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Variable Selection for Tailoring TreatmentPowerPoint Presentation

Variable Selection for Tailoring Treatment

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Presentation Transcript

Outline

- Motivation
- Need for Variable Selection
- Characteristics of a Tailoring Variable
- A New Technique for Finding Tailoring Variables
- Comparisons
- Discussion

Simple Example

Nefazodone - CBASP Trial

Nefazodone

Randomization

Nefazodone + Cognitive

Behavioral Analysis

System of Psychotherapy

(CBASP)

50+ baseline covariates, both categorical and continuous

Simple Example

Nefazodone - CBASP Trial

Which variables in X are important for tailoring the treatment?

Optimization

- We want to select the treatment that “optimizes” R
- The optimal choice of treatment may depend on X

Optimization

- The optimal treatment(s) is given by
- The value of d is

Need for Variable Selection

- In clinical trials many pretreatment variables are collected to improve understanding and inform future treatment
- Yet in clinical practice, only the most informative variables for tailoring treatment can be collected.
- A combination of theory, clinical experience and statistical variable selection methods can be used to determine which variables are important.

Current Statistical Variable Selection Methods

- Current statistical variable selection methods focus on finding good predictors of the response
- Also need variables to help determine which treatment is best for which types of patients, e.g. tailoring variables
- Experts typically have knowledge on which variables are good predictors, but intuition about tailoring variables is often lacking

What is a Tailoring Variable?

- Tailoring variables help us determine which treatment is best
- Tailoring variables qualitatively interact with the treatment; different values of the tailoring variable result in different best treatments.
No Interaction Non-qualitative Interaction Qualitative interaction

Qualitative Interactions

- Qualitative interactions have been discussed by many within stat literature (e.g. Byar & Corle,1977; Peto, 1982; Shuster & Van Eys, 1983; Gail & Simon, 1985; Yusuf et al., 1991; Senn, 2001; Lagakos, 2001)
- Many express skepticism concerning validity of qualitative interactions when found in studies
- Our approach for finding qualitative interactions should be robust to finding spurious results

Qualitative Interactions

- We focus on two important factors
- The magnitude of the interaction between the variable and the treatment indicator
- The proportionof patients for whom the best choice of treatment changes given knowledge of the variable
big interaction small interaction big interaction

big proportion big proportion small proportion

Ranking Score S

- Ranking Score:
where

- S estimates the quantity described by Parmigiani (2002) as the value of information.

Ranking Score S

- Higher Sscorescorrespond to higher evidence of a qualitative interaction between X and A
- We use this ranking in a variable selection algorithm to select important tailoring variables.
- Avoid over-fitting in due to large number of X variables
- Consider variables jointly

Variable Selection Algorithm

- Select important predictors of R from (X, X*A) using Lasso
-- Select tuning parameter using BIC

- Select all X*A variables with nonzero S.
-- Use predictors from 1. to form linear regression estimator of to form S.

(using linear models)

Lasso

- Lasso on (X, A, XA) (Tibshirani, 1996)
- Lasso minimization criterion:
where Zi is the vector of predictors for patient i, λ is a penalty parameter

- Coefficient for A not penalized
- Value of λ chosen by Bayesian Information Criterion (BIC) (Zou, Hastie & Tibshirani, 2007)

- Lasso minimization criterion:

Variable Selection Algorithm

- Rank order (X, X*A)variables selected in steps 1 & 2 using a weighted Lasso
-- Weight is 1 if variable is not an interaction

-- Otherwise weight for kth interaction is

-- is a small positive number.

-- Produces a combined ranking of the selected (X, X*A)variables (say p variables).

Variable Selection Algorithm

- Choose between variable subsets using a criterion that trades off maximal value of information and complexity.
-- The ordering of the p variables creates p subsets of variables. Estimate the value of information for each of the p subsets

-- Select the subset, k with largest

Simulations

- Data simulated under wide variety of realistic decision making scenarios (with and without qualitative interactions)
- Used X from the CBASP study, generated new Aand R

- Compared:
- New method: S with variable selection algorithm
- Standard method: BIC Lasso on (X, A, XA)

- 1000 simulated data sets: recorded percentage of time each variable’s interaction with treatment was selected for each method

Simulation Results

* Over the total possible increase; 1000 data sets each of size 440

Simulation Results

- Pros: when the model contained qualitative interactions, the new method gave significant increases in expected response over BIC-Lasso
- Cons: the new method resulted in a slight increase in the number of spurious interactions over BIC-Lasso

Nefazodone - CBASP Trial

Aim of the Nefazodone CBASP trial – to compare efficacy of three alternate treatments for major depressive disorder (MDD):

- Nefazodone,
- Cognitive behavioral-analysis system of psychotherapy (CBASP)
- Nefazodone + CBASP
Which variables might help tailor the depression treatment to each patient?

Nefazodone - CBASP Trial

- For our analysis we used data from 440 patients with

Method Application and Confidence Measures

- When applying new method to real data it is desirable to have a measure of reliability and to control family-wise error rate
- We used bootstrap sampling to assess reliability
- On each of 1000 bootstrap samples:
- Run variable selection method
- Record the interaction variables selected

- Calculate selection percentages over bootstrap samples

- On each of 1000 bootstrap samples:

Error Rate Thresholds

- To help control family-wise error rate, compute the following inclusion thresholdsfor selection percentages:
- Repeat 100 times
- Permute interactions to remove effects from the data
- Run method on 1000 bootstrap samples of permuted data
- Calculate selection percentages over bootstrap samples

- Record largest selection percentage over the p interactions

- Permute interactions to remove effects from the data
- Threshold: (1-α)th percentile over 100 max selection percentages

- Repeat 100 times
- Select all interactions with selection percentage greater than threshold

Error Rate Thresholds

- When tested in simulations using new method, error rate threshold effectively controlled family-wise error rate
- This augmentation of bootstrap sampling and thresholding was also tested on BIC Lasso and effectively controlled family-wise error rate in simulations

Discussion

- This method provides a list of potential tailoring variables while reducing the number of false leads.
- Replication is required to confirm the usefulness of a tailoring variable.
- Our long term goal is to generalize this method so that it can be used with data from Sequential, Multiple Assignment, Randomized Trials as illustrated by STAR*D.

- Email Susan Murphy at [email protected] for more information!
- This seminar can be found at
http://www.stat.lsa.umich.edu/~samurphy/seminars/

ASA11.11.08.ppt

- Support: NIDA P50 DA10075, NIMH R01 MH080015 and NSF DMS 0505432
- Thanks for technical and data support go to
- 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’

Lasso Weighting Scheme

- Lasso minimization criterion equivalent to:
so smaller wj means greater importance

- Weights where
- vj = 1for predictive variables
- vj = for prescriptive variables

AGV Criterion

- For a subset of k variables, X{k} the Average Gain in Value ( AGV) criterion is
where

- The criterion selects the subset of variables with the maximum proportion of increase in E[R] per variable

Simulation Results (S-score)

×Qualitative Interaction

Spurious Interaction

×Qualitative Interaction

Non-qualitative Interaction

Spurious Interaction

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