Two-Stage Designs for Phase II Clinical Trials. by Chung-Li Tsai Advisor Kam-Fai Wong

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1 Two-Stage Designs for Phase II Clinical Trials by Chung-Li Tsai Advisor Kam-Fai Wong Institute of Statistics National University of Kaohsiung Kaohsiung, Taiwan 811, R.O.C. July 2006

2 Contents List of Figures ii List of Tables iii iv v 1 Introduction 1 2 Single-Stage Designs 4 3 Two-Stage Designs Optimal Two-Stage Designs Minimax Two-Stage Designs Two-Stage Designs of minimizing {(EN(p 0 ) + EN(p 1 ))/2} Two-Stage Designs of minimizing the maximum of EN(p 0 ) and EN(p 1 ) Compare with Above Criteria Control PET(p 0 ) Algorithm 19 5 Discussion 23 References 26 A Appendix: Single-Stage Designs 28 B Appendix: Two-Stage Designs with p 1 p 0 = C Appendix: Two-Stage Designs with p 1 p 0 = i

3 List of Figures 1 p EN(p) and p PET(p) Collocate four above designs with EN(p) ii

4 List of Tables Table 1: Three cases are belonged to S 1 (r 1, n 1 p 0 = 0.10, p 1 = 0.30, α = 0.10, β = 0.10) Table 2: Parts of Table A.1 with (p 0, p 1 ) = (0.05, 0.25) and (α, β) = (0.10, 0.10) 5 Table.3: Parts of Table B.2 with (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10) 9 Table.4: Parts of Table B.1 with (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.05, 0.20) 10 Table.5: Parts of Table B.2 with (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.05, 0.20) 10 Table.6: Parts of Table B.2 and B.3 with (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10) Table.7: Part of Table B.2 and B.4 with (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10) Table.8: Parts of Table B.1, B.2, B.3 and B.4 with (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.10, 0.10) Table.9: Parts of Table B.1 and B.5 with (p 0, p 1 ) = (0.30, 0.50) and (α, β) = (0.10, 0.10), (minimax) Table.10: Parts of Table B.2 and B.6 with (p 0, p 1 ) = (0.60, 0.80) and (α, β) = (0.10, 0.10), (optimal) Table.11: Parts of Table B.3 and B.7 with (p 0, p 1 ) = (0.60, 0.80) and (α, β) = (0.10, 0.10), (min{(en(p 0 ) + EN(p 1 ))/2} Table.12: Parts of Table B.4 and B.8 with (p 0, p 1 ) = (0.30, 0.50) and (α, β) = (0.10, 0.10), (min max{en(p 0 ), EN(p 1 )}) iii

5 ÏÞÞ$ð'Œ ¼0>0? ` œ ŠÝ (p 0, p 1, α, β) D3 /) S = {(r 1, n 1, r 2, n 2 ) r 1 n 1, r 2 n 2, n 1 N, (r 1, r 2, n 2 ) N {0}}, ÍS âxb 'ÝlIClIIý0^ Þ$ð'Œ N ÍÞ$ð'ŒKE TëÍÙŒ 5½ÎTøÍÍóEN(p)@ 3Ï $ðý^ PET(p) CÀø Íón = n 1 + n 2 Ãyî ëíùœ!ýãj ÿõ8etýt Þ$ð' Œ Simon31989OèŒÝÞãJoptimalõ minimax OptimalÞ$ð'ŒÎÃyEN(p 0)ÿ Õ minimaxþ$ð'œjî ÊnõEN(p 0 ) Íp 0 Î Í ÝW ^ Í Z3 ÍðŒÝãJì ÍXETÝt Þ$ð'Œ M"DÍ8þF n"þ Û@ ÏÞ@ Þ$ð'Œ Š iv

6 Two-Stage Designs for Phase II Clinical Trials Advisor: Dr. Kam-Fai Wong Institute of Statistics National University of Kaohsiung Student: Chung-Li Tsai Institute of Statistics National University of Kaohsiung ABSTRACT Two-stage designs play an important role in the study of new drugs especially toxicity is the endpoint of interest. Given (p 0, p 1, α, β), there exists a set S = {(r 1, n 1, r 2, n 2 ) r 1 n 1, r 2 n 2, n 1 N, (r 1, r 2, n 2 ) N {0}} including all the probable two-stage designs which are satisfied with the two probabilities of error (α, β). Every two-stage designs lead to give an expected sample size EN(p), a probability of early termination PET(p), and a total sample size n = n 1 + n 2. A suggested design can be searched from S once a criterion is given based on EN(p), PET(p), n. Optimal and minimax two-stage designs have been discussed in the literature (Simon, 1989). The optimal designs are based on EN(p 0 ), and the minimax can be considered for n and EN(p 0 ) where p 0 is a given success probability. We find different designs under distinct criteria and these designs give us dissimilar advantage. Several designs under distinct criteria are tabulated for our interested parameters. Keywords: clinical trials, phase II trials, two-stage design v

7 1 Introduction In drug development, clinical trials are often classified clearly as phase I, II, III, and IV. Four types of clinical trial designs are commonly described as follows : Phase I Trials (Clinical Pharmacology and Toxicity) The primary goal of phase I trials is often used to determine maximum tolerated dose (MTD) of the treatment and to define the toxicities of the treatment. And the results supply direction for determining appropriate dose for use in phase II. Phase II Trials (Initial Clinical Investigation for Treatment Effect) The main purpose of phase II trials is a type of design with safety and efficacy estimation. Phase II clinical trials can be divided into two parts IIA and IIB. The studies designed to evaluate dosing are referred to as phase IIA studies, and studies designed to determine the effectiveness of the drug are called phase IIB. Phase III Trials (Full-scale Evaluation of Treatment) The primary objectives of phase III trials are gathered the additional information about effectiveness and safety needed to evaluate the overall benefit-risk relationship of the drug and provided an adequate basis for physician labeling. Phase IV Trials (Postmarketing Surveillance) The purpose for conducting phase IV trials is to elucidate further the incidence of adverse reactions and determine the effect of a drug on morbidity of mortality. In practice, phase IV are usually considered useful market-oriented comparison studies against competitor products. In Neyman(1950, pp ), the accepting and rejecting a statistical hypothesis are very convenient and are well established. It is important, however, to keep their exact meaning in mind and to discard various additional implications which may be suggested by intuition. Thus, to accept a hypothesis H means only to take an action A rather than 1

