Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFalse Discovery Control for Pairwise Comparisons - An Asymptotic Solution to Williams, Jones and Tukeys Conjecture
84
0
0.0
(
0
)
Added by
Dennis Leung
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Under weak moment and asymptotic conditions, we offer an affirmative answer to whether the BH procedure (Benjamini and Hochberg, 1995) can control the false discovery rate in testing pairwise comparisons of means under a one-way ANOVA layout. Specifically, despite the fact that the two sample t-statistics do not exhibit positive regression dependency (Benjamini and Yekutieli, 2001), our result shows that the BH procedure can asymptotically control the directional false discovery rate as conjectured by Williams, Jones, and Tukey (1999). Such a result is useful for most general situations when the number of variables is moderately large and/or when idealistic assumptions such as normality and a balanced design are violated.
rate research
Read More
We are considered with the false discovery rate (FDR) of the linear step-up test $varphi^{LSU}$ considered by Benjamini and Hochberg (1995). It is well known that $varphi^{LSU}$ controls the FDR at level $m_0 q / m$ if the joint distribution of $p$-values is multivariate totally positive of order 2. In this, $m$ denotes the total number of hypotheses, $m_0$ the number of true null hypotheses, and $q$ the nominal FDR level. Under the assumption of an Archimedean $p$-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of $varphi^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $varphi^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $varphi^{LSU}$ is exactly equal to $m_0 q / m$, as in the case of stochastically independent $p$-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable $p$-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel copulae.
Multiple hypothesis testing, a situation when we wish to consider many hypotheses, is a core problem in statistical inference that arises in almost every scientific field. In this setting, controlling the false discovery rate (FDR), which is the expected proportion of type I error, is an important challenge for making meaningful inferences. In this paper, we consider the problem of controlling FDR in an online manner. Concretely, we consider an ordered, possibly infinite, sequence of hypotheses, arriving one at each timestep, and for each hypothesis we observe a p-value along with a set of features specific to that hypothesis. The decision whether or not to reject the current hypothesis must be made immediately at each timestep, before the next hypothesis is observed. The model of multi-dimensional feature set provides a very general way of leveraging the auxiliary information in the data which helps in maximizing the number of discoveries. We propose a new class of powerful online testing procedures, where the rejections thresholds (significance levels) are learnt sequentially by incorporating contextual information and previous results. We prove that any rule in this class controls online FDR under some standard assumptions. We then focus on a subclass of these procedures, based on weighting significance levels, to derive a practical algorithm that learns a parametric weight function in an online fashion to gain more discoveries. We also theoretically prove, in a stylized setting, that our proposed procedures would lead to an increase in the achieved statistical power over a popular online testing procedure proposed by Javanmard & Montanari (2018). Finally, we demonstrate the favorable performance of our procedure, by comparing it to state-of-the-art online multiple testing procedures, on both synthetic data and real data generated from different applications.
Differential privacy provides a rigorous framework for privacy-preserving data analysis. This paper proposes the first differentially private procedure for controlling the false discovery rate (FDR) in multiple hypothesis testing. Inspired by the Benjamini-Hochberg procedure (BHq), our approach is to first repeatedly add noise to the logarithms of the $p$-values to ensure differential privacy and to select an approximately smallest $p$-value serving as a promising candidate at each iteration; the selected $p$-values are further supplied to the BHq and our private procedure releases only the rejected ones. Moreover, we develop a new technique that is based on a backward submartingale for proving FDR control of a broad class of multiple testing procedures, including our private procedure, and both the BHq step-up and step-down procedures. As a novel aspect, the proof works for arbitrary dependence between the true null and false null test statistics, while FDR control is maintained up to a small multiplicative factor.
We consider the problem of ranking $n$ players from partial pairwise comparison data under the Bradley-Terry-Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendalls tau distance that measures the difference between two rank vectors by counting the number of
The Bradley-Terry model assigns probabilities for the outcome of paired comparison experiments based on strength parameters associated with the objects being compared. We consider different proposed choices of prior parameter distributions for Bayesian inference of the strength parameters based on the paired comparison results. We evaluate them according to four desiderata motivated by the use of inferred Bradley-Terry parameters to rate teams on the basis of outcomes of a set of games: invariance under interchange of teams, invariance under interchange of winning and losing, normalizability and invariance under elimination of teams. We consider various proposals which fail to satisfy one or more of these desiderata, and illustrate two proposals which satisfy them. Both are one-parameter independent distributions for the logarithms of the team strengths: 1) Gaussian and 2) Type III generalized logistic.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
