Exact tests for association between alleles at arbitrary numbers of loci

Associations between allelic frequencies, within and between loci, can be tested for with an exact test. The probability of the set of multi-locus genotypes in a sample, conditional on the allelic counts, is calculated from multinomial theory under the hypothesis of no association. Alleles are then permuted and the conditional probability calculated for the permuted genotypic array. The proportion of arrays no more probable than the original sample provides the significance level for the test. An algorithm is provided for counting genotypes efficiently in the arrays, and the powers of the test presented for various kinds of association. The powers for the case when associations are generated by admixture of several populations suggest that exact tests are capable of detecting levels of association that would affect forensic calculations to a significant extent.     

Exact and Asymptotic Power Determination For Dose‐Response Studies With Ordinal Response

In this article, we describe a conditional score test for detecting a monotone dose‐response relationship with ordinal response data. We consider three different versions of this test: asymptotic, conditional exact, and mid‐P conditional score test. Exact and asymptotic power formulae based on these tests will be studied. Asymptotic sample size formulae based on the asymptotic conditional score test will be derived. The proposed formulae are applied to a vaccination study and a developmental toxicity study for illustrative purposes. Actual significance level and exact power properties of these tests are compared in a small empirical study. The mid‐P conditional score test is observed to be the most powerful test with actual significance level close to the pre‐specified nominal level.     

Unconditional versions of several tests commonly used in the analysis of contingency tables

By focusing on a confidence interval for a nuisance parameter, Berger and Boos (1994, Journal of the American Statistical Association 89, 1012-1016) proposed new unconditional tests. In particular, they showed that, for a 2 x 2 table, this procedure generally was more powerful than Fisher's exact test. This paper utilizes and extends their approach to obtain unconditional tests for combining several 2 x 2 tables and testing for trend and homogeneity in a 2 x K table. The unconditional procedures are compared to the conditional ones by reanalyzing some published biomedical data.     

Exact tests of goodness of fit of log-linear models for rates

We propose Metropolis-Hastings sampling methods for estimating the exact conditional p-value for tests of goodness of fit of log-linear models for mortality rates and standardized mortality ratios. We focus on two-way tables, where the required conditional distribution is a multivariate noncentral hypergeometric distribution with known noncentrality parameter. Two examples are presented: a 2 x 3 table, where the exact results, obtained by enumeration, are available for comparison, and a 9 x 7 table, where Monte Carlo methods provide the only feasible approach for exact inference.     

A comparison of exact, mid-P, and score tests for matched case-control studies

A recently developed algorithm for generating the distribution of sufficient statistics for conditional logistic models can be put to a twofold use. First, it provides an avenue for performing inference for matched case-control studies that does not rely on the assumption of a large sample size. Second, joint distributions generated by this algorithm can be used to make comparisons of various inferential procedures that are free from Monte Carlo sampling errors. In this paper, these two features of the algorithm are utilized to compare small-sample properties of the exact, mid-P value, and score tests for a conditional logistic model with two unmatched binary covariates. Both uniparametric and multiparametric tests, performed at a nominal significance level of .05, were studied. It was found that the actual significance levels of the mid-P test tend to be closer to the nominal level when compared with those of the other two tests.     

An incomplete enumeration algorithm for an exact test of Hardy–Weinberg proportions with multiple alleles

Testing of Hardy–Weinberg proportions (HWP) with asymptotic goodness-of-fit tests is problematic when the contingency table of observed genotype counts has sparse cells or the sample size is low, and exact procedures are to be preferred. Exact p-values can be (1) calculated via computational demanding enumeration methods or (2) approximated via simulation methods. Our objective was to develop a new algorithm for exact tests of HWP with multiple alleles on the basis of conditional probabilities of genotype arrays, which is faster than existing algorithms. We derived an algorithm for calculating the exact permutation significance value without enumerating all genotype arrays having the same allele counts as the observed one. The algorithm can be used for testing HWP by (1) summation of the conditional probabilities of occurrence of genotype arrays with smaller probability than the observed one, and (2) comparison of the sum with a nominal Type I error rate α. Application to published experimental data from seven maize populations showed that the exact test is computationally feasible and reduces the number of enumerated genotype count matrices about 30% compared with previously published algorithms.     

An Exact Unconditional test for the Hardy-Weinberg Equilibrium

An exact test based on the unconditional distribution of a test statistic for the Hardy-Weinberg equilibrium is introduced. This test is usually more powerful and requires less extensive tabulations compared with the (ordinary) exact conditional test. Tables of critical values are given for N = 5(1)100 and α =·10, ·05, ·01. Sample sizes required to attain a given power with the unconditional and conditional exact tests are provided.     

