1. Experiments and Dynamic Treatment Regimes S.A. Murphy Univ. of Michigan Yale: November, 2005

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 and Aliasing Examples

4. 4 Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing with ongoing subject need. Mimic Clinical Practice. Dynamic Treatment Regimes

5. 5 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

6. 6 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.

7. 7 Challenges in Experimentation

8. 8 Dynamic Treatment Regimes (review) Constructing decision rules is a multi-stage decision problem in which the system dynamics are unknown. Analysis methods for observational data dominate statistical literature (Murphy, Robins, Moodie & Richardson, Tsiatis) Better data provided by sequential multiple assignment randomized trials: randomize at each decision point— à la full factorial.

9. 9 Reality

10. 10 Challenges in Experimentation Dynamic Treatment Regimes are multi-component treatments: many possible components decision options for improving patients are often different from decision options for non-improving patients, multiple components employed simultaneously medications, adjunctive treatments, delivery mechanisms, behavioral contingencies, staff training, monitoring schedule……. Future: series of screening/refining, randomized trials prior to confirmatory trial --- à la Fisher/Box

11. 11 Screening experiments (review) 1)Goal is to eliminate inactive factors (e.g. components) and inactive effects. 2)Each factor at 2 levels 3)Screen marginal causal effects 4)Design experiment using working assumptions concerning the negligibility of certain effects. (Think ANOVA) 5)Designs and analyses permit one to determine aliasing (caused by false working assumptions) 6)Minimize formal assumptions

12. 12 Screening experiments When the goal is to construct/optimize a dynamic treatment regime can we design screening experiments using working assumptions concerning the marginal causal effects and provide an analysis method that permits the determination of the aliasing??

13. 13 Defining the Effects

14. 14 Defining the stage 2 effects Two decisions (two stages): Define effects involving T 2 in an ANOVA decomposition of

15. 15 Defining the stage 1 effects (T 1 )

16. 16 Defining the stage 1 effects

17. 17 Defining the stage 1 effects Define Define effects involving only T 1 in an ANOVA decomposition of

18. 18 Why marginal, why uniform? Define effects involving only T 1 in an ANOVA decomposition of 1)The defined effects are causal. 2)The defined effects are 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.

19. 19 Why marginal, 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

20. 20 An Aside: Ideally you’d like to replace by (X 2 is a vector of intermediate outcomes) in defining the effects of T 1.

21. 21 Use an ANOVA-like decomposition: Representing the effects

22. 22 where Causal effects: Nuisance parameters: and

23. 23 General Formula New ANOVA Z 1 matrix of stage 1 treatment columns, Z 2 is the matrix of stage 2 treatment columns, Y is a vector Classical ANOVA

24. 24 Aliasing {Z 1, Z 2 } is determined by the experimental design The defining words (associated with an fractional factorial experimental design) identify common columns in the collection {Z 1, Z 2 } ANOVA

25. 25 Aliasing ANOVA Consider designs with a shared column in both Z 1 and Z 2 only if the column in Z 1 can be safely assumed to have a zero η coefficient or if the column in Z 2 can be safely assumed to have a zero β, α coefficient. The defining words provide the aliasing in this case.

26. 26 Simple Examples

27. 27 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 dynamic treatment regimes) The budget permits 16 cells --16 simple dynamic treatment regimes. Simple Example

28. 28 Design: 1=M 2 M 1 ECT=A 2 M 1 ECT M 1 E C T A 2 =M 2 - - - - + - - - + - - - + - - - - + + + - + - - - - + - + + - + + - + - + + + - + - - - - + - - + + + - + - + + - + + - + + - - + + + - + - + + + - - + + + + +

29. 29 Assumptions A 2 C, A 2 T, M 2 E, M 2 T and CE along with the main effects in stage 1 and 2 are of primary interest. Working Assumption: All remaining causal effects are likely negligible. Formal Assumption: Consider designs for which a shared column in Z 1 and Z 2 occurs only if the column in Z 1 can be safely assumed to have a zero η coefficient (concerns interactions of stage 1 factors with R) or if the column in Z 2 can be safely assumed to have a zero β/α coefficient (stage 2 effects).

30. 30 Design 1 No formal assumptions. 1=M 1 ECT The design column for A 2 /M 2 is crossed with stage 1 design. EC is aliased with M 1 T. The interaction EC is of primary interest and the working assumption was that M 1 T is negligible. A 2 C is aliased with A 2 M 1 ET. The interaction A 2 C is of primary interest and the working assumption was that A 2 M 1 ET is negligible.

31. 31 Design 1 I=M 1 ECT Screening model: The estimator estimates the sum of the effects of CE and M 1 T.

32. 32 Design 2 Formal assumption: No three way and higher order stage 2 causal effects & no four way and higher order effects involving R and stage 1 factors. 1=M 2 M 1 ECT=A 2 M 1 ECT M 2 T and A 2 T are aliased with M 1 CE; the interaction M 2 T (A 2 T) is of primary interest and the working assumption was that M 1 CE is negligible. M 2 M 1 T is negligible so CE is not aliased.

33. 33 Design 2 Screening model: The estimator estimates the sum of the effects of M 2 T and of M 1 CE

34. 34 Design 3 Formal assumption: No four way and higher order causal effects & no three way and higher order effects involving R and first stage factors. I=M 2 M 1 ECT=A 2 M 1 ECT CE aliased with both A 2 M 1 T and with M 2 M 1 T

35. 35 Design 3 Screening model: The estimator estimates the sum of the effects of CE and A 2 M 1 T

36. 36 Discussion In classical screening experiments we Screen marginal causal effects Design experiment using working assumptions concerning the negligibility of the effects. Designs and analyses permit one to determine aliasing Minimize formal assumptions We can do this as well when screening for multi-stage decision problems!

37. 37 Discussion Compare this to using observational studies to construct dynamic treatment regimes –Uncontrolled selection bias (causal misattributions) –Uncontrolled aliasing. Secondary analyses would assess if variables collected during treatment should enter decision rules. This seminar can be found at: http:// www.stat. lsa.umich.edu/~samurphy/seminars/Yale11.05.ppt

38. 38 Reality

39. 39 Conceptual Model The meaning of T 2 depends on R