Multiple Hypotheses Testing with Weights Benjamini, Yoav; Hochberg, Yosef
Scandinavian journal of statistics,
September 1997, Letnik:
24, Številka:
3
Journal Article
Recenzirano
In this paper we offer a multiplicity of approaches and procedures for multiple testing problems with weights. Some rationale for incorporating weights in multiple hypotheses testing are discussed. ...Various type-I error-rates and different possible formulations are considered, for both the intersection hypothesis testing and the multiple hypotheses testing problems. An optimal per family weighted error-rate controlling procedure a lá Spjøtvoll (1972) is obtained. This model serves as a vehicle for demonstrating the different implications of the approaches to weighting. Alternative approaches to that of Holm (1979) for family-wise error-rate control with weights are discussed, one involving an alternative procedure for family-wise error-rate control, and the other involving the control of a weighted family-wise error-rate. Extensions and modifications of the procedures based on Simes (1986) are given. These include a test of the overall intersection hypothesis with general weights, and weighted sequentially rejective procedures for testing the individual hypotheses. The false discovery rate controlling approach and procedure of Benjamini & Hochberg (1995) are extended to allow for different weights.
We present two new families of two-sided nonequivariant confidence intervals for the mean θ of a continuous, unimodal, symmetric random variable. Compared with the conventional symmetric equivariant ...confidence interval, they are shorter when the observation is small, and restrict the sign of θ for smaller observations. One of the families, a modification of Pratt's construction of intervals with minimal expected length when θ = 0, is longer than the conventional symmetric interval when |X| is large and has longer expected length when |θ| is large. The other family gives the conventional symmetric interval when |X| is large, with a change to the proximal endpoint when |X| is small. Its expected length is smaller than that of the conventional symmetric interval when |θ| is small, larger for an intermediate range of |θ|, and approaches that of the conventional interval for large |θ|. This slight modification of the conventional two-sided interval has most of the power advantage of a one-sided interval, but short length. Neither procedure requires that a preferred direction be specified in advance. The constants that determine the procedures can be found for symmetrically distributed statistics using any software package that includes the cumulative distribution function and inverse cumulative distribution function of the statistic, along with a root finder. We present tables of constants needed to apply the procedures for normally and t-distributed test statistics, and give an application to employment discrimination litigation.
The problem of identifying the lowest dose level for which the mean response differs from that at the zero dose level is considered. A general framework for stepwise testing procedures that use ...contrasts among the dose level means is proposed. Using this framework, several new procedures are derived. These and some existing procedures, including that of Williams (1971, Biometrics 27, 103-117; 1972, Biometrics 28, 519-531), are compared analytically and by an extensive simulation study for the normal theory balanced one-way layout case. It is pointed out that the procedures based on the so-called step and basin contrasts proposed by Ruberg (1989, Journal of American Statistical Association 84, 816-822) have excessively high type I familywise error rates (FWEs) and, hence, they should not be used. Some findings of the simulation study are as follows: For monotone dose mean configurations, Williams' procedure and two step-down test procedures based on Helmert and linear contrasts offer the best performance. For nonmonotone dose mean configurations, the performance of Williams' procedure does degrade somewhat, but the other two procedures are still the best. For more complex designs, a simple step-down test procedure that uses any α-level tests (not necessarily t-tests) to compare each dose level with the zero dose level controls the FWE and is the only alternative available, but its power is rather low, especially under nonmonotone configurations. Step-up procedures are generally dominated by step-down procedures when the same contrasts are used although the differences are not great.
Consider two independent random variables X and Y. The functional R = Pr(X < Y) or λ = Pr(X < Y) - Pr(Y < X) is of practical importance in many situations, including clinical trials, genetics, and ...reliability. In this paper several approaches to estimation of λ when X and Y are presented in discretized (Categorical) form are analyzed and compared. Asymptotic formulas for the variance of the estimators are derived; use of the bootstrap to estimate variances is also discussed. Computer simulations indicate that the choice of the best estimator depends on the value of λ, the underlying distribution, and the sparseness of the data. It is shown that the bootstrap provides a robust estimate of variance. Several examples are treated.
