Multiple hypotheses testing is a core problem in statistical inference and arises in almost every scientific field. Given a sequence of null hypotheses @math , Benjamini and Hochberg benjamini1995controlling introduced the false discovery rate (FDR) criterion, which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls FDR below a pre-assigned significance level. They also proposed a different criterion, called mFDR, which does not control a property of the realized set of tests; rather it controls the ratio of expected number of false discoveries to the expected number of discoveries. In this paper, we propose two procedures for multiple hypotheses testing that we will call "LOND" and "LORD". These procedures control FDR and mFDR in an . Concretely, we consider an ordered --possibly infinite-- sequence of null hypotheses @math where, at each step @math , the statistician must decide whether to reject hypothesis @math having access only to the previous decisions. To the best of our knowledge, our work is the first that controls FDR in this setting. This model was introduced by Foster and Stine alpha-investing whose alpha-investing rule only controls mFDR in online manner. In order to compare different procedures, we develop lower bounds on the total discovery rate under the mixture model and prove that both LOND and LORD have nearly linear number of discoveries. We further propose adjustment to LOND to address arbitrary correlation among the @math -values. Finally, we evaluate the performance of our procedures on both synthetic and real data comparing them with alpha-investing rule, Benjamin-Hochberg method and a Bonferroni procedure.
There has been significant interest over the last two years in developing hypothesis testing procedures for high-dimensional regression, especially in conjunction with sparsity-seeking methods. Procedures for computing @math -values of low-dimensional coordinates were developed in @cite_0 @cite_12 @cite_2 @cite_17 @cite_10 . Sequential and selective inference methods were proposed in @cite_14 @cite_1 @cite_18 . Methods to control FDR were put forward in @cite_6 @cite_22 .
@cite_18 @cite_14 @cite_22 @cite_1 @cite_6 @cite_0 @cite_2 @cite_10 @cite_12 @cite_17
2109177042 2047028564 2014100575 2097209031
In many fields of science, we observe a response variable together with a large number of potential explanatory variables, and would like to be able to discover which variables are truly associated with the response. At the same time, we need to know that the false discovery rate (FDR) - the expected fraction of false discoveries among all discoveries - is not too high, in order to assure the scientist that most of the discoveries are indeed true and replicable. This paper introduces the knockoff filter, a new variable selection procedure controlling the FDR in the statistical linear model whenever there are at least as many observations as variables. This method achieves exact FDR control in finite sample settings no matter the design or covariates, the number of variables in the model, or the amplitudes of the unknown regression coefficients, and does not require any knowledge of the noise level. As the name suggests, the method operates by manufacturing knockoff variables that are cheap - their construction does not require any new data - and are designed to mimic the correlation structure found within the existing variables, in a way that allows for accurate FDR control, beyond what is possible with permutation-based methods. The method of knockoffs is very general and flexible, and can work with a broad class of test statistics. We test the method in combination with statistics from the Lasso for sparse regression, and obtain empirical results showing that the resulting method has far more power than existing selection rules when the proportion of null variables is high. In multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory, if not incorrect, if the prediction vectors are not orthogonal. Proposed is an estimation procedure based on adding small positive quantities to the diagonal of X′X. Introduced is the ridge trace, a method for showing in two dimensions the effects of nonorthogonality. It is then shown how to augment X′X to obtain biased estimates with smaller mean square error. 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. We propose a method for constructing p-values for general hypotheses in a high-dimensional linear model. The hypotheses can be local for testing a single regression parameter or they may be more global involving several up to all parameters. Furthermore, when considering many hypotheses, we show how to adjust for multiple testing taking dependence among the p-values into account. Our technique is based on Ridge estimation with an additional correction term due to a substantial projection bias in high dimensions. We prove strong error control for our p-values and provide sufficient conditions for detection: for the former, we do not make any assumption on the size of the true underlying regression coefficients while regarding the latter, our procedure might not be optimal in terms of power. We demonstrate the method in simulated examples and a real data application.