Generalized Pólya urn Designs with Null Balance

Abstract

In this paper we propose a class of sequential urn designs based on generalized Pólya urn (GPU) models for balancing the allocations of two treatments in sequential clinical trials. In particular, we consider a GPU model characterized by a 2 × 2 random addition matrix with null balance (i.e. null row sums) and replacement rule depending upon the urn composition. Under this scheme, the urn process has a Markovian structure and can be regarded as a random extension of the classical Ehrenfest model. We establish almost sure convergence and asymptotic normality for the frequency of treatment allocations and show that in some peculiar cases the asymptotic variance of the design admits a natural representation based on the set of orthogonal polynomials associated with the corresponding Markov process.

Randomized maximum-contrast selection: Subagging for large-scale regression

“…In contrast, our focus is the support recovery in large scale and potentially high dimensional problems. Other examples of randomized schemes that satisfy Condition 1, include a Balanced Pólya urn scheme (Antognini and Giannerini, 2007) and a coupling of Poisson and Multinomial distribution; the first includes a Pólya urn scheme with a strategy guaranteeing that each ball color is represented at least once whereas the second scheme includes a randomized game of throwing n balls into N urns, and repeating the throws until all the balls are in urns.…”

Randomly Reinforced Urn Designs with Prespecified Allocations

“…We want to construct a response-adaptive design, described in terms of an urn model, targeting any optimal, fixed asymptotic allocation, in order to compare these designs with others studied in the literature. A large class of response-adaptive randomized designs is based on urn models, a classical tool to guarantee a randomized device (see Rosenberger (2002) and Zhang et al (2006)), to balance the allocations (see Baldi Antognini and Giannerini (2007)), or to construct designs which asymptotically assign all subjects to the best treatment (see Flournoy et al (2012)). The two-color, randomly reinforced urn (RRU) introduced in Durham and Yu (1990), extended to the multi-color case in Durham et al (1998), and studied in Muliere et al (2006), Aletti et al (2009Aletti et al ( ), (2012, and May and Flournoy (2009), is a randomized device able to asymptotically allocate subjects to the optimal treatment; see Muliere et al (2006).…”

Randomly Reinforced Urn Designs with Prespecified Allocations

“…We want to construct a response-adaptive design, described in terms of an urn model, targeting any optimal, fixed asymptotic allocation, in order to compare these designs with others studied in the literature. A large class of response-adaptive randomized designs is based on urn models, a classical tool to guarantee a randomized device (see Rosenberger (2002) and Zhang et al (2006)), to balance the allocations (see Baldi Antognini and Giannerini (2007)), or to construct designs which asymptotically assign all subjects to the best treatment (see Flournoy et al (2012)). The two-color, randomly reinforced urn (RRU) introduced in Durham and Yu (1990), extended to the multi-color case in Durham et al (1998), and studied in Muliere et al (2006), Aletti et al (2009), (2012), and May and Flournoy (2009), is a randomized device able to asymptotically allocate subjects to the optimal treatment; see Muliere et al (2006).…”