8 B. This does not mean that we necessarily believe that the hypothesis H is true. Also, if the application of the rule of inductive behavior rejects H, this means only that the rule prescribes action B and does not mean that we believe the H is false. In many studies especially in the phase II clinical trial, researchers may interest in deciding to have further investigation or to terminate the study based on the probability of success. In most of the cases, they wish to warrant further investigation if the probability of success is large enough or to end the study if the probability of success is too small. Usually two success probabilities p 0 and p 1 (p 0 < p 1 ) are specified so that and max P (to have further investigation p) α p p 0 max P (to terminate the sduty p) β, p p 1 where α, β are two given values corresponding to the probabilities of making inaccurate decisions. This can be done by considering the simplest hypothesis H 0 : p = p 0 versus H A : p = p 1 with rejection region R = {the number of success among n patients > c} which satisfy P (R p 0 ) α and P (R p 1 ) 1 β. Therefore, rejecting H 0 implies to have further investigation. Accepting H 0 implies to have early termination. It is noteworthy that we do not well define what is an inaccurate decisions if the probability of success is between p 0 and p 1 due to the limit of our knowledge. In fact, to reject H 0 means we have more than (1 α) 100% confidence that the probability of success is larger than p 0. To accept H 0 means we have more than (1 β) 100% confidence that the probability of success is smaller than p 1. In this thesis, when we say to accept H 0 means to decide early termination. When we say to reject H 0 means to have further investigation. Later, we review the single-stage designs in section 2. In section 3, we study the two-stage designs and discuss different 2

9 designs under distinct criteria. Then an algorithm of searching the best two-stage designs under different criteria is introduced in section 4. Finally, we give a discussion and final remarks in section 5. 3

10 2 Single-Stage Designs Suppose there are n 1 individuals in the study. Let X i = I(the ith individual is success), n 1 i = 1,..., n 1, and T 1 = X i. Clearly, T 1 has a binomial distribution B(n 1, p). Specifically, P (T 1 = x) = ( ) n1 p x (1 p) n1 x, x for x = 0, 1, 2,..., n 1. Let the rejection region R = {T 1 > r 1 }, then P (R p) = Given (p 0, p 1, α, β), there exists a set n 1 i=r 1 +1 ( n1 i ) p i (1 p) n 1 i. S 1 (r 1, n 1 p 0, p 1, α, β) = {(r 1, n 1 ) r 1 n 1, n 1 N, r 1 N {0}} such that for all (r 1, n 1 ) S 1 (r 1, n 1 p 0, p 1, α, β) P (R p 0 ) α and P (R p 1 ) 1 β. The optimal single-stage design is defined as (r 1, n 1 ) S 1 (r 1, n 1 p 0, p 1, α, β), which has the smallest value of n 1. Table 1: Three cases are belonged to S 1 (r 1, n 1 p 0 = 0.10, p 1 = 0.30, α = 0.10, β = 0.10) p 0 p 1 α β r 1 n 1 α β For example, (4, 25), (5, 29) and (6, 33) are belonged to S 1 (r 1, n 1 p 0 = 0.10, p 1 = 0.30, α = β = 0.10) since the probabilities of making inaccurate decisions, α and β, are smaller than α = β = 0.10 (see Table 1). As we can see, (4, 25) has the smallest total sample size n 1 among them. In fact, (4, 25) has the smallest value of n 1 among (r 1, n 1 ) S 1 (r 1, n 1 p 0 = 0.10, p 1 = 0.30). Therefore, given p 0 = 0.10, p 1 = 0.30 and α = β = 0.10, 4

11 the optimal single-stage design suggests to recruit 25 individuals in the study. That is, there are more than 4 individuals give positive responses, then reject H 0. Otherwise, accept H 0. To simplify the notation, we define (r s, n s ) = arg min n 1 S 1 (r 1, n 1 p 0, p 1, α, β) which has the smallest total sample size n 1 for all (r 1, n 1 ) S 1 (r 1, n 1 p 0, p 1, α, β). We give the result of the optimal single-stage designs with p 1 p 0 = 0.20 and p 1 p 0 = 0.15 in Table A.1 and A.2. For each (p 0, p 1 ), the optimal single-stage designs with (α, β) = (0.10, 0.10), (0.05, 0.20), (0.05, 0.10), and (0.05, 0.05), respectively, are given in four rows. The results which are tabulated include the critical point of decision r 1 and the total sample size n 1. Two exact probabilities of error (α, β ) are also on them. For example, the first row in Table A.1 means a single-stage design with p 0 = 0.05, p 1 = 0.25 and α = β = Table 2: Parts of Table A.1 with (p 0, p 1 ) = (0.05, 0.25) and (α, β) = (0.10, 0.10) p 0 p 1 α β r 1 n 1 α β Table A.1 The optimal single-stage design contains 20 individuals. If two or fewer positive responses are found, the study is suggested to terminate. Conversely, if more than two positive responses are seen in the trial, we conclude to go for further study. In this case, the two exact probabilities of error are α = and β = 0.091, respectively. 5

