TESTS OF HIGH-DIMENSIONAL COMPOSITIONAL DATA

The human microbiome plays a crucial role in human health and diseases. How to comprehensively analyze microbiome data and explore the relationship between microor- ganisms and human health is a hot topic. High-dimensional compositional data are an im- portant type of microbiome data. By performing statistical inference on high-dimensional compositional data, we can profoundly investigate the potential relationships between human microbiome data and health and diseases. This paper focuses on the hypothesis testing problems of high-dimensional compositional data, including one-sample and two- sample mean and covariance testing. We propose brand new testing methods that address the issue of existing test statistics not being applicable to the sum-to-one constraint of high-dimensional compositional data.

For mean testing, the existing max-type test statistics are mainly designed for sparse high-dimensional compositional data. However, for data with weak signals and high den- sity, the test power significantly decreases. Sum-type test statistics are more suitable for dense data testing, but most of the existing sum-type test statistics are based on the as- sumption of independent component models and are not applicable to high-dimensional compositional data. Therefore, we modify the existing sum-type tests to make them ap- plicable to one-sample and two-sample mean testing of high-dimensional compositional data under more general conditions. Furthermore, we demonstrat the asymptotic inde- pendence of the sum-type test statistic and the max-type test statistic, and subsequently propose a max-sum combination test statistic that can handle both sparse and dense data. We establish the asymptotic distribution of these test statistics under the null hypothesis and the power analysis under the alternative hypothesis. Both theoretical derivations and numerical simulations indicate that the proposed max-sum test performs robustly regard- less of the sparsity of the data.

Then we consider the spherical test for the covariance matrix of high-dimensional compositional data. We adopt the classical multivariate analysis method, John’s test statis- tic, and modify it to be applicable to high-dimensional compositional data. To derive the asymptotic distribution of the modified John’s test statistic, we generalize the central limit theorem for the sample covariance matrix linear spectral statistic of independent compo- nent data, making it also applicable to cases with a degenerate population covariance matrix, including high-dimensional compositional data. Meanwhile, numerical simulations also show that our modified John’s test statistic maintains a good power while controlling the empirical test size.

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