Asymptotic theory of the quadratic assignment procedure for dyadic data analysis

TL;DR

This paper develops an asymptotic theory for the quadratic assignment procedure (QAP) used in dyadic data analysis, providing theoretical foundations and guidance for its proper application in social and medical sciences.

Contribution

It formulates network models underlying QAP and derives asymptotic distributions, establishing a theoretical basis for QAP's statistical properties.

Findings

Studentized statistics in QAP are robust and exact under strong null hypotheses.

Asymptotic permutation distributions are characterized for various QAP tests.

The theory supports proper use of QAP in dyadic data analysis.

Abstract

The quadratic assignment procedure (QAP) is a popular tool for analyzing dyadic data in medical and social sciences. To test the association between two dyadic measurements represented by two symmetric matrices, QAP calculates the p-value by permuting the units, or equivalently, by permuting the rows and columns of one 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 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.