Evaluating multiple treatment courses in clinical trials

Evaluating multiple treatment courses in clinical trials Peter F. Thall, Randall E. Millikan and HisGuang Sung Presented by: Ying Ding 01/22/2007 1 Outline • Introduction • Prostate Cancer Trial • Probability Models  Non-parametric model (*)  Parametric model • Simulation Study • Summary 2 Introduction • Evaluating multiple treatment courses:  a single response (Yj) to a single treatment (Tj) Yj is usually a binary variable identify only one treatment for each patient  Evaluate outcome-adaptive, multi-course treatment strategies and specify which treatment to give in each course Tj = (T1, T2, … , Tj), Yj = (Y1, Y2, … , Yj) with Tj chosen based on Yj-1 and Tj-1 for j>=2 3 Introduction • Motivation for adaptive strategy:  Reflect actual clinical practice In oncology, a patient’s treatment often involves multiple courses of chemotherapy.  Increase the amount of information per patient • Goal of this article:  Provide a general statistical framework for randomized multi-course clinical trial  Inference: selection of one best treatment, selection of a best ordered pair of treatments 4 The Prostate Cancer Trial • Background:  Prostate-specific antigen (PSA) is a useful marker for prostate cancer  PSA decline is used as a surrogate endpoint in clinical trials • Four candidate chemotherapy regimens:  PEE, KD, CVD and PEC • Some important definitions:      Treatment: a particular regimen given in a particular course Therapy: the entire sequence of treatments given to a patient Patient Success with trt t: two consecutive successful courses with t Patient Failure: a total of two unsuccessful courses regardless of trt Set of Acceptable Treatments: exclude any previous treatment that was unsuccessful 5 The Prostate Cancer Trial • Adaptive treatment assignment rule:  First course: randomly select one treatment from the four candidate treatments with equal probability  Second treatment: if the treatment in the first course is successful, then it is given in the second course; if not, the patient is randomized fairly among the treatments in the acceptable treatment set  Stop the trial as soon as we have either Patient Success or Patient Failure 6 The Prostate Cancer Trial An example of adaptive treatment assignment for a patient: 1st course 2nd course 3rd course 4th course outcome STOP STOP S PEE F S PEE F S KD F STOP STOP KD F STOP KD F STOP S STOP S KD S F Patient Success Patient Success Patient Failure Patient Failure Patient Success Patient Failure Patient Failure 7 Probability Models • First, a general probability model will be presented • Followed by a non-parametric model: Multinomial model (*) • Finally, the paper gives a parametric model: Regressive logistic probability model (I will not talk about it) 8 General Probability Model • Some important notations  Tj: the treatment for the jth course of chemotherapy  Yj: the outcome of the treatment for the jth course 1, jth course is successful  Y j   th 0, j course is unsuccessful      Tj = (T1, T2, … , Tj), Yj = (Y1, Y2, … , Yj) Su: success with treatment u on a given course Fu: failure with treatment u on a given course FuStSt = [Y1=0, Y2=Y3=1, T1=u, T2=T3=t] 9 General Probability Model • Overall Success (in 3 different ways)  SuSu, FuStSt or SuFuStSt • Overall Failure (in 4 different ways)  FuFt, FuStFt, SuFuFt or SuFuStFt • Some additional notations:  Aj: set of acceptable treatments at course j  tj(t) = Pr{Tj = t}, t  Aj  qj = qj (Yj-1; Tj) = Pr {Yj =1 | Yj-1; Tj} Pr {Yj=yj|Yj-1; Tj} = qjyj (1- qj)1-yj 10 General Probability Model • Likelihood Equation  The probability of a particular sequence of outcomes, yr, and treatments, tr, through r courses of treatment is: r y 1 y q j (1  q j ) t j (t j ) Lr = Pr {Yr = yr, Tr = tr} =  j 1  The full likelihood equation is: n L =  Li ,c j j here i indicate the patients’ number; ci indicate the total number of courses for the ith patient i 1 i 11 Multinomial models • Definitions  St*: patient success with treatment t  pt: the probability that “patient success with treatment t”  A single best treatment : The treatment that is associated with the largest pt, i.e. arg max p t t 12 Multinomial Models • Important properties of pt:  pt involves all of the treatments in the trial  pA in a trial of treatments {A, B, C, D} may be quite different from pA in a trial of treatments {A, E, F, G}.  In the common “naive” setting, pt is defined depending on t alone, regardless of the other treatment: pt = Pr {St} • Advantage of the current definition of pt:  Use all the information in (Y, T), fit the multi-course structure  Provides a basis for constructing desirable treatment combinations 13 Multinomial Models • An alternative goal: Find the “best” ordered pair of treatments (u, t)  treatment u is given initially, if u is unsuccessful, treatment t is given  the probability of patient success with (u, t) is where and  Find the pair (u, t) which maximize 14 Multinomial Models • Some notations (preparation for MLEs):  nu: the number of patients treated initially with u  Xu=|SuSu|: the number of patients who succeed in their first two courses with treatment u  nt|u: the number of patients treated initially with u who fail either the first or second course and are then treated with t  • MLEs         (u, t )   u  (1   u )  t|u 15 Multinomial Models • MLEs (continue)  Recall that  In our prostate cancer trial, 4 candidate treatments are available, so there are totally 4*3=12 ordered pairs  So we can rewrite pt as:  MLE of pt is   1 1 pt     (1   u )  t|u 4 t 12 u , u t   16 Multinomial Models • An example: consider a trial of 3 treatments {1, 2, 3}, The overall patient success probability is  u ,t  (u , t ) / 6  which is larger than the probability 0.50 of initial success with treatment 1 alone! 17 Multinomial Models • Continuation of the example The best ordered pair of treatment is (1, 3) with probability:  (1,3)  1  (1  1 )3|1  0.7 which is larger than the probability of success with using treatment 1 alone. 18 Simulation Study • Focus on selecting one best treatment • More related definitions/notations 3 probabilities characterizing patient outcome through the first two courses:     pt can be calculated based on the above 3 probabilities 19 Simulation Study • Two scenarios, each has 4 candidate treatments available. • Treatment 4 is the best in both scenarios. 20 Simulation Study • Probabilities of selection as best in terms of • Probabilities of correctly selecting treatment 4 are shown in bold type 21 Simulation Study • Simulation was used to determine the sample size (for achieving some certain power). n = 92, 124 and 156 were determined to ensure that probabilities of correctly selecting treatment 4 as best under scenario 1 by RLM1 are respectively, >= 0.80, 0.85, and 0.90. • Selecting the best treatment is statistically easier under scenario 1. Since p4 – p3 = 0.044 under scenario 1, while p4 – p3 = 0.028 under scenario 2. 22 Summary • A statistical framework has been provided for the case where:  Therapy consists multiple courses  Tj is chosen adaptively based on (Tj-1, Yj-1), where j>=2 • Strategy to select the best treatment has been proposed • Strategy to select treatment sequences has also been illustrated • Simulation Study has shown that:  Desirable powers can be achieved with moderate sample sizes under different clinical scenarios  The probability of selecting the best treatment correctly is greater in the adaptive strategy compared to the naïve method. 23 Discussion Any Questions? Thank you for your attention! 24