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Covariate-Adjusted Response-Adaptive RandomizationDesigns for Phase III Survival Trials
Alex Sverdlov1 Yevgen Ryeznik2
1Bristol-Myers Squibbemail: alex.sverdlov@bms.com
2Kharkov National University of Economics, Ukraineemail: yevgen.ryeznik@gmail.com
November 8, 2010
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 1 / 26
Outline
1 Introduction
2 CARA Randomization for Survival TrialsProposed designsOperating characteristicsComparison with the balanced randomization design
3 Discussion
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 2 / 26
Introduction
Aspects of a Clinical Trial Design
Logistical: • Study cost • Study duration
Ethical: • Overall Safety • Min. treatment
failures • Min. “adverse”
experience
Statistical: • Randomization • Blinding • Balance • Precision in
estimation • Power
Aspects of a Clinical Trial
Design
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 3 / 26
Introduction
Aspects of a Clinical Trial Design
In our presentation it is assumed that multiple objectives are pursued,including:
Randomized experiment (to mitigate experimental biases)
Maximizing inferential aspects (e.g., power of the hypothesis test)
Maximizing “ethical” aspects
minimizing expected total hazard in the study
minimizing the number of inferior treatment assignments
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 4 / 26
Introduction
Covariates in the Design of a Clinical Trial
Why adjust randomization for covariates?
In most phase III comparative trials, study subjects are heterogeneous
Important baseline covariates may have strong impact on responses to a model
Classes of randomization procedures that incorporate covariate data:
Stratified randomization
Covariate-adaptive randomization
Covariate-adjusted response-adaptive randomization
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 5 / 26
Introduction
Randomization Procedures that Account for Covariates
1970 1980 1990 2000 2010
Stratified randomization
Covariate‐adaptive (CA) randomization
Optimal designs for linear models
Covariate‐adjusted response‐adaptive (CARA) randomization
1978 Wei’s UD 1984 Smith’s BCD
1974‐75 “Minimization” (Taves, Pocock & Simon)
1993 Bayesian biased coin (Ball et al) 1982 Atkinson’s BCD
1974 Stratification (M. Zelen)
1971 Efron’s BCD
2001 Treatment Effect Mappings (Rosenberger et al; Bandyopadhyay & Biswas)
2005‐06 Bayesian CARA randomization (Thall & Wathen; Cheung et al.)
2007 General CARA framework (L‐X Zhang et al)
2010 ‐ Ongoing active research
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 6 / 26
Introduction
CARA Randomization
Two treatments: A and B
n patients enter the trial sequentially and must be randomized to either A or B
Response Yk ∼ fk(yk |θk , z), where k = A,B
Let Tm,Ym,Zm denote, respectively, the history of m treatment assignments,responses, and covariates
Patient (m + 1) enters the trial with covariate vector zm+1 and is randomized to Awith probability
φm+1 = Pr(Tm+1 = A|Tm,Ym,Zm, zm+1), m ≥ 2m0
The main purpose of CARA randomization is to balance the competing objectives of
allocating greater number of study patients to the superior treatment, achieving high
statistical efficiency in estimating treatment effects, and maintaining randomization
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 7 / 26
Introduction
CARA Randomization for Survival Trials
R
TRT A = red TRT B = blue
Patient id
1
2
3
5
6
7
...
n
A or B?
4 months and Died
1. Look at the history preceding subject 7:
Subject T Y Z1 Z2 … Zp
1 A 42 B 103 A 2.54 A 35 B 4*6 B N/A
2. Estimate treatment effects (A and B) based on dataaccrued thus far
3. Assign the new patient to treatment A with probabilityconditional on the accrued data, and the covariate vectorof the patient:
φ7 = Pr(T7=A|T6, Y6 Z1,…,Z6, Z7)
QUESTION: How to select φ7?
