This is a chapter of the forthcoming Handbook of Multiple Testing. We consider a variety of model selection strategies in a high-dimensional setting, where the number of potential predictors p is large compared to the number of available observations n. In particular modifications of information criteria which are suitable in case of p > n are introduced and compared with a variety of penalized likelihood methods, in particular SLOPE and SLOBE. The focus is on methods which control the FDR in terms of model identification. Theoretical results are provided both with respect to model identification and prediction and various simulation results are presented which illustrate the performance of the different methods in different situations.
Scalar-on-function linear models are commonly used to regress functional predictors on a scalar response. However, functional models are more difficult to estimate and interpret than traditional linear models, and may be unnecessarily complex for a data application. Hypothesis testing can be used to guide model selection by determining if a functional predictor is necessary. Using a mixed effects representation with penalized splines and variance component tests, we propose a framework for testing functional linear models with responses from exponential family distributions. The proposed metho…
Hypothesis Testing for Nonlinear and Interactive Effects Using Additive Gaussian Processes
Department of Biostatistics, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校生物统计系）；Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校环境科学与工程系）
A number of biomedical problems require performing many hypothesis tests, with an attendant need to apply stringent thresholds. Often the data take the form of a series of predictor vectors, each of which must be compared with a single response vector, perhaps with nuisance covariates. Parametric tests of association are often used, but can result in inaccurate type I error at the extreme thresholds, even for large sample sizes. Furthermore, standard two-sided testing can reduce power compared to the doubled $p$-value, due to asymmetry in the null distribution. Exact (permutation) testing is a…
The so-called gut-brain axis has stimulated extensive research on microbiomes. One focus is to assess the association between certain clinical outcomes and the relative abundances of gut microbes, which can be presented as sub-compositional data in conformity with the taxonomic hierarchy of bacteria. Motivated by a study for identifying the microbes in the gut microbiome of preterm infants that impact their later neurobehavioral outcomes, we formulate a constrained integrative multi-view regression, where the neurobehavioral scores form multivariate response, the sub-compositional microbiome d…
Large-scale hypothesis testing has become a ubiquitous problem in high-dimensional statistical inference, with broad applications in various scienfitic disciplines. One relevant application is constituted by imaging mass spectrometry (IMS) association studies, where a large number of tests are performed simultaneously in order to identify molecular masses that are associated with a particular phenotype, e. g., a cancer subtype. Mass spectra obtained from Matrix-assisted laser desorption/ionization (MALDI) experiments are dependent, when considered as statistical quantities. False discovery pro…
Discrete but ordered covariates are quite common in applied statistics, and some regularized fitting procedures have been proposed for proper handling of ordinal predictors in statistical modeling. In this study, we show how quadratic penalties on adjacent dummy coefficients of ordinal predictors proposed in the literature can be incorporated in the framework of generalized additive models, making tools for statistical inference developed there available for ordinal predictors as well. Motivated by an application from neonatal medicine, we discuss whether results obtained when constructing con…
Neural networks are powerful predictive models, but they provide little insight into the nature of relationships between predictors and outcomes. Although numerous methods have been proposed to quantify the relative contributions of input features, statistical inference and hypothesis testing of feature associations remain largely unexplored. We propose a permutation-based approach to testing that uses the partial derivatives of the network output with respect to specific inputs to assess both the significance of input features and whether significant features are linearly associated with the …
This paper presents a goodness-of-fit test for parametric regression models with scalar response and directional predictor, that is, a vector on a sphere of arbitrary dimension. The testing procedure is based on the weighted squared distance between a smooth and a parametric regression estimator, where the smooth regression estimator is obtained by a projected local approach. Asymptotic behavior of the test statistic under the null hypothesis and local alternatives is provided, jointly with a consistent bootstrap algorithm for application in practice. A simulation study illustrates the perform…
In the high dimensional regression analysis when the number of predictors is much larger than the sample size, an important question is to select the important variable which are relevant to the response variable of interest. Variable selection and the multiple testing are both tools to address this issue. However, there is little discussion on the connection of these two areas. When the signal strength is strong enough such that the selection consistency is achievable, it seems to be unnecessary to control the false discovery rate. In this paper, we consider the regime where the signals are b…
