UMP(U)‐Tests for a Binomial Parameter: A Paradox

We consider uniformly most powerful (UMP) as well as uniformly most powerful unbiased (UMPU) tests and their non‐randomized versions for certain hypotheses concerning a binomial parameter. It will be shown that the power function of a UMP(U)‐test based on sample size n can coincide on the entire parameter space with the power function of the corresponding test based on sample size n + 1. A complete characterization of this paradox will be derived. Apart some exceptional cases for two‐sided tests and equivalence tests the paradox appears if and only if a test based on sample size n is non‐randomized.     

A note on P-values for two-sided tests

Based on uniformly most powerful unbiased (UMPU) tests for two-sided hypotheses and a short note in Lehmann (1959) on critical levels for randomized tests, Meulepas (1998, 1999) proposed (two-tailed) P -values taking into account the randomization constant(s) of the UMPU-tests. While UMPU-tests need an extra uniform observation if randomization is required, the P -values proposed by Meulepas need no extra uniform observation. At first glance, his idea looks very promising in order to define a suitable and powerful P -value. Unfortunately, such P -values are generally too conservative.     

UMPU and Alternative Tests for Association in 2 × 2 Tables

The use of the uniformly most powerful among the unbiased (UMPU) test was recently suggested for the study of gametic association between two polymorphic loci as an alternative to the Fisher's exact test (Zapata and Alvarez, 1997, Annals of Human Genetics 61, 71-77). However, the proposed test is not UMPU for two-sided alternatives. In this study, we present the UMPU test, discuss criticisms against the use of randomized tests, and compare the power of several tests. We show that, in many practical cases, the use of the UMPU test is less than desirable and propose the alternative adjusted-more extreme tables (A-MET) and the equal-tails (ET) tests. We suggest that some of the general arguments against the use of randomized tests can be alleviated by a newly proposed extended p-value definition.     

Testing Average Equivalence — Finding a Compromise Between Theory and Practice

Recently, Brown , Hwang , and Munk (1998) proposed and unbiased test for the average equivalence problem which improves noticeably in power on the standard two one‐sided tests procedure. Nevertheless, from a practical point of view there are some objections against the use of this test which are mainly adressed to the ‘unusual’ shape of the critical region. We show that every unbiased test has a critical region with such an ‘unusual’ shape. Therefore, we discuss three (biased) modifications of the unbiased test. We come to the conclusion that a suitable modification represents a good compromise between a most powerful test and a test with an appealing shape of its critical region. In order to perform these tests figures are given containing the rejection region. Finally, we compare all tests in an example from neurophysiology. This shows that it is beneficial to use these improved tests instead of the two one‐sided tests procedure.     

A Two-Tailed P-Value for Fisher's Exact Test

A two-tailed P-value is presented for a significance test in two by two contingency tables. There is no extraneous quasi-observation such as is needed in the exact randomized uniformly most powerful unbiased (UMPU) test of the hypothesis of independence. The proposed P-value can never exceed unity and is always two-tailed, unlike other P-values proposed in the literature     

A modified revisited Haseman-Elston method to further improve power

The original and revisited Haseman-Elston methods are simple robust methods to detect linkage, but neither is uniformly optimal in terms of power. In this report, we propose a simple modification of the revisited Haseman-Elston method that retains the simplicity and robustness properties, but increases its power. We demonstrate theoretically that the modification can be more powerful than the optimally weighted Haseman-Elston method when the sibship mean can be correctly specified. We then examine the properties of this modification by simulation when the sibship mean is replaced by its best linear unbiased predictor. The simulation results indicate that this modification maintains good control over type I error, even in the case of larger sibships, and that the empirical power of this modification is similar to that of the optimally weighted Haseman-Elston method in most cases.     

