Multivariate extensions of McNemar's test

This article considers global tests of differences between paired vectors of binomial probabilities, based on data from two dependent multivariate binary samples. Difference is defined as either an inhomogeneity in the marginal distributions or asymmetry in the joint distribution. For detecting the first type of difference, we propose a multivariate extension of McNemar's test and show that it is a generalized score test under a generalized estimating equations (GEE) approach. Univariate features such as the relationship between the Wald and score tests and the dropout of pairs with the same response carry over to the multivariate case and the test does not depend on the working correlation assumption among the components of the multivariate response. For sparse or imbalanced data, such as occurs when the number of variables is large or the proportions are close to zero, the test is best implemented using a bootstrap, and if this is computationally too complex, a permutation distribution. We apply the test to safety data for a drug, in which two doses are evaluated by comparing multiple responses by the same subjects to each one of them.     

Genomic scans detect signatures of selection along a salinity gradient in populations of the intertidal seaweed Fucus serratus on a 12 km scale

Detecting natural selection in wild populations is a central challenge in evolutionary biology and genomic scans are an important means of detecting allele frequencies that deviate from neutral expectations among marker loci. We used nine anonymous and 15 EST-linked microsatellites, 362 AFLP loci, and several neutrality tests, to identify outlier loci when comparing four populations of the seaweed Fucus serratus spaced along a 12km intertidal shore with a steep salinity gradient. Under criteria of at least two significant tests in at least two population pairs, three EST-derived and three anonymous loci revealed putative signatures of selection. Anonymous locus FsB113 was a consistent outlier when comparing least saline to fully marine sites. Locus F37 was an outlier when comparing the least saline to more saline areas, and was annotated as a polyol transporter/putative mannitol transporter - an important sugar-alcohol associated with osmoregulation by brown algae. The remaining loci could not be annotated using six different data bases. Exclusion of microsatellite outlier loci did not change either the degree or direction of differentiation among populations. In one outlier test, the number of AFLP outlier loci increased as the salinity differences between population pairs increased (up to 14); only four outliers were detected with the second test and only one was consistent with both tests. Consistency may be improved with a much more rigorous approach to replication and/or may be dependent upon the class of marker used.     

Confidence Limits on the Ratio of the Mean Failure Times Based on Paired Exponential Observations

This paper discusses interval estimation for the ratio of the mean failure times on the basis of paired exponential observations. This paper considers five interval estimators: the confidence interval using an idea similar to Fieller's theorem (CIFT), the confidence interval using an exact parametric test (CIEP), the confidence interval using the marginal likelihood ratio test (CILR), the confidence interval assuming no matching effect (CINM), and the confidence interval using a locally most powerful test (CIMP). To evaluate and compare the performance of these five interval estimators, this paper applies Monte Carlo simulation. This paper notes that with respect to the coverage probability, use of the CIFT, CILR, or CIMP, although which are all derived based on large sample theory, can perform well even when the number of pairs n is as small as 10. As compared with use of the CILR, this paper finds that use of the CIEP with equal tail probabilities is likely to lose efficiency. However, this loss can be reduced by using the optimal tail probabilities to minimize the average length when n is small (<20). This paper further notes that use of the CIMP is preferable to the CIEP in a variety of situations considered here. In fact, the average length of the CIMP with use of the optimal tail probabilities can even be shorter than that of the CILR. When the intraclass correlation between failure times within pairs is 0 (i.e., the failure times within the same pair are independent), the CINM, which is derived for two independent samples, is certainly the best one among the five interval estimators considered here. When there is an intraclass correlation but which is small (<0.10), the CIFT is recommended for obtaining a relatively short interval estimate without sacrificing the loss of the coverage probability. When the intraclass correlation is moderate or large, either the CILR or the CIMP with the optimal tail probabilities is preferable to the others. This paper also notes that if the intraclass correlation between failure times within pairs is large, use of the CINM can be misleading, especially when the number of pairs is large.     

