Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Oftentimes, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test and also to a test obtained from a modified profile likelihood function. Our results generalize those in Zucker et al. (Journal of the Royal Statistical Society B, 2000, 62, 827-838) by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report numerical evidence which shows that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed.
We present a brief overview of the methods for making statistical inference (testing statistical hypotheses, construction of confidence and/or prediction intervals and regions) about linear functions of the fixed effects and/or about the fixed and random effects simultaneously, in conventional simple linear mixed model. The presented approach is based on solutions from the Henderson's mixed model equations.
University of California San Diego（加州大学圣迭戈分校）；KU Leuven（鲁汶大学）
Many scientific and engineering challenges -- ranging from pharmacokinetic drug dosage allocation and personalized medicine to marketing mix (4Ps) recommendations -- require an understanding of the unobserved heterogeneity in order to develop the best decision making-processes. In this paper, we develop a hypothesis test and the corresponding p-value for testing for the significance of the homogeneous structure in linear mixed models. A robust matching moment construction is used for creating a test that adapts to the size of the model sparsity. When unobserved heterogeneity at a cluster level…
This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio statistics and Barndorff-Nielsen's adjusted signed likelihood ratio statistics and we conduct a simula…
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 …
An important feature of linear mixed models and generalized linear mixed models is that the conditional mean of the response given the random effects, after transformed by a link function, is linearly related to the fixed covariate effects and random effects. Therefore, it is of practical importance to test the adequacy of this assumption, particularly the assumption of linear covariate effects. In this paper, we review procedures that can be used for testing polynomial covariate effects in these popular models. Specifically, four types of hypothesis testing approaches are reviewed, i.e. R tes…
In this paper, the problem of error control of stepwise multiple testing procedures is considered. For two-sided hypotheses, control of both type 1 and type 3 (or directional) errors is required, and thus mixed directional familywise error rate control and mixed directional false discovery rate control are each considered by incorporating both types of errors in the error rate. Mixed directional familywise error rate control of stepwise methods in multiple testing has proven to be a challenging problem, as demonstrated in Shaffer (1980). By an appropriate formulation of the problem, some new s…
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…
We consider three problems in high-dimensional Gaussian linear mixed models. Without any assumptions on the design for the fixed effects, we construct an asymptotic $F$-statistic for testing whether a collection of random effects is zero, derive an asymptotic confidence interval for a single random effect at the parametric rate $\sqrt{n}$, and propose an empirical Bayes estimator for a part of the mean vector in ANOVA type models that performs asymptotically as well as the oracle Bayes estimator. We support our results with numerical simulations and provide comparisons with oracle estimators. …
Generalized linear mixed models (GLMMs) are used to model responses from exponential families with a combination of fixed and random effects. For variance components in GLMMs, we propose an approximate restricted likelihood ratio test that conducts testing on the working responses used in penalized quasi-likelihood estimation. This presents the hypothesis test in terms of normalized responses, allowing for application of existing testing methods for linear mixed models. Our test is flexible, computationally efficient, and outperforms several competitors. We illustrate the utility of the propos…
We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464], Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847--860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias prob…
Fan et al. (2015) recently introduced a remarkable method for increasing asymptotic power of tests in high-dimensional testing problems. If applicable to a given test, their power enhancement principle leads to an improved test that has the same asymptotic size, uniformly non-inferior asymptotic power, and is consistent against a strictly broader range of alternatives than the initially given test. We study under which conditions this method can be applied and show the following: In asymptotic regimes where the dimensionality of the parameter space is fixed as sample size increases, there ofte…
The paper estimates the Chernoff rate for the efficiency of quantum hypothesis testing. For both joint and separable measurements, approximate bounds for the rate are given if both states are mixed and exact expressions are derived if at least one of the states is pure. The efficiency of tests with separable measurements is found to be close to the efficiency of tests with joint measurements. The results are illustrated by a test of quantum entanglement.
