Variable Selection for Optimal Decision Making Lacey Gunter University of Michigan Statistics Department Michigan Student Symposium for Interdisciplinary Statistical Sciences 2007 Joint work with Ji Zhu and Susan Murphy

Outline Background on 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 Decision making applications consist of 3 components, the observations X = (X 1, X 2,…, X p ), the action, A, and the response, R A policy, π, maps observations, X, to actions, A Policies compared via expected mean response, V π = E π [R], called the Value of π (Sutton & Barto,1998) Goal: find a policy, π *, for which

Simple Example Simple decision making 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 Current variable selection techniques were developed for optimal prediction of R Good predictive variables can be useful in decision making, but are only a small part of the puzzle Need variables that help determine which actions are optimal, 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 baseline observations in X, along with an action, A, and a response, 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 for a qualitative variable Interaction Factor: Proportion Factor:

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 2. Rank variables in X using score U j 3. Use some predictive variable selection method on main effects from step 1, A, and top k ranked interactions from step 2 4. Choose tuning parameter value such that the total subset of main effects and interactions lead to a policy with the highest estimated value

Simulation Data simulated under wide variety of realistic decision making scenarios (with and without qualitative interactions) Compared: New ranking method, U j using variable selection algorithm Standard technique: Lasso (Tibshirani, 1996) on (X, A, X  A) with penalty parameter chosen by cv on prediction error 1000 repetitions: recorded percentage of time each interaction was selected for each method

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 = 681 patients, p = 64 observation variables in X, actions, A = Nefazodone or A = Nefazodone + Cognitive psychotherapy (CBASP), Response, 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

Future Work New algorithm is promising Working to determine inclusion thresholds Extend algorithm to select variables for multiple time point decision making problems

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 DA 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’