Effect of imbalance in using stratified block randomization in clinical trials Anisimov, Vladimir GlaxoSmithKline, Research Statistics Unit New Frontiers Science Park (South), Third Avenue Harlow, Essex CM19 5AW, United Kingdom E-mail: Vladimir.V.Anisimov@gsk.com Randomization is an essential part of clinical trials design. It is carried out with the purpose of allocating randomly patients to treatments, preserving blindness and achieving balance in the number of patients on treatment arms. The choice of randomization affects the power of statistical tests and the amount of drug supply required to satisfy patient demand. The talk is concentrated on the analysis of centre-stratified block-permuted randomization. An analytic approach for evaluating imbalance in the number of patients on treatment arms using a newly developed patient recruitment model is proposed and the impact of imbalance on the power of a study is investigated. Properties of imbalance Consider a multicentre study with n patients, N centres and two treatments, a and b. Assume that it is planned to recruit an equal number, n/2, of patients on each treatment arm. Consider a centre-stratified block randomization – in each centre a randomization is provided independently by randomly permuted blocks of size K with equal proportions of patients within each block. For exam- ple, if K = 4 there are 6 possibilities for different permuted blocks: {(a, a, b, b); (a, b, a, b); (a, b, b, a); (b, a, a, b); (b, a, b, a); (b, b, a, a)}. If the number of patients in a particular centre is not a multiple of K, then the last block in this centre is incomplete. The incomplete block may contain an unequal number of patients on treatment arms, which causes imbalance in this centre. In a multicentre study many incomplete blocks may cause imbalance between the total number of patients on treatment arms and this may affect the power of the study. Let n i be the total number of patients recruited by centre i, n ij be the number of patients in centre i on treatment j and η i = n ia - n ib (imbalance in centre i). Denote by n * j the total number of patients on treatment j and let Δ = n * a - n * b be the total imbalance between the numbers of patients on treatment arms. Then Δ= N  i=1 η i . (1) As the imbalance is calculated using the numbers of patients recruited by different centres, we need first to consider a proper patient recruitment model reflecting the behaviour of real data. We suggest modeling patient recruitment processes in different centres as Poisson processes with the rates viewed as a sample from a gamma distributed population [1-3]. This model accounts for the natural variation of recruitment in time and recruitment rates between different centres and it has been validated using many real GSK trials with a large enough number of centres (> 20). Statement 1. For large N the imbalance Δ is well approximated by a normal distribution with mean zero and variance s 2 (n,N,K,α)N , where s 2 (·) is determined by the block size K, the values n, N , and the shape parameter α of the recruitment model and can be calculated numerically for any recruitment scenario. Proof. Let us first calculate the distribution of the number of patients n i recruited by a particular centre i. Assume that in centre i the number of patients recruited up to time t follows a Poisson process with rate λ i , where the rates {λ i ,i =1, .., N } are viewed as a sample from a gamma BULLETIN of THE INTERNATIONAL STATISTICAL INSTITUTE - LXII (2007) - 5938 -