A straightforward multiallelic significance test for the Hardy-Weinberg equilibrium law

Much forensic inference based upon DNA evidence is made assuming Hardy-Weinberg Equilibrium (HWE) for the genetic loci being used. Several statistical tests to detect and measure deviation from HWE have been devised, and their limitations become more obvious when testing for deviation within multiallelic DNA loci. The most popular methods-Chi-square and Likelihood-ratio tests-are based on asymptotic results and cannot guarantee a good performance in the presence of low frequency genotypes. Since the parameter space dimension increases at a quadratic rate on the number of alleles, some authors suggest applying sequential methods, where the multiallelic case is reformulated as a sequence of "biallelic" tests. However, in this approach it is not obvious how to assess the general evidence of the original hypothesis; nor is it clear how to establish the significance level for its acceptance/rejection. In this work, we introduce a straightforward method for the multiallelic HWE test, which overcomes the aforementioned issues of sequential methods. The core theory for the proposed method is given by the Full Bayesian Significance Test (FBST), an intuitive Bayesian approach which does not assign positive probabilities to zero measure sets when testing sharp hypotheses. We compare FBST performance to Chi-square, Likelihood-ratio and Markov chain tests, in three numerical experiments. The results suggest that FBST is a robust and high performance method for the HWE test, even in the presence of several alleles and small sample sizes.     

Simultaneous inference for ratios of linear combinations of general linear model parameters

Consider a general linear model with p -dimensional parameter vector beta and i.i.d. normal errors. Let K(1), ..., K(k ), and L be linearly independent vectors of constants such that L(T)beta not equal 0. We describe exact simultaneous tests for hypotheses that Ki(T)beta/L(T)beta equal specified constants using one-sided and two-sided alternatives, and describe exact simultaneous confidence intervals for these ratios. In the case where the confidence set is a single bounded contiguous set, we describe what we claim are the best possible conservative simultaneous confidence intervals for these ratios - best in that they form the minimum k -dimensional hypercube enclosing the exact simultaneous confidence set. We show that in the case of k = 2, this "box" is defined by the minimum and maximum values for the two ratios in the simultaneous confidence set and that these values are obtained via one of two sources: either from the solutions to each of four systems of equations or at points along the boundary of the simultaneous confidence set where the correlation between two t variables is zero. We then verify that these intervals are narrower than those previously presented in the literature.     

A note on one-sided tests with multiple endpoints

Testing problems with multivariate one-sided alternative hypotheses are common in clinical trials with multiple endpoints. In the case of comparing two treatments, treatment 1 is often preferred if it is superior for at least one of the endpoints and not biologically inferior for the remaining endpoints. Bloch et al. (2001, Biometrics57, 1039-1047) propose an intersection-union test (IUT) for this testing problem, but their test does not utilize the appropriate multivariate one-sided test. In this note we modify their test by an alternative IUT that does utilize the appropriate one-sided test. Empirical and graphical evidence show that the proposed test is more appropriate for this testing problem.     

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.     

A Closed-form Jackknife Solution for the Behrens-Fisher Problem

A simple, closed-form jackknife estimate of the actual variance of the Mann-Whitney-Wilcoxon statistic, as opposed to the standard permutational variance under the test's null hypothesis has been derived which permits avoiding anticonservative performance in the presence of heterosce-dasticity. The formulation given allows modifications of the exponential scores test, of censored data tests by Gehan (1965), Peto & Peto (1977) and Prentice (1978), of tests for monotonic τ association by Kendall (1962) and for tests of ordered k-sample hypotheses. A Monte Carlo study supports recommendations for the jackknife procedures, but also shows their limited advantages in exponential scores and censored data versions. Thus, the paper extends results by Fligner & Policello (1981).     

On the Theory for RYANS Universal Multiple Comparison Procedure with Treatment of Ties in the Ranks of Samples

The proposal of RYAN (1960) for multiple comparison procedures has not become well known despite of its universal applicability. Only his idea of adjusted levels has been used in a modified manner, cf. EINOT and GABRIEL (1975), WELSCH (1977), RAMSEY (1978). This article re-introduces the procedure and demonstrates its broad applicability also as a nonparametric method. Above all an exact mathematical foundation is aspired. The case of ties in the ranks of samples has been taken into consideration.     

A note on the measurement of redundancy

Plant Ecology - The present paper points out a simpler method for computing the redundancy coefficient proposed by Orlóci (1975) using the inverse of the variance-covariance matrix. It is...     

Comparison of Resubstitution,Data Splitting,the Bootstrap,and the Jackknife as Methods for Estimating Validity Indices of New Marker Tests: A Monte Carlo Study

Estimating the diagnostic efficiency of marker tests on the basis of the training set is an intricate problem of discriminant analysis for which no analytical solution exists. The paper outline the problem, describes various popular estimation procedures and presents the results of computer simulations comparing the estimators with respect to both bias and variance.     

A note on familywise error rate for a primary and secondary endpoint

Hung et al. (2007) considered the problem of controlling the type I error rate for a primary and secondary endpoint in a clinical trial using a gatekeeping approach in which the secondary endpoint is tested only if the primary endpoint crosses its monitoring boundary. They considered a two-look trial and showed by simulation that the naive method of testing the secondary endpoint at full level α at the time the primary endpoint reaches statistical significance does not control the familywise error rate at level α. Tamhane et al. (2010) derived analytic expressions for familywise error rate and power and confirmed the inflated error rate of the naive approach. Nonetheless, many people mistakenly believe that the closure principle can be used to prove that the naive procedure controls the familywise error rate. The purpose of this note is to explain in greater detail why there is a problem with the naive approach and show that the degree of alpha inflation can be as high as that of unadjusted monitoring of a single endpoint.     

Multiple Tests for Different Sets of Variables Using a Data‐Driven Ordering of Hypotheses,with an Application to Gene Expression Data

A multiple parametric test procedure is proposed, which considers tests of means of several variables. The single variables or subsets of variables are ordered according to a data‐dependent criterion and tested in this succession without alpha‐adjustment until the first non‐significant test. The test procedure needs the assumption of a multivariate normal distribution and utilizes the theory of spherical distributions. The basic version is particularly suited for variables with approximately equal variances. As a typical example, the procedure is applied to gene expression data from a commercial array.     

A note on the evaluation of fluctuation experiments

Fluctuation analysis has emerged as a valuable tool for the measurement of mutation rates in single-cell populations. In this paper, we show how to make fuller use of the information supplied by the outcome of a fluctuation experiment. We shall extend Lea and Coulson's theory of the Luria-Delbrück distribution so that it accounts for residual mutation, reduced plating efficiency of mutants, and phenotypic lag, and establish a unifying method for the evaluation of fluctuation experiments in these cases and discuss its limitations. It will be proved that not all factors that might influence the distribution of mutant colonies in a fluctuation experiment can, in effect, be determined simultaneously. Nevertheless, it will be shown that the fluctuation-analytic approach to the measurement of mutation rates may retain its value in comparison with (or may even be superior to) alternative methods. Finally, we give some numerical examples to illustrate our results.     

A note on an exact test and confidence interval for competition and overlap effects

A method has been proposed that gives an exact test and confidence interval for the coefficient of overlap and competition effects. The method is based on Tukey's statistic of nonadditivity. The method is very simple to adopt and can be implemented by regression analysis and analysis of variance programmes in some important cases.