1. The Probability of Being in Response Function and it Applications By Wei Yann Tsai Dept. of Biostatistics Columbia U. Joint works with Dr. John Crowley and Dr. XiaoDong Luo

2. 1.Introduction and Background 2.Estimators and Their Properties 3.Simulation 4.Real Data Analysis

3. A trial of two chemotherapeutic agents against testicular cancer (denoted A and ABV) by the Eastern Cooperative Oncology Group

6. The Win Ratio Pocock, S. J., Ariti, C. A., Collier, T. J., & Wang, D. (2012). The win ratio: a new approach to the analysis of composite endpoints in clinical trials based on clinical priorities. European heart journal, 33(2), 176-182. Luo, X., Tian, H., Mohanty, S., & Tsai, W. Y. (2015). An alternative approach to confidence interval estimation for the win ratio statistic. Biometrics, 71, 139-145

7. NC RP RL PG NC: No Change RP: Response RL: Relapse PG: Progress t2t2 t1t1 t1t1 (1) (2) (3) (4) h 12 (t 2 ) h 13 (t 1 ) h 24 (t 1, t 2 )

8. Nonhomogeneous Markov Model: h 24 (t 1, t 2 )=h 24 (t 1 ) Semi-Markov Model: h 24 (t 1, t 2 )=h 24 (t 1 -t 2 ) Homogeneous Markov Model: h 12 (t 2 )=h 12 h 13 (t)=h 13 h 24 (t 1, t 2 )=h 24

9. Bivariate Survival Times T 1 : Relapse Time or Progress Time T 2 : Response Time C : Censoring Time Observed Data Y 1 = min(T 1,C): Observed Survival Time Y 2 = min(T 1,T 2,C): Observed Response Time δ 1 = I(Y 1 =T 1 ) δ 2 = I(Y 2 =T 2 )

10. xxxxxx T1T1 T2T2 T 1 =T 2

11. The Probability of Being in Response Function P(t)=Pr(T 2 ≤t≤T 1 )

12. NC RP RL PG NC: No Change RP: Response RL: Relapse PG: Progress t2t2 t1t1 t1t1 (1) (2) (3) (4) h 12 (t 2 ) h 13 (t 1 ) h 24 (t 1, t 2 )

13. xxxxxx T1T1 T2T2 T 1 =T 2 t

15. xxxxxx T1T1 T2T2 t

16. The Probability of Being in Response Function P(t)=Pr(T 2 ≤ t ≤ T 1 ) P(t)=Pr(Y 2 ≤ t ≤ Y 1,δ 2 =1)/Pr(C ≥ t) P(t)=Pr(T 1 ≥ t) - Pr(T 0 > t) where T 0 =min(T 1,T 2 ) is the K-M estimator of the survival function of random variable A. 1

17. Temkin (1978) assume the Nonhomogeneous Markov model to get following Semi-nonparametric estimator Where

18. Theorem 1. If there exists a positive E such that P(t) > E for 0 < a ≤ t ≤ b and S T1 (a) = 1, then is consistent for t ≤ b even when the nonhomogeneous Markov assumption fails.

20. Simulation: T 1 (death Time): Weibull (alpha=2.0, lambda=0.04) T 2 (response time): Weibull (alpha=1.5, lambda=0.05) C(censoring time): Weibull ( alpha=1.5, lambda=0.06) Sample size: 1000 Simulation size: 1000

24. Data from the Stanford heart transplant study illustrate another possible application. The function P ( t ) is the probability of being alive with a transplanted heart at time t after acceptance.

26. A randomized trial of two treatments (standard treatment and standard treatment plus Thalidomide) for patients with myeloma was conducted by the Myeloma Institute for Research and Therapy (MIRT). 334(323) patients in the control(treatment) arm 157(133) had died after partial response, 104(142) partially responded without yet dying, 56(33) died without partial response and 28(15) had not yet died nor responded.