12 3 Two-Stage Designs Sometime, researchers prefer to recruit fewer individuals at the beginning of the trial especially toxicity is the endpoint of interest. If there are too fewer positive responses obviously, the clinical trial is terminated and conclude to accept H 0. If there are many positive responses, the clinical trial is also terminated but conclude to reject H 0. However, if the number of positive responses are fallen in a range, researchers will re-size individuals until decision of either accept or reject H 0 can be made. This leads to a two-stage design. Denote the number of individuals are studied in the first and the second stage by n 1 and n 2, respectively. Let r 1 and r 2 be the critical values needed to declare decision. Specifically, if r 1 or fewer positive responses are observed among the n 1 individuals in the first stage, we stop the trial and accept H 0. If more than r 1 + r 2 successes among the n 1 individuals in the first stage, we also stop the trial but reject H 0. Otherwise accrual continues to a total of n = n 1 + n 2 individuals. We give a general procedure of two-stage designs as follows: The first stage, n 1 if X i r 1, stop in the first stage and accept H 0 ; n 1 if X i > r 1 + r 2, stop in the first stage and reject H 0 ; n 1 if r 1 < X i r 1 + r 2, go to the second stage, The second stage, if if n 1 +n 2 n 1 +n 2 X i r 1 + r 2, accept H 0 ; X i > r 1 + r 2, reject H 0, That is, the rejection region is n 1 n 1 R = { X i > r 1 + r 2 or r 1 < X i r 1 + r 2 6 and n 1 +n 2 X i > r 1 + r 2 }.

13 Hence, the probability of reject H 0 with success probability p is P (R p) = 1 [B(r 1 ; p, n 1 ) + min(n 1,r) x=r 1 +1 b(x; p, n 1 )B(r x; p, n 2 )], (1) where b and B are the probability mass function and the cumulative density function of the binomial distribution, respectively, and r = r 1 + r 2. Similarly, given (p 0, p 1, α, β), there exists a set S 2 (r 1, n 1, r 2, n 2 p 0, p 1, α, β) = {(r 1, n 1, r 2, n 2 ) r 1 n 1, r 2 n 2, n 1 N, (r 1, r 2, n 2 ) N {0}} including all the probable two-stage designs which are satisfied with the two probabilities of error (α, β). As we can see, the total sample size of a two-stage design is either n 1 or n due to the number of positive responses in the first stage. Therefore, the expected total sample size n is related to the true probability of success. Let EN(p) be the expected sample size when response rate is equal to p. Then, EN(p) = n 1 + (1 PET(p))n 2 = n 1 (PET(p)) + n(1 PET(p)), where PET(p) is the probability of early termination after the first stage when response rate is p. At the end of the first stage, we will terminate the trial early and accept H 0 if r 1 or fewer positive responses are observed. We have enough evidence for making a decision to reject H 0 and the trial is unnecessary into the second stage when n 1 X i > r. As a result, PET(p) is given by B(r 1 ; p, n 1 ) + 1 B(r; p, n 1 ), if n 1 > r PET(p) = B(r 1 ; p, n 1 ), otherwise. Simon (1989) did not allow early termination for a high response rate and the probability of early termination was defined as PET(p) = B(r 1, p, n 1 ). 7

14 In other words, early acceptance of the drug is not permitted. Simon (1989) noted that when the drug has substantial activity ( p 1 ), there is often interesting in studying additional patients in order to estimate the proportion, extent and durability of response. However, it may not sufficiently improve efficiency of estimation for all scenarios, especially when size of n 2 is small related to n 1. For example, based on likelihood function L(p x) = ( ) n1 p x (1 p) n1 x = ˆp = x = x. x n 1 If we let p ˆp 0.05 under 95% confidence level, we need about n 1 = 385. In this thesis, we allow to have early acceptance of the drug. Later, we will review optimal and minimax two-stage designs which are given by Simon (1989) in section 3.1 and 3.2, respectively. In section 3.3 and 3.4, we consider two criteria such as minimize {(EN(p 0 ) + EN(p 1 ))/2} and minimize the maximum of EN(p 0 ) and EN(p 1 ). We compare these four designs in section 3.5. Finally, we control PET(p 0 ) 0.8 in section

15 3.1 Optimal Two-Stage Designs The optimal two-stage design is the two-stage design that the expected sample size is minimized when the response rate is p 0. It can be expressed as (r 1o, n 1o, r o, n o ) = arg min S 2 (r 1, n 1, r, n p 0, p 1, α, β) EN(p 0 ) which has the smallest expected sample size among (r 1, n 1, r, n) belong to S 2 (r 1, n 1, r, n p 0, p 1, α, β). For example, consider the case of (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10). Table.3: Parts of Table B.2 with (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10) p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) Table /12 5/ B.2 The first stage includes 12 individuals. If one or fewer than one positive responses are observed, the trial is suggested to stop in the first stage and accept H 0. If more than one positive responses are observed, the trial is also suggested to stop in the first stage but conclude to reject H 0. Otherwise accrual continues to a total of 35 individuals. When response rate p = p 0, the expected sample size EN(p 0 ) = is really the smallest and smaller than that of the optimal single-stage design which has sample size n 1 = 25. However, if the true response rate p = p 1, the expected sample size EN(p 1 ) = which is larger than that of the optimal single-stage design. In fact, the true response rate is unknown. 9

16 3.2 Minimax Two-Stage Designs The minimax two-stage design is defined to minimize the total sample size n first and then minimize the expected sample size EN(p 0 ) later. Specifically, we represent it as (r 1m, n 1m, r m, n m ) = arg min min S 2 (r 1, n 1, r, n p 0, p 1, α, β). EN(p 0 ) n For example, consider the case of (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.05, 0.20). Table.4: Parts of Table B.1 with (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.05, 0.20) p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) Table /18 10/ B.1 When response rate p = p 0, the expected sample size EN(p 0 ) = is smaller than the total sample size n 1 = 35 of the optimal single-stage design. Furthermore, the total sample size n = 33 is also smaller than that of the optimal single-stage design. In some cases, the minimax design may be more attractive than that with the minimum expected sample size. Consider the same case with optimal two-stage design as Table.4. Table.5: Parts of Table B.2 with (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.05, 0.20) p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) Table /13 12/ B.2 The optimal design has an expected sample size under H 0 of and a total sample size of 43 (see Table.5). The minimax two-stage design has an expected sample size of and a total sample size of 33 (see Table.4). If the accrual rate is only ten patients per year, it could take 1 year longer to complete the optimal design than the minimax design. 10

