Screening Experiments for Developing Dynamic Treatment Regimes S.A. Murphy At ICSPRAR January, 2008

2 Dynamic Treatment Regimes Are individually tailored treatments, with treatments changing with ongoing subject assessments. The development of a dynamic treatment regime is a multi-stage decision problem

3 Two stages of treatment for each individual Observation available at j th stage Treatment action (vector) at j th stage Primary outcome Y is a specified summary of actions and observations

4 A two stage dynamic treatment regime is a vector of decision rules, one per stage used to select the treatment actions, Next: using experiments to construct decision rules.

5 Experiments Currently the state of the art is to randomize the subject at each of multiple stages –usually one factor per stage. Dynamic treatment regimes are multi-component treatments: many possible factors (e.g. 5-6) Future: series of screening/refining, randomized trials prior to confirmatory trial --- à la Fisher/Box

6 Screening experiments (review) 1)Goal is to eliminate inactive factors (e.g. components) and inactive effects. 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) 6)Minimize assumptions

7 Stage 1 Factors: T 1 ={ A, B, C, D}, each with 2 levels Stage 1 outcome: 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

8 Screening experiments The fact that R, an outcome of stage 1 factors, determines the stage 2 factors + the fact that there are 2 stages of treatment for each subject is makes things interesting! Can we design screening experiments using working assumptions concerning higher order effects & determine the aliasing & provide an analysis method?

9 Reality

10 Defining the stage 2 effects via Potential Outcomes Simple case: two stages of treatment with one factor at stage 1, a early measure of response, R, at the end of stage 1 and two factors at stage 2 (one used if R=1, the other used if R=0) and a primary outcome Y. The potential outcomes are Define effects involving T 2 in a saturated linear model for

11 The stage 2 effects and data Suppose the factors T 1 and T 2 are randomized. Assume consistency (Robins, 1997) then

12 Defining the stage 2 effects Define (factorial) effects involving t 2 in a saturated linear model

13 Defining the stage 2 effects

14 Defining the stage 1 effects (T 1 )

15 Defining the stage 1 effects Lesson: Do not condition on R to define stage 1 effects

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

17 Defining the stage 1 effects In general when {T 1, T 2 } are randomized G-computation Formula

18 Why marginal, why uniform? Define effects involving only T 1 via 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 a discrete uniform distribution.

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

20 where Causal effects: Nuisance parameters: and

21 General Formula More than one factor per stage Nonparametric 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

22 Aliasing (Identification of Effects) When a fractional factorial design produces the data, the effect parameters in are not all identifiable. As in classical screening experiments the defining words pinpoint which effects can be identified. The above is not immediately obvious.

23 Aliasing (Identification of Effects) The rows of {Z 1, Z 2 } are determined by the experimental design; each column in Z 1 is associated with a stage 1 effect and similarly each column in Z 2 is associated with a stage 2 effect. Fractional factorial designs lead to common columns in {Z 1, Z 2 }; these columns are given by the defining words.

24 Aliasing (Identification of Effects) Lemma: Suppose E[R|T 1 ]. Suppose the defining words indicate a shared column in both Z 1 and Z 2. If either the column in Z 1 can be assumed to have a zero η coefficient or the column in Z 2 can be assumed to have a zero β, α coefficient, then the defining words provide the aliasing.

25 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 nonresponders}, each with 2 levels (2 6 = 64 simple dynamic treatment regimes) Simple Example for Two Stages

26 Two Stage Design: I=ABCDF 2 =ABCDG 2 A B C D F 2 =G

27 Assumptions for this Design We use the “1=ABCDF 2 =ABCDG 2 ” 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 nuisance effects involving R and stage 1 factors are negligible (R-p)ABCD, (R-p)ABC, (R-p)ABD …--- these are all η parameters.

28 Aliasing for this Design Stage 1 three way interactions and stage 2 two way interactions are aliased (e.g. ABC=DF 2, ABC=DG 2, etc.) The Stage 1 four way interaction and stage 2 main effects are aliased (e.g. ABCD=F 2, ABCD=G 2 ). 1=ABCDF 2 =ABCDG 2

29 Statistical 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.

30 Interesting Result in Simulations In simulations 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.

31 Discussion Two competitors to this approach: –Using nonrandomized comparisons to construct dynamic treatment regimes Uncontrolled selection bias (causal misattributions) Uncontrolled aliasing. –Expert opinion Secondary analyses would assess if other variables collected during treatment should enter decision rules.

32 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: lsa.umich.edu/~samurphy/seminars/ICPRAR01.08.ppt