1. Experiments and Dynamic Treatment Regimes S.A. Murphy Univ. of Michigan PSU, October, 2005 In Honor of Clifford C. Clogg

2. 2 Joint work with –Derek Bingham (Simon Fraser) –Linda Collins (PennState) And informed by discussions with –Vijay Nair (U. Michigan) –Bibhas Chakraborty (U. Michigan) –Vic Strecher (U. Michigan)

3. 3 Outline Dynamic Treatment Regimes Challenges in Experimentation Defining Effects Estimating Effects Simple Example

4. 4 Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing with ongoing subject need. Mimic Clinical Practice. High variability across patients in response to any one treatment Relapse is likely without either continuous or intermittent treatment for a large proportion of people. What works now may not work later Exacerbations in disorder may occur if there are no alterations in treatment

5. 5 The Big Questions What is the best sequencing of treatments? What is the best timings of alterations in treatments? What information do we use to make these decisions?

6. 6 k Decisions on one individual Observation made prior to j th decision point Treatment at j th decision point Primary outcome Y is a specified summary of decisions and observations

7. 7 A dynamic treatment regime is a vector of decision rules, one per decision where each decision rule inputs the available information and outputs a recommended treatment decision.

8. 8 Long Term Goal : Construct decision rules that lead to a maximal mean Y. An example of a decision rule is: stop treatment if otherwise maintain on current treatment.

9. 9 Challenges in Experimentation 1)Dynamic Treatment Regimes are multi-component treatments Multiple decision points through time Different kinds of decisions Decision options for improving patients are often different from decision options for non- improving patients Delivery mechanisms, behavioral contingencies, staff training, monitoring schedule……. 2)Constructing decision rules is a multi-stage decision problem

10. 10 Proposed Solutions to Challenges Dynamic Treatment Regimes are Multi-component Treatments series of screening/refining, randomized trials prior to confirmatory trial (MOST)--- à la G. Box! Multistage Decisions sequential multiple assignment randomized trials (SMART): randomize at each decision point— à la full factorial.

11. 11 Challenges in Experimentation 3)In the screening experiment, resources are scarce relative to the number of interesting treatment components/factors. Implementing many cells of a full factorial is very expensive. Consider designs that are similar to balanced fractional factorials. To do this you must define the effects.

12. 12 Defining the Effects

13. 13

14. 14 Conceptual Model

15. 15 Defining the stage 2 effects Two decisions (two stages): Define effects involving T 2 in an ANOVA decomposition of

16. 16 Defining the stage 1 effects

17. 17 Defining the stage 1 effects

18. 18 Defining the stage 1 effects Define Define effects involving only T 1 in an ANOVA decomposition of

19. 19 Why uniform? Define effects involving only T 1 in an ANOVA decomposition of 1)The defined effects are causal. 2)The defined effects are marginal -- consistent with tradition in experimental design for screening. –The main effect for one treatment factor is defined by marginalizing over the remaining treatment factors using an uniform distribution.

20. 20 Why uniform? 2)The defined effects are marginal consistent with tradition in experimental design for screening. –The main effect for one treatment factor is defined by marginalizing over the remaining factors using an uniform distribution. When there is no R, the main effect for treatment T 1 is

21. 21 Why uniform? 3)If R were always equal to 1 then the proposal is equivalent to defining both stage 2 and stage 1 effects in an ANOVA decomposition of T 2 denotes the treatment options when R=1.

22. 22 An Aside: Ideally you’d like to replace by (X 2 is a vector of intermediate outcomes) in defining the effects of T 1.

23. 23 Use an ANOVA-like decomposition: Representing the effects

24. 24 Stage 1 effects: so the interesting stage 1 and stage 2 effects are contained in the same decomposition.

25. 25 where Causal effects: Nuisance effects:

26. 26 Estimating the Effects

27. 27 Estimating the effects Two decisions (two stages): Four cells corresponding to (T 1,T 2 )= (1,1), (1,-1), (-1,1), (-1,-1). For R=1, cells are unequal in size and similarly for R=0. Proposal: Estimate stage 2 effects using cell means

28. 28 Proposal: Estimate stage 2 effects using cell means. This yields the same estimators as a weighted regression analysis in which an individual in the i th cell is weighted by where p i is the proportion of responders (R=1) in the i th cell. Estimating the stage 2 effects

29. 29 Proposal: Estimate stage 2 effects using weighted regression with Y as the outcome variable. The advantage is that the design matrix is orthogonal with respect to the weights. –The alias structure is easily determined using standard design of experiments techniques. –The estimators of the stage 2 effects are the same regardless of whether you include nuisance effects in the regression. Estimating the stage 2 effects

30. 30 Proposal: Use a regression with the residual as the dependent variable, and regressors equal to the stage 1 treatment factors (here T 1 ). Why? Estimating the stage 1 effects

31. 31 Proposal: Estimate stage 1 effects using the outcome The advantage is that the design matrix is orthogonal (if more than one first stage treatment). –The alias structure is easily determined using standard design of experiments techniques. –The estimators of the stage 1 effects are the same regardless of how many of these effects you choose to include in the regression. Estimating the stage 1 effects

32. 32 Simple Example

33. 33 Five Factors: M 1, E, C, T, A 2 (only for R=1), M 2 (only for R=0), each with 2 levels (2 6 =64 simple treatment strategies) If three way and higher order interactions are likely negligible choose a design that aliases these higher order interactions with main and two way effects. Simple Example

34. 34 Simple Example M 1 E C T A 2 =M 2 - - - - + - - - + - - - + - - - - + + + - + - - - - + - + + - + + - + - + + + - + - - - - + - - + + + - + - + + - + + - + + - - + + + - + - + + + - - + + + + +

35. 35 Discussion In the screening experiment the goal is to ascertain which decisions (“factors”) need further investigation; these are not confirmatory experiments. Some fractional factorial experiments will result in aliasing between causal effects and the nuisance effects. Using these experiments requires assumptions based on design principles such as effect hierarchy and effect heredity. It is unclear what kinds of secondary analyses are possible if the experiment is a fractional factorial. This seminar can be found at: http:// www.stat. lsa.umich.edu/~samurphy/seminars/PSUStatistics1 0-05.ppt

36. 36 To define effects of treatment factors at first and second stages use the ANOVA-like decomposition: where To design an experiment we make assumptions concerning the negligibility of these effects. Summary