A review of multivariate permutation tests: findings and trends

The permutation test is a widely recognized and frequently used nonparametric hypothesis test, notable for its minimal reliance on assumptions compared to parametric tests. It has found applications in many fields, particularly in multivariate analysis. Since its introduction in the 1930s, permutation tests have been extensively examined both theoretically and empirically. This article provides the results of a comprehensive and systematic review of the literature, focusing on different aspects of multivariate permutation tests. Key articles published in international journals from 2010 onwards have been analyzed, classifying them into four main research strands: data, model, test and issues. These strands were further subdivided into more specific categories. The state of the art and significant developments in this field are summarized, followed by a discussion on future research challenges and trends, offering guidance for the design and development on new approaches.