  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, we 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, we 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.
  Keywords: Asymptotic distribution, Chi-squared-type mixture, Nonparametric
test, Pseudo-$F$ test, Similarity matrix.
