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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 µB and µW are the mean responses, H0 : µB = µW versus H1 : µB > µW

• 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);
• δn: sequence of treatment allocations: −

→

• δn = 1 if we extract a black → n-th patient to B;

δn = 0 if we extract a white → n-th patient to W;

• (YB(n), YW(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 δnYB(n) + (1 − δn)YW(n) balls of the same color of the ball extracted.

• Zn: proportion of black balls at time n

− → conditionally to the past information, δn+1 is Bernoulli(Zn).

• 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, NB(n), NW(n), converges a.s. to infinity.
• The proportion of black balls Zn, that is the conditional probability of allocating B, converges

almost surely to a r.v. Z∞ ∈ [0, 1].

• If µB > µW (H1 hypothesis), then Z∞ is 1

− → NB(n)/n converges to 1 − → ρ = 1: optimal property!

• If µB = µW (H0 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: H0 : µB = µW versus H1 : µB > µW

Let the adaptive estimators of the mean: ˆ YB(n) = n

i=1 δiYB(i)

NB(n) and ˆ YW(n) = n

i=1(1 − δi)YW(i)

NW(n) . and the usual test statistic: ζ0(n) = ˆ YB(n) − ˆ YW(n)

• σB2

NB(n) + σW 2 NW(n)

• Problem: asymptotic normality of estimators and asymptotic distribution of the test

statistic ζ0(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.

• Default design:

             Sample size trial: n. Allocation: n/2 patients to B and n/2 patients to W. Z-test significance: α. Power=1 − β for a relevant clinical difference µB − µW ≥ δ0 > 0.

• RRU-design:

             Sample size trial: n∗. Allocation: NB(n∗) (r.v.) patients to B and NW(n∗) (r.v.) patients to W. Z-test significance: α. Power=1 − β and NW(n∗) < n/2 for µB − µW ≥ δ0 > 0. RESULTS: we choose RRU-designs iff moderate to large values of δ0.

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 H0 : µB = µW, the rate of NB(n) and of NW(n) is n.
• 2. Under H1 : µB > µW, the rate of NB(n) is n: NB(n)/n → 1;

the rate of NW(n) is nµW /µB: l´ ım

n→+∞

NW(n) nµW /µB = η2, a.s., with P(0 < η2 < ∞) = 1. THEOREM 2: asymptotic normality of adaptive estimators. Under either H0 and H1, the joint vector

• NB(n)
• ˆ

YB(n) − µB

• /σB,
• NW(n)
• ˆ

YW(n) − µW

• /σW
• converges in distribution to a standard Gaussian vector.

Remark: wa can’t deduce for Theorem 2 the asymptotic normality of the test statistic ζ0(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 ζ0(n).

• 1. Under H0 : µB = µW, ζ0(n) is asymptotically normal.
• 2. Under H1 : µB > µW, ζ0(n) is a specific mixture of normal distributions. In fact, the condi-

tional distribution of ζ0(n) given the random variable η2, is asymptotically normal with mean equal to √ nµB/µW η µB − µW σW and unit variance. Consequence: under H1, the non-centrality parameter is φ(n) = µB − µW

• σB2

NB(n) + σW 2 NW(n) ≈ √ nµW /µB η2 µB − µW σW .

• Remark. η2 is not constant (simulations). Study of the mixing distribution η2: work in

progress (MPS (2007)). → Under H1, ζ0(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
• f the urn Zinfinity under the null hypothesis and the distribution of η2.
• Utility function. Can we find an optimal utility function?
• Sequential analysis. Sequential stopping rules and sequential tests.