Statistical Methods for Analyzing Sequentially Randomized Trials

1. Statistical Methods for Analyzing Sequentially Randomized Trials Bembom, O., & Van der Laan, M. J. (2007). J Natl Cancer Inst, 99, 1577-82. Presented by: C. Akunna Emeremni 12/03/2010

2. Outline • Sequentially Randomized Trials • The Prostate Cancer Trial • G-Computation Algorithm • Inverse Probability of Treatment Weighting • Summary

3. Sequentially Randomized Trials(SRT): • An SRT is a randomized experiment in which the exposure value at each successive visit t is randomly assigned with known randomization probabilities (bounded away from 0 and 1) that by design may depend on a subjects past exposure and covariate history through t • Generalized identifiable conditions hold true.

4. Motivation for an SRT: • General goal of any sequentially randomized trial (SRT) is to examine different treatment regimes to determine which regime yields the most favorable outcomes over time. • Reflective of common medical practice. • Selection of best ordered pair of treatments instead of a best treatment.

5. The Prostate Cancer Trial: • Novel trial design utilizing sequential randomization based on intermediate patient outcomes. • Four candidate drug regimens: • CVD KA/VE TEC TEE • Examine overall success rates of different adaptive treatment regimes.

6. The Prostate Cancer Trial: • Overall Success: two consecutive successful responses • Overall Failure: two cumulative unsuccessful responses • Stopping Rules: Stop trial when either an overall success or an overall failure has occurred.

7. The Prostate Cancer Trial: • Example Adaptive Treatment Assignment (CVD, TEE regime) STOP: Pt Success STOP: Pt Success S S CVD TEE S S STOP: Pt Failure F F CVD TEE F STOP: Pt Failure F S STOP: Pt Success TEE S TEE F STOP: Pt Failure F STOP: Pt Failure

8. The Prostate Cancer Trial: • 3 ways to get overall Patient Success • SS, FSS, SFSS (2 consecutive successes) • 4 ways to get overall patient failure • FF, FSF, SFF, SFSF. (2 cumulative failures)

9. Goal of the Analysis: • Find the distribution of the response Y if the entire population were subjected to a particular regime g, for all regimes. • Chapter 23 LDA showed why and how the distribution of the potential response can be derived from the observed data using: • G computation Algorithm • Inverse Probability of Treatment Weighting.

10. Notation and Definitions: • Treatment decisions are made at discrete times t0,….,tK • The observed data for each individual is time ordered as L0 ,A0, L1, A1, ……….LK, AK, Y where: • L0 is the baseline covariates • A0denotes treatment assigned at time t0 • Lidenotes covariate information collected between ti-1and ti • Aidenotes treatment assigned at time tiamong the set of potential treatments • The over-bar notation is used to denote the history of time-dependent variables • Y denotes the outcome of interest, measured after time tK

11. G-computation (Overview) • The observed data is O = (L0 ,A0, L1, A1, ……….LK, AK, Y ) • Using the law of total probability we can write the joint distribution of the observed data as: pO(o) = pL0 (l0) x pA0|L0,{a0|l0)} x pL1|L0 ,,A0{l1|l0, a0} X …….. x pAk|Lk ,,Ak-1{ak|lk, ak-1} X pY|Lk, Ak{y|lk, ak)} • The G computation algorithm ignores the treatment assignment distribution • Thus implementation is the same irrespective of whether the data were from a sequentially randomized trial or an observational study

12. G-comp of the Cancer data: • Focus on overall success S ignoring intermediate outcomes Sj • 12 different dynamic regimes

13. G-comp of the Cancer data:

14. G-comp of the Cancer data: • CVD-TEC regime • Need only the distribution of overall success given treatment and covariate history. • 15% experienced overall success on CVD • Of the 85% who failed and where randomized to TEC, 17% experienced overall success. G-computation estimate of overall success = 0.15 + (0.85)(0.17) = 0.29

