Hypothesis Testing and Dynamic Treatment Regimes S.A. Murphy, L. Gunter & B. Chakraborty ENAR March 2007

2 Outline Dynamic treatment regimes Constructing and addressing questions regarding an optimal dynamic treatment regime Why and when non-regular? A Solution Simulation Results.

3 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

4 k Stages History available at j th stage “Reward” following j th stage (r j is a known function) Primary Outcome:

5 Goal : Construct decision rules that input information in the history at each stage and output a recommended decision; these decision rules should lead to a maximal mean Y. The dynamic treatment regime is the sequence of decision rules:

6 In the future we employ the actions determined by the decision rules: An example of a simple decision rule is: alter treatment at time j if otherwise maintain on current treatment; S j is a summary of the history, H j.

7 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

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 Constructing and Addressing Questions Regarding an Optimal Dynamic Treatment Regime

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 centered to have conditional mean zero.

12 (k=2) Dynamic Programming

13 Approximate for S', S vector summaries of the history and A Simple Version of Q-Learning –binary actions Stage 2 regression: Use least squares with outcome, Y, and covariates to obtain Set Stage 1 regression: Use least squares with outcome, and covariates to obtain

14 Decision Rules:

15 Why non-regular?

16 Non-regularity

17 When do we have non-regularity?

18 A Soft-Max Solution

19 A Soft-Max Solution

20 Distributions for Soft-Max

21 Regularized Q-Learning (binary actions) Set Stage 1 regression: Use least squares with outcome, and covariates to obtain

22 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

23 Interpretation of λ

24 Proposal

25 Proposal

26 Proposal

27 Simulation

28 P[β 2 T S 2 =0]=1 β 1 (∞)=β 1 (0)=0 Test Statistic Nominal Type 1 based on Error= *.024 * (1)Nonregularity results in low Type 1 error (2)Additional smoothing due to use of is useful.

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

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

31 Test Statistic Type 1 Error=.05 based on (1) The price is that the null hypothesis is altered. P[β 2 T S 2 =0]=.25 β 1 (∞)=0, β 1 (0)=-.25

32 Discussion We replace the hypothesis test concerning a non- regular parameter, β 1 (∞) by a hypothesis test concerning a near-by regular parameter β 1 (λ * ). This is work in progress—limited theoretical results are available. If you let increase with the sample size you again end up with a non-regular problem (convergence to limiting distribution is locally non-uniform).

33 Discussion Robins (2004) proposes several conservative confidence intervals for β 1. Ideally to decide if the two 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.

34 This seminar can be found at: seminars/ENAR0307.ppt me with questions or if you would like a copy!