0 DRecruitment Phase
Trial Duration
4
10 months and Died
2.5 months and Died
4 months and lost to follow-up (Censored)
3 months and Died
10 months and Died, but these data N/A at subject’s 7 randomization
covariates
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 8 / 26
Introduction
CARA Randomization for Survival Trials
Existing work:
1 Cheung YK et al. Continuous Bayesian adaptive randomization based on eventtimes with covariates. Statistics in Medicine 2006; 25:55-70.
2 Bandyopadhyay U et al. A covariate-adjusted adaptive design for two-stage clinical
trials with survival data. Statistica Neerlandica 2010; 64(2):202-226.
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 9 / 26
CARA Randomization for Survival Trials
Objectives
1 Propose new CARA randomization procedures for survival intervention trials withexponential regression model with treatment-covariate interactions
2 Simulate the operating characteristics of the proposed procedures and compare
them with the traditional balanced randomization design in terms of
Power and Type I errorVariability of allocation proportionsNumber of inferior/superior treatment assignments
Number of deaths and total hazard
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 10 / 26
CARA Randomization for Survival Trials
Model Description
Consider a survival intervention trial comparing two treatments, A and B.
Trial has a fixed duration D and a fixed recruitment period R < D
Patient arrival times are uniform over (0,R)
Tk =survival time, exponential with mean λk = exp(θ′kz), k = A,B
C =censoring time, uniform over (0,D)
Observed response tk = min(Tk ,C ,D − R) and δk = 1{tk=Tk}
It is assumed that sufficient amount (60% or more) of response data is accrued
during the recruitment phase to apply adaptation for responses
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 11 / 26
CARA Randomization for Survival Trials Proposed designs
Proposed CARA Design I
Step 1. Derive an optimal allocation for a model without covariates (e.g. Zhangand Rosenberger, 2007):
nA
nB=
√λ3A/εA√λ3B/εB
.
Step 2. Use covariate-adjusted version of the optimal allocation as the target:
πA(θA,θB , z) =
√λ3A(z)/εA(z)√
λ3A(z)/εA(z) +
√λ3B(z)/εB(z)
,
where λk(z) = exp(θ′kz) and εk(z) = Pr(Tk ≤ C |θk , z).
Step 3. Based on data from m patients, obtain (θ̂m,A, θ̂m,B). Then allocatepatient (m + 1) with covariate zm+1 to treatment A with probability
φm+1 = πA(θ̂m,A, θ̂m,B , zm+1).
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 12 / 26
CARA Randomization for Survival Trials Proposed designs
Proposed CARA Design II
Assign patient (m + 1) with covariate zm+1 to treatment A with probability
φm+1 =Fk ·
{d(A, θ̂m,A, zm+1)
}1/γ
∑Bk=A Fk ·
{d(k, θ̂m,k , zm+1)
}1/γ,
where FA = {1 + exp((θ̂m,B − θ̂m,A)′zm+1)}−1 is the hazard ratio (B vs. A),FB = 1− FA, and
d(k,θk , z) = z′M−1m,kzεk(z)
is the directional derivative of the D-optimal criterion log |M−1|.
