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Adaptive Designs for U-shaped ( Umbrella) Dose-Response. Yevgen Tymofyeyev Merck & Co. Inc September 12, 2008. Outline. Utility function Gradient design Normal Dynamic Linear Models Two stage design Comparison and conclusions. Application Scope.

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adaptive designs for u shaped umbrella dose response

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

Yevgen Tymofyeyev

Merck & Co. Inc

September 12, 2008

  • Utility function
  • Gradient design
  • Normal Dynamic Linear Models
  • Two stage design
  • Comparison and conclusions
application scope
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
examples utility function
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
example of utility function cont
Example of Utility function (cont.)

Try to modify utility function in order to reduce its variance while preserving the “bulk” structure.

adaptive design applicable for dr maximization
Adaptive Design Applicable for ∩ DR maximization
  • Frequent adaptation:

Adaptations are made after each cohort of subjects’ responses, data driven

    • Gradient design
      • Assume ∩ 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
case study
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)
Illustration of the Design

Current cohort

Next cohort










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.

update rule
Update rule
  • Let dose j and j+1 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 > 0.3 then next dose pair (j+1,j+2), i.e. "move up“
    • If T < -0.3 then next dose pair ( j-1, j), i.e. "move down“
    • Otherwise, next dose pair ( j, j+1), i.e. “stay”
    • If not possible to “move” dose pair, ( j=1 or j=K-1), change pair’s randomization probabilities from 1:1 to 2:1 (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
inference after adaptive allocation
Inference after Adaptive Allocation
  • Compare PLB to each dose using Dunnett’s 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.
  • The standard parallel 4 arms design would require 4*64 subjects to provide 80% to detect difference from placebo 0.5 with no adjustment for multiplicity.
    • The adaptive design uses 3*64 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
  • 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
Graphical Structure of DLM





θ t-1


θ t+1

Task is to estimate the state vector θ=(θ1,…, θK)

Idea: At each dose a straight line is fitted.

The slope of the line changes by adding an evolution noise, Berry et. al. (2002)








implementation details
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
model specification details
Model Specification (details)
  • ‘Measurements’ errors ε 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 θ as suggested in West and Harrison (1999)
bayesian decision analysis for dose allocation and optimal stopping ref to miller et al 2006
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 θ posterior
    • Select next dose that maximizes the utility function
  • Bayesian decision approach for early stopping
Example of NDLM Dose allocation: utility = response variance at dose of interest (most effective safe dose)
discussion as for ndlm use
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
design description
Design Description
  • N – fixed total sample size; K – number of treatment arms including placebo
  • 1st stage (pilot):
    • Equal allocation of r*N 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
combination test
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)
closed testing procedure
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 α under all possible response configuration of other arms
  • H1, H2 null hypothesis to test

Use local level α test for H1, H2 and H12=H1∩H2

  • Closure principle :

The test at multiple level α Reject Hj , j = 1, 2

If H12 and Hj are rejected at local level α.

test of the intersection hypothesis
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 < p2 < 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)
comparison of 2 stage and gradient kw designs for a 4 treatment arms study
Comparison of 2-stage and Gradient (KW) Designs for a 4 Treatment Arms Study

* Marks the starting dose pair for KW design

comments on posch bauer method
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.
conclusions as for two stage design
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
  • 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, 18:1833-1848
  • 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):581–589, 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 flexible group sequential designs with adaptive treatment selection. Stat. Medicine, 24:3697-3714.
  • Smith M, Jones I, Morris M, Grieve A, Ten K (2006) Implementation of a Bayesian adaptive design in a proof of concept study. Pharmaceutical Statistics 5: 39-50
  • West M and Harrison P (1997) Bayesian Forecasting and Dynamic Models (2nd edn). Springer: New York.