A NEW TOOL FOR COMPARING ADAPTIVE DESIGNS; A POSTERIORI EFFICIENCY - PowerPoint PPT Presentation

A NEW TOOL FOR COMPARING ADAPTIVE DESIGNS; A POSTERIORI EFFICIENCY Jos´ e A. Moler, Universidad P´ ublica de Navarra. Nancy Flournoy, University of Missouri.

Statistical model in a clinical trial � in a clinical trial patients arrive sequentially and patients are allocated in different treatments or doses. � For the n th patient we consider the following notation: Y n : observed response of the patient L : number of different doses or treatments. K : number of different covariates observed in each patient x n = ( δ n 1 , . . . , δ nL , F n 1 , . . . , F nK ) � �� � COVARIATES � 1 , i th treatment; δ ni = 0 , otherwise.

Statistical model in a clinical trial � Finally, we consider the model E [ Y n | x 1 , . . . , x n ] = η ( x 1 , . . . , x n , β ) ւ ց Y n = x n β + ε n π n := P ( Y n = 1 | x 1 , . . . , x n ) = F ( x 1 , . . . , x n , β ) � Design matrix up to the n th patient   x 1  .  . A n = .     x n � N nj = � n k =1 δ kj : number of patients allocated in treatment j up to the n th patient.

ALLOCATION IN CLINICAL TRIALS Accrued information up to the n th patient. F n = σ ( Y j , δ j , F j : j ≤ n ) � How to allocate the n th patient depending on the accrued information up to the ( n − 1) th patient: π nj := P ( δ nj = 1 |F n − 1 ) .

ALLOCATION IN CLINICAL TRIALS Depending on the information needed to allocate the present patient, we distinguish: • Non-adaptive design : the present allocation does NOT DEPEND on the accrued information. Example: complete randomization with two treatments π nj := 1 2 , ∀ n

ALLOCATION IN CLINICAL TRIALS Depending on the information needed to allocate the present patient, we distinguish: • Adaptive design (-non-response-driven) : the present allocation depends only on the past allocations:  1 / 2 N n − 1 , 1 / ( n − 1) = 1 / 2    π n 1 = 2 / 3 , N n − 1 , 1 / ( n − 1) < 1 / 2    1 / 3 N n − 1 , 1 / ( n − 1) > 1 / 2 Efron’s biased coin design [Rosenberger and Lachin (2002)]

ALLOCATION IN CLINICAL TRIALS Depending on the information needed to allocate the present patient, we distinguish: • Response-driven adaptive design : the present allocation depends on the past allocations and on the past responses. Different goals: .- A targeted allocation: [Hu and Zhang, Annals of Statistics (2004)] � Y k δ k 1 π n 1 = G ( N n 1 /n, ˆ ρ ) , ρ = ˆ N n 1 .-Ethical issues: skewing the allocation to the treatment with best performance. Play-The-Winner (PTW)[Wei and Durham, JASA (1979)] π n 1 = X n − 1 , 1 , X n − 1 , 1 : proportion of balls of type 1 in an urn. [Extension: Moler, Plo , San Miguel, Statistics and Probability letters (2006)]

Simulation: 100 clinical trials with 100 patients. Responses: N (0 , 8 , 1) , N (0 , 3 , 1)

ADAPTIVE REGRESSION √ Linear adaptive regression Y n = x n β + ε n = β 1 δ n 1 + · · · + β L δ nL + β L +1 F n 1 + · · · + β L + K F nK + ε n Properties of ORDINARY LEAST SQUARES-OLS • ˆ β n,OLS → β a.s. when λ min ( A ′ n A n ) /log ( λ max ( A ′ n A n )) → ∞ . → I , then R n (ˆ • A n = R ′ n R n and R n B − 1 β n,OLS − β ) → N ( 0 , σ 2 I ) n UNDER ANY ASSUMPTION ON THE PATIENT RESPONSE Lai, T.L. and Wei, C.Z. The Annals of Statistics (1982).

ADAPTIVE GENERALIZED LINEAR MODELS √ Generalized linear adaptive regression (logistic link) π n log ( ) = x n β + ε n 1 − π n √ n (ˆ • ˆ β n,MLE → β , [ P ] β n,MLE − β ) → N (0 , Σ) Provided that n 1 � E [ x ′ i x i π i (1 − π i ) |F i − 1 ] → Σ − 1 l´ ım n n →∞ i =1 Rosenberger, Hu. Statistics and Probability Letters (2002) Rosenberger, Durham, Flournoy. J. Stat. Plan. Inference (1997)

EXAMPLE Consider a clinical trial WITH 2 TREATMENTS AND NO COVARIATES where the following assumption holds: [A1] for each treatment i , { Z ni } n ≥ 1 is a sequence of identically distributed random variables, such that µ i = E [ Z ni ] , σ 2 i = V ar [ Z ni ] > 0 and Z ni is independent of the past history of the trial and of the treatment actually assigned. ⇓ It can be proved that Y n = x n β + ε n = β 1 δ n 1 + β 2 δ n 2 + ε n where { ε n } is a sequence of martingale differences.

� Information matrix:   N n 1 0 . . . 0    0 N n 2 . . . 0  • A ′   n A n = . . . ...   . . . . . .     0 0 . . . N nL � V ar (ˆ β n | δ 1 , . . . , δ n ) = σ 2 ( A ′ n A n ) − 1 . Strong consistency and central limit theorems for ˆ β n, OLS if N n /n → π > 0 a.s.

STATEMENT OF THE PROBLEM We apply an adaptive design to allocate patients in a clinical trial and formulate the model E [ Y n | x 1 , . . . , x n ] = η ( x 1 , . . . , x n , β ) Targets • In a phase I or phase II clinical trial the target is to estimate a percentile of the dose response curve. • In a phase III clinical trial the target is to compare the behavior of several treat- ments. QUESTION • In the literature, many adaptive designs have been studied but how to rank them with respect to the degree of achievement of the specific target?

