1. Hypothesis Testing and Dynamic Treatment Regimes S.A. Murphy Schering-Plough Workshop May 2007 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

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

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

4. 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 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 dynamic treatment regime is the sequence of two decision rules: k=2 Stages

6. 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. A j is a randomized action with known randomization probability. binary actions with P[A j =1]=P[A j =-1]=.5

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8. 8 Sequential, Multiple Assignment Randomized Studies CATIE (2001) Treatment of Psychosis in Schizophrenia STAR*D (2003) Treatment of Depression Tummarello (1997) Treatment of Small Cell Lung Cancer (many, for many years, in this field) Oslin (on-going) Treatment of Alcohol Dependence Pellman (on-going) Treatment of ADHD

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10. 10 Outline Dynamic treatment regimes Constructing a dynamic treatment regime Non-regularity & an adaptive solution Example/Simulation Results.

11. 11 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 1 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 1. Do the same for the two augments.

12. 12 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 1 non-remitters went on to have a remission in stage 2; these people have an dependent variable equal to 0 in the myopic analysis. the remission or lack of remission in stage 2 may be partially attributable to the stage 1 treatment. Patching together the separate analyses of the stages requires unnecessary causal assumptions.

13. 13 Need an alternative for the stage 1 dependent variable What should the value of the stage 1 dependent variable be for those that do not remit and move to stage 2? We should not use a stage 1 dependent variable of Y=1 for those people who remit in stage 2. We should not use an stage 1 dependent variable of Y=0 for those people who remit in stage 2. The dependent variable should be something in between.

14. 14 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.

15. 15 There is a regression for each stage. A Simple Version of Q-Learning – Stage 2 regression: Regress Y on to obtain Stage 1 regression: Regress on to obtain

16. 16 for patients entering stage 2: is the estimated probability of remission in stage 2 as a function of variables that may include or be affected by stage 1 treatment. is the estimated probability of remission assuming the “best” treatment is provided at stage 2 (note max in formula). will be the dependent variable in the stage 1 regression for patients moving to stage 2

17. 17 A Simple Version of Q-Learning – Stage 2 regression, (using Y as dependent variable) yields Stage 1 regression, (using as dependent variable) yields

18. 18 Decision Rules:

19. 19 Outline Dynamic treatment regimes Constructing a dynamic treatment regime Non-regularity & an adaptive solution Example/Simulation Results.

20. 20 Non-regularity

21. 21 Non-regularity

22. 22 Non-regularity – Replace hard-max by soft-max

23. 23 A Soft-Max Solution

24. 24 Distributions for Soft-Max

25. 25 To conduct inference concerning β 1 Set Stage 1 regression: Use least squares with outcome, and covariates to obtain

26. 26 Interpretation of λ Future treatments are assigned with equal probability, λ=0 Optimal future treatment is assigned, λ=∞ Future treatment =1 is assigned with probability Estimator of Stage 1 Treatment Effect when

27. 27 Proposal

28. 28 Proposal

29. 29 Outline Dynamic treatment regimes Constructing a dynamic treatment regime Non-regularity & an adaptive solution Example/Simulation Results.

30. 30 STAR*D Regression at stage 1: S 1 = ((1-Aug), Aug, Aug*Qids) X 1 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 β 1 coefficients as these are used to form the decision rule at stage 1.

31. 31 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.

32. 32 Stage 1 Augment Treatments bbb

33. 33 = means not significant in two sided test at.05 level < means significant in two sided test at.05 level

34. 34 Simulation

35. 35 P[β 2 T S 2 =0]=1 β 1 (∞)=β 1 (0)=0 Test Statistic Nominal Type 1 based on Error=.05.045.039.025 * (1)Nonregularity results in low Type 1 error (2) Adaptation due to use of is useful.

36. 36 P[β 2 T S 2 =0]=1 β 1 (∞)=β 1 (0)=.1 Test Statistic Power based on.15.13.09 (1)The low Type 1 error rate translates into low power

37. 37 Test Statistic Power based on.05.11.12 (1) Averaging over the future is not a panacea P[β 2 T S 2 =0]=0 β 1 (∞)=.125, β 1 (0)=0

38. 38 Test Statistic Type 1 Error=.05 based on.57.16.05 (1) Insufficient adaptation in “small” samples. P[β 2 T S 2 =0]=.25 β 1 (∞)=0, β 1 (0)=-.25

39. 39 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.

40. 40 Discussion Robins (2004) proposes several conservative confidence intervals for β 1. 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.

41. 41 This seminar can be found at: http://www.stat.lsa.umich.edu/~samurphy/ seminars/Harvard0507.ppt Email me with questions or if you would like a copy! samurphy@umich.edu

42. 42 STAR*D Regression at stage 2: α 2 T S 2 ' + β 2 S 2 A 2 S 2 ' =(1,X 2, (1-Aug)*A 1, Aug*A 1, Aug*A 1 *Qids), (X 2 is a vector of variables available at or prior to stage 2) S 1 = 1 Decision rule: Choose TCP if, otherwise offer Mirtazapine + Venlafaxine XR

43. 43 Switch-.11(.07)-1.6 Augment.47(.25)1.9 Augment*QIDS 2 -.04(.02)-2.3 Stage 1 Coefficients