Sequential, response-adaptive designs driven by a randomly - PowerPoint PPT Presentation

Sequential, response-adaptive designs driven by a randomly reinforced urn model Caterina May, Universit` a del Piemonte Orientale (Italy) joint work with Nancy Flournoy, University of Missouri (USA) MODA8 2007, Almagro (Ciudad Real) June 4, 2007

Response-adaptive designs for clinical trials 1. Collecting evidence for determining the superior treatment between (two) treatments B and W . → Inferential goal − For instance: if µ B and µ W are the mean responses, H 0 : µ B = µ W versus H 1 : µ B > µ W 2. Biasing, along the experiment, the allocation probabilities toward the superior treatment. → Ethical goal − ρ = target allocation = asymptotic proportion of patients allocated to treatment B Remark. Usually, in literature, ρ is a constant in (0 , 1) and is determined on some optimality criteria which is a function of the unknown parameters of outcomes.

Response-adaptive designs driven by a randomly reinforced urn ( RRU ) (THE IDEA) Target an asymptotic proportion of patients allocated to the superior treatment ρ = 1 In our knowledge, asymptotic theory for this ”singular” case has not been still developed → we need a different approach respect of the usual proposed in literature to study − asymptotic properties and the performance of this design. Remark. In RRU -designs, the responses to treatments can be either discrete or continuous. Model for binary responses: DFL (1998).

Main references May, C. Flournoy, N. (2007). Asymptotics in response-adaptive designs generated by a two- color, randomly reinforced urn. MOX-Report , n. 09/2007. Paganoni, A.M., Secchi, P. (2007). A numerical study for comparing two response-adaptive designs for continuous treatment effects, Statistical Methods and Applications . In press . Aletti, G., May, C. and Secchi, P. (2007). On the Distribution of the limit proportion for a two-color, randomly reinforced urn with equal reinforcement distributions. To appear on Advances in Applied Probability , 37, n.3 Muliere, P., Paganoni, A.M. and Secchi, P. (2006). A randomly reinforced urn, Journal of Statistical Planning and Inference , 136, 1853-1874. Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a favourable response. Canad. J. Statist. , 26 , no. 3, 479–495.

OUTLINE RRU model description. Basic properties. Evaluation of the performance through numerical simulations (PS(2007)). Exact rate of converge to infinity of the number of patients allocated to each treatment. Asymptotic distribution of mean responses estimators and of the test statistic. Conclusions.

RRU description - ( b, w ) : urn initial composition ( b ≥ 0 , w ≥ 0 ); � δ n = 1 if we extract a black → n -th patient to B ; - δ n : sequence of treatment allocations: − → δ n = 0 if we extract a white → n -th patient to W ; - ( Y B ( n ) , Y W ( n )) : sequence of (potential) responses to treatments (i.i.d. vectors where compo- nents have positive and bounded distributions); At every n we reinforce the urn with δ n Y B ( n ) + (1 − δ n ) Y W ( n ) balls of the same color of the ball extracted. - Z n : proportion of black balls at time n → conditionally to the past information, δ n +1 is Bernoulli ( Z n ) . − Remark. We can use a suitable utility function u to transform the responses 1. to have pos- itive and bounded reinforcements 2. when we want to express a different preference between treatments then the order on the mean responses.

Some preliminary properties - The number of patients allocated to EACH treatment, N B ( n ) , N W ( n ) , converges a.s. to infinity. - The proportion of black balls Z n , that is the conditional probability of allocating B , converges almost surely to a r.v. Z ∞ ∈ [0 , 1] . - If µ B > µ W ( H 1 hypothesis), then Z ∞ is 1 → N B ( n ) /n converges to 1 − → ρ = 1 : optimal property! − - If µ B = µ W ( H 0 hypothesis), Z ∞ has not point masses in [0 , 1] . Particular case: when responses have the same distribution, studied in AMS (2007).