EXAMPLES OF ADAPTIVE DESIGNS PHASE I or PHASE II PHASE III biased coin design randomized play the winner rule k in a row design drop the looser rule group up and down designs Hu and Zhang (2004) designs narayama design Melfi-page-geraldes designs (2005) Continual reassessment method (CRM) Randomization designs (Atkinson (2002))) Ivanova and Flournoy (2006) Rosenberger and Lachin (2002)

THEORY OF OPTIMAL DESIGNS For a clinical trial with L treatments or doses, a design is � � 1 . . . L ξ : = . p 1 . . . p L Given a statistical model E [ Y n | x 1 , . . . , x n ] = η ( x 1 , . . . , x n , β ) We denote the information matrix as M ( ξ n , β ) and for a linear model M ( ξ n ) = A ′ n A n .

THEORY OF LINEAR OPTIMAL DESIGNS • A criterion function is a convex (concave) function ϕ that takes values in the space of information matrices. • The optimal design ξ ∗ = argmin ξ ϕ ( M ( ξ )) • Examples of criteria function: D-optimal: ϕ ( M ( ξ n )) = | A ′ n A n | D C -optimal: ϕ ( M ( ξ n )) = | C t ( A ′ n A n ) − 1 C | E-optimal: ϕ ( M ( ξ n )) = λ max ( A ′ n A n ) G-optimal: ϕ ( M ( ξ n )) = trace (( A ′ n A n ) − 1 ) n A n ) − 1 c = V ar ( c ˆ c-optimal: ϕ ( M ( ξ n )) = c t ( A ′ β )

A new tool to solve the problem Consider an adaptive design: { δ n } ֌ STOCHASTIC PROCESS and generates, for each realization, a design � � 1 . . . L ξ n := . N n 1 /n . . . N nL /n So that the design matrix is random   δ 11 . . . δ 1 L F 11 . . . F 1 K    δ 21 . . . δ 2 L F 21 . . . F 2 K    A n = . . . . ... ...   . . . . . . . .     δ n 1 . . . δ nL F n 1 . . . F nK

A new tool to solve the problem • Consider a criterion function ϕ . ϕ ( A ′ n A n ) ֌ stochastic process • Let ξ ∗ be the optimal design for a convex criterion function ϕ . We define ϕ ( ξ ∗ ) A-POSTERIORI EFFICIENCY: PE n := ϕ ( ξ n | N n ) . (1) � � 1 ME n := E N n [ PE n ] = ϕ ( ξ ∗ ) E N n MEAN A-POSTERIORI EFFICIENCY: . (2) ϕ ( ξ n )

INTERPRETATION • 0 ≤ PE n ≤ 1 When PE n = 1 , the adaptive design has generated the optimal design. When PE n = r means � Our adaptive design has generated a realization such that for each patient we lose an efficiency 1 − r . � We need a sample size equal to n/r times to reach the optimal value. � n (1 − r ) total information loss in terms of patients.

Example: homocedasticity, two trials. Y n = β 1 δ n 1 + β 2 δ n 2 + ε n where { ε n } is a sequence of martingale differences and E [ ε 2 n | δ 1 , . . . , δ n ] = σ 2 . •   N n 1 0 . . . 0    0 N n 2 . . . 0  V ar (ˆ A ′ β n | δ 1 , . . . , δ n ) = σ 2 ( A ′ n A n ) − 1 .   n A n = . . . . ...   . . . . . .     0 0 . . . N nL

Example: homocedasticity, two treatments • We consider the optimal design ϕ ( ξ n ) = V ar [ˆ β 1 n − ˆ β 2 n | δ 1 , . . . , δ n ] � � 1 = (1 , − 1)( A ′ n A n ) − 1 − 1 = σ 2 ( 1 1 + ) (3) N n 1 N n 2 • � � ϕ ( ξ ∗ ) = 4 σ 2 1 2 ξ ∗ = min ξ ϕ ( ξ ) = �→ n . 1 / 2 1 / 2

Example Consider an adaptive design such that N n 1 /n → π 1 . 4 σ 2 /n PE n = σ 2 (1 /N n 1 + 1 /N n 2 ) = 4[ N n 1 /n − ( N n 1 /n ) 2 ] π 1 (1 − π 1 ) + N n 1 /n − π 1 − (( N n 1 /n ) 2 − π 2 � � = 4 1 ) ME n = 4 π 1 (1 − π 1 ) − 4 V ar [ N n 1 n ] . � This is valid without assumptions on the response distribution.

Cuadro 1: For n = 10 , 25 , 50 patients: average of allocations to Treatment 1 (n times sample variance of allocations to Treatment 1 . n = 10 n = 25 n = 50 [A] Efron’s design 0.46 (0.037) 0.49 (0.025) 0.49 (0.010) [B] Ehrenfest model* ( w = 10) 0.50 (0.062) 0.50 (0.025) 0.50 (0.012) [C] Smith’s design 0.55 (0.034) 0.52 (0.027) 0.51 (0.025) [D] General Efron 0.54 (0.045) 0.52 (0.047) 0.52 (0.037) [E] Atkinson design 0.55 (0.067) 0.52 (0.055) 0.51 (0.052) [F] Wei’s urn (1, 3) 0.55 (0.096) 0.52 (0.090) 0.51 (0.090) [G] Complete Randomization 0.50 (0.248) 0.50 (0.253) 0.50 (0.247) * Exact values of nV ar [ N n 1 /n ]. Example: comparison of design-adaptive designs with π 1 = 1 / 2 � We obtain a similar graphic to Atkinson (2002):