Investigation of up-and-down strategies for isotonic dose-finding Anastasia Ivanova Department of Biostatistics UNC at Chapel Hill, USA aivanova@bios.unc.edu 1
Migraine headache trial Phase IIa dose selection trial for migraine headache treatment (Hall et al., 2005) Primary outcome was headache severity. Positive response was defined as the reduction of headache severity from baseline of severe or moderate headache pain to mild or no headache pain 2 hours after treatment. The target dose was defined as the lowest dose with response rate of at least 60%. 2
Group design used in migraine study Assign 4 subjects at a time to a dose of the drug. Start with the lowest dose. • If 0,1,2/4 responses are observed, increase the dose; • If 3/4 or 4/4 responses are observed, decrease the dose. This is UD(4,2,3). 3
Group design in migraine study: correct target? The response rate actually targeted by the group design was computed by Hall et al. (2005) using quick calculation as (2/4 + 3/4)/2 = 0.625 For a group design applied to an increasing dose-response curve most of the assignments will be concentrated around a certain dose . The probability of response at this dose, Γ , can be calculated by solving the equation (Ivanova et al., 2007). Pr{decrease the dose} = Pr{increase the dose} Pr{increase the dose} = Pr{Bin(4, Γ ) � 2} = (1 - Γ ) 4 + 4 Γ (1 - Γ ) 3 + 6 Γ 2 (1 - Γ ) 2 Pr {decrease the dose} = Pr{Bin(4, Γ ) � 3} = 4 Γ 3 (1 - Γ ) + Γ 4 4 Γ Γ = 0.612 . Γ Γ
Group design in migraine study: the only group design? Another group design with cohorts of size 4, UD(4,1,4), that can be used • If 0,1/4 responses are observed, increase the dose; • If 2,3/4 responses are observed, repeat the dose; • If 4/4 responses are observed, decrease the dose The target quantile is Γ = 0.625, therefore this design, Design 2, can also be used in the migraine trial. 5 Which of the two designs is better?
Looking for the best non-parametric design Better design = better quality of estimation of the target dose. The strategy is first to find the optimal allocation for estimating the dose of interest. The optimal allocation depends on the true model for dose- toxicity relationship. For wide varieties of models and midrange of Γ the optimal design is the one assigning all the subjects to the MTD (Mats, Rosenberger, Flournoy 1998). Based on simulations for 10000 scenarios, for isotonic estimator the best allocation allocates the most subjects to the target dose. 6
How to compare allocations for group designs ? Giovagnoli and Pintacuda (1998) suggested comparing designs based on 1) “peakedness” of the stationary distribution 2) the rate of convergence to the stationary distribution For two designs with stationary distributions � 1 and � 2 with the same mode, they say � 1 is “ more peaked ”, if it grows more quickly to the left of the mode and decreases more quickly to the right of the mode. The rate of convergence is determined by the second largest eigenvalue in the absolute value of the transition matrix ( cf. Mira, 2001). 7
How to compare group designs ? Declare design 1 to be better than design 2 for a particular dose-toxicity scenario if design 1’s stationary distribution is more peaked and it converges faster to the stationary distribution. Bortot and Giovagnoli (2005) showed that for Biased Coin (BC) designs (Durham and Flournoy, 1994) BC( � 1 ) is better than BC( � 2 ) for every scenario if � 1 ≥ � 2 8
Comparing two group designs Consider Design 1 UD(6,1,2) Γ = 0.26 Design 2 UD(6,0,3) Γ = 0.25 Comparison is difficult because there are scenarios where the two designs have modes at different doses. We considered 10000 scenarios where designs have the same mode and computed proportion of scenarios where design 1 is better as far as rate of convergence 0.01 design 1 is better as far as peakedness 0.00 That is, design 2 is overall better than design 1 but not for ALL the scenarios. 9
How to compare group designs ? Analysis of other pairs of group designs lead to the following conclusions: 1) There was not a single pair of group designs where design 1 was better than design 2 as far as both measures in ALL scenarios. 2) Group design where the dose is never repeated was overwhelmingly worse than “adjacent” group design. For example, UD(6,1,2) is worse than UD(6,0,3). 3) For the rest of the pairs, no conclusion was reached. 10
Which group design is better for migraine study? Design 1, UD(6,1,2), used in migraine study • If 0,1,2/4 responses are observed, increase the dose; • If 3,4/4 responses are observed, decrease the dose is not as good as Design 2, UD(6,0,3), • If 0,1/4 responses are observed, increase the dose; • If 2,3/4 responses are observed, repeat the dose; • If 4/4 responses are observed, decrease the dose 11
