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 j th stage Action at j th stage (usually a treatment)

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 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 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 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

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 β 1 Set Stage 1 regression: Use least squares with outcome, and covariates to obtain

19 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

20 Proposal

21 Proposal

22 STAR*D Regression at stage 1: S 1 '=(1, X 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.

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

28 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

29 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

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

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 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.

33 This seminar can be found at: seminars/ICSA0708.ppt me with questions or if you would like a copy!

34 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

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