T-Statistic-based Up&Down Design for Dose-Finding Competes Favorably with Bayesian 4-parameter Logistic Design

Transcription

1 T-Statistic-based Up&Down Design for Dose-Finding Competes Favorably with Bayesian 4-parameter Logistic Design James A. Bolognese, Cytel Nitin Patel, Cytel Yevgen Tymofyeyef, Merck Inna Perevozskaya, Wyeth Jeff Palmer, Millenium Acknowledgement: Vipul Suru and Patrick Tai of Cytel provided valuable programming support

2 OUTLINE T-statistic-based Up&Down Design for Dose-Finding Competes Favorably with Bayesian 4-parameter Logistic Design JSM2008 Presentation by Nitin Patel, et al. - Comparison based on true underlying 4-parameter logistic (4PL) dose-response curves Review summary of findings JSM2009 Current Presentation - Comparison based on true underlying NON-4PL dose-response curves Bayesian 4PL adaptive design T-Statistic Up&Down adaptive design Simulation Spec s Findings Summary / Conclusions / Recommendations 8/17/09 Confidential Cytel

3 Simulation Comparison of Bayesian & T-Stat Dose-Adaptive Designs (Patel N, et al. JSM2008) Concluding Remarks Based on simulated data from 4-parameter logistic dose response models: The two methods are comparable in terms of their performance criteria Bayesian method Better MSE of estimate of mean response at target dose can be much better when the number of possible doses is large relative to sample size an important advantage is ability to use informative priors T-Stat implementation for an actual trial is easier Adaptive randomization can be implemented by switching between pre-generated randomization lists Frequentist methods are familiar to trial statisticians, clinicians, and managers Future work study non-4-parameter logistic dose-response curves 3 3

4 Bayesian Design Uses 4-parameter logistic model as parametric approximation of response curve Assumes flat (non-informative) prior distribution for parameters of logistic curve model for mean response and σ 2 4 4

5 Bayesian Adaptive Allocation 1. Specify dose assignments to subjects in first cohort 2. Use responses of all previous cohorts to compute posterior distribution 3. Compute an effectiveness criterion for each possible dose if assigned to next subject. 4. Use effectiveness criterion to assign doses to subjects in next cohort 5. Repeat 2 nd to 4 th steps until the trial sample size is reached 5 5

6 Effectiveness Criterion The expected decrease in variance of the mean response at the target dose is estimated for each possible dose assignment using the most recent posterior distribution. Doses are assigned to subjects in the next cohort with probability proportional to the square root of this expected decrease 6 6

7 T-statistic method of Ivanova, et al. (Stat in Med, 2008) At each look the next cohort is randomized with a fixed number of subjects on placebo and the remaining subjects at the dose level of the previous cohort or one level above or below it. The decision of whether to stay at the previous level, go up, or go down is made by computing a t-statistic defined by: Where n 1 is the number of subjects assigned to placebo so far with mean response and n j is the total (cumulative) number of subjects assigned so far to the dose level of the previous cohort with mean response. S is the pooled standard deviation estimate, and D is the target level of response (difference from placebo). 7 7

8 T-Stat Design Up&Down Rule If - ψ < T j < ψ assign next cohort dose that is the same as the previous cohort If T j - ψ, assign dose to next cohort that is one level above that of the previous cohort (unless previous dose was highest, in which case assign same dose) If T j ψ assign dose to next cohort that is one level below that of the previous cohort (unless previous dose was lowest, in which case assign same dose) For our simulations we used ψ =

9 Seven Dose Response Curves (Y-axis response is difference from placebo) DR_1 Linear DR_2 Umbrella DR_3 Sigmoid Emax DR_4 Emax DR_5 Emax low DR_6 Explicit DR_7 Null Case Target Response 9 9

10 Seven Dose Response Curves (Y-axis response is difference from placebo) Target Response DR_1 Linear DR_2 Umbrella DR_3 Sigmoid Emax DR_4 Emax DR_5 Emax low DR_6 Explicit DR_7 Null Case 10 10

11 Simulation Scenarios Error SD = 1.3 (Effect Sizes of Targets: ~0.4, ~0.7) Sample sizes (drug:placebo~2:1 and 3:1) = 135, 200, 320 Number of equally sized cohorts: 12 to 32, depending on sample size Target doses = Dose with mean response 0.5 and 0.9 (each run separately) Active drug patients in first cohort of each simulation assigned Dose replications of trials for each scenario 11 11

12 Performance Criteria 1. Power to reject null hypothesis of no difference from placebo to demonstrate proof-of-concept (trend test using linear regression) 2. Probability of selecting dose that is at or near target dose. Dose with mean closest to target level of response, selected using isotonic regression fit to response data 3. Mean Square Error and Bias in estimation of mean response at the target dose (maximum likelihood estimates from isotonic regression fit) 4. Number of Patients assigned to dose with est. mean nearest target and next nearest (i.e., at or adjacent to target dose) 12 12