Tests for establishing compatibility of an observed genotype distribution with Hardy-Weinberg equilibrium in the case of a biallelic locus

The classical chi(2)-procedure for the assessment of genetic equilibrium is tailored for establishing lack rather than goodness of fit of an observed genotype distribution to a model satisfying the Hardy-Weinberg law, and the same is true for the exact competitors to the large-sample procedure, which have been proposed in the biostatistical literature since the late 1930s. In this contribution, the methodology of statistical equivalence testing is adopted for the construction of tests for problems in which the assumption of approximate compatibility of the genotype distribution actually sampled with Hardy-Weinberg equilibrium (HWE) plays the role of the alternative hypothesis one aims to establish. The result of such a construction highly depends on the choice of a measure of distance to be used for defining an indifference zone containing those genotype distributions whose degree of disequilibrium shall be considered irrelevant. The first such measure proposed here is the Euclidean distance of the true parameter vector from that of a genotype distribution with identical allele frequencies being in strict HWE. The second measure is based on the (scalar) parameter of the distribution first introduced into the present context by Stevens (1938, Annals of Eugenics 8, 377-383). The first approach leads to a nonconditional test (which nevertheless can be carried out in a numerically exact way), the second to an exact conditional test shown to be uniformly most powerful unbiased (UMPU) for the associated pair of hypotheses. Both tests are compared in terms of the exact power attained against the class of those specific alternatives under which HWE is strictly satisfied.     

Detection of Cyclic Trends in Incidence Data Using the Efficiency Score Method

In studies involving a cyclic regularity, researchers usually have a good working knowledge regarding the peak time in the cycle. Capitalizing on this information, we derive the asymptotically uniformly most powerful unbiased test for detecting a cyclic trend using the likelihood score, and present the asymptotic power function of the test and the approximate formula for sample size. Numerical studies demonstrate great advantages of the proposed test over the standard test in terms of power and sample size. Asymptotic power of the score test is satisfactorily close to actual power. We also generalize this method so that it is applicable for incidence data from unequally spaced intervals or risk populations of unequal size.     

Detecting Heterogeneity of Substitution along DNA and Protein Sequences

Regions of differing constraint, mutation rate or recombination along a sequence of DNA or amino acids lead to a nonuniform distribution of polymorphism within species or fixed differences between species. The power of five tests to reject the null hypothesis of a uniform distribution is studied for four classes of alternate hypothesis. The tests explored are the variance of interval lengths; a modified variance test, which includes covariance between neighboring intervals; the length of the longest interval; the length of the shortest third-order interval; and a composite test. Although there is no uniformly most powerful test over the range of alternate hypotheses tested, the variance and modified variance tests usually have the highest power. Therefore, we recommend that one of these two tests be used to test departure from uniformity in all circumstances. Tables of critical values for the variance and modified variance tests are given. The critical values depend both on the number of events and the number of positions in the sequence. A computer program is available on request that calculates both the critical values for a specified number of events and number of positions as well as the significance level of a given data set.     

No BLUE among phylogenetic estimators

 Multivariate analysis is a branch of statistics that successfully exploits the powerful tools of linear algebra to obtain a fairly comprehensive theory of estimation. The purpose of this paper is to explore to what extent a linear theory of estimation can be developed in the context of coalescent models used in the analysis of DNA polymorphism. We consider a large class of coalescent models, of which the neutral infinite sites model is one example. In the process, we discover several limitations of linear estimators that are quite distinct from those in the classical theory. In particular, we prove that there does not exist a uniformly BLUE (best linear unbiased estimator) for the scaled mutation parameter, under the assumptions of the neutral model of evolution. In fact, we show that no linear estimator performs uniformly better than the Watterson (1975) method based on the total number of segregating sites. For certain coalescent models, the segregating-sites estimator is actually optimal. The general conclusion is the following. If genealogical information is useful for estimating the rate of evolution, then there is no optimal linear method. If there is an optimal linear method, then no information other than the total number of segregating sites is needed. Received: 29 July 1998 / Revised version: 9 October 1998     

Robust Tests for Candidate‐Gene Association Using Case‐Parents Trios with a Multi‐Allelic Marker

In case‐parents trios design, the association between a multi‐allelic candidate‐gene and a disease can be detected by using maximum of score tests (max‐score) when the mode of inheritance is known. We apply the maximum of the max‐score statistics and the maximum of likelihood ratio statistics when the genetic model is unknown and examine their robust properties compared to max‐score statistics. The simulation results demonstrate that the two maximum robust tests are more efficacious and robust across all genetic models compared with the three max‐score tests. Moreover, in most situations, the maximum of the max‐score tests seems to be more powerful than the maximum of the likelihood ratio tests. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparing Trials with Multiple Outcomes: The Multivariate One-Sided Hypothesis with Unknown Covariances

The paper deals with a problem arising for tests in clinical trials. The outcomes of a standard and a new treatment to be compared are multivariate normally distributed with common but unknown covariance matrix. Under the null hypothesis the means of the outcomes are equal, under the alternative the new treatment is assumed to be superior, i.e. the means are larger without further quantification. For known covariance matrix there is a variety of tests for this problem. Some of these procedures can be extended to the case of unknown covariances if one is willing to accept a bias. There is, however, also an efficient unbiased test. The paper contains some numerical comparisons of these different procedures and takes a look on the minimax properties of the unbiased test.     