The 2 x 2 matched-pairs trial: exact unconditional design and analysis

An exact unconditional method for the design and analysis of the 2 x 2 matched-pairs trial is presented. Unlike the exact conditional method, which is based on only the number of discordant pairs, the exact unconditional method uses the total number N of sampled pairs. This unconditional test, based on a simple Z statistic, yields sample sizes that are generally smaller than those produced by the exact conditional test for the cases tabulated herein, namely one-sided alpha = .01, .025, and .05 along with 80% and 90% power. Moreover, it is found to be uniformly more powerful than the latter for all the combinations of parameters considered in this paper, namely alpha = .01, .025, and .05, and N = 10(1)200. The method is illustrated by assessing various design options of an in vitro study of the effectiveness of antimicrobial agents. Some numerical examples of matching efficiency are also given.     

The Exact Test for Cytonuclear Disequilibria

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.     

Order‐restricted Scores Test for the Evaluation of Population‐based Case–control Studies when the Genetic Model is Unknown

The Cochran–Armitage (CA) linear trend test for proportions is often used for genotype‐based analysis of candidate gene association. Depending on the underlying genetic mode of inheritance, the use of model‐specific scores maximises the power. Commonly, the underlying genetic model, i.e. additive, dominant or recessive mode of inheritance, is a priori unknown. Association studies are commonly analysed using permutation tests, where both inference and identification of the underlying mode of inheritance are important. Especially interesting are tests for case–control studies, defined by a maximum over a series of standardised CA tests, because such a procedure has power under all three genetic models. We reformulate the test problem and propose a conditional maximum test of scores‐specific linear‐by‐linear association tests. For maximum‐type, sum and quadratic test statistics the asymptotic expectation and covariance can be derived in a closed form and the limiting distribution is known. Both the limiting distribution and approximations of the exact conditional distribution can easily be computed using standard software packages. In addition to these technical advances, we extend the area of application to stratified designs, studies involving more than two groups and the simultaneous analysis of multiple loci by means of multiplicity‐adjusted p‐values for the underlying multiple CA trend tests. The new test is applied to reanalyse a study investigating genetic components of different subtypes of psoriasis. A new and flexible inference tool for association studies is available both theoretically as well as practically since already available software packages can be easily used to implement the suggested test procedures.     

A cautionary note on exact unconditional inference for a difference between two independent binomial proportions

Fisher's exact test for comparing response proportions in a randomized experiment can be overly conservative when the group sizes are small or when the response proportions are close to zero or one. This is primarily because the null distribution of the test statistic becomes too discrete, a partial consequence of the inference being conditional on the total number of responders. Accordingly, exact unconditional procedures have gained in popularity, on the premise that power will increase because the null distribution of the test statistic will presumably be less discrete. However, we caution researchers that a poor choice of test statistic for exact unconditional inference can actually result in a substantially less powerful analysis than Fisher's conditional test. To illustrate, we study a real example and provide exact test size and power results for several competing tests, for both balanced and unbalanced designs. Our results reveal that Fisher's test generally outperforms exact unconditional tests based on using as the test statistic either the observed difference in proportions, or the observed difference divided by its estimated standard error under the alternative hypothesis, the latter for unbalanced designs only. On the other hand, the exact unconditional test based on the observed difference divided by its estimated standard error under the null hypothesis (score statistic) outperforms Fisher's test, and is recommended. Boschloo's test, in which the p-value from Fisher's test is used as the test statistic in an exact unconditional test, is uniformly more powerful than Fisher's test, and is also recommended.     

Unconditional inferences on the difference of two proportions in cross-sectional studies

In order to analyse a 2 x 2 table it is usual to perform inferences (hypothesis test or interval of confidence) on the difference d = p2 - p1 between two independent proportions. To this end it has been customary to adopt the Fisher conditional method, but nowadays the unconditional method of Barnard is increasingly adopted. However, all the present unconditional inferences are based on a double-binomial model. This article performs these inferences - exact and asymptotic - under a multinomial model, which is the appropriate one when the data proceed from a cross-sectional survey. At http://www.ugr.es/-bioest/SG_ASO.EXE there is a program for performing the said unconditional tests that may be copied.     

A Comparison of the Three Conditional Exact Tests in Two‐way Contingency Tables Using the Unconditional Exact Power

The conditional exact tests of homogeneity of two binomial proportions are often used in small samples, because the exact tests guarantee to keep the size under the nominal level. The Fisher's exact test, the exact chi‐squared test and the exact likelihood ratio test are popular and can be implemented in software StatXact. In this paper we investigate which test is the best in small samples in terms of the unconditional exact power. In equal sample cases it is proved that the three tests produce the same unconditional exact power. A symmetry of the unconditional exact power is also found. In unequal sample cases the unconditional exact powers of the three tests are computed and compared. In most cases the Fisher's exact test turns out to be best, but we characterize some cases in which the exact likelihood ratio test has the highest unconditional exact power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An exact test for Hardy-Weinberg and multiple alleles

Algorithms for generating the exact distribution of a finite sample drawn from a population in Hardy-Weinberg equilibrium are given for multiple alleles. The finite sampling distribution is derived analogously to Fisher's 2 X 2 exact distribution and is equivalent to Levene's conditional finite sampling distribution for Hardy-Weinberg populations. The algorithms presented are fast computationally and allow for quick alternatives to standard methods requiring corrections and approximations. Computation time is on the order of a few seconds for three-allele examples and up to 2 minutes for four-allele examples on an IBM 3081 machine.     

