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Traditional statistical and machine learning methods assume identical distribution for the training and test data sets. This assumption, however, is often violated in real applications, particularly in health care research, where the training data~(source) may underrepresent specific subpopulations in the testing or target domain. Such disparities, coupled with censored observations, present significant challenges for investigators aiming to make predictions for those minority groups. This paper focuses on target-independent learning under covariate shift, where we study multicalibration for survival probability and restricted mean survival time, and propose a black-box post-processing boosting algorithm designed for censored survival data. Our algorithm, leveraging the pseudo observations, yields a multicalibrated predictor competitive with propensity scoring regarding predictions on the unlabeled target domain, not just overall but across diverse subpopulations. Our theoretical analysis for pseudo observations relies on functional delta method and $p$-variational norm. We further investigate the algorithm's sample complexity and convergence properties, as well as the multicalibration guarantee for post-processed predictors. Our theoretical insights reveal the link between multicalibration and universal adaptability, suggesting that our calibrated function performs comparably to, if not better than, the inverse propensity score weighting estimator. The performance of our proposed methods is corroborated through extensive numerical simulations and a real-world case study focusing on prediction of cardiovascular disease risk in two large prospective cohort studies. These empirical results confirm its potential as a powerful tool for predictive analysis with censored outcomes in diverse and shifting populations.

This paper considers the problem of regression analysis with random covariance matrix as outcome and Euclidean covariates in the framework of Fr\'echet regression on the Bures-Wasserstein manifold. Such regression problems have many applications in single cell genomics and neuroscience, where we have covariance matrix measured over a large set of samples. Fr\'echet regression on the Bures-Wasserstein manifold is formulated as estimating the conditional Fr\'echet mean given covariates $x$. A non-asymptotic $\sqrt{n}$-rate of convergence (up to $\log n$ factors) is obtained for our estimator $\hat{Q}_n(x)$ uniformly for $\left\|x\right\| \lesssim \sqrt{\log n}$, which is crucial for deriving the asymptotic null distribution and power of our proposed statistical test for the null hypothesis of no association. In addition, a central limit theorem for the point estimate $\hat{Q}_n(x)$ is obtained, giving insights to a test for covariate effects. The null distribution of the test statistic is shown to converge to a weighted sum of independent chi-squares, which implies that the proposed test has the desired significance level asymptotically. Also, the power performance of the test is demonstrated against a sequence of contiguous alternatives. Simulation results show the accuracy of the asymptotic distributions. The proposed methods are applied to a single cell gene expression data set that shows the change of gene co-expression network as people age.

Ridge regression with random coefficients provides an important alternative to fixed coefficients regression in high dimensional setting when the effects are expected to be small but not zeros. This paper considers estimation and prediction of random coefficient ridge regression in the setting of transfer learning, where in addition to observations from the target model, source samples from different but possibly related regression models are available. The informativeness of the source model to the target model can be quantified by the correlation between the regression coefficients. This paper proposes two estimators of regression coefficients of the target model as the weighted sum of the ridge estimates of both target and source models, where the weights can be determined by minimizing the empirical estimation risk or prediction risk. Using random matrix theory, the limiting values of the optimal weights are derived under the setting when $p/n \rightarrow \gamma$, where $p$ is the number of the predictors and $n$ is the sample size, which leads to an explicit expression of the estimation or prediction risks. Simulations show that these limiting risks agree very well with the empirical risks. An application to predicting the polygenic risk scores for lipid traits shows such transfer learning methods lead to smaller prediction errors than the single sample ridge regression or Lasso-based transfer learning., Comment: 16 pages, 5 figures

The human microbiome has an important role in determining health. Mediation analyses quantify the contribution of the microbiome in the causal path between exposure and disease; however, current mediation models cannot fully capture the high dimensional, correlated, and compositional nature of microbiome data and do not typically accommodate dichotomous outcomes. We propose a novel approach that uses inverse odds weighting to test for the mediating effect of the microbiome. We use simulation to demonstrate that our approach gains power for high dimensional mediators, and it is agnostic to the effect of interactions between the exposure and mediators. Our application to infant gut microbiome data from the New Hampshire Birth Cohort Study revealed a mediating effect of 6-week infant gut microbiome on the relationship between maternal prenatal antibiotic use during pregnancy and incidence of childhood allergy by 5 years of age.

