1. Hypothesis Testing and Adaptive Treatment Strategies S.A. Murphy SCT May 2007

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

3. 3 Outline Adaptive treatment strategies Constructing and addressing questions regarding an optimal adaptive treatment strategy A solution to non-regularity? Example using STAR*D.

4. 4 Adaptive treatment strategies 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 j th stage Action at j th stage (usually a treatment)

5. 5 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 adaptive treatment strategy is the sequence of two decision rules: k=2 Stages

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

7. 7

8. 8 A natural approach: Myopic Decisions Evaluate each stage of treatment in isolation; the dependent variable is 1 if remission in that stage, 0 otherwise. In stage 3 there are two treatment actions for those who prefer a switch in treatment (Mirtazapine or Nortriptyline) and two treatment actions for those who prefer an augment (Lithium or Thyroid). Compare the two switches in treatment according to the remission rate achieved by end of stage 3. Do the same for the two augments.

9. 9 Need an alternative This is not a good idea if we want to evaluate the sequence of treatments (e.g. adaptive treatment strategies). Some of the stage 3 non-remitters went on to have a remission in stage 4; these people have an dependent variable equal to 0 in the myopic analysis. the remission or lack of remission in stage 4 may be partially attributable to the stage 3 treatment. Patching together the separate analyses of the stages requires unnecessary causal assumptions.

10. 10 Need an alternative for the stage 3 dependent variable What should the value of the stage 3 dependent variable be for those that move to stage 4? We should not use a stage 3 dependent variable of Y=1 for those people who remit in stage 4. We should not use an stage 3 dependent variable of Y=0 for those people who remit in stage 4. The dependent variable should be something in between.

11. 11 Regression-based methods for constructing decision rules Q-Learning (Watkins, 1989) (a popular method from computer science) A-Learning or optimal nested structural mean model (Murphy, 2003; Robins, 2004) The first method is an inefficient version of the second method when each stages’ covariates include the prior stages’ covariates and the actions are coded to have conditional mean zero.

12. 12 There is a regression for each stage. A Simple Version of Q-Learning – Stage 4 regression: Regress Y on to obtain Stage 3 regression: Regress on to obtain

13. 13 for patients entering stage 4: is the estimated probability of remission in stage 4 as a function of variables that may include or be affected by stage 3 treatment. is the estimated probability of remission assuming the “best” treatment is provided at stage 4 (note max in formula). will be the dependent variable in the stage 3 regression for patients moving to stage 4

14. 14 A Simple Version of Q-Learning – Stage 4 regression, (using Y as dependent variable) yields Stage 3 regression, (using as dependent variable) yields

15. 15 Decision Rules:

16. 16 Non-regularity

17. 17 Non-regularity

18. 18 Non-regularity – Replace hard-max by soft-max

19. 19 STAR*D Stages 3 & 4 Regression at stage 3: α 3 T S 3 ' + β 3 T S 3 A 3 S 3 ' = (1, X 3 ) X 3 is a vector of variables available at or prior to stage 3 S 3 = ((1-Aug), Aug, Aug*Qids 2 ) We are interested in the β 3 coefficients as these are used to form the decision rule at stage 3.

20. 20 STAR*D Stages 3 & 4 Decision Rule at stage 3: 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.

21. 21 STAR*D Stages 3 & 4 Regression at stage 4: α 4 T S 4 ' + β 4 S 4 A 4 S 4 ' =(1,X 4, (1-Aug)*A 3, Aug*A 3, Aug*A 3 *Qids 2 ), (X 4 is a vector of variables available at or prior to stage 4, Aug is 1 if patient preference is augment and 0 otherwise) S 4 = 1 Decision rule: Choose TCP if, otherwise offer Mirtazapine + Venlafaxine XR

22. 22 Switch-.11(.07)-1.6 Augment.47(.25)1.9 Augment*QIDS 2 -.04(.02)-2.3 Stage 3 Coefficients

23. 23

24. 24 = means not significant in two sided test at.05 level < means significant in two sided test at.05 level

25. 25 Discussion We replace the hypothesis test concerning a non- regular parameter, β 3 by a hypothesis test concerning a near-by regular parameter. These multi-stage regression methods need to be generalized to survival analysis. This is work in progress!

26. 26 Discussion Robins (2004) proposes several conservative confidence intervals for β 3. Ideally to decide if the two stage 3 treatments are equivalent, we would evaluate whether the choice of stage 3 treatment influences the mean outcome resulting from the use of the adaptive treatment strategy. We did not do this here. Constructing “evidence-based” strategies is of great interest in clinical research and there is much to be done by statisticians.

27. 27 This seminar can be found at: http://www.stat.lsa.umich.edu/~samurphy/ seminars/SCT0507.ppt Email me with questions or if you would like a copy! samurphy@umich.edu

28. 28