Subject allocation for group designs Response rate = (0.3,0.6,0.6,0.6,0.6,0.6,0.6), sample size = 72 28.0 30 30 Design 1 Design 2 25 25 18.7 20 20 15 15 10 10 5 5 0 0 12 d1 d2 d3 d4 d5 d6 d7 d1 d2 d3 d4 d5 d6 d7
How to compare group designs for moderate sample sizes ? Since our interest was in moderate sample sizes, we compared group designs as follows: 1) Fix the number of cohorts n 2) For a dose-toxicity scenario, compute the expected proportion of subjects assigned to the MTD in a trial with n cohorts by averaging the elements of the matrix ( I + P ++ P n )/( n +1) in the column corresponding to the location of the MTD, where P is design’s transition matrix for the scenario. 3) Generate 10000 scenarios and compute the average of the measure in 2) 13
Which group design is better? The best designs according to the moderate sample size measure: Design 1 UD(6,1,2) Γ = 0.26 Design 2 UD(6,0,3) Γ = 0.25 Proportion of scenarios where design 1 is better as far as rate of convergence 0.01 design 1 is better as far as peakedness 0.00 design 1 is better as far as moderate n 0.26 14
Which group design is better? One can find the best group design for every cohort size n and target Γ . For example, for n = 12 and Γ = 0.25, the designs are UD (12,2,4), UD (12,1,5) and UD (12,0,6) UD (12,2,4) � UD (12,1,5) , it was better in 0.30 of scenarios UD (12,1,5) � UD (12,0,6), it was better in 0.82 of scenarios Therefore the best design for n = 12 and Γ = 0.25 is UD (12,1,5) 15
Cumulative cohort design For example, for Γ = 0.25, the best group designs for each n = 3,…, 25 are UD (3,0,2), UD (4,0,2), UD (5,0,2), UD (6,0,3), UD (7,1,3), UD (8,1,3), UD (9,1,4), UD (10,1,4), UD (11,1,4), UD (12,1,5), UD (13,2,5), UD (14,2,5), UD (15,2,6), UD (16,2,6), UD (17,2,6), UD (18,2,7), UD (19,3,7), UD (20,3,7), UD (21,3,8), UD (22,3,8), UD (23,3,8), UD (24,4,8), and UD (25,4,9). Ivanova et al., 2007 used the sequence of the best group design to construct the cumulative cohort design. 16
Cumulative cohort design Interestingly, the cumulative cohort design for Γ = 0.25 can be written in a very simple form: If the observed response rate is less than 0.25-0.09=0.16 , increase • the dose; • If the observed response rate is within 0.09 form 0.25 , repeat the dose; If the observed response rate is higher than 0.25+0.09=0.34 , • decrease the dose. The range of observed response rate for which the dose is repeated depends on the target quantile: use target � 0.15 if target is close to 0.5 use target � 0.09 or less if target is close to 0.2 or 0.8 17
Subject allocation for group and cumulative cohort designs Response rate = (0.3,0.6,0.6,0.6,0.6,0.6,0.6), sample size = 72 44.7 Cumulative Design 2 40 40 Cohort 28.0 30 30 20 20 10 10 0 0 18 d1 d2 d3 d4 d5 d6 d7 d1 d2 d3 d4 d5 d6 d7
Optimal cumulative cohort type design The method of selecting the best group design first and then using it in the CCD does not necessarily yields the best design of the type: Γ - � , Γ Γ + � ) repeat the dose if estimated rate is in ( Γ Γ Γ Γ Γ and change otherwise. To find the best design of this type we ran it on 10000 scenarios to find the optimal � for given Γ and number of doses. Optimal � was very close to the one used in CCD. 19
Continual reassessment method (CRM) • The CRM is a Bayesian design proposed in 1990 by O’Quigley, Pepe and Fisher. The CRM requires a working model for the dose-toxicity relationship to be specified. • In the CRM, no pre-specified set of doses is required, and patients are assigned one at a time. The next assignment is to the dose with the estimated toxicity rate equal to � . • If doses are chosen from a pre-specified ordered set d 1 < …< d K , the possible working model is F ( � , d i ) = b i � , where F ( � , d i ) is the probability of toxicity at dose d i , b i is a set of pre- defined constants, and parameter � is to be estimated (O’Quigley et al., 20 1990).
CRM: convergence Cheung and Chappell (2001) noticed that CRM was not converging in 1 out of 5 models they took from O’Quigley et al. (1990) n = 25 n = 48 Toxicity scenarios CRM CRM I 0.05 0.10 0.20 0.30 0.50 0.70 0.48 0.62 II 0.30 0.40 0.52 0.61 0.76 0.87 0.92 0.98 III0.05 0.06 0.08 0.11 0.19 0.34 0.42 0.51 IV0.06 0.08 0.12 0.18 0.40 0.71 0.54 0.67 V 0.00 0.00 0.03 0.05 0.11 0.22 0.29 0.36 Conditions for CRM convergence can be found in Shen and O’Quigley (1996) and Cheung and Chappell (2002). 21
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