The Genotype-Tissue Expression (GTEx) project collects samples from multiple human tissues to study the relationship between genetic variation or single nucleotide polymorphisms (SNPs) and gene expression in each tissue. However, most existing eQTL analyses only focus on single tissue information. In this paper, we develop a multi-tissue eQTL analysis that improves the single tissue cis-SNP gene expression association analysis by borrowing information across tissues. Specifically, we propose an empirical Bayes regression model for SNP-expression association analysis using data across multiple tissues. To allow the effects of SNPs to vary greatly among tissues, we use a mixture distribution as the prior, which is a mixture of a multivariate Gaussian distribution and a Dirac mass at zero. The model allows us to assess the cis-SNP gene expression association in each tissue by calculating the Bayes factors. We show that the proposed estimator of the cis-SNP effects on gene expression achieves the minimum Bayes risk among all estimators. Analyses of the GTEx data show that our proposed method is superior to traditional simple regression methods in terms of predicting accuracy for gene expression levels using cis-SNPs in testing data sets. Moreover, we find that although genetic effects on expression are extensively shared among tissues, effect sizes still vary greatly across tissues.

Motivated by a range of applications, we study in this paper the problem of transfer learning for nonparametric contextual multi-armed bandits under the covariate shift model, where we have data collected on source bandits before the start of the target bandit learning. The minimax rate of convergence for the cumulative regret is established and a novel transfer learning algorithm that attains the minimax regret is proposed. The results quantify the contribution of the data from the source domains for learning in the target domain in the context of nonparametric contextual multi-armed bandits. In view of the general impossibility of adaptation to unknown smoothness, we develop a data-driven algorithm that achieves near-optimal statistical guarantees (up to a logarithmic factor) while automatically adapting to the unknown parameters over a large collection of parameter spaces under an additional self-similarity assumption. A simulation study is carried out to illustrate the benefits of utilizing the data from the auxiliary source domains for learning in the target domain., Comment: Accepted to the Annals of Statistics

Physical activity has long been shown to be associated with biological and physiological performance and risk of diseases. It is of great interest to assess whether the effect of an exposure or intervention on an outcome is mediated through physical activity measured by modern wearable devices such as actigraphy. However, existing methods for mediation analysis focus almost exclusively on mediation variable that is in the Euclidean space, which cannot be applied directly to the actigraphy data of physical activity. Such data is best summarized in the form of an histogram or density. In this paper, we extend the structural equation models (SEMs) to the settings where a density is treated as the mediator to study the indirect mediation effect of physical activity on an outcome. We provide sufficient conditions for identifying the average causal effects of density mediator and present methods for estimating the direct and mediating effects of density on an outcome. We apply our method to the data set from the iCOMPARE trial that compares flexible duty-hour policies and standard duty-hour policies on interns' sleep related outcomes to explore the mediation effect of physical activity on the causal path between flexible duty-hour policies and sleep related outcomes.

This paper studies the problem of statistical inference for genetic relatedness between binary traits based on individual-level genome-wide association data. Specifically, under the high-dimensional logistic regression models, we define parameters characterizing the cross-trait genetic correlation, the genetic covariance and the trait-specific genetic variance. A novel weighted debiasing method is developed for the logistic Lasso estimator and computationally efficient debiased estimators are proposed. The rates of convergence for these estimators are studied and their asymptotic normality is established under mild conditions. Moreover, we construct confidence intervals and statistical tests for these parameters, and provide theoretical justifications for the methods, including the coverage probability and expected length of the confidence intervals, as well as the size and power of the proposed tests. Numerical studies are conducted under both model generated data and simulated genetic data to show the superiority of the proposed methods. By analyzing a real data set on autoimmune diseases, we demonstrate its ability to obtain novel insights about the shared genetic architecture between ten pediatric autoimmune diseases.

Existing high-dimensional statistical methods are largely established for analyzing individual-level data. In this work, we study estimation and inference for high-dimensional linear models where we only observe "proxy data", which include the marginal statistics and sample covariance matrix that are computed based on different sets of individuals. We develop a rate optimal method for estimation and inference for the regression coefficient vector and its linear functionals based on the proxy data. Moreover, we show the intrinsic limitations in the proxy-data based inference: the minimax optimal rate for estimation is slower than that in the conventional case where individual data are observed; the power for testing and multiple testing does not go to one as the signal strength goes to infinity. These interesting findings are illustrated through simulation studies and an analysis of a dataset concerning the genetic associations of hindlimb muscle weight in a mouse population.