13 Power (%), 0.5 target, Doses 2,4,6,8 design Sample size: Placebo size: Design: Bayes T-Stat Bayes T-Stat Bayes T-Stat Linear SigmoidEmax Emax Explicit Umbrella Emax Low Null Case * * * difference>2 * alpha-level inflated somewhat; should be 2.5% 13

14 % at or near 0.5 target, Doses 2,4,6,8 design Sample size: Placebo size: Design: Bayes T-Stat Bayes T-Stat Bayes T-Stat Linear SigmoidEmax Emax Explicit difference>2 Umbrella Emax Low Null Case

15 #Patients allocated to doses for 0.5 target, Doses 2,4,6,8 design (Total N=135) Design dose_0 dose_2 dose_4 dose_6 dose_8 Bayesian Linear T-Stat Bayesian Sigmoid T-Stat Bayesian Emax T-Stat Bayesian Explicit T-Stat Bayesian Umbrella T-Stat Bayesian EmaxLow T-Stat Bayesian Null Case T-Stat more at target dose, Target Dose Bolded 15

16 Power (%), 0.9 target, Doses 1 to 8 design Sample size Placebo size Bayes T-Stat Bayes T-Stat Bayes T-Stat Linear SigmoidEmax Emax Explicit Umbrella Emax Low Null Case 4.1* * * 3.4* difference>2 * alpha-level inflated somewhat; should be 2.5% 16

17 % at or near 0.9 target, Doses 1 to 8 design Sample size Placebo size Design Bayes T-Stat Bayes T-Stat Bayes T-Stat Linear SigmoidEmax Emax Explicit Umbrella Emax Low Null Case difference>2 17

18 #Patients allocated to doses for 0.9 target, Doses 1 to 8 design (Total N=135) Design D0 D1 D2 D3 D4 D5 D6 D7 D8 Bayesian Linear T-Stat Bayesian Sigmoid T-Stat Bayesian Emax T-Stat Bayesian Explicit T-Stat Bayesian Umbrella T-Stat Bayesian EmaxLow T-Stat Bayesian Null Case T-Stat more at target dose, Target Dose Bolded

19 Main Conclusions (T=T-Stat; B=Bayesian) Target: Doses: 0.5 2,4,6, to ,4,6, to 8 Power % at or near target B & T generally similar; T-Stat somewhat inflated type I error for 4 doses B & T generally similar T-Stat mostly better Each better in some cases; most gen. similar Bias T-Stat better at Dose 8 T-Stat better at Dose 8 & target dose T-Stat better at Dose 8 & target dose T-Stat better at Doses 7 & 8, & at target dose MSE T-Stat better at Dose 8 T-Stat better at Dose 8 & target dose No clear winner #@target dose T-Stat generally assigns more than Bayesian B & T generally similar 19 19

20 Concluding Remarks Our simulations show that the two adaptive methods demonstrate useful performance criteria Bayesian 4-parameter logistic Design Better if model is closer to S-shaped or linear Better to model response across dose-range an important advantage could be its ability to use informative priors T-Stat Up&Down Design Implementation for an actual trial is easier Better properties at extremes of dose-range Better properties at doses with targeted level of response Choice of Design driven by particular study objectives and span of potential true underlying dose-response curves assumed

21 References 1. Patel N, Bolognese J, Perevozskaya I. (2008 Joint Statistics Meetings, Denver, CO, USA) Comparing a Bayesian approach with a frequentist t-statistic method in adaptive dose finding trials Ivanova A, Bolognese J, and Perevozskaya I. (2008) Adaptive design based on t-statistic for dose-response trials. Statistics in Medicine, 27:

22 BACK-UP SLIDES FOLLOW THIS ONE 8/17/09 Confidential Cytel

23 Single dose-adaptive design can replace typical PoC and PIIa dose-ranging trials Tradi&onal Phase II Program 2N* 5N 4N PoC (Ib/IIa) (High Dose vs. Placebo) Dose Finding Defini;ve Dose Response (if needed) Phase III Phase II with Dose Adap&ve PoC Trial 3 4N^ 4N PoC (Some Doses vs. Placebo) Defini;ve Dose Response (if needed) Phase III *N = # subjects/group for desired precision in PoC trial ^ 2N, if fu&lity realized Replace 2 trials with 1 4N fewer subjects; less &me

24 Motivation Merck and Cytel collaboration over past three years to develop CytelSim, a soft ware tool for design of adaptive dose-finding clinical trials Two methods for PhII dose finding trials Bayesian method 4 parameter logistic model for mean dose response By Scott Berry for Cytel Frequentist method T-Statistic to adapt dose allocation By Ivanova, et al. (2008)

25 Four parameter logistic curve Where µ is the mean response at dose d β, δ, θ, τ are the four parameters of the logistic curve β=minimum, β+δ=maximum θ=inflection point (ED50) τ=slope parameter 25 25

26 Seven Dose Response Curves (Y-axis response is difference from placebo) DR_1 Linear DR_2 Umbrella DR_3 Sigmoid Emax DR_4 Emax DR_5 Emax low DR_6 Explicit DR_7 Null Case Target Response dose closest to target 4- doses 8- doses 2,4,6,8 1 to 8 26 DR_ ID <2 < <2 <1 6 >8 >8 7 26

27 Seven Dose Response Curves (Y-axis response is difference from placebo) Target Response DR_1 Linear DR_2 Umbrella DR_3 Sigmoid Emax DR_4 Emax DR_5 Emax low DR_6 Explicit DR_7 Null Case dose closest to target 4- doses 8- doses 2,4,6,8 1 to 8 DR_ ID ,8 4, >8 > >8 >

28 Simulation Results Two cases summarized on following slides (sometimes for representative sample size) Design with Doses 2,4,6,8 targeted to response = 0.5 Design with Doses 1 to 8 targeted to response = 0.9 By Sample size or for N=135 Cohort sizes & pbo:active ratios similar so averaged Other cases summarized in Main Conclusions above Design with Doses 1 to 8 targeted to response = 0.5 Design with Doses 2,4,6,8 targeted to response =

29 Bias for 0.5 target case, Doses 2,4,6,8 design Placebo size 45 Sample size 135 Design dose_0 dose_2 dose_4 dose_6 dose_8 Bayesian Linear T-Stat ratio(b/ttest) Bayesian Sigmoid T-Stat Emax ratio(b/ttest) Bayesian Emax T-Stat ratio(b/ttest) Bayesian Explicit T-Stat ratio(b/ttest) bias>0.1, Target Dose Bolded 29

30 Bias for 0.5 target case, Doses 2,4,6,8 design Placebo size 45 Sample size 135 Design dose_0 dose_2 dose_4 dose_6 dose_8 Bayesian Um T-Stat brella ratio(b/ttest) Bayesian Emax T-Stat Low ratio(b/ttest) Bayesian Null T-Stat Case ratio(b/ttest) bias>0.1, Target Dose Bolded 30

31 MSE for 0.5 target case, Doses 2,4,6,8 design Placebo size 45 Total Sample size 135 Design dose_0 dose_2 dose_4 dose_6 dose_8 Bayesian Linear T-Stat ratio(b/ttest) Bayesian Sigmoid T-Stat Emax ratio(b/ttest) Bayesian Emax T-Stat ratio(b/ttest) Bayesian Explicit T-Stat ratio(b/ttest) MSE>0.1, Target Dose Bolded

32 MSE for 0.5 target case, Doses 2,4,6,8 design Placebo size 45 Total Sample size 135 Design dose_0 dose_2 dose_4 dose_6 dose_8 Bayesian Um T-Stat brella ratio(b/ttest) Bayesian Emax T-Stat Low ratio(b/ttest) Bayesian Null T-Stat Case ratio(b/ttest) MSE>0.1, Target Dose Bolded 32

33 Bias for 0.9 target case, Doses 1 to 8 design Placebo N=45 Total N=135 Design D0 D1 D2 D3 D4 D5 D6 D7 D8 Bayesian Linear T-Stat ratio(b/t) Bayesian Sigmoid T-Stat ratio(b/t) Bayesian Emax T-Stat ratio(b/t) Bayesian Explicit T-Stat ratio(b/t) bias>0.1, Target Dose Bolded 33

34 Bias for 0.9 target case, Doses 1 to 8 design Placebo N=45 Total N=135 Design D0 D1 D2 D3 D4 D5 D6 D7 D8 Bayesian Umbrela T-Stat ratio(b/t) Bayesian EmxLow T-Stat ratio(b/t) Bayesian Null T-Stat ratio(b/t) bias>0.1, Target Dose Bolded 34

35 MSE for 0.9 target case, Doses 1 to 8 design Placebo size 45 Total Sample size 135 Design D0 D1 D2 D3 D4 D5 D6 D7 D8 Bayesian Linear T-Stat ratio(b/t) Bayesian Sigmoid T-Stat ratio(b/t) Bayesian Emax T-Stat ratio(b/t) Bayesian Explicit T-Stat ratio(b/t) bias>0.1, Target Dose Bolded 35

36 MSE for 0.9 target case, Doses 1 to 8 design Placebo size 45 Total Sample size 135 Design D0 D1 D2 D3 D4 D5 D6 D7 D8 Bayesian Umbrela T-Stat ratio(b/t) Bayesian EmxLow T-Stat ratio(b/t) Bayesian Null T-Stat ratio(b/t) MSE>0.1, Target Dose Bolded 36