It is ‘common knowledge’ that closed multiple test procedures are typically better than others in terms of power. In the first part of this paper we prove that closed procedures constitute a complete ...class in the class of all coherent procedures, and, also, that
α-exhaustive procedures constitute a minimal complete class there. However, closed procedures are not always ‘monotone in p-values’ and are often computationally inefficient. Thus, in the second part of this paper we first investigate conditions for a closed procedure to be ‘monotone in p-values’, and second, we shed further light on the possibility of reducing a given closed procedure to a step-down sequentially rejective procedure. We give a sufficient condition for a closure based on some generalized
Simes (1986), (Biometrika 73, 751–754) type tests to be ‘monotone in p-values’ and clarify
Hochberg and Tamhane (1987), (Multiple Comparison Procedures, Wiley, New York) general result on the existence of a shortcut in families satisfying the free combination condition. We comment on the interrelationship between ‘stepwise’ (step-down and step-up), ‘monotone in p-values’, and ‘sequentially rejective’ procedures, in families satisfying the ‘free combinations condition’ as well as in families involving ‘logically related’ hypotheses.
Shaffer (
J. Amer. Statist. Assoc.
81 (1986) 826–831) gave simple and more powerful modifications of Holm's (
Scand. J. Statist.
6 (1979) 65–70) procedure to families of Logically Related Hypotheses ...(LRH). Since then, several procedures more powerful than Holm's (based on Simes (
Biometrika
73 (1986) 751–754)) have been introduced, with partial or no discussion of their improvements in cases of LRH. In this article, simple and more powerful modifications of these procedures (analogous to Shaffer's modifications) for general problems of LRH are given. These modified procedures are more powerful than the modified Holm's procedure in general problems of LRH. The various modifications are demonstrated and their superiority over existing procedures is indicated for some LRH. Another extension concerns one-sided tests with normal correlated test statistics. Some analysis of the type-I error-rate associated with Simes' type procedures in this case indicates satisfactory control. A class of problems suitable for applications of a Simes' type procedure in view of the given results, is indicated.
The search for particle-like dark matter with meV-to-GeV masses has developed rapidly in the past few years. We summarize the science case for these searches, the recent progress, and the exciting ...upcoming opportunities. Funding for Research and Development and a portfolio of small dark matter projects will allow the community to capitalize on the substantial recent advances in theory and experiment and probe vast regions of unexplored dark-matter parameter space in the coming decade.
We discuss the consequences of the quantum uncertainty on the spectrum of the electron emitted by the \(\beta\)-processes of a tritium atom bound to a graphene sheet. We analyze quantitatively the ...issue recently raised in Cheipesh et al., Phys. Rev. D 104, 116004 (2021), and discuss the relevant time scales and the degrees of freedom that can contribute to the intrinsic spread in the electron energy. We perform careful calculations of the potential between tritium and graphene with different coverages and geometries. With this at hand, we propose possible avenues to mitigate the effect of the quantum uncertainty.
We present two new families of two-sided nonequivariant confidence intervals for the mean theta of a continuous, unimodal, symmetric random variable. Compared with the conventional symmetric ...equivariant confidence interval, they are shorter when the observation is small, and restrict the sign of theta for smaller observations.
The PTOLEMY transverse drift filter is a new concept to enable precision analysis of the energy spectrum of electrons near the tritium beta-decay endpoint. This paper details the implementation and ...optimization methods for successful operation of the filter. We present the first demonstrator that produces the required magnetic field properties with an iron return-flux magnet. Two methods for the setting of filter electrode voltages are detailed. The challenges of low-energy electron transport in cases of low field are discussed, such as the growth of the cyclotron radius with decreasing magnetic field, which puts a ceiling on filter performance relative to fixed filter dimensions. Additionally, low pitch angle trajectories are dominated by motion parallel to the magnetic field lines and introduce non-adiabatic conditions and curvature drift. To minimize these effects and maximize electron acceptance into the filter, we present a three-potential-well design to simultaneously drain the parallel and transverse kinetic energies throughout the length of the filter. These optimizations are shown, in simulation, to achieve low-energy electron transport from a 1 T iron core (or 3 T superconducting) starting field with initial kinetic energy of 18.6 keV drained to <10 eV (<1 eV) in about 80 cm. This result for low field operation paves the way for the first demonstrator of the PTOLEMY spectrometer for measurement of electrons near the tritium endpoint to be constructed at the Gran Sasso National Laboratary (LNGS) in Italy.