Quantification of microbial interactions from 16S rRNA and meta-genomic sequencing data is difficult due to their sparse nature, as well as the fact that the data only provides measures of relative abundance. In this paper, we propose using copula models with mixed zero-beta margins for estimation of taxon-taxon interactions using the normalized microbial relative abundances. Copulas allow for separate modeling of the dependence structure from the margins, marginal covariate adjustment, and uncertainty measurement. Our method shows that a two-stage maximum likelihood approach provides accurate estimation of the model parameters. A corresponding two-stage likelihood-ratio test for the dependence parameter is derived. Simulation studies show that the test is valid and more powerful than tests based upon Pearson's and rank correlations. Furthermore, we demonstrate that our method can be used to build biologically meaningful microbial networks based on the data set of the American Gut Project.

High-throughput sequencing technology allows us to test the compositional difference of bacteria in different populations. One important feature of human microbiome data is that it often includes a large number of zeros. Such data can be treated as being generated from a two-part model that includes a zero point-mass. Motivated by analysis of such non-negative data with excessive zeros, we introduce several truncated rank-based two-group and multi-group tests for such data, including a truncated rank-based Wilcoxon rank-sum test for two-group comparison and two truncated Kruskal-Wallis tests for multi-group comparison. We show both analytically through asymptotic relative efficiency analysis and by simulations that the proposed tests have higher power than the standard rank-based tests, especially when the proportion of zeros in the data is high. The tests can also be applied to repeated measurements of compositional data via simple within-subject permutations. In a simple before-and-after treatment experiment, the within-subject permutation is similar to the paired rank test. However, the proposed tests handle the excessive zeros, which leads to a better power. We apply the tests to the analysis of a gut microbiome data set to compare the microbiome compositions of healthy and pediatric Crohn's disease patients and to assess the treatment effects on microbiome compositions. We identify several bacterial genera that are missed by the standard rank-based tests.

Repeated observations have become increasingly common in biomedical research and longitudinal studies. For instance, wearable sensor devices are deployed to continuously track physiological and biological signals from each individual over multiple days. It remains of great interest to appropriately evaluate how the daily distribution of biosignals might differ across disease groups and demographics. Hence these data could be formulated as multivariate complex object data such as probability densities, histograms, and observations on a tree. Traditional statistical methods would often fail to apply as they are sampled from an arbitrary non-Euclidean metric space. In this paper, we propose novel non-parametric graph-based two-sample tests for object data with repeated measures. A set of test statistics are proposed to capture various possible alternatives. We derive their asymptotic null distributions under the permutation null. These tests exhibit substantial power improvements over the existing methods while controlling the type I errors under finite samples as shown through simulation studies. The proposed tests are demonstrated to provide additional insights on the location, inter- and intra-individual variability of the daily physical activity distributions in a sample of studies for mood disorders.

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.

Genome-wide association studies (GWAS) have identified thousands of genetic variants associated with complex traits, and some variants are shown to be associated with multiple complex traits. Genetic covariance between two traits is defined as the underlying covariance of genetic effects and can be used to measure the shared genetic architecture. The data used to estimate such a genetic covariance can be from the same group or different groups of individuals, and the traits can be of different types or collected based on different study designs. This paper proposes a unified regression-based approach to robust estimation and inference for genetic covariance of general traits that may be associated with genetic variants nonlinearly. The asymptotic properties of the proposed estimator are provided and are shown to be robust under certain model mis-specification. Our method under linear working models provides a robust inference for the narrow-sense genetic covariance, even when both linear models are mis-specified. Numerical experiments are performed to support the theoretical results. Our method is applied to an outbred mice GWAS data set to study the overlapping genetic effects between the behavioral and physiological phenotypes. The real data results reveal interesting genetic covariance among different mice developmental traits.