17 3.3 Two-Stage Designs of minimizing {(EN(p 0 ) + EN(p 1 ))/2} In section 3.1, we always consider acting as the situation that the response probability is p 0. In fact, we do not know what the true response rate p is. We may have p 0 with 50% and p 1 with 50%. Here, we make an adjustment to consider minimizing {(EN(p 0 ) + EN(p 1 ))/2}. It can be expressed as (r 1a, n 1a, r a, n a ) = arg min {(EN(p 0 )+EN(p 1 ))/2} S 2(r 1, n 1, r, n p 0, p 1, α, β). This design takes advantage of EN(p 1 ) than optimal two-stage design. For example, consider the case of (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10). Table.6: Parts of Table B.2 and B.3 with (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10) p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) Table /12 5/ B.2 1/16 4/ B.3 If the true response probability is p 1, the expected sample size of the optimal twostage design EN(p 1 ) = is bigger than the total sample of the optimal single-stage design which has the sample size n 1 = 25. However, the two stage design that minimize {(EN(p 0 ) + EN(p 1 ))/2} will have EN(p 0 ) = and EN(p 1 ) = which are both smaller than n 1 = 25. Although the expected sample size EN(p 0 ) for this design (EN(p 0 ) = 20.21) is larger than that of the optimal two-stage design (EN(p 0 ) = 19.83), it is only increasing by 1.9%. If we only control EN(p 0 ) but ignore EN(p 1 ), the twostage designs which we found may be give a large value of EN(p 1 ). Sometime, we can find a more robust design by small adjustment. Here, we only consider minimizing {(EN(p 0 ) + EN(p 1 ))/2}. In fact, it can be done by considering the prior distribution of the success probability p. 11

18 3.4 Two-Stage Designs of minimizing the maximum of EN(p 0 ) and EN(p 1 ) In some situation, we are interested in controlling the risk for different success probability simultaneously. This leads to the design that minimize the maximum value of EN(p) for some p, which form the formal minimax design we are familiar. In section 3.2, the minimax two-stage design given by Simon, in fact, is minimin. In this section, we consider minimizing the maximum of EN(p 0 ) and EN(p 1 ). It can be expressed as (r 1mm, n 1mm, r mm, n mm ) = arg min max{en(p 0 ),EN(p 1 )} S 2(r 1, n 1, r, n p 0, p 1, α, β). For example, consider the case of (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10). Table.7: Part of Table B.2 and B.4 with (p 0, p 1 ) = (0.10, 0.30) and (α, β) = (0.10, 0.10) p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) Table /12 5/ B. 2 2/18 4/ B.4 If the true response probability is p 1, the expected sample size of the optimal twostage design EN(p 1 ) = is bigger than the total sample of the optimal single-stage design which has the sample size n 1 = 25. However, the two-stage design that minimize the maximum two values of EN(p 0 ) and EN(p 1 ) will have EN(p 0 ) = and EN(p 1 ) = which are both smaller than n 1 = 25. Although the expected sample size EN(p 0 ) for this design (EN(p 0 ) = 19.90) is larger than that of the optimal two-stage design (EN(p 0 ) = 19.83), it is only increasing by 0.4%. The expected sample size of the criterion minimizing the maximum two values of EN(p 0 ) and EN(p 1 ) is surely smaller than the total sample size of the optimal single-stage design. 12

19 3.5 Compare with Above Criteria In this section, we compare these four designs by an example as follows. Consider the case of (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.10, 0.10) Table.8: Parts of Table B.1, B.2, B.3 and B.4 with (p 0, p 1 ) = (0.20, 0.40) and (α, β) = (0.10, 0.10) r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) Criteria 3/19 10/ minimax 3/17 10/ optimal 6/25 10/ minimize {(EN(p 0 ) + EN(p 1 ))/2} 6/27 10/ minimize max{en(p 0 ), EN(p 1 )} Figure 1: p EN(p) and p PET(p) Four criteria (p EN(p)) Simon (p EN(p)) minimax optimal min[(en(p 0 )+EN(p 1 ))/2] min max[en(p 0 ),EN(p 1 )] Four criteria (p PET(p)) Simon (p PET(p)) We can see the curve p EN(p), p [0, 1], that the criterion minimizing the maximum of EN(p 0 ) and EN(p 1 ) has smooth but the optimal has quake. The minimax and the criterion minimizing the maximum of EN(p 0 ) and EN(p 1 ) have same design under PET(p) 13

20 which given by Simon. The optimal and the criterion minimizing {(EN(p 0 ) + EN(p 1 ))/2} also have same design under PET(p) which given by Simon. In these cases, we find some patterns as follows (see Figure 1). Minimax versus another three designs If p (0.35, 0.75), the EN(p) of optimal two-stage design is always bigger than minimax. If p (0.20, 0.60), the EN(p) of the criterion minimizing {(EN(p 0 )+EN(p 1 ))/2} is always smaller than minimax. If p (0.25, 0.65), the EN(p) of the criterion minimizing the maximum of EN(p 0 ) and EN(p 1 ) is always smaller than minimax. Optimal versus two designs which given by criteria minimizing {(EN(p 0 )+EN(p 1 ))/2} and minimizing the maximum of EN(p 0 ) and EN(p 1 ) If p (0.30, 0.65), the EN(p) of optimal two-stage design is always bigger than the criterion minimizing{(en(p 0 ) + EN(p 1 ))/2}. If p (0.30, 0.65), the EN(p) of optimal two-stage design is always bigger than the criterion minimizing the maximum of EN(p 0 ) and EN(p 1 ). Minimize {(EN(p 0 ) + EN(p 1 ))/2} versus minimize the maximum of EN(p 0 ) and EN(p 1 ) If p (0.30, 0.65), the EN(p) of the criterion minimizing {(EN(p 0 )+EN(p 1 ))/2} is always bigger than the criterion minimizing the maximum two values of EN(p 0 ) and EN(p 1 ). If p (0.35, 0.75), the EN(p) of optimal two-stage design is the biggest. If p 0.30 or p 0.75, the EN(p) of optimal two-stage design is the smallest. 14

