2 ALC1 Som Anx OCD ALC2

Inclusion Thresholds Based on previous plots, which variables should we select? Need inclusion thresholds Idea: remove effect of X on R from data, then run algorithm to determine maximum percentage of selections this tells us the noise threshold variables with percentages above this threshold are selected

Inclusion Thresholds Do 100 times Randomly assign the observed rewards to different subjects given a particular action Run the methods on new data Record the variables that were selected by each method Threshold: largest percentage of time a variable was selected over the 100 iterations

Thresholds for Depression Study We should disregard any interactions selected 6% of the time or less when using either method

Threshold on Results New method U includes 2 indicator variables for Alcohol problems and Somatic Anxiety Score Standard Lasso includes 39 variables! ALC2 ALC1 Som Anx

Future Work Extend algorithm to select variables for multiple time points How best to do this? What rewards to use at each time point Do we need to adjust the distribution of our X based on prior actions What order should variable selection be done

Other Issues To Think About Do we need to account for variability in our estimate of E[R| X j, A=a] over different X j Can we reasonably estimate the value of a derived policy from a fixed data set collected under random actions when the number of time points gets larger? Any other issues?

References & Acknowledgements For more information see: L. Gunter, J. Zhu, S.A. Murphy (2007). Variable Selection for Optimal Decision Making. Technical Report, Department of Statistics, University of Michigan. This work was partially supported by NIH grants: R21 DA019800,K02 DA15674,P50 DA10075 Technical and data support A. John Rush, MD, Betty Jo Hay Chair in Mental Health at the University of Texas Southwestern Medical Center, Dallas Martin Keller and the investigators who conducted the trial `A Comparison of Nefazodone, the Cognitive Behavioral- analysis System of Psychotherapy, and Their Combination for Treatment of Chronic Depression’

Addressing Concerns Many Biostat literature discourage looking for qualitative interactions and are very skeptical when new interactions are found, why is this? Qualitative interactions are hard to find, have small effects Too many people fishing without disclosing Strict entry criteria for most clinical trials, thus small variability in X precludes looking at avoid looking at interesting subgroups How are we addressing these concerns? Testing new algorithms in multiple settings where no qualitative interactions exist

No Interaction: What can we expect? No Qualitative Interactions No relationship between (X, A, X*A) and R Main effects of X only Main effects of X & moderate effect of A only Everything but qualitative interactions

Estimating the Value 1. Fit selected variables into chosen estimator, Ê 2. Estimate optimal policy: 3. Estimate Value of by:

Estimating the Value (2 time points) 1. Estimate of the optimal policy: 2. Estimate Value of by: