Adaptive Designs for U-shaped ( Umbrella) Dose-Response

1
Adaptive Designs for U-shaped ( Umbrella)
Dose-Response

• Yevgen Tymofyeyev
• Merck Co. Inc
• September 12, 2008

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Outline

• Utility function
• Gradient design
• Normal Dynamic Linear Models
• Two stage design
• Comparison and conclusions

3
Application Scope

• Proof-of-Concept and (or) Dose-Ranging Studies
  when
• Dose-response can not be assumed monotonic but
  rather uni-modal
• Efficacy and safety are considered combined by
  means of some utility function

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Examples Utility Function

• Utility efficacy Coefficient AE_rate 2nd
  order_term
• Utility is evaluated at each dose level
• Another way to define is by means of a table
• Example below

Drug Efficacy vs. PLB AE vs. PBO AE vs. PBO AE vs. PBO AE vs. PBO
Drug Efficacy vs. PLB 0 10 20 30
-2 0.3 -0.2 -0.8 -1.4
-1 1.6 0.9 0.1 -0.7
0 2.9 1.9 1.0 0.1
1 4.2 3.0 1.9 0.8
2 5.4 4.1 2.8 1.5
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Example of Utility function (cont.)
Try to modify utility function in order to reduce
its variance while preserving the bulk
structure.
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Example (cont.)
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Adaptive Design Applicable for n DR maximization

• Frequent adaptation
• Adaptations are made after each cohort of
  subjects responses, data driven
• Gradient design
• Assume n shape dose-response
• NDLM
• No assumption for DR shape, but rather on
  smoothness of DR (dose levels are in order)
• Two stage design
• No assumptions on treatment ordering
• Inference is done at the adaptation point

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Gradient Design for Umbrella Shaped Dose-Response

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Case Study

• Therapeutic area Neuroscience
• Outcome composite score derived from several
  tests
• Objective to maximize number of subjects
  assigned to the dose with the highest mean
  response, the peak dose
• improve power for placebo versus the peak dose
  comparison
• Assumption monotonic or (uni-modal)
  dose-response
• Proposed Method Adaptive design that uses
  Kiefer-Wolfowitz (1951) procedure for finding
  maximum in the presence of random variability in
  the function evaluation as proposed by Ivanova
  et. al.(2008)

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Illustration of the Design
Current cohort
Next cohort
Doses
1
2
3
4
1
2
3
4
Active pair of levels
At given point of the study, subjects are
randomized to the levels of the current dose pair
and placebo only. The next pair is obtained by
shifting the current pair according to the
estimated slope.
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Update rule

• Let dose j and j1 constitute the current
  dose pair.
• Use isotonic (unimodal) regression or quadratic
  regression fitted locally to estimate responses
  at all dose levels using all available data
• Compute T
• If T gt 0.3 then next dose pair (j1,j2), i.e.
  "move up
• If T lt -0.3 then next dose pair ( j-1, j), i.e.
  "move down
• Otherwise, next dose pair ( j, j1), i.e. stay
• If not possible to move dose pair, ( j1 or
  jK-1), change pairs randomization probabilities
  from 11 to 21 (the extreme dose of the pair get
  twice more subjects)
• Modification of this rule (including
  different cutoffs for T) are possible but logic
  is similar

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Inference after Adaptive Allocation

• Compare PLB to each dose using Dunnetts
  adjustment for multiplicity (?)
• Compare PLB to the dose with the maximum number
  of subjects assigned (expect inflated type I
  error)
• Trend Tests are not straightforward since an
  umbrella shape response is possible.

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Simulation Results
True Dose Response Scenarios True Dose Response Scenarios True Dose Response Scenarios True Dose Response Scenarios Power Power Dose Allocation in percents of total sample size (average over simulations) Dose Allocation in percents of total sample size (average over simulations) Dose Allocation in percents of total sample size (average over simulations) Dose Allocation in percents of total sample size (average over simulations)
True Dose Response Scenarios True Dose Response Scenarios True Dose Response Scenarios True Dose Response Scenarios Plb vs all doses (Dunnett) Plb vs max alloc dose Dose Allocation in percents of total sample size (average over simulations) Dose Allocation in percents of total sample size (average over simulations) Dose Allocation in percents of total sample size (average over simulations) Dose Allocation in percents of total sample size (average over simulations)
Plb D1 D2 D3 Plb vs all doses (Dunnett) Plb vs max alloc dose Plb D1 D2 D3
S0 .5 .5 .5 .5 2 6 40 19 24 17
S1 0 .1 .25 .5 72 82 40 4 21 35
S2 0 .1 .5 .25 65 76 40 12 27 21
S3 0 .5 .25 .1 64 75 40 30 21 9
S4 0 0 0 .5 77 82 40 2 19 38
S5 0 0 .5 0 65 78 40 14 29 18
S6 0 .5 0 0 66 77 40 32 20 8
S7 0 .5 .5 .5 87 92 40 14 23 22
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Discussion

• The standard parallel 4 arms design would require
  464 subjects to provide 80 to detect difference
  from placebo 0.5 with no adjustment for
  multiplicity.
• The adaptive design uses 364 subjects and
  results in 75-92 power depending on scenario.
• (25 sample size reduction)
• Limitations
• Somewhat excessive variability for S2 and S3 for
  some DR scenarios
• Logistical complexity due to the large number of
  adaptations

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Normal Dynamic Linear Models
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Introduction

• How to pool information across dose levels in
  dose-response analysis ?
• Solution Normal Dynamic Linear Model (NDLM)
• Bayesian forecaster
• parametric model with dynamic unobserved
  parameters
• forecast derived as probability distributions
• provides facility for incorporation expert
  information
• Refer to West and Harrison (1999)
• NDLM idea filter or smooth data to estimate
  unobserved true state parameters

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Graphical Structure of DLM
?0
etc
?t
? t-1
?2
?1
? t1

Task is to estimate the state vector ?(?1,, ?K)
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Dynamic Linear Models
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Idea At each dose a straight line is fitted. The
slope of the line changes by adding an evolution
noise, Berry et. al. (2002)
4
3
2
5
6
1
Dose
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Local Linear Trend Model for Dose Response (ref.
to Berry (2002) et al.)
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Implementation Details

• Miller et. al.(2006)
• MCMC method
• Smith et. al. (2006) provide WinBugs code
• West and Harrison (1999)
• An algebraic close form solution that computes
  posterior distribution of the state parameters

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Model Specification (details)

• Measurements errors e can be easily estimated
  as the residual from ANOVA type model with dose
  factor
• Specification of the variance for the state
  equation (evolution of slope) is difficult
• MLE estimate form state equation (works well for
  large sample sizes, not for small)
• Other options (i) use prior for Wt and update
  it later on (ii) discount factor for posterior
  variance of ? as suggested in West and Harrison
  (1999)

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NDLM fit, 200 subjects
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Example of Biased NDLM Estimates
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NDLM versus Simple Mean
Dose allocation Dose allocation Dose allocation Dose allocation Dose allocation Dose allocation Response MSE Ratio (mean / NDLM) at doses Response MSE Ratio (mean / NDLM) at doses Response MSE Ratio (mean / NDLM) at doses Response MSE Ratio (mean / NDLM) at doses Response MSE Ratio (mean / NDLM) at doses Response MSE Ratio (mean / NDLM) at doses
plb 1 2 3 4 5 plb 1 2 3 4 5
10 10 10 10 10 10 1.3 2.2 2.3 2.2 2.16 1.27
5 5 10 10 15 15 1.4 2.8 2 2.2 1.88 1.19
15 15 10 10 5 5 1.2 1.9 2.3 2.1 2.7 1.33
5 10 25 10 5 5 1.5 2.1 1.5 2.3 3.01 1.36
30 6 6 6 6 6 1.1 2.9 2.4 2.2 2.26 1.32
5 5 20 20 5 5 1.4 3 1.6 1.6 2.8 1.34
30 0 0 5 20 5 1.1 NaN NaN 1.1 0.12 0.2
30 1 1 5 19 4 1 4.4 4.1 1.9 1.24 1.36
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Bayesian Decision Analysis for Dose Allocation
and Optimal Stopping (Ref. to Miller et. al.(2006)

• Sampling from the posterior of the state vector
  allows to program dose allocation as a decision
  problem using a utility function approach (not to
  confuse with efficacy safety utility)
• Use the utility function that reflects the key
  parameter of the dose/response curve, e.g.
• the posterior variance of the mean response at
  the ED95
• variance at the most effective safe dose
• Utility function is evaluated by MC method by
  sampling from ? posterior
• Select next dose that maximizes the utility
  function
• Bayesian decision approach for early stopping

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Bayesian Decision Analysis (Cont.) Utility
function for dose allocation (details)
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Example of NDLM Dose allocation utility
response variance at dose of interest (most
effective safe dose)
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Discussion as for NDLM use

• Potentially very broad scope applications
• Flexible dose response learner including
  arbitrary DR relationship
• possible incorporation of longitudinal modeling
• Framework for Bayesian decision analysis
• Adaptive dose allocation
• Early stopping for futility or efficacy
• Probabilistic statements regarding features
  derived from the modeled dose-response

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Two Stage Design
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Design Description

• N fixed total sample size K number of
  treatment arms including placebo
• 1st stage (pilot)
• Equal allocation of rN subjects to all arms
• Analysis to select the best (compared to placebo)
  arm
• 2nd stage (confirmation)
• Equal allocation of (1-r)N subjects to the
  selected arm and placebo
• Final inference by combing responses from both
  stages (one-sided testing)
• Posch Bauer method is used to control type 1
  error in the strong sense
• BAUER KIESER 1999, HOMMEL, 2001, POSCH ET AL.
  2005

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Adaptive Closed Test Scheme for 3 treatments Arm
and Placebo
Hypoth. p-value 1st Stage p-value 2nd Stage C(p,q) lt a
(1) p1 q1 x
(2) p2    
(3) p3    
       