21 We also find that under Simon s condition, these four curves are always non-decreasing (see Figure 2). In fact, the true response rate is unknown. The optimal two-stage design is not defined obviously. We find different designs under distinct criteria and these designs can give us dissimilar advantage. 15

22 3.6 Control PET(p 0 ) 0.80 The ethical imperative for early termination occurs when the drug has low activity especially toxicity is the endpoint of interest. If we only control minimizing the expected sample size EN(p 0 ), the trial also has high probability into the second stage under the drugs with toxicity. Therefore, we think that we consider the PET(p 0 ) 0.80 first and minimize the expected sample size EN(p 0 ) later. There exists a set S 2(r 1, n 1, r 2, n 2 p 0, p 1, α, β, PET(p 0 ) 0.80) S 2 (r 1, n 1, r 2, n 2 p 0, p 1, α, β) = {(r 1, n 1, r 2, n 2 ) r 1 n 1, r 2 n 2, n 1 N, (r 1, r 2, n 2 ) N {0}} Then, it can be expressed as min S EN(p 0 2(r 1, n 1, r 2, n 2 p 0, p 1, α, β, PET(p 0 ) 0.80) ) We collocate four above designs with EN(p 0 ) by four cases. Case 1: Consider the case of (p 0, p 1 ) = (0.30, 0.50) and (α, β) = (0.10, 0.10) Table.9: Parts of Table B.1 and B.5 with (p 0, p 1 ) = (0.30, 0.50) and (α, β) = (0.10, 0.10), (minimax) r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) minimax 7/28 15/ PET(p 0 ) /37 15/ PET(p 0 ) 0.80 Case 2: Consider the case of (p 0, p 1 ) = (0.60, 0.80) and (α, β) = (0.10, 0.10) Table.10: Parts of Table B.2 and B.6 with (p 0, p 1 ) = (0.60, 0.80) and (α, β) = (0.10, 0.10), (optimal) r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) optimal 6/11 26/ PET(p 0 ) /27 24/ PET(p 0 )

23 Case 3: Consider the case of (p 0, p 1 ) = (0.60, 0.80) and (α, β) = (0.10, 0.10) Table.11: Parts of Table B.3 and B.7 with (p 0, p 1 ) = (0.60, 0.80) and (α, β) = (0.10, 0.10), (min{(en(p 0 ) + EN(p 1 ))/2}) r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) min{(en(p 0 ) + EN(p 1 ))/2} 9/16 25/ PET(p 0 ) /27 24/ PET(p 0 ) 0.80 Case 4: Consider the case of (p 0, p 1 ) = (0.30, 0.50) and (α, β) = (0.10, 0.10) Table.12: Parts of Table B.4 and B.8 with (p 0, p 1 ) = (0.30, 0.50) and (α, β) = (0.10, 0.10), (min max{en(p 0 ), EN(p 1 )}) r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) min max{en(p 0 ), EN(p 1 )} 8/30 15/ PET(p 0 ) /34 16/ PET(p 0 ) 0.80 Figure 2: Collocate four above designs with EN(p) minimax optimal min[(en(p 0 )+EN(p 1 ))/2] 30 PET(p 0 ) 0.00 PET(p 0 ) min max[en(p 0 ),EN(p 1 )] From Figure 3, we find these curves which are always smooth under controlling PET(p 0 ) And under this condition, it brings a advantage which reduces the risks 17

24 into the second stage if the drugs with toxicity. If the true response rate is p 0, in case 1, the first row in Table.9, the trial is early terminated in the first stage with probability The second row in Table.9, the trial is early terminated in the first stage with probability If the true response rate is p 1, in case 1, the first row in Table.9, the trial is early terminated in the first stage with probability The second row in Table.9, the trial is early terminated in the first stage with probability Whether the true response rate is p 0 or p 1 under PET(p 0 ) 0.80, the trial is early terminated in the first stage with probability If the drugs with toxicity, the condition is very useful. 18

25 4 Algorithm Although S 2 (r 1, n 1, r, n p 0, p 1, α, β) has infinite probable combination, we can make an order to the elements in S 2 (r 1, n 1, r, n p 0, p 1, α, β). Here, the elements in S 2 (r 1, n 1, r, n p 0, p 1, α, β) are rank by n 1 followed with n, r 1 and r. It is trivial that EN(p) n 1 for all p [0, 1]. Therefore, only finite many elements in S 2 (r 1, n 1, r, n p 0, p 1, α, β) need to be checked. In this section, we give an algorithm to the criteria of minimizing EN(p 0 ). The idea can be extended to all the other criteria. Consider the formula (1) f p (n, r, n 1, r 1 ) = B(r 1 ; p, n 1 ) + min(n 1,r) x=r 1 +1 some properties we have found are as follows: b(x; p, n 1 )B(r x; p, n 2 ), Property 1. f p (n, r, n 1, r 1 ) f p (n, r, 0, 0). Proof : f p (n, r, 0, 0) = P ( = = n x i r) r n 1 P ( x i = j, j=0 r 1 j=0 r 1 j=0 n 1 P ( x i = j, n x i r) n x i r) + n 1 P ( x i = j) + n 1 = P ( x i r 1 ) + = f p (n, r, n 1, r 1 ) min(r,n 1 ) j=r 1 +1 min(r,n 1 ) j=r 1 +1 min(r,n 1 ) j=r 1 +1 n 1 P ( x i = j, n 1 P ( x i = j, n 1 P ( x i = j, n x i r) n x i r) n x i r) Property 2. f p (n + 1, r, n 1, r 1 ) f p (n, r, n 1, r 1 ). n 1 Proof : f p (n + 1, r, n 1, r 1 ) = P ( x i r 1 ) + n 1 P ( x i r 1 ) + = f p (n, r, n 1, r 1 ) min(r,n 1 ) j=r 1 +1 min(r,n 1 ) j=r 1 +1 n 1 n+1 P ( x i = j, x i r) n 1 P ( x i = j, n x i r) 19

