Abstract
In biomedical research, multiple endpoints are commonly analyzed in “omics” fields like genomics, proteomics and metabolomics. Traditional methods designed for low-dimensional data either perform poorly or are not applicable when analyzing high-dimensional data whose dimension is generally similar to, or even much larger than, the number of subjects. The complex biochemical interplay between hundreds (or thousands) of endpoints is reflected by complex dependence relations. The aim of the paper is to propose tests that are very suitable for analyzing omics data because they do not require the normality assumption, are powerful also for small sample sizes, in the presence of complex dependence relations among endpoints, and when the number of endpoints is much larger than the number of subjects. Unbiasedness and consistency of the tests are proved and their size and power are assessed numerically. It is shown that the proposed approach based on the nonparametric combination of dependent interpoint distance tests is very effective. Applications to genomics and metabolomics are discussed.
Acknowledgement
We are very grateful to Prof. Dr. H. Shen for kindly providing the second data set analyzed in Section 5.
Conflict of Interest: The author has declared no conflict of interest.
A Appendix
Theorem 1
The FReuclid test is unbiased for testing H0 : 𝛍 = 0 against H1 : 𝛍 ≠ 0.
Proof. We consider the following additive model, which is equivalent to the location difference setting considered in Section 2,
where Vs are independent and identically distributed multivariate random variables with 0 location and Σ variance-covariance matrix with no infinite elements. Note that Vs are independent among themselves but their p components can be dependent. Let
denote the pooled sample under the null hypothesis and let
denote the pooled sample under the alternative hypothesis. Define similarly
The FReuclid test rejects for large values of its statistic, therefore to prove unbiasedness we have to show that the FReuclid test statistic is stochastically larger when μ ≠ 0, ie under H1, than when μ = 0, ie under H0, as shown by Theorem 3 in Pesarin and Salmaso (2010) p. 138.
Theorem 1 in Marozzi (2015a) shows that the Meuclid test is unbiased. Therefore when μ ≠ 0 the Meuclid test statistic is stochastically smaller than when μ = 0. It follows that
note that Meuclid test rejects for small values of its statistic. Of course it is also
ie the FReuclid test statistic is stochastically larger under H1 than under H0. This result completes the proof. QED □
Theorem 2
The FReuclid test is consistent for testing H0 : 𝛍 = 0 against H1 : 𝛍 ≠ 0.
Proof. From Theorem 2 in Marozzi (2015a) it follows that the Meuclid test is consistent, that is
where N → ∞ means that m, n → ∞ with
and that
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