Transcript Slide 1

Adaptive Designs for U-shaped
( Umbrella) Dose-Response
Yevgen Tymofyeyev
Merck & Co. Inc
September 12, 2008
Outline

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Utility function
Gradient design
Normal Dynamic Linear Models
Two stage design
Comparison and conclusions
Application Scope
Proof-of-Concept and (or) Dose-Ranging
Studies when
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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
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Utility = efficacy + Coefficient * AE_rate + 2nd order_term
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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
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
Example of Utility function (cont.)
Utility surface linear
approximation
Utility surface (quadratic)
ica
cy
AE
rate
Eff
rate
Eff
ica
cy
Utility
Utility
AE
Try to modify utility function in order to reduce its variance while
preserving the “bulk” structure.
Example (cont.)
1 2 3 4 5 6
Comparison of Utilities
Utility
Utility simple approximation
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
3
1
3
2
3
3
3
4
3
5
3
6
3
7
3
8
3
9
3
10
3
11
2
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
9
2
10
2
11
8
6
4
2
0
-2
8
6
4
2
0
-2
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
11
8
6
4
2
0
-2
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
dose
1 2 3 4 5 6
1 2 3 4 5 6
Adaptive Design Applicable for ∩ DR
maximization
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Frequent adaptation:
Adaptations are made after each cohort of subjects’
responses, data driven
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Gradient design
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NDLM
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Assume ∩ shape dose-response
No assumption for DR shape, but rather on “smoothness” of
DR (dose levels are in order)
Two stage design
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No assumptions on treatment ordering
Inference is done at the adaptation point
Gradient Design for
Umbrella Shaped DoseResponse
Case Study
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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
Doses
1
Next cohort
2
3
4
Active pair
of levels
1
2
3
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.
4
Update rule
T
(ˆ j 1  ˆ j )
ˆ 2 (1 / n j 1  1 / n j )
Let dose j and j+1 constitute the current dose pair.
1. Use isotonic (unimodal) regression or quadratic regression fitted
locally to estimate responses at all dose levels using all available
data
2. Compute T
i. If T > 0.3 then next dose pair (j+1,j+2), i.e. "move up“
ii. If T < -0.3 then next dose pair ( j-1, j), i.e. "move down“
iii. 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
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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.
Simulation Results
Power
True Dose Response
Scenarios
Dose Allocation in
percents of total
sample size (average
over simulations)
Plb vs all
doses
(Dunnett)
Plb vs
max
alloc
dose
Plb
D1
D2
D3
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
0.1
0.2
0.3
0.4
Dose-Response (S1): 0, 0.1, 0.25, 0.5; P=0.817
0.0
0.0
0.1
0.2
0.3
0.4
Dose-Response (S0): 0.5, 0.5, 0.5, 0.5; P=0.053
1
2
3
4
1
2
3
4
Dose-Response (S2): 0, 0.1, 0.5, 0.25; P=0.761
Dose-Response (S3): 0, 0.5, 0.25, 0; P=0.752
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
Doses
0.4
Doses
1
2
3
Doses
4
1
2
3
Doses
4
Discussion

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.
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The adaptive design uses 3*64 subjects and results in 7592 % power depending on scenario.
(25% sample size reduction)
Limitations:
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Somewhat excessive variability for S2 and S3 for some DR
scenarios
Logistical complexity due to the large number of
adaptations
Normal Dynamic Linear Models
Introduction
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How to pool information across dose levels in doseresponse analysis ?
Solution: Normal Dynamic Linear Model (NDLM)
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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
θ0
 y1,1 
 