Blockwise missing data occurs frequently when we integrate multisource or multimodality data where different sources or modalities contain complementary information. In this paper, we consider a high-dimensional linear regression model with blockwise missing covariates and a partially observed response variable. Under this framework, we propose a computationally efficient estimator for the regression coefficient vector based on carefully constructed unbiased estimating equations and a blockwise imputation procedure, and obtain its rate of convergence. Furthermore, building upon an innovative projected estimating equation technique that intrinsically achieves bias-correction of the initial estimator, we propose a nearly unbiased estimator for each individual regression coefficient, which is asymptotically normally distributed under mild conditions. Based on these debiased estimators, asymptotically valid confidence intervals and statistical tests about each regression coefficient are constructed. Numerical studies and application analysis of the Alzheimer's Disease Neuroimaging Initiative data show that the proposed method performs better and benefits more from unsupervised samples than existing methods., Comment: V2: 40 pages, 2 figures. Accepted at Statistica Sinica

This paper studies the high-dimensional mixed linear regression (MLR) where the output variable comes from one of the two linear regression models with an unknown mixing proportion and an unknown covariance structure of the random covariates. Building upon a high-dimensional EM algorithm, we propose an iterative procedure for estimating the two regression vectors and establish their rates of convergence. Based on the iterative estimators, we further construct debiased estimators and establish their asymptotic normality. For individual coordinates, confidence intervals centered at the debiased estimators are constructed. Furthermore, a large-scale multiple testing procedure is proposed for testing the regression coefficients and is shown to control the false discovery rate (FDR) asymptotically. Simulation studies are carried out to examine the numerical performance of the proposed methods and their superiority over existing methods. The proposed methods are further illustrated through an analysis of a dataset of multiplex image cytometry, which investigates the interaction networks among the cellular phenotypes that include the expression levels of 20 epitopes or combinations of markers.

Transfer learning for high-dimensional Gaussian graphical models (GGMs) is studied with the goal of estimating the target GGM by utilizing the data from similar and related auxiliary studies. The similarity between the target graph and each auxiliary graph is characterized by the sparsity of a divergence matrix. An estimation algorithm, Trans-CLIME, is proposed and shown to attain a faster convergence rate than the minimax rate in the single study setting. Furthermore, a debiased Trans-CLIME estimator is introduced and shown to be element-wise asymptotically normal. It is used to construct a multiple testing procedure for edge detection with false discovery rate control. The proposed estimation and multiple testing procedures demonstrate superior numerical performance in simulations and are applied to infer the gene networks in a target brain tissue by leveraging the gene expressions from multiple other brain tissues. A significant decrease in prediction errors and a significant increase in power for link detection are observed.

This paper considers the estimation and prediction of a high-dimensional linear regression in the setting of transfer learning, using samples from the target model as well as auxiliary samples from different but possibly related regression models. When the set of "informative" auxiliary samples is known, an estimator and a predictor are proposed and their optimality is established. The optimal rates of convergence for prediction and estimation are faster than the corresponding rates without using the auxiliary samples. This implies that knowledge from the informative auxiliary samples can be transferred to improve the learning performance of the target problem. In the case that the set of informative auxiliary samples is unknown, we propose a data-driven procedure for transfer learning, called Trans-Lasso, and reveal its robustness to non-informative auxiliary samples and its efficiency in knowledge transfer. The proposed procedures are demonstrated in numerical studies and are applied to a dataset concerning the associations among gene expressions. It is shown that Trans-Lasso leads to improved performance in gene expression prediction in a target tissue by incorporating the data from multiple different tissues as auxiliary samples.

The human gastrointestinal tract is an environment that hosts an ecosystem of microorganisms essential to human health. Vital biological processes emerge from fundamental inter- and intra-species molecular interactions that influence the assembly and composition of the gut microbiota ecology. Here we quantify the complexity of the ecological relationships within the human infant gut microbiota ecosystem as a function of the information contained in the nonlinear associations of a sequence of increasingly-specified maximum entropy representations of the system. Our paradigm frames the ecological state, in terms of the presence or absence of individual microbial ecological units that are identified by amplicon sequence variants (ASV) in the gut microenvironment, as a function of both the ecological states of its neighboring units and, in a departure from standard graphical model representations, the associations among the units within its neighborhood. We characterize the order of the system based on the relative quantity of statistical information encoded by high-order statistical associations of the infant gut microbiota., Comment: 21 pages, 3 figures

Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general structured principal subspace estimation problem which includes as special cases non-negative PCA/SVD, sparse PCA/SVD, subspace constrained PCA/SVD, and spectral clustering. General minimax lower and upper bounds are established to characterize the interplay between the information-geometric complexity of the structural set for the principal subspaces, the signal-to-noise ratio (SNR), and the dimensionality. The results yield interesting phase transition phenomena concerning 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 minimax rates of convergence for those problems, including the previous unknown optimal rates for non-negative PCA/SVD, sparse SVD and subspace constrained PCA/SVD.