26 Property 3. f p (n, r + 1, n 1, r 1 ) f p (n, r, n 1, r 1 ). n 1 Proof : f p (n, r + 1, n 1, r 1 ) = P ( x i r 1 ) + n 1 = P ( x i r 1 ) + min(r,n 1 ) + j=r 1 +1 min(r,n 1 ) j=r 1 +1 min(r,n 1 ) j=r 1 +1 n 1 P ( x i = j, = f p (n, r, n 1, r 1 ) + f p (n, r, n 1, r 1 ) min(r,n 1 ) j=r 1 +1 n 1 P ( x i = j, n 1 P ( x i = j, n x i = r + 1) n 1 P ( x i = j, n x i r + 1) n x i r) n x i = r + 1) Property 1 tells us that if f p1 (n, r, 0, 0) = β + ɛ, we are impossible to find any twostage design (r 1, n 1, r, n) satisfy f p1 (n, r, n 1, r 1 ) β. If f p0 (n, r 1, 0, 0) < 1 α, then any two-stage design (r1, n 1, r, n ) satisfy f p0 (n, r, n 1, r1) 1 α and f p1 (n, r, n 1, r1) β will imply n > n (Property 3). Furthermore, given (r 1, n 1 ), we can adjust (r, n) so that f p0 (n, r, n 1, r 1 ) 1 α and f p1 (n, r, n 1, r 1 ) β (Properties 2 and 3). We give an algorithm which uses by above properties in the following: Step 1: Given (p 0, p 1, α, β). Step 2: Find n s, r s satisfy f p0 (n s, r s, 0, 0) 1 α ɛ 1 and f p1 (n s, r s, 0, 0) β + ɛ 2, where ɛ 1, ɛ 2 > 0. Step 3: Let CN = n s, Cr = r s, Cn 1 = 0, Cr 1 = 0 and CEN =. Step 4: Let n 1 = 1, r 1 = 0, n = n s, r = r s and calculate PET(p 0 ). Step 5: If n 1 + (n s n 1 )[1 PET(p 0 )] CEN, we stop searching and output the recent optimal. Else, reset n = n s + 1, r = r and go to Step 6. Step 6: If f p0 (n, r, n 1, r 1 ) 1 α, go to Step 8. Else, reset r = r + 1 and go to Step 7. Step 7: If r n 1, go to Step 6. Else, reset n = n + 1, r = r and go to Step 6. 20

27 Step 8: If f p1 (n, r, n 1, r 1 ) β, calculate EN(p 0 ) based on n, r, n 1, r 1 and go to Step 11. Else, reset n = n + 1 and go to Step 9. Step 9: If n n max, where n max CEN n 1 [ 1 PET(p 0 ) + n 1], go to Step 6. Else, reset r 1 = r and go to Step 10. Step 10: If r 1 = n 1, reset n 1 = n 1 + 1, r 1 = 0 and go to Step 3. Else, go to Step 3. Step 11: If EN(p 0 ) CEN, record (n, r, n 1, r 1 ), reset r 1 = r and go to Step 10. Else, reset r 1 = r and go to Step

28 Start Find n, r s s satisfy f ( ) p ns, rs,0,0 1 α 1 0 ε and p ( ns, rs,0,0) β + 2 f ε ε 0 1 ε 1, 2 > CEN n + C,Cn = n,cr = r,cn = 0,Cr1 = s s s 1 = 0 Let n 1 = 1,r1 = 0,n = n,r = Calcuate PET p ( ) s r s 0 n = ns + 1, r = r No Yes EN( p 0 ) CEN Output * f p ( nr, n, ) 1 α, r Yes f p ( n, r, n ) β, r No n = n + 1 No Yes Yes n n max r = r + 1 ( ) Calculate EN p Based on n, r, n 0, r 1 1 No 1 = r1 + r 1 Yes r n 1 No No EN( p 0 ) CEN r 1 = n 1 No n = n + 1 r = r Yes Record n, r, n, r 1 1 Yes n 1 = n 1 + 1, r 1 = 0 22

29 5 Discussion One aspect of the Simon design rests with the concept that if the response rate is high, it might be in the best interest of the individuals not to close the study for significance after the first stage. This strategy seems to put more individuals on effective treatments, perhaps at the expense of doing fewer studies and/or delaying the release of promising results to the public. In fact, the true response rate p we do not know. If we only consider the true response rate p 0, it is improper. Simon optimal two-stage designs have a defect which the expected sample size EN(p 1 ) may be bigger than the total sample size of the optimal single-stage design. Therefore, we do an adjustment that we consider minimizing {(EN(p 0 )+EN(p 1 ))/2}. This consideration can really reduce the expected sample size EN(p 1 ). However, the expected sample size EN(p 1 ) is not always smaller than the total sample size of the optimal single-stage design. Then, we consider another criterion minimizing the maximumof EN(p 0 ) and EN(p 1 ) that the expected sample size EN(p) is surely smaller than the total sample size of the optimal single-stage design. Simon minimax designs are minimin in fact. We consider minimizing the maximum of EN(p 0 ) and EN(p 1 ) which is the minimax we are familiar. Under controlling PET(p 0 ) 0.80, we can prevent individuals from drugs with toxicity into the second stage. If the study has to take long time, choosing two-stage designs could be wasted double time than optimal single-stage design. It is not smart. But if the drugs with toxicity, it is reasonable to use two-stage designs. When to use optimal single-stage or two-stage designs is not fixed. This is because of the proper designs being considered the case. How to estimate the response probability p in two-stage designs is important. Suppose the number of patients n 1 and n 2 are studied in the first and second stage design, respectively. Let Y 1 be the number of successes with response probability p in the first stage. Not only Y 1 or fewer than r 1 but also n 1 > r are observed, the study is termi- 23