Testing hypothesis on the mean responses: H 0 : µ B = µ W versus H 1 : µ B > µ W Let the adaptive estimators of the mean: � n � n i =1 δ i Y B ( i ) i =1 (1 − δ i ) Y W ( i ) ˆ ˆ Y B ( n ) = and Y W ( n ) = . N B ( n ) N W ( n ) and the usual test statistic: Y B ( n ) − ˆ ˆ Y W ( n ) ζ 0 ( n ) = � σ B 2 σ W 2 N B ( n ) + N W ( n ) - Problem: asymptotic normality of estimators and asymptotic distribution of the test statistic ζ 0 ( n ) . In literature, results on the asymptotic normality when ρ constant ∈ (0 , 1) (for instance, Melfi and Page (2000)). - In literature, comparison between the performance (power) of procedures with same target allocation ρ ∈ (0 , 1) (Hu and Rosenberger (2003), Zhang and Rosenberger (2006)).

Numerical studies (PS(2007)) Comparison of the performance of a RRU -design with a default, non adaptive design, through numerical simulations. Normal responses are considered.  Sample size trial: n .      Allocation: n/2 patients to B and n/2 patients to W .  - Default design: Z-test significance: α .     Power= 1 − β for a relevant clinical difference µ B − µ W ≥ δ 0 > 0 .    Sample size trial: n ∗ .     Allocation: N B ( n ∗ ) (r.v.) patients to B and N W ( n ∗ ) (r.v.) patients to W .   - RRU -design: Z-test significance: α .     Power= 1 − β and N W ( n ∗ ) < n/ 2 for µ B − µ W ≥ δ 0 > 0 .   RESULTS: we choose RRU -designs iff moderate to large values of δ 0 .

Theoretical results (MF(2007) THEOREM 1: exact rate of convergence (a.s.) to infinity of the number of patients allocated to each treatment. 1. Under H 0 : µ B = µ W , the rate of N B ( n ) and of N W ( n ) is n . 2. Under H 1 : µ B > µ W , the rate of N B ( n ) is n : N B ( n ) /n → 1 ; the rate of N W ( n ) is n µ W /µ B : N W ( n ) n µ W /µ B = η 2 , a.s., with P (0 < η 2 < ∞ ) = 1 . l´ ım n → + ∞ THEOREM 2: asymptotic normality of adaptive estimators. Under either H 0 and H 1 , the joint vector �� � � � � � ˆ ˆ � N B ( n ) Y B ( n ) − µ B /σ B , N W ( n ) Y W ( n ) − µ W /σ W converges in distribution to a standard Gaussian vector. Remark: wa can’t deduce for Theorem 2 the asymptotic normality of the test statistic ζ 0 ( n ) using Slutsky’s Theorem as we could do in the case ρ constant ∈ (0 , 1) .

TOOL for Theorems 2 and 3: stable convergence for martingale vectors (see Heyde (1997)). THEOREM 3: asymptotic distribution of the test statistic ζ 0 ( n ) . 1. Under H 0 : µ B = µ W , ζ 0 ( n ) is asymptotically normal. 2. Under H 1 : µ B > µ W , ζ 0 ( n ) is a specific mixture of normal distributions. In fact, the condi- tional distribution of ζ 0 ( n ) given the random variable η 2 , is asymptotically normal with mean √ n µ B /µ W η µ B − µ W equal to and unit variance. σ W Consequence: under H 1 , the non-centrality parameter is √ µ B − µ W η 2 µ B − µ W n µ W /µ B � φ ( n ) = . ≈ � σ W σ B 2 σ W 2 N B ( n ) + N W ( n ) Remark. η 2 is not constant (simulations). Study of the mixing distribution η 2 : work in progress (MPS (2007)). → Under H 1 , ζ 0 ( n ) is NOT asymptotically normal.

Conclusions: further developments - Response-adaptive designs. Research on response-adaptive designs, considering the ethical target allocation ρ = 1 . Can we find a more suitable test statistic? - Asymptotic theory. Open problems on the RRU -model, for instance the limit composition of the urn Z i nfinity under the null hypothesis and the distribution of η 2 . - Utility function. Can we find an optimal utility function? - Sequential analysis. Sequential stopping rules and sequential tests.