Asymptotic normality of urn models for clinical trials with delayed response

Response-adaptive design involves the sequential selection of design points chosen depending on the outcomes at previously selected design points. The response-adaptive design has been extensively studied in the literature; see Rosenberger (1996), Flournoy and Rosenberger (1995) and Hu and Ivanova (2004) for details. An important family of adaptive designs can be developed from the generalized Friedman’s urn (GFU) model; see Athreya and Karlin (1968) and Rosenberger (2002). It is also called the generalized Pólya urn (GPU) model in the literature. The model is described as follows. Consider an urn containing balls of K types, representing K ‘treatments’ in a clinical trial. Initially the urn contains Y0 1⁄4 (Y01, . . . , Y0K) balls, where Y0k denotes the number of balls of type k, k 1⁄4 1, . . . , K. At stage i, i 1⁄4 1, . . . , n, a ball is drawn from the urn and replaced. If the ball is of type k, then treatment k is assigned to the ith patient, i 1⁄4 1, . . . , n. We then wait to observe a random variable i, the response of the treatment by patient i. After that, an additional Dk,q(i) balls of type q, q 1⁄4 1, . . . , K, are added to the urn, where Dk,q(i) is some function of i. This procedure is repeated for n stages. After n stages, the urn composition is denoted by the row vector Yn 1⁄4 (Yn1, . . . , YnK ), where Ynk represents the number of balls of type k in the urn after the nth addition of balls. This relation can be written as the recursive formula