(1,2) p12 q12 x
(1,3) p13 q13 x
(1,2,3) p123 q123 x
Multiplicity
Combination
Intersection
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Combination Test

• Let p and q be respective p-values for testing
  any null hypothesis H from stage 1 and 2
  respectively
• Note Type I error is controlled if, under H0, p
  and q are independent and Unif.(0,1)
• E.g. Two different sets of subjects at stages 1
  and 2
• Combination function (inverse normal)

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Closed Testing Procedure

• To ensure strong control of type I error
• Probability of selecting a treatment arm that is
  no better than placebo and concluding its
  superiority is less than given a under all
  possible response configuration of other arms
• H1, H2 null hypothesis to test
• Use local level a test for H1, H2 and
  H12H1nH2
• Closure principle
• The test at multiple level a Reject Hj , j
  1, 2
• If H12 and Hj are rejected at local level a.

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Test of the Intersection Hypothesis

• 1st state
• Bonferroni test
• p12 min 2 min p1, p2, 1, p13- similar
• p123 min 3 min p1, p2, p3, 1
• Simes test (1986)
• Let p1 lt p2 lt p3
• p12 min 2p1, p2 p13 min 2p1,p3
• p123 min 3 p1, 3/2 p2, p3
• 2nd stage
• B/c just one dose is selected
• q(1,2) q(1,3) q(1,2,3) q(1)

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Optimal Split of the Total Sample Size into Two
Stages
True response True response True response True response Prop. of total sample size used in 1st Stage, r Probability of rejecting H0 Prob. Of selecting Max Dose
D1 D2 D3 D4 Prop. of total sample size used in 1st Stage, r Probability of rejecting H0 Prob. Of selecting Max Dose
0.25 0.5 0.75 1 0.26 0.8165 0.5149
0.25 0.5 0.75 1 0.33 0.828 0.5483
0.25 0.5 0.75 1 0.4 0.83 0.5706
0.25 0.5 0.75 1 0.46 0.8194 0.5965
0.25 0.5 0.75 1 0.53 0.8134 0.6099
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Comparison of 2-stage and Gradient (KW) Designs
for a 4 Treatment Arms Study
Scenario True Response True Response True Response True Response Probability of Rejecting H0 Probability of Rejecting H0 Prob. of Selecting Max Dose Prob. of Selecting Max Dose
Scenario PLB D1 D2 D3 two-stage KW two-stage KW
1 0 0.00 0.00 0.00 0.021 0.024 NA NA
2 0 0.00 0.00 1.00 0.819 0.646 0.93 0.74
3 0 0.00 0.50 1.00 0.799 0.742 0.80 0.74
4 0 0.33 0.66 1.00 0.810 0.740 0.68 0.60
5 0 0.50 0.50 1.00 0.787 0.689 0.70 0.57
6 0 1.00 0.50 0.00 0.799 0.818 0.80 0.83
7 0 0.50 1.00 0.50 0.788 0.805 0.77 0.87
8 0 0.33 1.00 0.66 0.810 0.818 0.68 0.81
Marks the starting dose pair for KW design
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Comments on Posch Bauer Method

• Very flexible method
• several combination functions and methods for
  multiplicity adjustments are available
• Permits data dependent changes
• sample size re-estimation
• arm dropping
• several arms can be selected into stage 2
• furthermore, it is not necessary to pre-specify
  adaptation rule from stat. methodology point of
  view, but is necessary from regulatory
  prospective.

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Conclusions as for Two Stage Design

• Posch-Bauer method is
• Robust
• Strong control of alpha
• No assumptions on dose-response relation
• Powerful
• Simple implementation Just single interim
  analysis

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References

• Berry D, Muller P, Grieve A, Smith M, Park T,
  Blazek R, Mitchard N, Krams M (2002) Adaptive
  Bayesian designs for dose-ranging trials. In Case
  studies in Bayesian statistics V. Springer
  Berlin pp. 99-181
• Bauer P, Kieser M. (1999). Combining different
  phases in the development of medical treatments
  within a single trial. Statistics in Medicine,
  181833-1848
• Ivanova A, Lie K, Snyder E, Snavely D.(2008) An
  Adaptive Crossover Design for Identifying the
  Dose with the Best Efficacy/Tolerability Profile
  (in preparation)
• Hommel G. Adaptive modifications of hypotheses
  after an interim analysis. Biometrical Journal,
  43(5)581589, 2001
• Muller P, Berry D, Grieve A, Krams M. A Bayesian
  Decision-Theoretic Dose-Finding Trial (2006)
  Decision Analysis 3(4) 197-207
• Posch M, Koenig F, Brannath W, Dunger-Baldauf C,
  Bauer P (2005). Testing and estimation in
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  243697-3714.
• Smith M, Jones I, Morris M, Grieve A, Ten K
  (2006) Implementation of a Bayesian adaptive
  design in a proof of concept study.
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• West M and Harrison P (1997) Bayesian Forecasting
  and Dynamic Models (2nd edn). Springer New York.