Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

Exact tests for one sample correlated binary data

In this paper we developed exact tests for one sample correlated binary data whose cluster sizes are at most two. Although significant progress has been made in the development and implementation of the exact tests for uncorrelated data, exact tests for correlated data are rare. Lack of a tractable likelihood function has made it difficult to develop exact tests for correlated binary data. However, when cluster sizes of binary data are at most two, only three parameters are needed to characterize the problem. One parameter is fixed under the null hypothesis, while the other two parameters can be removed by both conditional and unconditional approaches, respectively, to construct exact tests. We compared the exact and asymptotic p-values in several cases. The proposed method is applied to real-life data.     

Conditional Rejection Probabilities of Student's t‐test and Design Adaptations

For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t‐test and show that it can be non monotone in the interim mean of the data and non monotone in the non‐centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t‐test and give a numerical example. Additionally, the conditional rejection probability given the interim t‐statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unconditional Exact Tests for Equivalence or Noninferiority for Paired Binary Endpoints

Problems of establishing equivalence or noninferiority between two medical diagnostic procedures involve comparisons of the response rates between correlated proportions. When the sample size is small, the asymptotic tests may not be reliable. This article proposes an unconditional exact test procedure to assess equivalence or noninferiority. Two statistics, a sample-based test statistic and a restricted maximum likelihood estimation (RMLE)-based test statistic, to define the rejection region of the exact test are considered. We show the p-value of the proposed unconditional exact tests can be attained at the boundary point of the null hypothesis. Assessment of equivalence is often based on a comparison of the confidence limits with the equivalence limits. We also derive the unconditional exact confidence intervals on the difference of the two proportion means for the two test statistics. A typical data set of comparing two diagnostic procedures is analyzed using the proposed unconditional exact and asymptotic methods. The p-value from the unconditional exact tests is generally larger than the p-value from the asymptotic tests. In other words, an exact confidence interval is generally wider than the confidence interval obtained from an asymptotic test.     

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.     

Point and Interval Estimation in the Combination of Bioassay Results with Unequal Variances

Symmetric parallel‐line biological assays involve the estimation of (log) relative potencies. The class of p(≥ 2) combination of symmetric parallel line bioassays are considered in this study. A large sample test for the equality of the several potencies is developed. An estimator and a confidence interval are proposed for the common relative potency parameter. The asymptotic distribution of the proposed test‐statistic under the null hypothesis as well as under a contagious hypothesis is derived.     

Testing for Treatment Effects on Subsets of Endpoints

Multiple endpoints are tested to assess an overall treatment effect and also to identify which endpoints or subsets of endpoints contributed to treatment differences. The conventional p‐value adjustment methods, such as single‐step, step‐up, or step‐down procedures, sequentially identify each significant individual endpoint. Closed test procedures can also detect individual endpoints that have effects via a step‐by‐step closed strategy. This paper proposes a global‐based statistic for testing an a priori number, say, r of the k endpoints, as opposed to the conventional approach of testing one (r = 1) endpoint. The proposed test statistic is an extension of the single‐step p‐value‐based statistic based on the distribution of the smallest p‐value. The test maintains strong control of the FamilyWise Error (FWE) rate under the null hypothesis of no difference in any (sub)set of r endpoints among all possible combinations of the k endpoints. After rejecting the null hypothesis, the individual endpoints in the sets that are rejected can be tested further, using a univariate test statistic in a second step, if desired. However, the second step test only weakly controls the FWE. The proposed method is illustrated by application to a psychosis data set.     

A Test for Homogeneity of Odds Ratios in Ordered 2 × 2 Tables

Consider K ordered 2 × 2 contingency tables. A new test of the null hypothesis that the odds ratios of these tables are equal vs the alternative hypothesis that the odds ratios are nondecreasing, is recommended. The test is exact (non‐asymptotic), is easily carried out (software is available), and has other favorable properties. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient exact p-value computation for small sample, sparse, and surprising categorical data.

A major obstacle in applying various hypothesis testing procedures to datasets in bioinformatics is the computation of ensuing p-values. In this paper, we define a generic branch-and-bound approach to efficient exact p-value computation and enumerate the required conditions for successful application. Explicit procedures are developed for the entire Cressie-Read family of statistics, which includes the widely used Pearson and likelihood ratio statistics in a one-way frequency table goodness-of-fit test. This new formulation constitutes a first practical exact improvement over the exhaustive enumeration performed by existing statistical software. The general techniques we develop to exploit the convexity of many statistics are also shown to carry over to contingency table tests, suggesting that they are readily extendible to other tests and test statistics of interest. Our empirical results demonstrate a speed-up of orders of magnitude over the exhaustive computation, significantly extending the practical range for performing exact tests. We also show that the relative speed-up gain increases as the null hypothesis becomes sparser, that computation precision increases with increase in speed-up, and that computation time is very moderately affected by the magnitude of the computed p-value. These qualities make our algorithm especially appealing in the regimes of small samples, sparse null distributions, and rare events, compared to the alternative asymptotic approximations and Monte Carlo samplers. We discuss several established bioinformatics applications, where small sample size, small expected counts in one or more categories (sparseness), and very small p-values do occur. Our computational framework could be applied in these, and similar cases, to improve performance.     