Uniformly minimum variance unbiased estimation of gene diversity

Gene diversity is an important measure of genetic variability in inbred populations. The survival of species in changing environments depends on, among other factors, the genetic variability of the population. In this communication, I have derived the uniformly minimum variance unbiased estimator of gene diversity. The proposed estimator of gene diversity does not assume that the inbreeding coefficient is known. I have also provided the approximate variance of this estimator according to Fisher's method. In addition, I have developed a numerical resampling-based method for obtaining variances and confidence intervals based on the maximum likelihood estimator and the uniformly minimum variance unbiased estimator. Efficiency in estimation of the gene diversity based on these two estimators is discussed. In accordance with the simulation results, I found that the uniformly minimum variance estimator developed in this report is more accurate for estimation of gene diversity than the maximum likelihood estimator.     

Methods and criteria for deciding whether specific-locus mutation-rate data in mice indicate a positive,negative, or inconclusive result

The binomial approximation of the UMPU (uniformly most powerful unbiased) test for the equality of 2 binomial proportions is shown to be a highly accurate and easily applied method for testing the hypothesis that a given mouse specific-locus mutation frequency is not higher than the spontaneous mutation frequency (43 mutations in 801 406 offspring, for males). Critical sample sizes have been calculated that show at a glance whether P < 0.05.The first hypothesis that the mutation frequency (induced + spontaneous) of treated mice is not higher than the spontaneous mutation frequency is combined with the second hypothesis that the induced mutation frequency of treated mice is no less than 4 times the historical-control mutation frequency to produce a multiple decision procedure with 4 possible decisions: inconclusive result, negative result, positive result, and weak mutagen. Critical sample sizes for the second hypothesis, also with P < 0.05, are combined with those for the first hypothesis into a grid that permits rapid evaluation of data according to these criterea. The justification for using these criteria in reaching decisions, assuming a high level of exposure has been given, is the practical necessity of rapidly determining which chemicals are potent mutagens.Positive results can become apparent in relatively small samples. Larger samples, of at least 11 166 offspring, are required to obtain a negative result. If samples of 18 000 are routinely collected (unless positive results are found earlier), 75% of tests of chemicals that are non-mutagens will give a negative result. If the question being asked is not whether a chemical induces gene mutations but, rather, whether the exposure received by humans causes any important risk from gene mutations, a much smaller sample size may be acceptable, under certain conditions.A comparison of the relative efficiencies of the specific-locus test (for gene mutations and small deficiencies) and the heritable-translocation test (for transmissible chromosome rearrangements), in detecting the same proportional increases over the spontaneous frequencies of their respective types of genetic damage, shows that less work is involved in reaching a conclusive result in the specific-locus test. Proposed specific-locus tests using biochemical markers are at a considerable statistical disadvantage compared with the standard test (using 7 visible markers) for which there is available a very large historical control showing a very low mutation rate.     

Explicite Formulae of Exact Tests in Mixed Balanced ANOVA-models

As an alternative to SATTERTHWAITE's approximative test strongly unbiased tests of BARTLETT-SCHEFFÉ type are considered to test hypotheses about fixed and random effects in mixed and random ANOVA-models with equal cell frequencies. Explicite formulae are given to test hypotheses in the usual two- and three-way classifications and the four-way cross-classification. These formulae also can be used in the multivariate case.     

