High-dimensional logistic regression is widely used in analyzing data with binary outcomes. In this paper, global testing and large-scale multiple testing for the regression coefficients are considered in both single- and two-regression settings. A test statistic for testing the global null hypothesis is constructed using a generalized low-dimensional projection for bias correction and its asymptotic null distribution is derived. A minimax lower bound for the global testing is established, which shows that the proposed test is asymptotically minimax optimal. For testing the individual coefficients simultaneously, multiple testing procedures are proposed and shown to control the false discovery rate (FDR) and falsely discovered variables (FDV) asymptotically. Simulation studies are carried out to examine the numerical performance of the proposed tests and their superiority over existing methods. The testing procedures are also illustrated by analyzing a metabolomics study that investigates the association between fecal metabolites and pediatric Crohn's disease and the effects of treatment on such associations.
The problems of global testing and large-scale simultaneous testing for high-dimensional linear regression have been studied by @cite_11 and more recently by @cite_21 . An inverse regression approach is used and testing procedures are constructed based on coordinate-wise standardized test statistics. However, unlike linear regression, the outcome variable in logistic regression is binary and does not follow a joint multivariate normal distribution along with the covariates. Thus the inverse regression approach therein cannot be directly applied to logistic regression problems. In the Markov random field setting, @cite_24 and @cite_5 constructed pivotal test statistics based on the de-biased LDP estimators for node-wise logistic regressions with binary covariates. Nevertheless, the results for sparse high-dimensional logistic regression models with general continuous covariates remain unknown. Other related problems include joint testing and false discovery rate control for high-dimensional multivariate regression and testing for high-dimensional precision matrices and Gaussian graphical models , where the inverse regression approach and de-biasing are also used in the construction of the test statistics. Such statistics are then used for testing the global null with extreme value type asymptotic null distributions or to perform multiple testing that controls the false discovery rate.
@cite_24 @cite_5 @cite_21 @cite_11
2206289079 2063751559 2588527329 2014100575
We study parameter estimation and asymptotic inference for sparse nonlinear regression. More specifically, we assume the data are given by y = f(x > ) + , where f is nonlinear. To recover , we propose an ‘1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. In addition, we provide an ecient algorithm that provably converges to a stationary point. We also access the uncertainty of the obtained estimator. Specifically, based on any stationary point of the objective, we construct valid hypothesis tests and confidence intervals for the low dimensional components of the high-dimensional parameter . Detailed numerical results are provided to back up our theory. Model organisms and human studies have yielded increasing empirical evidence that interactions among genes contribute broadly to genetic variation of complex traits. In the presence of gene-gene interactions, the dimensionality of the feature space becomes extremely high relative to the sample size. This poses a significant methodological challenge in the identification of gene-gene interactions. In this paper, by using a Gaussian graphical model framework, we translate the problem of identifying gene-gene interactions associated with a binary trait D into an inference problem on the difference of two high-dimensional precision matrices that summarize the conditional dependence network structures of the genes. We propose a procedure for testing the differential network globally, which is particularly powerful against sparse alternatives. In addition, a multiple testing procedure with false discovery rate control is developed to infer the specific structure of the differential network. Theoretical justification is provided to ensure the validity of the proposed tests, and optimality results are derived under sparsity assumptions. Through a simulation study we demonstrate that the proposed tests maintain the desired error rates under the null hypothesis and have good power under the alternative hypothesis. The methods are applied to a breast cancer gene expression study. In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. Motivated by genetic sequencing association studies for rare variant effects, we investigate the complexity of the hypothesis testing problem when the design matrix is sparse. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear regression. We derive the detection boundary as a function of two components: a design matrix sparsity index and signal strength, each of which is a function of the sparsity of the alternative. For any alternative, if the design matrix sparsity index is too high, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices with the sparsity index that is not too high, our results are parallel to those in the Gaussian case. In this context, we derive detection boundaries for both dense and sparse regimes. For the dense regime, we show that the generalized likelihood ratio is rate optimal; for the sparse regime, we propose an extended Higher Criticism Test and show it is rate optimal and sharp. We illustrate the finite sample properties of the theoretical results using simulation studies.