Abstract #300653

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JSM 2003 Abstract #300653
Activity Number: 30
Type: Contributed
Date/Time: Sunday, August 3, 2003 : 2:00 PM to 3:50 PM
Sponsor: Biometrics Section
Abstract - #300653
Title: Exact Moments and Probabilities for the Two-Color Wei's Urn Randomization Model
Author(s): Neal Oden*+ and Matthew J. McIntosh
Companies: EMMES Corporation and The EMMES Corporation
Address: 401 N. Washington St., Rockville, MD, 20850-1737,
Keywords: randomization ; Eulerian numbers ; clininical trials ; urn model ; permutations ; early balance
Abstract:

We concentrate on a restricted randomization (adaptive biased coin) design by Wei (1977, 1978). A realization of Wei's (a, b) urn model starts with a black and a white balls in an urn. On each step of the realization, a ball is randomly drawn and its color used to assign a treatment to the next trial participant. Once the assignment is made, the ball and b balls of the opposite color are returned to the urn, and the next realization step is taken. The probability of white on any given step thus depends on the degree of imbalance between the colors in the urn. Clinical trials are actively using Wei's urn randomization scheme. We present a simple closed form for the exact variance of the number of white balls, given that n balls have been drawn. Even for large n, this is considerably better than the previously published asymptotic variance approximation. We also illustrate the link between the (1,1) urn model and Euler's triangle, which gives a closed-form expression for the probability that a given number of subjects are randomized to a treatment arm. We generalize this probability expression to the (a, b) urn model, and also present the exact autocovariance.

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