### Distance-Based Regression Analysis for Measuring Associations

SHI Yuke1,2, ZHANG Wei1, LIU Aiyi3, LI Qizhai1,2

1. 1. LSC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. Biostatistics and Bioinformatics Branch, Eunice Kennedy Shriver National Institute of Child Health;Human Development, National Institutes of Health, Bethesda, MD 20847, USA
• Received:2022-01-24 Revised:2022-02-13 Online:2023-01-25 Published:2023-02-09
• Supported by:
This work was partially supported by Beijing Natural Science Foundation under Grant No. Z180006.

SHI Yuke, ZHANG Wei, LIU Aiyi, LI Qizhai. Distance-Based Regression Analysis for Measuring Associations[J]. Journal of Systems Science and Complexity, 2023, 36(1): 393-411.

Distance-based regression model, as a nonparametric multivariate method, has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of interest in genetic association studies, genomic analyses, and many other research areas. Based on it, a pseudo-$F$ statistic which partitions the variation in distance matrices is often constructed to achieve the aim. To the best of our knowledge, the statistical properties of the pseudo-$F$ statistic has not yet been well established in the literature. To fill this gap, the authors study the asymptotic null distribution of the pseudo-$F$ statistic and show that it is asymptotically equivalent to a mixture of chi-squared random variables. Given that the pseudo-$F$ test statistic has unsatisfactory power when the correlations of the response variables are large, the authors propose a square-root $F$-type test statistic which replaces the similarity matrix with its square root. The asymptotic null distribution of the new test statistic and power of both tests are also investigated. Simulation studies are conducted to validate the asymptotic distributions of the tests and demonstrate that the proposed test has more robust power than the pseudo-$F$ test. Both test statistics are exemplified with a gene expression dataset for a prostate cancer pathway.
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