The Logical Consistency of Simultaneous Agnostic Hypothesis Tests.

Simultaneous hypothesis tests can fail to provide results that meet logical requirements. For example, if A and B are two statements such that A implies B, there exist tests that, based on the same data, reject B but not A. Such outcomes are generally inconvenient to statisticians (who want to communicate the results to practitioners in a simple fashion) and non-statisticians (confused by conflicting pieces of information). Based on this inconvenience, one might want to use tests that satisfy logical requirements. However, Izbicki and Esteves shows that the only tests that are in accordance with three logical requirements (monotonicity, invertibility and consonance) are trivial tests based on point estimation, which generally lack statistical optimality. As a possible solution to this dilemma, this paper adapts the above logical requirements to agnostic tests, in which one can accept, reject or remain agnostic with respect to a given hypothesis. Each of the logical requirements is characterized in terms of a Bayesian decision theoretic perspective. Contrary to the results obtained for regular hypothesis tests, there exist agnostic tests that satisfy all logical requirements and also perform well statistically. In particular, agnostic tests that fulfill all logical requirements are characterized as region estimator-based tests. Examples of such tests are provided

Rethinking Hypothesis Tests

Null Hypothesis Significance Testing (NHST) have been a popular statistical tool across various scientific disciplines since the 1920s. However, the exclusive reliance on a p-value threshold of 0.05 has recently come under criticism; in particular, it is argued to have contributed significantly to the reproducibility crisis. We revisit some of the main issues associated with NHST and propose an alternative approach that is easy to implement and can address these concerns. Our proposed approach builds on equivalence tests and three-way decision procedures, which offer several advantages over the traditional NHST. We demonstrate the efficacy of our approach on real-world examples and show that it has many desirable properties

Logically-consistent hypothesis testing and the hexagon of oppositions

Although logical consistency is desirable in scientific research, standard statistical hypothesis tests are typically logically inconsistent. To address this issue, previous work introduced agnostic hypothesis tests and proved that they can be logically consistent while retaining statistical optimality properties. This article characterizes the credal modalities in agnostic hypothesis tests and uses the hexagon of oppositions to explain the logical relations between these modalities. Geometric solids that are composed of hexagons of oppositions illustrate the conditions for these modalities to be logically consistent. Prisms composed of hexagons of oppositions show how the credal modalities obtained from two agnostic tests vary according to their threshold values. Nested hexagons of oppositions summarize logical relations between the credal modalities in these tests and prove new relations

Covariate-assisted ranking and screening for large-scale two-sample inference

Two-sample multiple testing has a wide range of applications. The conventionalpractice first reduces the original observations to a vector of p-values and then chooses a cutoffto adjust for multiplicity. However, this data reduction step could cause significant loss ofinformation and thus lead to suboptimal testing procedures.We introduce a new framework fortwo-sample multiple testing by incorporating a carefully constructed auxiliary variable in inferenceto improve the power. A data-driven multiple-testing procedure is developed by employinga covariate-assisted ranking and screening (CARS) approach that optimally combines the informationfrom both the primary and the auxiliary variables. The proposed CARS procedureis shown to be asymptotically valid and optimal for false discovery rate control. The procedureis implemented in the R package CARS. Numerical results confirm the effectiveness of CARSin false discovery rate control and show that it achieves substantial power gain over existingmethods. CARS is also illustrated through an application to the analysis of a satellite imagingdata set for supernova detection