γ = 0 → most efficient (D-optimal) design
γ =∞ → most “ethical” (Treatment Effect Mapping) design
γ = 0.25 → “tradeoff” design
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 13 / 26
CARA Randomization for Survival Trials Operating characteristics
Simulation Study
5 competing randomization procedures
Pocock and Simon’s procedure (PS)Covariate-adjusted Zhang-Rosenberger optimal target (ZR)“Most ethical” design with γ =∞ (TEM)“Most efficient” design with γ = 0 (eff)
“Tradeoff” design with γ = 0.25 (tradeoff)
Covariate structure: 3 independent covariates
Gender ∼ Bernoulli(p = 0.5)Age ∼ Uniform[18, 75]
Cholesterol level ∼ Normal(µ = 200, σ = 30)
Trial as in Zhang and Rosenberger (2007): R = 1, D = 1.5936 (a patient recruited
at time 0 with mean survival time= 1 has 50% chance of either die or being
censored)
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 14 / 26
CARA Randomization for Survival Trials Operating characteristics
Simulation Assumptions
Table: 3 experimental scenarios
Scenario Population Population Hazard Ratio Hazard Ratio Hazard RatioMean A Mean B B vs. A B vs. A B vs. A
(Males) (Females)Null 1.00 1.00 1.00 1.00 1.00
Alternative I 1.00 1.55 0.65 0.35 0.94Alternative II 1.00 0.73 1.37 0.75 2.00
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 15 / 26
CARA Randomization for Survival Trials Operating characteristics
Simulation Assumptions
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Null caseAlternative I − MaleAlternative I − FemaleAlternative II − MaleAlternative II − Female
Mean Hazard Ratio B vs. A
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 16 / 26
CARA Randomization for Survival Trials Operating characteristics
Simulation Details
Age and Cholesterol Level are discretized into 2 and 3 levels, respectively, forPocock-Simon implementation
First m = 80 patients are randomized to treatments using Pocock-Simon’sprocedure to accrue data for estimating model parameters
Next, patients are assigned to treatment groups using CARA randomization
Sample size is chosen empirically such that Pocock-Simon’s procedure results in∼ 90% power under a given alternative
Priority queue data structures were utilized to account for staggered entries anddelayed responses, and a continuous monitoring scheme for updating history ofresponses was implemented
5,000 replications for each experimental scenario in R
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 17 / 26
CARA Randomization for Survival Trials Comparison with the balanced randomization design
Simulation Results
Figure: Allocation proportion NA(n)/n
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Alternative II
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 18 / 26
CARA Randomization for Survival Trials Comparison with the balanced randomization design
Simulation Results
Table: Number of patients on Treatment A (SD)
HR Procedure
Scenario n (B vs A) PS ZR TEM eff tradeoffNull 200 1.00 100 (1) 100 (6) 100 (6) 100 (4) 100 (5)Alternative I† 200 0.65 100 (1) 76 (5) 84 (5) 100 (4) 86 (5)Alternative II†† 325 1.37 163 (1) 177 (8) 171 (8) 162 (5) 170 (6)† B is “superior”†† A is “superior”
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 19 / 26
CARA Randomization for Survival Trials Comparison with the balanced randomization design
Simulation Results
Figure: Average numbers (+SD) of males and females on Treatment A0
1020
3040
5060
Num
ber
of M
ales
on
A
PS ZR TEM eff tradeoff
Null case Average Number of Males on A (+SD)
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Alternative II Average Number of Males on A (+SD)
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Alternative II Average Number of Females on A (+SD)
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 20 / 26
CARA Randomization for Survival Trials Comparison with the balanced randomization design
Simulation Results
Table: Average number of males and females on Treatment A (SD)
HR Procedure
Scenario Gender (B vs A) PS ZR TEM eff tradeoffNull Male 1.00 50 (4) 50 (5) 50 (5) 50 (5) 50 (5)
Female 1.00 50 (4) 50 (5) 50 (5) 50 (5) 50 (5)
Alternative I Male 0.35† 50 (4) 28 (4) 35 (4) 52 (5) 38 (4)Female 0.94† 50 (4) 48 (5) 49 (5) 48 (5) 48 (5)
Alternative II Male 0.75† 81 (5) 65 (6) 71(7) 84 (6) 73 (6)Female 2.00†† 81 (5) 112 (8) 100 (8) 78 (6) 98 (7)
