Zhao Wenle, Weng Yanqiu
Department of Medicine, Medical University of South Carolina, Charleston, SC, USA.
J Stat Plan Inference. 2011 Jan 1;141(1):474-478. doi: 10.1016/j.jspi.2010.06.023.
Open label and single blinded randomized controlled clinical trials are vulnerable to selection bias when the next treatment assignment is predictable based on the randomization algorithm and the preceding assignment history. While treatment predictability is an issue for all constrained randomization algorithms, deterministic assignments are unique to permuted block randomization. Deterministic assignments may lead to treatment predictability with certainty and selection bias, which could inflate the type I error and hurts the validity of trial results. It is important to accurately evaluate the probability of deterministic assignments in permuted block randomization, so proper protection measures can be implemented. For trials with number of treatment arms T = 2 and a balance block size B = 2m, Matts and Lachin indicated that the probability of deterministic assignment is 1m+1. For more general situations, with T ≥ 2 and a block size B=∑j=1Tmj, Dupin-Spriet provided a formula, which can be written as 1B∑j=1T∑i=1mj∏k=1imj-k+1B-k+1. This formula involves extensive calculation in evaluation. In this paper, we simplified this formula to 1B∑j=1TmjB-mj+1 for general scenarios and 1B-m+1 for trials with a balanced allocation. Through mathematical induction we show the equivalence of the formulas. While the new formula is numerically equivalent to Dupin-Spriet's formula, the simple format not only is easier for evaluation, but also is clearer in describing the impact of parameters T and m(i) on the probability of deterministic assignments.
当根据随机化算法和先前的分配历史可以预测下一个治疗分配时，开放标签和单盲随机对照临床试验容易受到选择偏倚的影响。虽然治疗可预测性是所有受限随机化算法都存在的问题，但确定性分配是置换块随机化所特有的。确定性分配可能会必然导致治疗可预测性和选择偏倚，这可能会夸大I型错误并损害试验结果的有效性。准确评估置换块随机化中确定性分配的概率很重要，这样才能实施适当的保护措施。对于治疗组数量(T = 2)且平衡块大小(B = 2m)的试验，马茨和拉钦指出确定性分配的概率为(\frac{1}{m + 1})。对于更一般的情况，当(T≥2)且块大小(B = \sum_{j = 1}^{T}m_j)时，迪潘 - 斯普里特给出了一个公式，可写成(\frac{1}{B}\sum_{j = 1}^{T}\sum_{i = 1}^{m_j}\frac{\prod_{k = 1}^{i}m_j - k + 1}{B - k + 1})。这个公式在评估时涉及大量计算。在本文中，我们将这个公式简化为一般情况下的(\frac{1}{B}\sum_{j = 1}^{T}\frac{m_j}{B - m_j + 1})以及平衡分配试验的(\frac{1}{B - m + 1})。通过数学归纳法我们证明了这些公式的等价性。虽然新公式在数值上与迪潘 - 斯普里特的公式等价，但简单的形式不仅更易于评估，而且在描述参数(T)和(m(i))对确定性分配概率的影响方面也更清晰。
