Constructing Dynamic Treatment Regimes & STAR*D

1. Constructing Dynamic Treatment Regimes &STAR*D S.A. Murphy ICSA June 2008

2. Collaborators • Lacey Gunter • A. John Rush • Bibhas Chakraborty

3. Outline • Dynamic treatment regimes • Constructing a dynamic treatment regime • Non-regularity & an adaptive solution • Example/Simulation Results.

4. Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing according to patient outcomes. Operationalize clinical practice. k Stages for one individual Observation available at jth stage Action at jth stage (usually a treatment)

5. k=2 Stages Goal: Construct decision rules that input information available at each stage and output a recommended decision; these decision rules should lead to a maximal mean Y where Y is a function of The dynamic treatment regime is the sequence of two decision rules:

6. Data for Constructing the Dynamic Treatment Regime: Subject data from sequential, multiple assignment, randomized trials. At each stage subjects are randomized among alternative options. Aj is a randomized action with known randomization probability. binary actions with P[Aj=1]=P[Aj=-1]=.5

8. Regression-based methods for constructing decision rules • Q-Learning (Watkins, 1989) (a popular method from computer science) • Optimal nested structural mean model (Murphy, 2003; Robins, 2004; I like the term A-learning) • When using linear models, the first method is an inefficient version of the second method when each stages’ covariates include the prior stages’ covariates and the actions are centered to have conditional mean zero.

9. A Simple Version of Q-Learning – There is a regression for each stage. • Stage 2 regression: Regress Y on to obtain • Stage 1 regression: Regress on to obtain

10. for patients entering stage 2: • is the estimated probability of remission in stage 2 as a function of patient history (includes past treatment and variables affected by stage 1 treatment). • is the estimated probability of remission assuming the “best” treatment is provided at stage 2 (note max in formula). • is the dependent variable in the stage 1 regression for patients moving to stage 2

11. A Simple Version of Q-Learning – • Stage 2 regression, (using Y as dependent variable) yields • Stage 1 regression, (using as dependent variable) yields

12. Decision Rules:

13. Non-regularity

14. Non-regularity

15. Non-regularity– • Replace hard-max • by soft-max

16. A Soft-Max Solution

17. Distributions for Soft-Max

18. To conduct inference concerning β1 • Set • Stage 1 regression: Use least squares with outcome, • and covariates to obtain

19. Interpretation of λ Estimator of Stage 1 Treatment Effect when

20. Proposal

21. Proposal

22. STAR*D • Regression at stage 1: • S1'=(1, X1) • S1= ((1-Aug), Aug, Aug*Qids) • X1 is a vector of variables available at or prior to stage 1, Aug is 1 if patient preference is augment and 0 otherwise • We are interested in the β1 coefficients as these are used to form the decision rule at stage 1.

23. STAR*D • Decision Rule at stage 1: • If patient prefers a Switch then • if offer Mirtazapine, otherwise offer Nortriptyline. • If patient prefers an Augment then • if offer Lithium, otherwise offer Thyroid Hormone.

24. Stage 1 Augment Treatments bbb

25. = means not significant in two sided test at .05 level < means significant in two sided test at .05 level

26. Simulation

27. P[β2TS2=0]=1 β1(∞)=β1(0)=0 Test Statistic Nominal Type 1 based on Error=.05 • Nonregularity results in low Type 1 error • Adaptation due to use of is useful.

28. P[β2TS2=0]=1 β1(∞)=β1(0)=.1 Test Statistic Power based on • The low Type 1 error rate translates into low power

29. P[β2TS2=0]=0 β1(∞)=.125, β1(0)=0 Test Statistic Power based on • Averaging over the future is not a panacea

30. P[β2TS2=0]=.25 β1(∞)=0, β1(0)=-.25 Test Statistic Type 1 Error=.05 based on • Insufficient adaptation in “small” samples.

31. Discussion • We replace the test statistic based on an estimator of a non-regular parameter by an adaptive test statistic. • This is work in progress—limited theoretical results are available. • The use of the bootstrap does not allow to increase too fast.

32. Discussion • Robins (2004) proposes several conservative confidence intervals for β1. • Ideally to decide if the stage 1 treatments are equivalent, we would evaluate whether the choice of stage 1 treatment influences the mean outcome resulting from the use of the dynamic treatment regime. We did not do this here. • Constructing “evidence-based” regimes is of great interest in clinical research and there is much to be done by statisticians.

33. This seminar can be found at: http://www.stat.lsa.umich.edu/~samurphy/ seminars/ICSA0708.ppt Email me with questions or if you would like a copy! samurphy@umich.edu

34. STAR*D • Regression at stage 2: α2TS2' + β2S2A2 • S2' =(1,X2, (1-Aug)*A1, Aug*A1, Aug*A1*Qids), • (X2 is a vector of variables available at or prior to stage 2) • S1 = 1 • Decision rule: Choose TCP if , otherwise offer Mirtazapine + Venlafaxine XR

35. Stage 1 Coefficients