SAMSI Tutorial on Dynamic Treatment Regimes by Anastasios Tsiatis

1. SAMSI Tutorial on Dynamic Treatment Regimes by Anastasios Tsiatis Dr. Gong Tang Wentao Feng Sachiko Miyahara

2. Introduction Goal of Physicians: To give treatment to patients over time that will result in as favorable a clinical outcome as possible.

3. Introduction (Cont.) When a new patient comes to a physician’s office, the physician needs to make many decisions such as: - Treatment Choice - Dose - When to switch => Complex and often difficult to know

4. Goal of This Presentation To find the distribution of the responses, based on different treatment regimes, using observed data from: • a controlled intervention study • an observational study

5. Notations For time point j= 0 to k, Lj = covariate information collected between time tj-1 and tj Aj = treatment assigned at time tj Y = Outcome

6. Notations (Cont.) L0 L1 L2 Lk …… Ak Y …… A0 A1 A2 t0 t1 t2 tk

7. Notations (cont.) = (L0, … Lj) The history of time dependent covariates = (A0, … Aj) The history of time dependent treatment decisions

8. Treatment Regimes What is a treatment regime? an algorithm which dictates how each patient in the population treated possibly based on intervening covariate information. In formula: g(tj, ) = aj where and

9. Treatment Regimes Example Example: HIV Study Let L1j = CD4 counts aj = 1 to give antiretroviral therapy 0 to not to give the therapy The treatment regime: g(tj, ) = I(CD4j <= 200)

10. Methods What are the methods to estimate the distribution of Y for various g from the observed data? - G-computation algorithm - Inverse Probability Weighting Need to consider: 1. Concept of Potential Outcomes 2. Three assumptions

11. Potential Outcomes • Denoted as Y*(g) • Y*( ) is the potential outcome of a randomly selected individual in our population if he/she hypothetically received treatment a0 at time t0, a1 at time t1…ak at time tk • L*( ) is also referred to as potential outcome • Also called “Counterfactuals”

12. Potential Outcomes • The set of all potential outcomes denoted by: W ={L*0(g), L*1(g)… L*k(g), Y*(g)} where L*0(g) = L0 L*1(g) = L*1(g(t0, L0) … L*k(g) = L*k{g(t0, L0), …, g(tk-1, (g))} Y*(g) = Y*{g(t0, L0), …, g(tk-1, (g))}

13. Three Assumptions 1. Consistency Assumptions 2. Sequential randomization assumption 3. Identification assumption

14. 1. Consistency Assumptions • Assume: Y = Y*( ) Lk = L*( ) • In words, we assume that the potential outcome corresponds to observed outcome.

15. 2. Sequential Randomization Assumption = No Unmeasured Confounder Assumption • Assume: {W _||_ Aj | ( , ) for all j = 0,…k} • In words, conditioning on the history of time dependent treatments and covariate information up to time tj, the treatment Aj is independent of the set of potential outcomes

16. 3. Identification Assumption • Assume if every covariate-treatment history up to time tj that has a positive probability of observed, then there must be a positive probability that the corresponding treatment will be observed • Example: violated assumption case Lj shows an adverse event, so that no aj is given, then this assumption is violated.

17. Purpose: To derive the distribution of potential outcomes From observed data: For example, if the potential outcome Y* is survival time, we may be interested in estimating or mean { }

18. Inverse probability weighting

19. Inverse weighting Inverse probability weighting (continued) The probability that a patient received regime is: So,

20. Slide 27 of Tsiatis’s = Proof of consistency of inverse probability weighted estimator Consistency assumption =

21. Defined by regime specify models G-computation algorithm

22. Estimating procedure • Solve the estimating equation • to get the estimated parameters for the conditional distributions, • Then integrate out L’s to get the marginal distribution of • Compare the distribution of for different g’s.