In this paper, we consider a single-index mixed model with longitudinal data. A new set of estimating equations is proposed to estimate the single-index coefficient. The link function is estimated by using the local linear smoothing. Asymptotic normality is established for the proposed estimators. Also, the estimator of the link function achieves optimal convergence rates; and the estimators of variance components have root-$n$ consistency. These results facilitate the construction of confidence regions/intervals and hypothesis testing for the parameters of interest. Some simulations and an ap…
Conformal testing is a way of testing the IID assumption based on conformal prediction. The topic of this note is computational evaluation of the performance of conformal testing in a model situation in which IID binary observations generated from a Bernoulli distribution are followed by IID binary observations generated from another Bernoulli distribution, with the parameters of the distributions and changepoint unknown. Existing conformal test martingales can be used for this task and work well in simple cases, but their efficiency can be improved greatly.
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…
Indian Institute of Science（印度科学学院）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；MPI-SWS（马克斯·普朗克软件系统研究所）
We study nonzero-sum hypothesis testing games that arise in the context of adversarial classification, in both the Bayesian as well as the Neyman-Pearson frameworks. We first show that these games admit mixed strategy Nash equilibria, and then we examine some interesting concentration phenomena of these equilibria. Our main results are on the exponential rates of convergence of classification errors at equilibrium, which are analogous to the well-known Chernoff-Stein lemma and Chernoff information that describe the error exponents in the classical binary hypothesis testing problem, but with pa…
Recently Hui et al. (2018) use F tests for testing a subset of random effect, demonstrating its computational simplicity and exactness when the first two moment of the random effects are specified. We extended the investigation of the F test in the following two aspects: firstly, we examined the power of the F test under non-normality of the errors. Secondly, we consider bootstrap counterparts to the F test, which offer improvement for the cases with small cluster size or for the cases with non-normal errors.
We consider the problems of hypothesis testing and model comparison under a flexible Bayesian linear regression model whose formulation is closely connected with the linear mixed effect model and the parametric models for SNP set analysis in genetic association studies. We derive a class of analytic approximate Bayes factors and illustrate their connections with a variety of frequentist test statistics, including the Wald statistic and the variance component score statistic. Taking advantage of Bayesian model averaging and hierarchical modeling, we demonstrate some distinct advantages and flex…
The issue of variance components testing arises naturally when building mixed-effects models, to decide which effects should be modeled as fixed or random. While tests for fixed effects are available in R for models fitted with lme4, tools are missing when it comes to random effects. The varTestnlme package for R aims at filling this gap. It allows to test whether any subset of the variances and covariances corresponding to any subset of the random effects, are equal to zero using asymptotic property of the likelihood ratio test statistic. It also offers the possibility to test simultaneously …
When we use the normal mixture model, the optimal number of the components describing the data should be determined. Testing homogeneity is good for this purpose; however, to construct its theory is challenging, since the test statistic does not converge to the $χ^{2}$ distribution even asymptotically. The reason for such asymptotic behavior is that the parameter set describing the null hypothesis (N.H.) contains singularities in the space of the alternative hypothesis (A.H.). Recently, a $\it{Bayesian}$ theory for singular models was developed, and it has elucidated various problems of statis…
We propose a novel and easy-to-implement joint location-scale association testing procedure that can account for complex genetic architecture without explicitly modeling interaction effects, and is suitable for large-scale whole-genome scans and meta-analyses. We focus on Fisher's method and use it to combine evidence from the standard location test and the more recent scale test, and we describe its use for single-variant, gene-set and pathway association analyses.
As the sequencing costs are decreasing, there is great incentive to perform large scale association studies to increase power of detecting new variants. Federated association testing among different institutions is a viable solution for increasing sample sizes by sharing the intermediate testing statistics that are aggregated by a central server. There are, however, standing challenges to performing federated association testing. Association tests are known to be confounded by numerous factors such as population stratification, which can be especially important in multiancestral studies and in…
Mixed Gaussian Projections for Two-Sample Testing of Functional Data
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC (UFABC)（阿贝辛联邦大学数学、计算与认知中心）；Departamento de Matemática, Universidad de San Andrés–CONICET（圣安德烈斯大学数学系-阿根廷国家科学研究委员会）
Consider a large social network with possibly severe degree heterogeneity and mixed-memberships. We are interested in testing whether the network has only one community or there are more than one communities. The problem is known to be non-trivial, partially due to the presence of severe degree heterogeneity. We construct a class of test statistics using the numbers of short paths and short cycles, and the key to our approach is a general framework for canceling the effects of degree heterogeneity. The tests compare favorably with existing methods. We support our methods with careful analysis …