30 nated. Otherwise, the study is allowed to continue. The number of successes Y 2 with the same response probability p is observed in the second stage. We find the possible values of Y 1 which are truncated by r 1 and all of these range from r to min(n 1, r). The probability mass function of Y 1 is mentioned the truncated binomial distribution with response probability p. So, the probability mass function of Y 1 is ( ) n1 p y 1 (1 p) n 1 y 1 P (Y 1 = y 1 ) = min(n 1,r) k=r 1 +1 y 1 ( n1 k ) p k (1 p) n 1 k While Y 2 is a binomial distribution with same response probability p, and probability (2) mass function of Y 2 is P (Y 2 = y 2 ) = ( n2 y 2 ) p y 2 (1 p) n 2 y 2, (3) where y 2 = 0,..., n 2. Based on formula (2) and (3), we obtain the joint probability mass function of Y 1 and Y 2 is P (Y 1 = y 1, Y 2 = y 2 ) = P (Y 1 = y 1 )P (Y 2 = y 2 ) ( )( ) n1 n2 p y 1+y 2 (1 p) n 1+n 2 y 1 y 2 = y 1 y 2 min(n 1,r) k=r 1 +1 And the likelihood function can be expressed as ( n1 L(p) = P (Y 1 = y 1 )P (Y 2 = y 2 ), k ) p k (1 p) n 1 k. (4) then by (4), the above expression L(p) = min(n 1,r) k=r ) p k (1 p) n 1 k k ( n1 ( )( n1 n2 y 1 y 2 ) p y 1+y 2 (1 p) n 1+n 2 y 1 y 2. (5) It is hard to differentiate by (5), so we can easily differentiate the log likelihood function log L(p) = log min(n 1,r) k=r 1 +1 ( n1 log(1 p) + log k ) p k (1 p) n 1 k + (y 1 + y 2 ) log P + (n 1 + n 2 y 1 y 2 ) ( n1 y 1 ) + log ( n1 y 1 24 )

31 The first derivative of log L(p) with respective to p is min(n 1,r) ( ) n1 kp k 1 (1 p) n1 k k log L(p) k=r 1 +1 = p + y 1 + y 2 p min(n 1,r) k=r 1 +1 n 1 + n 2 y 1 y 2 1 p ( n1 k min(n 1,r) k=r 1 +1 ) p k (n 1 k)(1 p) n 1 k 1 k ( n1 ) p k (1 p) n 1 k Let (6) be equal to zero and solve this equation, but the solution does have close form. Hence, Newton-Raphson s algorithm can be considered to obtain numerical estimate of p. By the Newton-Raphson s iteration for p, we can get a numerical solution and it is the maximum likelihood estimator. (6) 25

32 References [1] Chang MN, Therneau TM, Wieand HS, Cha SS (1987): Designs for Group Sequential Phase II Clinical Trials. Biometrics, 43, [2] Colton T, McPherson K (1976): Two-Stage Plans Compared with Fixed-Sample- Size and Wald SPRT Plans. Journal of The American Statistical Association, 71, [3] Fleming TR (1982): One-Sample Multiple Testing Procedure for Phase II Clinical Trials. Biometrics, 38, [4] Herson J (1979): Predictive Probability Early Termination Plans for Phase II Clinical Trials. Biometrics, 35, [5] Jennison C (1987): Efficient Group Sequential Tests with Unpredictable Group Sizes. Biometrika, 74, [6] Jung SH, Lee Taiyeong, Kim KM and George SL (2004): Admissible Two-Stage Designs for Phase II Cancer Clinical Trials. Statistics In Medicine, 23, [7] Neyman J(1950): A First Course in Probability and Statistics. Henry Holt & Co., New York. [8] O Brien PC, Fleming TR (1979): A Multiple Testing Procedure for Clinical Trials. Biometrics, 35, [9] Schultz JR, Nichol FR, Elfring GL, Weed SD (1973): Multiple-Stage Procedures for Drug Screening. Biometics, 29, [10] Shuster J (2002): Optimal Two-Stage Designs for Single Arm Phase II Cancer Trials. Journal of Biopharmaceutical Statistics, 12,

33 [11] Simon R (1989): Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials, 10,

34 A Appendix: Single-Stage Designs Table A.1 Single-Stage Designs for p 1 p 0 = Table A.2 Single-Stage Designs for p 1 p 0 =

35 Table A.1 Single-Stage Designs for p 1 p 0 = 0.20 p 0 p 1 α β r 1 n 1 α β p 0 p 1 α β r 1 n 1 α β For each value of (p 0,p 1 ), the single-stage designs are given for four sets of the two probabilities of error (α,β). The first, second, third and four rows correspond to error probabilities limits (0.10, 0.10), (0.05, 0.20), (0.05, 0.10) and (0.05, 0.05). For each design, (α, β ) denote the two exact probabilities of error (α,β). Table A.2 Single-Stage Designs for p 1 p 0 = 0.15 p 0 p 1 α β r 1 n 1 α β p 0 p 1 α β r 1 n 1 α β For each value of (p 0,p 1 ), the single-stage designs are given for four sets of the two probabilities of error (α,β). The first, second, third and four rows correspond to error probabilities limits (0.10, 0.10), (0.05, 0.20), (0.05, 0.10) and (0.05, 0.05). For each design, (α, β ) denote the two exact probabilities of error (α,β). 29