Motivated by the problem of estimating the 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. This paper studies the estimation of the extreme values associated to the signal matrix $\Theta$, including its first and last columns, as well as their difference. Treating these estimation problems as compound decision problems, minimax rate-optimal estimators are constructed using the spectral column sorting method. Numerical experiments through simulated and synthetic microbiome metagenomic data are presented, showing the superiority of the proposed methods over the alternatives. The methods are illustrated by comparing the growth rates of gut bacteria between inflammatory bowel disease patients and normal controls.

We consider high dimensional $M$-estimation in settings where the response $Y$ is possibly missing at random and the covariates $\mathbf{X} \in \mathbb{R}^p$ can be high dimensional compared to the sample size $n$. The parameter of interest $\boldsymbol{\theta}_0 \in \mathbb{R}^d$ is defined as the minimizer of the risk of a convex loss, under a fully non-parametric model, and $\boldsymbol{\theta}_0$ itself is high dimensional which is a key distinction from existing works. Standard high dimensional regression and series estimation with possibly misspecified models and missing $Y$ are included as special cases, as well as their counterparts in causal inference using 'potential outcomes'. Assuming $\boldsymbol{\theta}_0$ is $s$-sparse ($s \ll n$), we propose an $L_1$-regularized debiased and doubly robust (DDR) estimator of $\boldsymbol{\theta}_0$ based on a high dimensional adaptation of the traditional double robust (DR) estimator's construction. Under mild tail assumptions and arbitrarily chosen (working) models for the propensity score (PS) and the outcome regression (OR) estimators, satisfying only some high-level conditions, we establish finite sample performance bounds for the DDR estimator showing its (optimal) $L_2$ error rate to be $\sqrt{s (\log d)/ n}$ when both models are correct, and its consistency and DR properties when only one of them is correct. Further, when both the models are correct, we propose a desparsified version of our DDR estimator that satisfies an asymptotic linear expansion and facilitates inference on low dimensional components of $\boldsymbol{\theta}_0$. Finally, we discuss various of choices of high dimensional parametric/semi-parametric working models for the PS and OR estimators. All results are validated via detailed simulations., Comment: 34 pages, 4 tables; (Supplement: 58 pages, 10 tables)

Motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies, we consider a permuted monotone matrix model $Y=\Theta\Pi+Z$, where the rows represent different samples, the columns represent contigs in genome assemblies and the elements represent log-read counts after preprocessing steps and Guanine-Cytosine (GC) adjustment. In this model, $\Theta$ is an unknown mean matrix with monotone entries for each row, $\Pi$ is a permutation matrix that permutes the columns of $\Theta$, and $Z$ is a noise matrix. This paper studies the problem of estimation/recovery of $\Pi$ given the observed noisy matrix $Y$. We propose an estimator based on the best linear projection, which is shown to be minimax rate-optimal for both exact recovery, as measured by the 0-1 loss, and partial recovery, as quantified by the normalized Kendall's tau distance. Simulation studies demonstrate the superior empirical performance of the proposed estimator over alternative methods. We demonstrate the methods using a synthetic metagenomics dataset of 45 closely related bacterial species and a real metagenomic dataset to compare the bacterial growth dynamics between the responders and the non-responders of the IBD patients after 8 weeks of treatment.

Gene expression and phenotype association can be affected by potential unmeasured confounders from multiple sources, leading to biased estimates of the associations. Since genetic variants largely explain gene expression variations, they can be used as instruments in studying the association between gene expressions and phenotype in the framework of high dimensional instrumental variable (IV) regression. However, because the dimensions of both genetic variants and gene expressions are often larger than the sample size, statistical inferences such as hypothesis testing for such high dimensional IV models are not trivial and have not been investigated in literature. The problem is more challenging since the instrumental variables (e.g., genetic variants) have to be selected among a large set of genetic variants. This paper considers the problem of hypothesis testing for sparse IV regression models and presents methods for testing single regression coefficient and multiple testing of multiple coefficients, where the test statistic for each single coefficient is constructed based on an inverse regression. A multiple testing procedure is developed for selecting variables and is shown to control the false discovery rate. Simulations are conducted to evaluate the performance of our proposed methods. These methods are illustrated by an analysis of a yeast dataset in order to identify genes that are associated with growth in the presence of hydrogen peroxide.