Considerable interest has recently been focused on studying multiple phenotypes simultaneously in both epidemiological and genomic studies, either to capture the multidimensionality of complex disorders or to understand shared etiology of related disorders. We seek to identify {\em multiple regulators} or predictors that are associated with multiple outcomes when these outcomes may be measured on very different scales or composed of a mixture of continuous, binary, and not-fully-observed elements. We first propose an estimation technique to put all effects on similar scales, and we induce spar…
For regression models, most of existing specification tests can be categorized into the class of local smoothing tests and of global smoothing tests. Compared with global smoothing tests, local smoothing tests can only detect local alternatives distinct from the null hypothesis at a much slower rate when the dimension of predictor vector is high, but can be more sensitive to oscillating alternatives. In this paper, we suggest a projection-based test to bridge between the local and global smoothing-based methodologies such that the test can benefit from the advantages of these two types of test…
University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
We study the new feature utility prediction problem: statistically testing whether adding a new feature to the data representation can improve predictive accuracy on a supervised learning task. In many applications, identifying new informative features is the primary pathway for improving performance. However, evaluating every potential feature by re-training the predictor with it can be costly. The paper describes an efficient, learner-independent technique for estimating new feature utility without re-training based on the current predictor's outputs. The method is obtained by deriving a con…
The research described in this paper is motivated by model checking for parametric single-index models with diverging number of predictors. To construct a test statistic, we first study the asymptotic property of the estimators of involved parameters of interest under the null and alternative hypothesis when the dimension is divergent to infinity as the sample size goes to infinity. For the testing problem, we study an adaptive-to-model residual-marked empirical process as the basis for constructing a test statistic. By modifying the approach in the literature to suit the diverging dimension s…
This paper develops a new approach to post-selection inference for screening high-dimensional predictors of survival outcomes. Post-selection inference for right-censored outcome data has been investigated in the literature, but much remains to be done to make the methods both reliable and computationally-scalable in high-dimensions. Machine learning tools are commonly used to provide {\it predictions} of survival outcomes, but the estimated effect of a selected predictor suffers from confirmation bias unless the selection is taken into account. The new approach involves construction of semi-p…
Determining a lack of association between an outcome variable and a number of different explanatory variables is frequently necessary in order to disregard a proposed model (i.e., to confirm the lack of an association between an outcome and predictors). Despite this, the literature rarely offers information about, or technical recommendations concerning, the appropriate statistical methodology to be used to accomplish this task. This paper introduces non-inferiority tests for ANOVA and linear regression analyses, that correspond to the standard widely used $F$-test for $\hatη^2$ and $R^{2}$, r…
Scientific theories can often be formulated using equality and order constraints on the relative effects in a linear regression model. For example, it may be expected that the effect of the first predictor is larger than the effect of the second predictor, and the second predictor is expected to be larger than the third predictor. The goal is then to test such expectations against competing scientific expectations or theories. In this paper a simple default Bayes factor test is proposed for testing multiple hypotheses with equality and order constraints on the effects of interest. The proposed…
Today, generalized linear mixed models are broadly used in many fields. However, the development of tools for performing simultaneous inference has been largely neglected in this domain. A framework for joint inference is indispensable to carry out statistically valid multiple comparisons of parameters of interest between all or several clusters. We therefore develop simultaneous confidence intervals and multiple testing procedures for empirical best predictors under generalized linear mixed models. In addition, we implement our methodology to study widely employed examples of mixed models, th…
We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^2$-distances - the new…
A common and important problem arising in the study of networks is how to divide the vertices of a given network into one or more groups, called communities, in such a way that vertices of the same community are more interconnected than vertices belonging to different ones. We propose and investigate a testing based community detection procedure called Extraction of Statistically Significant Communities (ESSC). The ESSC procedure is based on $p$-values for the strength of connection between a single vertex and a set of vertices under a reference distribution derived from a conditional configur…
In life-testing experiments, it is often of interest to predict unobserved future failure times based on observed early failure times. A point best linear unbiased predictor (BLUP) has been developed in this context by Kaminsky and Nelson (1975). In this article, we develop joint BLUPs of two future failure times based on early failure times by minimizing the determinant of the variance-covariance matrix of the predictors. The advantage of applying joint prediction is demonstrated by using a real data set. The non-existence of joint BLUPs in certain setups is also discussed.
