Simultaneous Analysis of Variance Tests Using Friedman's χ Statistics

This study develops simultaneous tests for shifts in marginal distributions pertaining to complete two-way multivariate data not having Gaussian errors. Gerig's (1969) statistics based on ranks are used, a special case yielding Friedman's (1937) χ Statistics statistics for the one-dimensional marginal distributions. Exact α-level procedures are not now available. Instead, approximate but conservative tests are constructed using available tables together with suitable probability inequalities. A variational result permits unlimited tests even when these are suggested by the data. The procedures are illustrated using data from field studies with peanuts; some extensions are noted; and selected percentage points are provided to support some of the tests.     

Applications and Tabulations of the Multivariate t Distribution with ρ = 0

Quantiles of the multivariate t distribution with ρ = 0 are tabulated. Some applications are discussed.     

On the Assessment of the Performance of Multiple Test Procedures

Based on the concept of the multiple level of significance two criteria for assessing the performance of multiple tests are proposed: The simultaneous power is defined as the probability of rejecting all false null hypotheses. The probability of a correct decision is defined as the probability of correctly rejecting all false null hypotheses and accepting all true ones. Both criteria are discussed for nonstagewise and stagewise procedures in case of independent test statistics. For the example of 5 independently and normally distributed test statistics the values of the two criteria are calcutated under reasonably simple alternatives.     

Note on Confidence Regions in Multiple Assays

This note is in continuation of the author's results on ’classical‘ confidence regions for relative potencies in multiple assays (Bennett, 1987). It is shown by the use of a Bonferroni inequality (e.g. Miller, ch. 2, 1980) that approximate confidence regions for the relative potencies {M_i} i = 1, …, p may also be obtained directly. A comparison with the classical regions is made for the case: p = 2 using Finney's example (1978, ch. 2).     

Estimating the null distribution to adjust observed confidence levels for genome-scale screening

In a novel approach to the multiple testing problem, Efron (2004, Journal of the American Statistical Association 99, 96-104; 2007a Journal of the American Statistical Association 102, 93-103; 2007b, Annals of Statistics 35, 1351-1377) formulated estimators of the distribution of test statistics or nominal p-values under a null distribution suitable for modeling the data of thousands of unaffected genes, nonassociated single-nucleotide polymorphisms, or other biological features. Estimators of the null distribution can improve not only the empirical Bayes procedure for which it was originally intended, but also many other multiple-comparison procedures. Such estimators in some cases improve the proposed multiple-comparison procedure (MCP) based on a recent non-Bayesian framework of minimizing expected loss with respect to a confidence posterior, a probability distribution of confidence levels. The flexibility of that MCP is illustrated with a nonadditive loss function designed for genomic screening rather than for validation. The merit of estimating the null distribution is examined from the vantage point of the confidence-posterior MCP (CPMCP). In a generic simulation study of genome-scale multiple testing, conditioning the observed confidence level on the estimated null distribution as an approximate ancillary statistic markedly improved conditional inference. Specifically simulating gene expression data, however, indicates that estimation of the null distribution tends to exacerbate the conservative bias that results from modeling heavy-tailed data distributions with the normal family. To enable researchers to determine whether to rely on a particular estimated null distribution for inference or decision making, an information-theoretic score is provided. As the sum of the degree of ancillarity and the degree of inferential relevance, the score reflects the balance conditioning would strike between the two conflicting terms. The CPMCP and other methods introduced are applied to gene expression microarray data.     

Generalized Maximum Range Tests for Pairwise Comparisons of Several Populations

A multiple comparison procedure (MCP) is proposed for the comparison of all pairs of several independent samples. This MCP is essentially the closed procedure with union-intersection tests based on given single tests Q_ij for the minimal hypotheses H_ij. In such cases where the α-levels of the nominal tests associated with the MCP can be exhausted, this MCP has a uniformly higher all pair power than any refined Bonferroni test using the same Q_ij. Two different general algorithms are described in section 3. A probability inequality for ranges of i.i.d. random variables which is useful for some algorithms is proved in section 4. Section 5 contains the application to independent normally distributed estimates and section 6 the comparisons of polynomial distributions by multivariate ranges. Further applications are possible. Tables of the 0.05-bounds for the tests of section 5 and 6 are enclosed.     

重复力估计值的概率分布及其假设检验和区间估计

本文用概率论的方法导出了重复力估计值的概率分布密度函数,并对重复力的假设检验和区间估计也进行了讨论。