† B is “superior”†† A is “superior”
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 21 / 26
CARA Randomization for Survival Trials Comparison with the balanced randomization design
Simulation Results
Table: Type I Error, Power†, and Total Number of Deaths (SD)
Procedure
Scenario Characteristic PS ZR TEM eff tradeoffNull Type I Error 0.054 0.054 0.053 0.054 0.052
D(n) (SD) 104 (7) 104 (7) 104 (7) 104 (7) 104 (7)Alternative I Power 0.928 0.886 0.919 0.926 0.924
D(n) (SD) 87 (7) 79 (7) 82 (7) 88 (7) 83 (7)Alternative II Power 0.882 0.833 0.860 0.882 0.861
D(n) (SD) 177 (9) 167 (9) 171 (9) 177 (9) 172 (9)
† Using test statistic T = (θ̂m,A − θ̂m,B )′(M̂−1m,A + M̂
−1m,B
)−1(θ̂m,A − θ̂m,B ), which is asymptotically χ2 with df=4− 1
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 22 / 26
Discussion
Conclusions
Several new CARA randomization designs were proposed
1 The proposed CARA procedures are more ethical than the balanced design:
For a trial of ≥ 200 patients, on average, 14− 16 more patients are allocatedto the “superior” treatment with negligible (< 1%) loss in powerWith treatment-covariate interactions, on average, greater number ofpatients receive the treatment which is “best” given their covariate profiles
Average number of deaths can be reduced with CARA randomization designs
2 Some of the proposed CARA procedures have good inferential properties:
Valid statistical inference (nominal Type I error is maintained)ZR procedure is 5% less powerful than the balanced design“Efficient” design is almost identical to the balanced design
TEM and “tradeoff” designs are very close to the balanced design in terms of
power
3 The proposed designs are more variable than the balanced randomization design
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 23 / 26
Discussion
Cautions/Limitations
CARA randomization is not relevant for long-term survival trials with shortrecruitment period
CARA randomization designs rely on the correctly specified parametric model
Number of covariates must be limited, since m.l.e.’s may converge very slowly due
to delayed responses
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 24 / 26
Discussion
Current/Future Work
Other parametric models (Weibull, log-logistic) and semi-parametric (Cox’sproportional hazards) model
Robustness to model misspecification
Different censoring schemes
Time-dependent covariates
Defining “optimality” given that patient covariate values are unknown in advance
Bayesian CARA randomization
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 25 / 26
Discussion
Some Useful References
Atkinson AC, Biwas A (2005) Bayesian adaptive biased-coin designs for clinical trials with normal responses, Biometrics 61,118-125.
Bandyopadhyay U, Biswas A, Bhattacharya R (2010) A covariate-adjusted adaptive design for two-stage clinical trials withsurvival data, Statistica Neerlandica 64 (2), 202-226.
Cheung YK, Lurdes YT, Wathen JK, Thall PF (2006) Continuous Bayesian adaptive randomization based on event times withcovariates, Statistics in Medicine 25, 55-70.
Hu, F and Rosenberger, WF (2006) The Theory of Response-Adaptive Randomization in Clinical Trials, New York, Wiley.
Rosenberger WF, Seshaiyer P (1997) Adaptive survival trials, Journal of Biopharmaceutical Statistics 7(4), 617-624.
Rosenberger WF, Sverdlov O (2008) Handling covariates in the design of clinical trials, Statistical Science 23(4), 404-419.
Thall PF, Wathen JK (2005) Covariate-adjusted adaptive randomization in a sarcoma trial with multi-stage treatments,Statistics in Medicine 24, 1947-1964.
Zhang L, Rosenberger WF (2007) Response-adaptive randomization for survival trials: the parametric approach, AppliedStatistics 56(2), 153-165.
Zhang L-X, Hu F, Cheung SH, Chan, WS (2007) Asymptotic properties of covariate-adjusted response-adaptive designs,The Annals of Statistics 35(3), 1166-1182.
Zhang L-X, Hu F (2009) A new family of covariate-adjusted response-adaptive designs and their properties, AppliedMathematics - A Journal of Chinese Universities 24(1), 1-13.
Alex Sverdlov and Yevgen Ryeznik ( Bristol-Myers Squibb email: alex.sverdlov@bms.com, Kharkov National University of Economics, Ukraine email: yevgen.ryeznik@gmail.com )CARA Randomization Designs November 8, 2010 26 / 26
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