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Register a new userLocal biplots for multi-dimensional scaling, with application to the microbiome
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Julia Fukuyama
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
Authors
Julia Fukuyama
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We present local biplots, a an extension of the classic principal components biplot to multi-dimensional scaling. Noticing that principal components biplots have an interpretation as the Jacobian of a map from data space to the principal subspace, we define local biplots as the Jacobian of the analogous map for multi-dimensional scaling. In the process, we show a close relationship between our local biplot axes, generalized Euclidean distances, and generalized principal components. In simulations and real data we show how local biplots can shed light on what variables or combinations of variables are important for the low-dimensional embedding provided by multi-dimensional scaling. They give particular insight into a class of phylogenetically-informed distances commonly used in the analysis of microbiome data, showing that different variants of these distances can be interpreted as implicitly smoothing the data along the phylogenetic tree and that the extent of this smoothing is variable.
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Mediation analysis has become an important tool in the behavioral sciences for investigating the role of intermediate variables that lie in the path between a randomized treatment and an outcome variable. The influence of the intermediate variable on the outcome is often explored using structural equation models (SEMs), with model coefficients interpreted as possible effects. While there has been significant research on the topic in recent years, little work has been done on mediation analysis when the intermediate variable (mediator) is a high-dimensional vector. In this work we present a new method for exploratory mediation analysis in this setting called the directions of mediation (DMs). The first DM is defined as the linear combination of the elements of a high-dimensional vector of potential mediators that maximizes the likelihood of the SEM. The subsequent DMs are defined as linear combinations of the elements of the high-dimensional vector that are orthonormal to the previous DMs and maximize the likelihood of the SEM. We provide an estimation algorithm and establish the asymptotic properties of the obtained estimators. This method is well suited for cases when many potential mediators are measured. Examples of high-dimensional potential mediators are brain images composed of hundreds of thousands of voxels, genetic variation measured at millions of SNPs, or vectors of thousands of variables in large-scale epidemiological studies. We demonstrate the method using a functional magnetic resonance imaging (fMRI) study of thermal pain where we are interested in determining which brain locations mediate the relationship between the application of a thermal stimulus and self-reported pain.
Tracking and estimating Daily Fine Particulate Matter (PM2.5) is very important as it has been shown that PM2.5 is directly related to mortality related to lungs, cardiovascular system, and stroke. That is, high values of PM2.5 constitute a public health problem in the US, and it is important that we precisely estimate PM2.5 to aid in public policy decisions. Thus, we propose a Bayesian hierarchical model for high-dimensional multi-type responses. By multi-type responses we mean a collection of correlated responses that have different distributional assumptions (e.g., continuous skewed observations, and count-valued observations). The Centers for Disease Control and Prevention (CDC) database provides counts of mortalities related to PM2.5 and daily averaged PM2.5 which are both treated as responses in our analysis. Our model capitalizes on the shared conjugate structure between the Weibull (to model PM2.5), Poisson (to model diseases mortalities), and multivariate log-gamma distributions, and we use dimension reduction to aid with computation. Our model can also be used to improve the precision of estimates and estimate values at undisclosed/missing counties. We provide a simulation study to illustrate the performance of the model, and give an in-depth analysis of the CDC dataset.
In microbiome studies, one of the ways of studying bacterial abundances is to estimate bacterial composition based on the sequencing read counts. Various transformations are then applied to such compositional data for downstream statistical analysis, among which the centered log-ratio (clr) transformation is most commonly used. Due to limited sequencing depth and DNA dropouts, many rare bacterial taxa might not be captured in the final sequencing reads, which results in many zero counts. Naive composition estimation using count normalization leads to many zero proportions, which makes clr transformation infeasible. This paper proposes a multi-sample approach to estimation of the clr matrix directly in order to borrow information across samples and across species. Empirical results from real datasets suggest that the clr matrix over multiple samples is approximately low rank, which motivates a regularized maximum likelihood estimation with a nuclear norm penalty. An efficient optimization algorithm using the generalized accelerated proximal gradient is developed. Theoretical upper bounds of the estimation errors and of its corresponding singular subspace errors are established. Simulation studies demonstrate that the proposed estimator outperforms the naive estimators. The method is analyzed on Gut Microbiome dataset and the American Gut project.
Modern microbiome compositional data are often high-dimensional and exhibit complex dependency among microbial taxa. However, existing approaches to analyzing microbiome compositional data either do not adequately account for the complex dependency or lack scalability to high-dimensionality, which presents challenges in appropriately incorporating the random effects in microbiome compositions in the resulting statistical analysis. We introduce a generative model called the logistic-tree normal (LTN) model to address this need. The LTN marries two popular classes of models -- the log-ratio normal (LN) and the Dirichlet-tree (DT) -- and inherits key benefits of each. LN models are flexible in characterizing covariance among taxa but lacks scalability to higher dimensions; DT avoids this issue through a tree-based binomial decomposition but incurs restrictive covariance. The LTN incorporates the tree-based decomposition as the DT does, but it jointly models the corresponding binomial probabilities using a (multivariate) logistic-normal distribution as in LN models. It therefore allows rich covariance structures as LN, along with computational efficiency realized through a Polya-Gamma augmentation on the binomial models at the tree nodes. Accordingly, Bayesian inference on LTN can readily proceed by Gibbs sampling. The LTN also allows common techniques for effective inference on high-dimensional data -- such as those based on sparsity and low-rank assumptions in the covariance structure -- to be readily incorporated. Depending on the goal of the analysis, LTN can be used either as a standalone model or embedded into more sophisticated hierarchical models. We demonstrate its use in estimating taxa covariance and in mixed-effects modeling. Finally, we carry out an extensive case study using an LTN-based mixed-effects model to analyze a longitudinal dataset from the DIABIMMUNE project.
Identifying which taxa in our microbiota are associated with traits of interest is important for advancing science and health. However, the identification is challenging because the measured vector of taxa counts (by amplicon sequencing) is compositi
onal, so a change in the abundance of one taxon in the microbiota induces a change in the number of sequenced counts across all taxa. The data is typically sparse, with zero counts present either due to biological variance or limited sequencing depth (technical zeros). For low abundance taxa, the chance for technical zeros is non-negligible. We show that existing methods designed to identify differential abundance for compositional data may have an inflated number of false positives due to improper handling of the zero counts. We introduce a novel non-parametric approach which provides valid inference even when the fraction of zero counts is substantial. Our approach uses a set of reference taxa that are non-differentially abundant, which can be estimated from the data or from outside information. We show the usefulness of our approach via simulations, as well as on three different data sets: a Crohns disease study, the Human Microbiome Project, and an experiment with spiked-in bacteria.
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