1. Planning Survival Analysis Studies of Dynamic Treatment Regimes Z. Li & S.A. Murphy UNC October, 2009

2. Dynamic treatment regimes are individually tailored treatments, with treatment type and dosage changing according to patient outcomes. Conceptualize treatment as a series of stages. 2 Stages for one individual Observation available at j th stage Action at j th stage (usually a treatment)

3. A dynamic treatment regime is the sequence of two decision rules: d 1 (X 1 ), d 2 (X 1,d 1,X 2 ) for selecting the actions in future. In planning survival analysis trials, the observation X 2 includes an indicator of response/non-response and whether the failure time has occurred.

4. Our Goal is to plan a sequential, multiple assignment, randomized trial (SMART). These are trials in which subjects are randomized among alternative options (the A j ’s are randomized) at each stage. Simple regimes: No X 1, for example, d 1 = 1 (tx coded 1) X 2 = R, an indicator of early signs of non- response, d 2 (1,R) is 0 if R=1 (tx coded 0) otherwise stay on current tx

5. SMART Precursors of the SMART design: CATIE (2001), STAR*D (2003), many in cancer SMART designs: Treatment of Alcohol Dependence (Oslin, data analysis; NIAAA) Treatment of ADHD (Pelham, data analysis; IES Treatment of Drug Abusing Pregnant Women (Jones, in field; NIDA) Treatment of Autism (Kasari, in field; Foundation) Treatment of Alcoholism (McKay, in field; NIAAA) Treatment of Prostate Cancer (Millikan, 2007)

6. ADHD (Pelham, PI) A 1 =0. Begin low dose medication 8 weeks Assess- Adequate response? Continue, reassess monthly; randomize if deteriorate A 2 =0 Increase dose of medication Random assignment: A 2 =1 Add BEMOD, medication dose remains stable No A 1 =1. Begin low-intensity BEMOD 8 weeks Assess- Adequate response? Continue, reassess monthly; randomize if deteriorate A 2 =1 Add medication; BEMOD remains stable Random assignment: A 2 =0 Increase intensity of BEMOD Yes No Random assignment:

7. Background Survival probabilities (and associated tests) –Lunceford et al. (2002) 3 weighted sample proportion estimators –Wahed and Tsiatis (2006) semiparametric efficient + implementable estimator –Miyahara and Wahed (2009) weighted Kaplan-Meier estimator. –Feng and Wahed (2009) sample size formulae based on a Lunceford et al. estimator –Guo and Tsiatis (2005) weighted cumulative hazard estimator Weighted version of the log rank test –Guo(2005) proposes weighted log rank test –Feng and Wahed (2008) weighted version of supremum log rank test and associated sample size formulae

8. Notation Suppose we decide to size the study to compare regimes (A 1, A 2 )= (1,1) versus (A 1, A 2 )= (0,1) Randomization probabilities are p 1, p 2 T 11, T 01 potential failure times under each regime T, S, C are the failure time, time to nonresponse, censoring time, respectively R is the nonresponse indicator, e.g. R=1 S≤min(T,C)

9. Test Statistics Weighted version of the Kaplan-Meier to test Weighted version of the log rank test to test Survival function Selected time point (usually end of study)

10. Weights are necessary to adjust for the trial design. Time independent weights (for regimes 11 and 01): Time dependent weights (potentially more efficient): Weights R=1 S≤min(T,C)

11. Weighted Kaplan-Meier (WKM) Estimator Time dependent weights (tWKM): -Asymptotically normal with mean and variance Can use the time independent weights (cWKM) as well. (j,k)=(1,1), (0,1) i th subject,

12. Weighted Log Rank Test (WLR) Time dependent weights (tWLR): where (j,k)=(1,1), (0,1) and Asymptotically normal under a local alternative, PH assumption, with mean, and variance Can use the time independent weights (cWLR) as well.

13. Sample Size Formulae Test based on WKM estimator: WLR test:

14. Challenges Variances are complex and depend on the joint distribution of the failure time T and the time to non-response, S. These two times are likely to be dependent. It may be hard to elicit information about this joint distribution in order to plan the trial.

15. Our Approach Use time independent weights in the sample size formulae (cWKM or cWLR). Express the variances in terms of the potential failure times under each regime, T jk, e.g. in terms of Replace variances with simpler upper bounds.

16. The Variances cWKM: cWLR:

17. Upper Bounds on Variances (Replace R by 1)

18. Sample Size Formulae Test based on cWKM: where cWLR:

19. Data Analysis Use potentially more powerful tests than that used for sample size calculation. Testing Test based on tWKM Test based on Lunceford 3 (Lunceford et al, 2002) Test based on Wahed and Tsiatis, (2006) implementable estimator, WT Testing tWLR

20. Simulation Proportional hazards for T 11 and T 01 Frank Copula model for potential outcomes (T jk, S j ) Weibull distributions for T jk and S j Compare with Feng and Wahed (2009) sample size formula: –Based on a weighted sample proportion estimator (the second estimator in Lunceford et al., 2002). –Assumed independence between T jk and S j to simplify variances.

21. Simulation Results for WKM (desired power is 80%)

23. Simulation Results for WLR (desired power is 80%)

24. Discussion Working assumptions used to size the study are the same as in the standard two arm study. Sample sizes are conservative, but the degree of conservatism depends on the percentage of subjects with R=1. cWLR yields smaller sample sizes than cWKM and needs less information, but power guarantees rely on proportional hazards assumption. These formulae can be easily generalized to more complex designs with the number of treatment options differing by both response status and prior treatment.

25. This seminar can be found at: http://www.stat.lsa.umich.edu/~samurphy/ seminars/UNC.10.2009.ppt Email Zhiguo Li or me with questions or if you would like a copy of the paper: zhiguo@umich.eduzhiguo@umich.edu or samurphy@umich.edusamurphy@umich.edu

26. Timing of movement between stages The timing of the stages may be fixed or may be an outcome of treatment. -----suppose the second stage is only for non- responders Fixed timing: Second stage starts at 8 weeks after entry into trial. Random timing: Second stage starts as soon as a nonresponse criterion is met.