A theoretic study of a distance-based regression model
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摘要:
The distance-based regression model has many applications in analysis of multivariate response regression in various fields,such as ecology,genomics,genetics,human microbiomics,and neuroimaging.It yields a pseudo F test statistic that assesses the relation between the distance (dissimilarity) of the subjects and the predictors of interest.Despite its popularity in recent decades,the statistical properties of the pseudo F test statistic have not been revealed to our knowledge.This study derives the asymptotic properties of the pseudo F test statistic using spectral decomposition under the matrix normal assumption,when the utilized dissimilarity measure is the Euclidean or Mahalanobis distance.The pseudo F test statistic with the Euclidean distance has the same distribution as the quotient of two Chi-squared-type mixtures.The denominator and numerator of the quotient are approximated using a random variable of the form ζx2d + rη,and the approximate error bound is given.The pseudo F test statistic with the Mahalanobis distance follows an F distribution.In simulation studies,the approximated distribution well matched the "exact" distribution obtained by the permutation procedure.The obtained distribution was further validated on H1N1 influenza data,aging human brain data,and embryonic imprint data.