Asymptotic theory of the quadratic assignment procedure for dyadic data analysis

The quadratic assignment procedure (QAP) is a popular tool for analysing dyadic data in medical and social sciences. To test the association between 2 dyadic measurements represented by 2 symmetric matrices, QAP calculates the p-value by permuting the units, or equivalently, by permuting the rows and columns of 1 matrix in the same way. Its extension to the regression setting, known as the multiple regression QAP, has also gained popularity, especially in psychometrics. However, the statistical theory for QAP has not been fully established in the literature. We fill the gap in this paper. We formulate the dyadic 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. Our asymptotic theory combines tasks (a) and (b), and suggests that we must use properly studentized statistics in QAP to ensure that it is finite-sample exact under the strong null hypothesis and preserves the asymptotic type I error rate under the weak null hypothesis.