Comparisons of site- and haplotype-frequency methods for detecting positive selection

In this report, we compare the differences between various site- and haplotype-frequency tests in their power to detect positive selection by doing computer simulations. Our results are the following. 1) Although haplotype-frequency tests that are conditional on the number of haplotypes (K) were developed for nonrecombining haplotypes, these tests are insensitive to recombination. Such tests, including the Ewens-Watterson (EW) test, can therefore be applied to recombining haplotypes. 2) Tests conditional on the number of segregating sites (S) become overly conservative in the presence of recombination. 3) The EW test is usually the most powerful test during the sweep phase, especially when the local recombination rate is high. 4) The "extended haplotype homozygosity" test relies heavily on the prior knowledge of the target of selection. With that knowledge, it is the most powerful test, whereas in the absence of this prior information, the test has little power. We also study the sensitivities of the haplotype-frequency tests to background selection and various demographic forces. We find that these tests are sensitive to some forces other than positive selection. To alleviate the problem of low specificity, compound tests, such as the DH test (Zeng et al. 2006), may be a solution. In the companion paper (Zeng K, Shi S, Wu C-I, in preparation), we use the EW test to devise 2 compound tests, which are more powerful in detecting positive selection than DH, but are also relatively insensitive to demography.     

A note on linkage analysis with affected sib triplets

Previously, it has been shown for affected sib pairs that the mean test is the uniformly (in theta) most powerful test in case of a multiplicative mode of inheritance and that the mean test is equivalent to parametric linkage analysis calculated under an assumed multiplicative mode of inheritance. Here, these two results are extended to samples consisting of affected sib triplets. For affected sib quadruplets, however, it is shown that these results are no longer valid.     

Assessing antiviral potency of anti-HIV therapies in vivo by comparing viral decay rates in viral dynamic models

A virologic marker, the number of HIV RNA copies or viral load, is currently used to evaluate anti-HIV therapies in AIDS clinical trials. This marker can be used to assess the antiviral potency of therapies, but is easily affected by noncompliance, drug resistance, toxicities, and other factors during the long-term treatment evaluation process. Recently it has been suggested to use viral dynamics to assess the potency of antiviral drugs and therapies, since viral decay rates in viral dynamic models have been shown to be related to the antiviral drug potency directly, and they need a shorter evaluation time. In this paper we first review the two statistical approaches for characterizing HIV dynamics and estimating viral decay rates: the individual nonlinear least squares regression (INLS) method and the population nonlinear mixed-effect model (PMEM) approach. To compare the viral decay rates between two treatment arms, parametric and nonparametric tests, based on the estimates of viral decay rates (the derived variables) from both the INLS and PMEM methods, are proposed and studied. We show, using the concept of exchangeability, that the test based on the empirical Bayes' estimates from the PMEM is valid, powerful and robust. This proposed method is very useful in most practical cases where the INLS-based tests and the general likelihood ratio test may not apply. We validate and compare various tests for finite samples using Monte Carlo simulations. Finally, we apply the proposed tests to an AIDS clinical trial to compare the antiviral potency between a 3-drug combination regimen and a 4-drug combination regimen. The proposed tests provide some significant evidence that the 4-drug regimen is more potent than the 3-drug regimen, while the naive methods fail to give a significant result.*To whom correspondence should be addressed.     

A mixed two-stage method for detecting interactions in genomewide association studies

Genomewide association studies (GWAS) are being conducted to unravel the genetic etiology of complex diseases, in which complex epistasis may play an important role. One-stage method in which interactions are tested using all samples at one time may be computationally problematic, may have low power as the number of markers tested increases and may not be cost-efficient. A common two-stage method may be a reasonable and powerful approach for detecting interacting genes using all samples in both two stages. In this study, we introduce an alternative two-stage method, in which some promising markers are selected using a proportion of samples in the first stage and interactions are then tested using the remaining samples in the second stage. This two-stage method is called mixed two-stage method. We then investigate the power of both one-stage method and mixed two-stage method to detect interacting disease loci for a range of two-locus epistatic models in a case-control study design. Our results suggest that mixed two-stage method may be more powerful than one-stage method if we choose about 30% of samples for single-locus tests in the first stage, and identify less than and equal to 1% of markers for follow-up interaction tests. In addition, we compare both two-stage methods and find that our two-stage method will lose power because we only use part of samples in both two stages.     

Drawbacks to Integer Scoring for Ordered Categorical Data

Linear rank tests are widely used when testing for independence against stochastic order in a 2 x J contingency table with two treatments and J ordered outcome levels. For this purpose, numerical scores are assigned, possibly by default, to the J outcome levels. When the choice of scores is not apparent, integer (equally spaced) scores are often assigned. We show that this practice generally leads to unnecessarily conservative tests. The use of slightly perturbed scores will result in a less conservative and uniformly more powerful test.