 y1, 2 
 
 
 y1,n1 
 y2,1 
 
 y2, 2 
 
 
 y2,n2 
θ1
θ2
 yt 1,1 


 yt 1, 2 




 yt 1,nt 1 
…etc…
θ t-1
 yt 1,1 


 yt 1, 2 




 yt 1,nt 1 
θt
Task is to estimate the state vector θ=(θ1,…, θK)
 yt 1,1 


 yt 1, 2 




 yt 1,nt 1 
θ t+1 …
Dynamic Linear Models
Idea: At each dose a straight line is fitted.
The slope of the line changes by adding an
evolution noise, Berry et. al. (2002)
1
2
3
Dose
4
5
6
Local Linear Trend Model for Dose
Response (ref. to Berry (2002) et al.)
Implementation Details
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Miller et. al.(2006)
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MCMC method
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Smith et. al. (2006) provide WinBugs code
West and Harrison (1999)
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An algebraic close form solution that computes
posterior distribution of the state parameters
Model Specification (details)
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‘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)
NDLM fit, 200 subjects
Emax
linearLogDose
10
Response
Response
-15
-10
-10
-10
-5
0
0
0
Response
5
10
10
20
20
15
flat
3
5
8
10
12
0
3
5
8
10
12
0
3
5
Dose
Dose
Dose
exponential
quadratic
logistic4
8
10
12
8
10
12
0
10
Response
Response
-10
10
-20
0
-10
-10
Response
0
20
20
10
30
0
0
3
5
8
Dose
10
12
0
3
5
8
Dose
10
12
0
3
5
Dose
13.5
13.0
12.5
True model
ndld, dose allocation: 30,30,30,30,30,30
ndld, dose allocation: 15,15,30,30,45,45
ndld, dose allocation: 45,45,30,30,15,15
12.0
Response
14.0
14.5
Example of Biased NDLM Estimates
0
3
5
8
Dose (mg)
10
12
NDLM versus Simple Mean
Response MSE Ratio
(mean / NDLM) at doses
Dose allocation
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
Root MSE ratios (isototic / NDLM) for 9 Profiles
Dose-response profile
root MSE ratio
0.0
0.2
0.4
0.6
0.8
1.0
step1
step2
step3
linear
exponential
logistic4
MSE ratio and response profile
1.0
0.5
1.0
0.5
flat
Emax
linearLogDose.ext
1.0
0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Dose
Equal dose allocation
0.2
0.4
0.6
0.8
1.0
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.
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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
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the posterior variance of the mean response at the ED95
variance at the most effective safe dose
Bayesian Decision Analysis (Cont.): Utility function for
dose allocation (details)
30
20
10
Allocation (%)
40
50
Example of NDLM Dose allocation: utility = response
variance at dose of interest (most effective safe dose)
0
3
5
8
Dose
10
12
Discussion as for NDLM use
Potentially very broad scope applications
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Flexible dose response learner including arbitrary DR
relationship
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Framework for Bayesian decision analysis
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possible incorporation of longitudinal modeling
Adaptive dose allocation
Early stopping for futility or efficacy
Probabilistic statements regarding features derived from
the modeled dose-response
Two Stage Design
Design Description
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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 (onesided testing)
 Posch –Bauer method is used to control type 1 error in
the strong sense

BAUER & KIESER 1999, HOMMEL, 2001, POSCH ET AL. 2005
Adaptive Closed Test Scheme for 3
treatments Arm and Placebo
Hypoth.
p-value
1st Stage
p-value
2nd Stage
C(p,q) < α
(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
Intersection
Combination
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)


n
n
1
1
1
1
C ( p, q )  1  
 (1  p ) 
 (1  q ) 
n1  n2
 n1  n2

  normalcdf
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
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1st state:
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Bonferroni test
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Simes test (1986);
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p12 = min{ 2 min { p1, p2}, 1}, p13- similar
p123 = min{ 3 min {p1, p2, p3}, 1}
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)
Optimal Split of the Total Sample Size
into Two Stages
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
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
Scenario
Comparison of 2-stage and Gradient
(KW) Designs for a 4 Treatment Arms
Probability of
Prob. of Selecting
Study True Response
Rejecting H0
Max Dose
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
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

Posch-Bauer method is

Robust




Strong control of alpha
No assumptions on dose-response relation
Powerful
Simple implementation; Just single interim
analysis
References
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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.