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Summary Phylogenetic association analysis plays a crucial role in investigating the correlation between microbial compositions and specific outcomes of interest in microbiome studies. However, existing methods for testing such associations have limitations related to the assumption of a linear association in high-dimensional settings and the handling of confounding effects. Hence, there is a need for methods capable of characterizing complex associations, including nonmonotonic relationships. This article introduces a novel phylogenetic association analysis framework and associated tests to address these challenges by employing conditional rank correlation as a measure of association. The proposed tests account for confounders in a fully nonparametric manner, ensuring robustness against outliers and the ability to detect diverse dependencies. The proposed framework aggregates conditional rank correlations for subtrees using weighted sum and maximum approaches to capture both dense and sparse signals. The significance level of the test statistics is determined by calibration through a nearest-neighbour bootstrapping method, which is straightforward to implement and can accommodate additional datasets when these are available. The practical advantages of the proposed framework are demonstrated through numerical experiments using both simulated and real microbiome datasets. 

Driven by a wide range of applications, several principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a uni- fied framework for the statistical analysis of a general structured principal subspace estima- tion problem which includes as special cases sparse PCA/SVD, non-negative PCA/SVD, subspace constrained PCA/SVD, and spectral clustering. General minimax lower and up- per bounds are established to characterize the interplay between the information-geometric complexity of the constraint set for the principal subspaces, the signal-to-noise ratio (SNR), and the dimensionality. The results yield interesting phase transition phenomena concern- ing the rates of convergence as a function of the SNRs and the fundamental limit for consistent estimation. Applying the general results to the specific settings yields the mini- max rates of convergence for those problems, including the previous unknown optimal rates for sparse SVD, non-negative PCA/SVD and subspace constrained PCA/SVD. 

Summary Motivated by the problem of estimating bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix model $$Y=\Theta\Pi+Z$$ where $$Y\in \mathbb{R}^{n\times p}$$ is observed, $$\Theta\in \mathbb{R}^{n\times p}$$ is an unknown approximately rank-one signal matrix with monotone rows, $$\Pi \in \mathbb{R}^{p\times p}$$ is an unknown permutation matrix, and $$Z\in \mathbb{R}^{n\times p}$$ is the noise matrix. In this article we study estimation of the extreme values associated with the signal matrix $$\Theta$$, including its first and last columns and their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column-sorting method. Numerical experiments on simulated and synthetic microbiome metagenomic data are conducted, demonstrating the superiority of the proposed methods over existing alternatives. The methods are illustrated by comparing the growth rates of gut bacteria in inflammatory bowel disease patients and control subjects. 

Summary Quantitative comparison of microbial composition from different populations is a fundamental task in various microbiome studies. We consider two-sample testing for microbial compositional data by leveraging phylogenetic information. Motivated by existing phylogenetic distances, we take a minimum-cost flow perspective to study such testing problems. We first show that multivariate analysis of variance with permutation using phylogenetic distances, one of the most commonly used methods in practice, is essentially a sum-of-squares type of test and has better power for dense alternatives. However, empirical evidence from real datasets suggests that the phylogenetic microbial composition difference between two populations is usually sparse. Motivated by this observation, we propose a new maximum type test, detector of active flow on a tree, and investigate its properties. We show that the proposed method is particularly powerful against sparse phylogenetic composition difference and enjoys certain optimality. The practical merit of the proposed method is demonstrated by simulation studies and an application to a human intestinal biopsy microbiome dataset on patients with ulcerative colitis.