15. Limitations of G-computation: • Requires the estimation of the distribution of each intermediate outcome Sj given each possible covariate and treatment history up to that time point. • i.e. we have to evaluate the distribution of all potential outcomes over all individuals. • Consider P(successful response) on {KA/VE CVD} 3 different ways to achieve success (SkA/VE SkA/VE), (FkA/VESCVDSCVD), (SkA/VEFkA/VESCVDSCVD)

16. Limitations of G-computation: • (SkA/VEFkA/VESCVDSCVD) • 21 individuals who failed on KA/VE were later assigned to CVD at time point 3 • None achieved success at time point 3 • Cannot estimate the probability of a successful response for this potential outcome. • G-computation often relies on simplifying assumptions which introduce bias into the estimates.

17. IPTW (Overview) • Overcomes the need for simplifying assumptions • The idea is to take the average of the response for individuals who received a particular regime g weighted by the inverse probability of having received that particular regime. • The IPTW weights allow the individual who received that treatment to account for him/herself as well as similar individuals who did not receive that treatment.

18. IPTW of the Cancer Data: • P(receive any 1st line trt)=1/4=0.25 • IPTW weight = 1/(1/4) =4 • P(receive any salvage trt) = 1/3=0.33 • IPTW weight =1/(1/3) = 3 • Improve the IPTW performance by using empirical proportions rather than the known randomization probabilities.

19. IPTW of the Cancer Data: • P(receive 1st line CVD trt)= 26/108 • IPTW weight = 108/26 • P(receive KA/VE as salvage after CVD)=10/22 • IPTW Weight = 22/10 For 4pts with overall 1st line success, weight them by 108/26 For remaining 5 with overall success on salvage trt, upweight them by an additional 22/10 Then normalize to the original observed sample size

20. IPTW of the Cancer Data: • 4*(108/26)= 16.61 [1st line success] • Normalize to observed sample size of 14 • 16.61*(14/108)=2.2 • 5* (108/26)*(22/10) = 45.69 [Salvage success] • Normalize to observed sample size of 14 • 45.69 * (14/108) =5.9 • Then # of total failures= 14- (2.2+5.9) = 5.9 • Est overall success rate = (2.2+5.9)/14= 0.58

21. IPTW of the Cancer Data: • Can create a pseudo population: The population where everyone followed that regime. • Assume exchangeability. i.e the exposed had they been unexposed would have experience the same distribution of outcomes as the exposed.

22. IPTW pseudo pop of the Cancer data • CVD- KA/VE regime again • Asume all 108 patients had been on CVD, would still expect the same 15% first line success rate (because of exchangebility) • 0.15(108)= 16.2 success on 1st line therapy. • Then .85(108) =91.8 persons would go on to KA/VE • Then expect the same salvage trt response rate of 0.50 i.e 0.50 (91.8)=45.9 persons succeed. • Total success rate = (16.2 + 45.9)/108 = 0.58

23. IPTW vs. G-computation • Identical estimates if IPTW uses the empirical estimates rather than randomization estimates and G-comp does not rely on simplifying assumptions • IPTW unlike G-comp is guaranteed to provide valid estimates in the absence of any additional assumptions.

24. Conclusion: Which regime is best? • TEC and TEE are good first line therapies. • CVD is the worst first line therapy. • Practical implication: consider TEC or TEE instead of CVD as the first line therapy for treatment naïve patients.

25. Conclusion: Which regime is best? • Salvage Success rates are in general low except for KA/VE after CVD. • The choice of salvage therapies after failure with a 1st line therapy other than CVD is unimportant • Best ordered regimes are d(CVD,KA/VE), d(TEE, CVD) and d(TEC, CVD)

26. References: • Bembom, O., & Van der Laan, M. J. (2007). Statistical Methods for Analyzing Sequentially Randomized Trials. J. Natl Cancer Inst , 1577-82.