An unconditional exact test for the Hardy-Weinberg equilibrium law: sample-space ordering using the Bayes factor

Much forensic inference based upon DNA evidence is made assuming that the Hardy-Weinberg equilibrium (HWE) is valid for the genetic loci being used. Several statistical tests to detect and measure deviation from HWE have been devised, each having advantages and limitations. The limitations become more obvious when testing for deviation within multiallelic DNA loci is attempted. Here we present an exact test for HWE in the biallelic case, based on the ratio of weighted likelihoods under the null and alternative hypotheses, the Bayes factor. This test does not depend on asymptotic results and minimizes a linear combination of type I and type II errors. By ordering the sample space using the Bayes factor, we also define a significance (evidence) index, P value, using the weighted likelihood under the null hypothesis. We compare it to the conditional exact test for the case of sample size n = 10. Using the idea under the method of chi(2) partition, the test is used sequentially to test equilibrium in the multiple allele case and then applied to two short tandem repeat loci, using a real Caucasian data bank, showing its usefulness.     

A note on the Wilcoxon-Mann-Whitney test for 2 X kappa ordered tables

Biological and medical investigations often use ordered categorical data. When two groups are to be compared and the data for the groups fall in three or more ordered categories, the Wilcoxon-Mann-Whitney (WMW) test uses information in the ordering to give a test that is usually powerful against shift alternatives. However, such applications of WMW often involve distributions for which extensive ties play an important role. Newly available computer programs for performing exact tests give deeper insights into the characteristics of the exact WMW distributions and the suitability of normal approximations. We offer practical advice, based on experience with published biomedical data sets and on numerical studies of hypothetical ordered tables, for the use of WMW and its normal approximations.     

An exact test for detecting inconsistency in readers interpretation over time in screening mammograms

Radiologists' interpretation on screening mammograms is measured by accuracy indices such as sensitivity and specificity. The hypothesis that radiologists' interpretation on screening mammograms is constant across time can be tested by measuring overdispersion. However, small sample sizes are problematic for the accuracy of asymptotic approaches. In this article, we propose an exact conditional distribution for testing overdispersion of the binomial assumption that is assumed for the accuracy indices. An exact p -value can be defined from the developed distribution. We also describe an algorithm for computing this exact test. This proposed method is applied to data from a study in reading screening mammograms in a population of US radiologists (Beam et al., 2003). The exact method is compared analytically with a currently available method based on large sample approximations.     

Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models

Inference for Dirichlet process hierarchical models is typicallyperformed using Markov chain Monte Carlo methods, which canbe roughly categorized into marginal and conditional methods.The former integrate out analytically the infinite-dimensionalcomponent of the hierarchical model and sample from the marginaldistribution of the remaining variables using the Gibbs sampler.Conditional methods impute the Dirichlet process and updateit as a component of the Gibbs sampler. Since this requiresimputation of an infinite-dimensional process, implementationof the conditional method has relied on finite approximations.In this paper, we show how to avoid such approximations by designingtwo novel Markov chain Monte Carlo algorithms which sample fromthe exact posterior distribution of quantities of interest.The approximations are avoided by the new technique of retrospectivesampling. We also show how the algorithms can obtain samplesfrom functionals of the Dirichlet process. The marginal andthe conditional methods are compared and a careful simulationstudy is included, which involves a non-conjugate model, differentdatasets and prior specifications.     

Normal Approximations of Exact Tests in Configural Frequency Analysis

The most common tests for types and antitypes in configural frequency analysis are normal approximations of exact tests. In the paper such statistics under the complete independence model and under the fixed margins model are discussed. It turns out that these test statistics are not acceptable when the number of simultaneously performed tests is large or when the expected frequencies are small. In these cases, the use of exact tests is advocated and some existing computer programs for such tests are indicated. A normal approximation based on the strong version of the De Moivre-Laplace limit theorem is also discussed. Empirical examples are given from longitudinal data describing psychological development of boys.     

An Exact Paired Rank Test

An exact rank test for two dependent samples based on overall mid‐ranks is discussed which can be applied to metric as well as to ordinal data. The exact conditional distribution of the test statistic given the observed vector of rank differences is determined. A recursion formula is given as well as a fast shift algorithm in SAS/IML code. Moreover, it is demonstrated that the paired rank test can be more powerful than other tests for paired samples by means of a simulation study. Finally, the test is applied to a psychiatric trial with longitudinal ordinal data.