Assessing Equivalence Tests with Respect to their Expected p‐Value

Monte‐Carlo simulation methods are commonly used for assessing the performance of statistical tests under finite sample scenarios. They help us ascertain the nominal level for tests with approximate level, e.g. asymptotic tests. Additionally, a simulation can assess the quality of a test on the alternative. The latter can be used to compare new tests and established tests under certain assumptions in order to determinate a preferable test given characteristics of the data. The key problem for such investigations is the choice of a goodness criterion. We expand the expected p‐value as considered by Sackrowitz and Samuel‐Cahn (1999) to the context of univariate equivalence tests. This presents an effective tool to evaluate new purposes for equivalence testing because of its independence of the distribution of the test statistic under null‐hypothesis. It helps to avoid the often tedious search for the distribution under null‐hypothesis for test statistics which have no considerable advantage over yet available methods. To demonstrate the usefulness in biometry a comparison of established equivalence tests with a nonparametric approach is conducted in a simulation study for three distributional assumptions.     

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.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

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.     

Confidence Interval for Rate Ratio in a 2 × 2 Table with Structural Zero: An Application in Assessing False-Negative Rate Ratio When Combining Two Diagnostic Tests

Summary . In this article, we consider problems with correlated data that can be summarized in a 2 × 2 table with structural zero in one of the off‐diagonal cells. Data of this kind sometimes appear in infectious disease studies and two‐step procedure studies. Lui (1998, Biometrics 54, 706–711) considered confidence interval estimation of rate ratio based on Fieller‐type, Wald‐type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false‐negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false‐negative rate ratio. Score test–based confidence interval construction for false‐negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test–based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre‐assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Pre-post Treatment Evaluation by Symmetry Testing in Square Contingency Table

Starting from an r × r contingency table derived from ordinal pre- and post treatment observations, the traditional Bowker symmetry test is suggested to be replaced by one of the following alternatives (1) a globals test of general improvement, (2) a set of simultaneous symmetry tests of differential improvement without assuming general improvement under H₀ and (3) a set of simultaneous symmetry tests beyond assuming general improvement under H₀. The tests are illustrated by a bivariate example of observing neuroticism and introversion binarily (+, -) before and after Rogerian psychotherapy in a sample of clients.     

A More Powerful Simultaneous Test Procedure in Configural Frequency Analysis

In the Configural Frequency Analysis (CFA) of KRAUTH and LIENERT (1973 a, b), overfrequented (or underfrequented) cells in multivariate contingency tables are identified by simultaneous binomial tests. As an alternative, finite and asymptotic tests are proposed, which are derived from the (exact conditional) generalized hypergeometrical distribution of the cell frequencies. These tests allow for considerably more powerful decisions than do the conservative binomial tests.     

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.     

A Monte Carlo Study to Evaluate the Performance of Selected Tests of Homogeneity in Sparse Contingency Tables

We address the problem of tests of homogeneity in two-way contingency tables in case-control studies when the case category is subdivided into k subcategories. In this situation, we have two cells with large frequencies and 2 X k cells with frequencies that become small as k increases. We propose two ad hoc statistics in which a statistic for the sparse cells is combined with a statistic for the cells with large frequencies. We will study these tests along with the Pearson test (using a chi-square approximation) in a Monte Carlo simulation study. Two sets of null hypothesis models and two sets of alternative hypothesis models are considered. The best test for the models considered is the usual Pearson test (using an approximate chi-square distribution) although the ad hoc models are more powerful under one alternative model considered.     

A Discordant-Sibship Test for Disequilibrium and Linkage: No Need for Parental Data

The sibship disequilibrium test (SDT) is designed to detect both linkage in the presence of association and association in the presence of linkage (linkage disequilibrium). The test does not require parental data but requires discordant sibships with at least one affected and one unaffected sibling. The SDT has many desirable properties: it uses all the siblings in the sibship; it remains valid if there are misclassifications of the affectation status; it does not detect spurious associations due to population stratification; asymptotically it has a chi2 distribution under the null hypothesis; and exact P values can be easily computed for a biallelic marker. We show how to extend the SDT to markers with multiple alleles and how to combine families with parents and data from discordant sibships. We discuss the power of the test by presenting sample-size calculations involving a complex disease model, and we present formulas for the asymptotic relative efficiency (which is approximately the ratio of sample sizes) between SDT and the transmission/disequilibrium test (TDT) for special family structures. For sib pairs, we compare the SDT to a test proposed both by Curtis and, independently, by Spielman and Ewens. We show that, for discordant sib pairs, the SDT has good power for testing linkage disequilibrium relative both to Curtis''s tests and to the TDT using trios comprising an affected sib and its parents. With additional sibs, we show that the SDT can be more powerful than the TDT for testing linkage disequilibrium, especially for disease prevalence >.3.     

On the comparison of the strength of morphological integration across morphometric datasets

Evolutionary morphologists frequently wish to understand the extent to which organisms are integrated, and whether the strength of morphological integration among subsets of phenotypic variables differ among taxa or other groups. However, comparisons of the strength of integration across datasets are difficult, in part because the summary measures that characterize these patterns (RV coefficient and r_PLS) are dependent both on sample size and on the number of variables. As a solution to this issue, we propose a standardized test statistic (a z‐score) for measuring the degree of morphological integration between sets of variables. The approach is based on a partial least squares analysis of trait covariation, and its permutation‐based sampling distribution. Under the null hypothesis of a random association of variables, the method displays a constant expected value and confidence intervals for datasets of differing sample sizes and variable number, thereby providing a consistent measure of integration suitable for comparisons across datasets. A two‐sample test is also proposed to statistically determine whether levels of integration differ between datasets, and an empirical example examining cranial shape integration in Mediterranean wall lizards illustrates its use. Some extensions of the procedure are also discussed.