36 B Appendix: Two-Stage Designs with p 1 p 0 = 0.20 Table B.1 Minimax Designs for p 1 p 0 = Table B.2 Optimal Designs for p 1 p 0 = Table B.3 Minimize {(EN(p 0 ) + EN(p 1 ))/2} with p 1 p 0 = Table B.4 Minimize the maximum of EN(p 0 ) and EN(p 1 ) with p 1 p 0 = Table B.5 Minimax Designs for p 1 p 0 = 0.20, PET(p 0 ) 0.8. Table B.6 Optimal Designs for p 1 p 0 = 0.20, PET(p 0 ) 0.8. Table B.7 Minimize {(EN(p 0 ) + EN(p 1 ))/2} with p 1 p 0 = 0.20, PET(p 0 ) 0.8. Table B.8 Minimize the maximum of EN(p 0 ) and EN(p 1 ) with p 1 p 0 = 0.20, PET(p 0 )

37 Table B.1 Minimax Designs for p 1 p 0 = 0.20 p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) /13 2/ /12 2/ /15 3/ /26 4/ /16 4/ /15 5/ /22 6/ /25 7/ /19 10/ /18 10/ /24 13/ /24 16/ /28 15/ /19 16/ /24 21/ /36 26/ /28 20/ /34 20/ /29 27/ /44 33/ /23 23/ /23 23/ /27 32/ /37 38/ /27 24/ /13 25/ /26 32/ /28 39/ /16 20/ /23 21/ /18 26/ /18 33/ For each value of (p 0,p 1 ), the designs are given for four sets of the two probabilities of error (α,β). The first, second, third and four rows correspond to error probabilities limits (0.10, 0.10), (0.05, 0.20), (0.05, 0.10) and (0.05, 0.05). The expected sample size EN(p 0 ) and EN(p 1 ) and probability of terminating the trial at the end of the first stage PET(p 0 ) and PET(p 1 ) if the response probability is p 0 and p 1, respectively. For each design, (α, β ) denote the two exact probabilities of error (α,β). 31

38 Table B.2 Optimal Designs for p 1 p 0 = 0.20 p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) /9 2/ /9 2/ /9 3/ /18 4/ /12 5/ /10 5/ /18 6/ /20 8/ /17 10/ /13 12/ /19 15/ /28 17/ /22 17/ /15 18/ /24 24/ /34 27/ /18 22/ /16 23/ /25 32/ /29 38/ /21 26/ /15 26/ /24 36/ /27 45/ /11 26/ /11 30/ /19 37/ /28 46/ /9 22/ /6 22/ /15 29/ /18 35/ For each value of (p 0,p 1 ), the designs are given for four sets of the two probabilities of error (α,β). The first, second, third and four rows correspond to error probabilities limits (0.10, 0.10), (0.05, 0.20), (0.05, 0.10) and (0.05, 0.05). The expected sample size EN(p 0 ) and EN(p 1 ) and probability of terminating the trial at the end of the first stage PET(p 0 ) and PET(p 1 ) if the response probability is p 0 and p 1, respectively. For each design, (α, β ) denote the two exact probabilities of error (α,β). 32

39 Table B.3 Minimize {(EN(p 0 ) + EN(p 1 ))/2} with p 1 p 0 = 0.20 p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) /11 2/ /9 2/ /15 3/ /19 4/ /16 4/ /15 5/ /19 6/ /25 7/ /25 10/ /18 10/ /30 13/ /39 16/ /30 15/ /19 16/ /24 21/ /39 26/ /33 20/ /17 21/ /29 27/ /39 35/ /17 24/ /16 24/ /27 32/ /37 38/ /16 25/ /13 25/ /14 33/ /28 39/ /9 22/ /6 22/ /18 26/ /18 35/ For each value of (p 0,p 1 ), the designs are given for four sets of the two probabilities of error (α,β). The first, second, third and four rows correspond to error probabilities limits (0.10, 0.10), (0.05, 0.20), (0.05, 0.10) and (0.05, 0.05). The expected sample size EN(p 0 ) and EN(p 1 ) and probability of terminating the trial at the end of the first stage PET(p 0 ) and PET(p 1 ) if the response probability is p 0 and p 1, respectively. For each design, (α, β ) denote the two exact probabilities of error (α,β). 33

40 Table B.4 Minimize the maximum of EN(p 0 ) and EN(p 1 ) with p 1 p 0 = 0.20 p 0 p 1 α β r 1 /n 1 r/n EN(p 0 ) EN(p 1 ) α β PET(p 0 ) PET(p 1 ) /11 2/ /9 2/ /17 3/ /21 4/ /18 4/ /18 5/ /22 6/ /29 7/ /27 10/ /26 10/ /35 13/ /43 16/ /30 15/ /31 16/ /43 21/ /54 26/ /35 20/ /34 20/ /46 27/ /57 33/ /34 23/ /33 23/ /46 32/ /56 38/ /31 24/ /13 25/ /41 32/ /50 39/ /23 20/ /23 21/ /18 26/ /38 33/ For each value of (p 0,p 1 ), the designs are given for four sets of the two probabilities of error (α,β). The first, second, third and four rows correspond to error probabilities limits (0.10, 0.10), (0.05, 0.20), (0.05, 0.10) and (0.05, 0.05). The expected sample size EN(p 0 ) and EN(p 1 ) and probability of terminating the trial at the end of the first stage PET(p 0 ) and PET(p 1 ) if the response probability is p 0 and p 1, respectively. For each design, (α, β ) denote the two exact probabilities of error (α,β). 34