The quadratic assignment procedure (QAP) is a popular tool for analyzing network data in medical and social sciences. To test the association between two network measurements represented by two symmetric matrices, QAP calculates the $p$-value by permuting the units, or equivalently, by simultaneously permuting the rows and columns of one matrix. Its extension to the regression setting, known as the multiple regression QAP, has also gained popularity, especially in psychometrics. However, the statistics theory for QAP has not been fully established in the literature. We fill the gap in this paper. We formulate the network models underlying various QAPs. We derive (a) the asymptotic sampling distributions of some canonical test statistics and (b) the corresponding asymptotic permutation distributions induced by QAP under strong and weak null hypotheses. Task (a) relies on applying the theory of U-statistics, and task (b) relies on applying the theory of double-indexed permutation statistics. The combination of tasks (a) and (b) provides a relatively complete picture of QAP. Overall, our asymptotic theory suggests that using properly studentized statistics in QAP is a robust choice in that it is finite-sample exact under the strong null hypothesis and preserves the asymptotic type one error rate under the weak null hypothesis.

翻译：二次分配过程（QAP）是医学和社会科学中分析网络数据的常用工具。为检验由两个对称矩阵表示的两种网络测量之间的关联性，QAP通过对单元进行置换（或等价地，同时置换其中一个矩阵的行和列）来计算$p$值。其在回归设定下的扩展，即多元回归QAP，也日益普及，尤其在心理测量学领域。然而，现有文献尚未完全建立QAP的统计理论基础。本文旨在填补这一空白。我们构建了各类QAP所基于的网络模型，并推导出：(a) 若干典型检验统计量的渐近抽样分布，以及(b) 在强原假设和弱原假设下由QAP导出的相应渐近置换分布。任务(a)依赖于U统计量理论的应用，任务(b)则依赖于双索引置换统计量理论的应用。结合(a)与(b)的结果，为QAP提供了相对完整的理论图景。总体而言，我们的渐近理论表明，在QAP中使用适当学生化的统计量是一种稳健的选择：其在强原假设下具有有限样本精确性，同时在弱原假设下能保持渐近第一类错误率。