1. Screening Experiments for Dynamic Treatment Regimes S.A. Murphy At ENAR March, 2008

2. 2 Two stages of treatment for each individual Stage 1: Treatment is {A=on, B=off, C=low, D=high} Observe if early responder (R=1) or early non- responder (R=0) Stage 2: If R=1, treatment is {F 2 =low} If R=0, treatment is {G 2 =high}

3. 3 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 (stage 2 treatment differs by outcomes observed during initial treatment) multiple components employed simultaneously medications, behavioral therapy, adjunctive treatments, delivery mechanisms, motivational therapy, staff training, monitoring schedule……. Future: series of screening/refining, randomized trials prior to confirmatory trial --- à la Fisher/Box

4. 4 Screening experiments (review) 1)Goal is to eliminate inactive factors (e.g. components). 1)Use if you can expect that many factors will be inactive but costly. 2)Use if concerned that an inactive factor will have a negative impact on other components’ effectiveness. 2)Each factor at 2 levels 3)Screen main effects and some interactions 4)Design experiment using working assumptions concerning the negligibility of certain higher order factorial effects. 5)Designs and analyses permit one to determine aliasing (caused by false working assumptions)

5. 5 Stage 1 Factors: T 1 ={ A, B, C, D}, each with 2 levels Stage 1 outcome: R in {0,1} Stage 2 Factors: T 2 = {F 2 --only if R=1, G 2 —only if R=0}, each with 2 levels Primary Outcome: Y continuous (2 6 = 64 simple dynamic treatment regimes) Simple Example for Two Stages

6. 6 Two Stage Design: 1=ABCDF 2 (1=ABCDG 2 ) A B C D F 2 =G 2 - - - - + - - - + - - - + - - - - + + + - + - - - - + - + + - + + - + - + + + - + - - - - + - - + + + - + - + + - + + - + + - - + + + - + - + + + - - + + + + +

7. 7 Screening experiments Can we: design screening experiments using working assumptions concerning higher order effects & determine the aliasing and provide an analysis method?

8. 8 Defining the stage 2 effects Simple case: two stages of treatment with only one factor at each stage, a early measure of response, R at the end of stage 1 and a primary outcome Y. The potential outcomes are Define effects involving T 2 in a saturated linear model for

9. 9 Defining the stage 2 effects

10. 10 Defining the stage 1 effects (T 1 )

11. 11 Defining the stage 1 effects

12. 12 Defining the stage 1 effects Define effects involving only T 1 in a saturated linear model Assuming consistency, the above is equal to when {T 1, T 2 } are independently randomized and T 2 has a discrete uniform distribution on {-1,1}.

13. 13 Defining the stage 1 effects The formula for the main effect of stage 1 treatment (in terms of potential outcomes) is

14. 14 Surprisingly both stage 1 and 2 effects can be represented in one (nonstandard) linear model: Representing the effects

15. 15 where Causal effects: Nuisance parameters: and

16. 16 Two Stage Design: 1=ABCDF 2 (1=ABCDG 2 ) A B C D F 2 =G 2 - - - - + - - - + - - - + - - - - + + + - + - - - - + - + + - + + - + - + + + - + - - - - + - - + + + - + - + + - + + - + + - - + + + - + - + + + - - + + + + +

17. 17 General Formula Saturated model (for both R=0, 1) Z 1 matrix of stage 1 factor columns, Z 2 is the matrix of stage 2 factor columns, Y is a vector, p is the vector of response rates Classical saturated factorial effects model

18. 18 Aliasing (Identification of Effects) In classical designs, the defining words immediately yield the aliasing. 1=ABCDF 2 means F 2 =ABCD or equivalently that the column in the matrix Z 2 associated with F 2 is the same as the column in matrix Z 1 associated with ABCD. In classical experimental design the consequence is that only the sum of the main effect of F 2 and the four way interaction, ABCD can be estimated. These two effects are aliased.

19. 19 Aliasing (Identification of Effects) Lemma. If the defining words indicate that there are common columns in the matrices Z 1 and Z 2 then under the assumption that either the associated η coefficient or the associated β, α coefficient is zero, the defining words provide the aliasing.

20. 20 Six Factors: Stage 1: T 1 ={ A, B, C, D}, each with 2 levels Stage 2: T 2 = {F 2 --only for stage 1 responders, G 2 --only for stage 1 non-responders}, each with 2 levels (2 6 = 64 simple dynamic treatment regimes) Simple Example for Two Stages

21. 21 Simple Example This design has defining words: 1=ABCDF 2 (1=ABCDG 2 ) The labels of the identical columns in Z 1, Z 2 are CD=ABF 2, ABC=DF 2, CDF 2 =AB………..

22. 22 Formal Assumptions for this Design We use this design if we can assume that all three way and higher order stage 2 effects are negligible (ABG 2, ABF 2, ABCG 2, ABCF 2, ….)—these are all β and α parameters. assume all four way and higher order effects involving R and stage 1 factors are negligible (R- p)ABCD, (R-p)ABC, (R-p)ABD …--- these are all η parameters. 1=ABCDF 2 =ABCDG 2

23. 23 Analysis for this design Recall that Many columns in Z 1, Z 2 are identical (hence the aliasing of effects). Eliminate all multiple copies of columns and label remaining columns as stage 1 (or stage 2) main and two-way interaction effects. Replace response rates in p by observed response rates. Fit model.

24. 24 Interesting Result in Simulations In simulations formal assumptions are violated. Response rates (probability of R=1) across 16 cells range from.55 to.73 Results are surprisingly robust to violations of assumptions. The maximal value of the correlation between 32 estimators of effects was.12 and average absolute correlation value is.03 Why? Binary R variables can not vary that much. If response rate is constant, then the effect estimators are uncorrelated as in classical experimental design.

25. 25 Discussion Compare this to using observational studies to construct dynamic treatment regimes –Uncontrolled selection bias (causal misattributions) –Uncontrolled aliasing. Secondary analyses would assess if other variables collected during treatment should be used in addition to R to determine stage 2 treatment.

26. 26 Joint work with –Derek Bingham (Simon Fraser) And informed by discussions with –Vijay Nair (U. Michigan) –Bibhas Chakraborty (U. Michigan) –Vic Strecher (U. Michigan) This seminar can be found at: http:// www.stat. lsa.umich.edu/~samurphy/seminars/ENAR03.08.ppt