In this paper, we propose a new method remMap -- REgularized Multivariate regression for identifying MAster Predictors -- for fitting multivariate response regression models under the high-dimension-low-sample-size setting. remMap is motivated by investigating the regulatory relationships among different biological molecules based on multiple types of high dimensional genomic data. Particularly, we are interested in studying the influence of DNA copy number alterations on RNA transcript levels. For this purpose, we model the dependence of the RNA expression levels on DNA copy numbers through m…
We consider parameter estimation, hypothesis testing and variable selection for partially time-varying coefficient models. Our asymptotic theory has the useful feature that it can allow dependent, nonstationary error and covariate processes. With a two-stage method, the parametric component can be estimated with a $n^{1/2}$-convergence rate. A simulation-assisted hypothesis testing procedure is proposed for testing significance and parameter constancy. We further propose an information criterion that can consistently select the true set of significant predictors. Our method is applied to autor…
We consider the problem of identifying significant predictors in large data bases, where the response variable depends on the linear combination of explanatory variables through an unknown link function, corrupted with the noise from the unknown distribution. We utilize the natural, robust and efficient approach, which relies on replacing values of the response variables by their ranks and then identifying significant predictors by using well known Lasso. We provide new consistency results for the proposed procedure (called ,,RankLasso") and extend the scope of its applications by proposing it…
In the fight against hard-to-treat diseases such as cancer, it is often difficult to discover new treatments that benefit all subjects. For regulatory agency approval, it is more practical to identify subgroups of subjects for whom the treatment has an enhanced effect. Regression trees are natural for this task because they partition the data space. We briefly review existing regression tree algorithms. Then we introduce three new ones that are practically free of selection bias and are applicable to two or more treatments, censored response variables, and missing values in the predictor varia…
In genomic studies, identifying biomarkers associated with a variable of interest is a major concern in biomedical research. Regularized approaches are classically used to perform variable selection in high-dimensional linear models. However, these methods can fail in highly correlated settings. We propose a novel variable selection approach called WLasso, taking these correlations into account. It consists in rewriting the initial high-dimensional linear model to remove the correlation between the biomarkers (predictors) and in applying the generalized Lasso criterion. The performance of WLas…
Model selection consistency in the high-dimensional regression setting can be achieved only if strong assumptions are fulfilled. We therefore suggest to pursue a different goal, which we call a minimal class of models. The minimal class of models includes models that are similar in their prediction accuracy but not necessarily in their elements. We suggest a random search algorithm to reveal candidate models. The algorithm implements simulated annealing while using a score for each predictor that we suggest to derive using a combination of the Lasso and the Elastic Net. The utility of using a …
We consider a linear regression model and propose an omnibus test to simultaneously check the assumption of independence between the error and the predictor variables, and the goodness-of-fit of the parametric model. Our approach is based on testing for independence between the residual obtained from the parametric fit and the predictor using the Hilbert--Schmidt independence criterion (Gretton et al. (2008)). The proposed method requires no user-defined regularization, is simple to compute, based merely on pairwise distances between points in the sample, and is consistent against all alternat…
Variable Selection in Functional Linear Quantile Regression for Identifying Associations between Daily Patterns of Physical Activity and Cognitive Function
Variable selection, also known as feature selection in machine learning, plays an important role in modeling high dimensional data and is key to data-driven scientific discoveries. We consider here the problem of detecting influential variables under the general index model, in which the response is dependent of predictors through an unknown function of one or more linear combinations of them. Instead of building a predictive model of the response given combinations of predictors, we model the conditional distribution of predictors given the response. This inverse modeling perspective motivate…
The Akaike information criterion (AIC) is commonly used to select a logistic regression model for optimal prediction of a binary response by a specified family of models. It however lacks a convincing method of prescribing a proper family of models using the desired predictors and their interaction effects. For an alternative approach to model selection, we propose a direct selection scheme which first identifies the indispensable regressors as main-effect predictors, then examines significant interaction effects between the selected predictors such that a logistic model is constructed. The tw…
Variable selection is an important statistical problem. This problem becomes more challenging when the candidate predictors are of mixed type (e.g. continuous and binary) and impact the response variable in nonlinear and/or non-additive ways. In this paper, we review existing variable selection approaches for the Bayesian additive regression trees (BART) model, a nonparametric regression model, which is flexible enough to capture the interactions between predictors and nonlinear relationships with the response. An emphasis of this review is on the capability of identifying relevant predictors.…
This paper presents Sparse Partitioning, a Bayesian method for identifying predictors that either individually or in combination with others affect a response variable. The method is designed for regression problems involving binary or tertiary predictors and allows the number of predictors to exceed the size of the sample, two properties which make it well suited for association studies. Sparse Partitioning differs from other regression methods by placing no restrictions on how the predictors may influence the response. To compensate for this generality, Sparse Partitioning implements a novel…
The spatial panel regression model has shown great success in modelling econometric and other types of data that are observed both spatially and temporally with associated predictor variables. However, model checking via testing for spatial correlations in spatial-temporal residuals is still lacking. We propose a general methodology for fast permutation testing of local and global indicators of spatial association. This methodology extends past statistics for univariate spatial data that can be written as a gamma index for matrix similarity to the multivariate and panel data settings. This inc…
We consider the joint sparse estimation of regression coefficients and the covariance matrix for covariates in a high-dimensional regression model, where the predictors are both relevant to a response variable of interest and functionally related to one another via a Gaussian directed acyclic graph (DAG) model. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying predictors. A variety of methods have been developed in recent years for Bayesi…
In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an $\operatorname {Exp}(1)$ asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first…
Selecting powerful predictors for an outcome is a cornerstone task for machine learning. However, some types of questions can only be answered by identifying the predictors that causally affect the outcome. A recent approach to this causal inference problem leverages the invariance property of a causal mechanism across differing experimental environments (Peters et al., 2016; Heinze-Deml et al., 2018). This method, invariant causal prediction (ICP), has a substantial computational defect -- the runtime scales exponentially with the number of possible causal variables. In this work, we show tha…