A Note on Interval Estimation of the Simple Difference in Data with Correlated Matched Pairs

When we employ cluster sampling to collect data with matched pairs, the assumption of independence between all matched pairs is not likely true. This paper notes that applying interval estimators, that do not account for the intraclass correlation between matched pairs, to estimate the simple difference between two proportions of response can be quite misleading, especially when both the number of matched pairs per cluster and the intraclass correlation between matched pairs within clusters are large. This paper develops two asymptotic interval estimators of the simple difference, that accommodate the data of cluster sampling with correlated matched pairs. This paper further applies Monte Carlo simulation to compare the finite sample performance of these estimators and demonstrates that the interval estimator, derived from a quadratic equation proposed here, can actually perform quite well in a variety of situations.     

A mixture-model approach for parallel testing for unequal variances

Testing for unequal variances is usually performed in order to check the validity of the assumptions that underlie standard tests for differences between means (the t-test and anova). However, existing methods for testing for unequal variances (Levene's test and Bartlett's test) are notoriously non-robust to normality assumptions, especially for small sample sizes. Moreover, although these methods were designed to deal with one hypothesis at a time, modern applications (such as to microarrays and fMRI experiments) often involve parallel testing over a large number of levels (genes or voxels). Moreover, in these settings a shift in variance may be biologically relevant, perhaps even more so than a change in the mean. This paper proposes a parsimonious model for parallel testing of the equal variance hypothesis. It is designed to work well when the number of tests is large; typically much larger than the sample sizes. The tests are implemented using an empirical Bayes estimation procedure which `borrows information' across levels. The method is shown to be quite robust to deviations from normality, and to substantially increase the power to detect differences in variance over the more traditional approaches even when the normality assumption is valid.     

Selection for brain size impairs innate,but not adaptive immune responses

Both the brain and the immune system are energetically demanding organs, and when natural selection favours increased investment into one, then the size or performance of the other should be reduced. While comparative analyses have attempted to test this potential evolutionary trade-off, the results remain inconclusive. To test this hypothesis, we compared the tissue graft rejection (an assay for measuring innate and acquired immune responses) in guppies (Poecilia reticulata) artificially selected for large and small relative brain size. Individual scales were transplanted between pairs of fish, creating reciprocal allografts, and the rejection reaction was scored over 8 days (before acquired immunity develops). Acquired immune responses were tested two weeks later, when the same pairs of fish received a second set of allografts and were scored again. Compared with large-brained animals, small-brained animals of both sexes mounted a significantly stronger rejection response to the first allograft. The rejection response to the second set of allografts did not differ between large- and small-brained fish. Our results show that selection for large brain size reduced innate immune responses to an allograft, which supports the hypothesis that there is a selective trade-off between investing into brain size and innate immunity.     

A STATISTICAL ANALYSIS OF DIFFERENCE TESTS WITH REPLICATIONS

This paper proposes a statistical method for difference tests with repetitions. Classical methods for difference tests are based upon the binomial distribution, and are not concerned with the number of repetitions per judge. But when more than one replication of a difference test is required, judgements from different judges are more independent than replicates from the same judge; these two cannot be combined in the classical methods. In this paper, we propose another approach that takes into account two points: the number of repetitions per judge, and the differences within subjects. Two examples are presented to illustrate this approach.     

Empirical evaluation of two-sample statistical tests for differences of stepping phase during insect walking

In order to determine which statistical tests can validly be applied to data that describe a temporal relationship between two or more repetitive movements by an animal, we evaluated empirically seven two-sample tests that seemed potentially useful: Student's t test, the Watson Williams test for means, the variance-ratio F test, the Watson Williams test for the concentration parameter k, the Wallraff test, the Mann Whitney test and the Watson U² test. Evaluations were carried out on the timing (phases) of bursts of muscular activity in one leg relative to those in another during free walking in cockroaches. Each statistical test was evaluated by dividing randomly a single parent set of data into two subsets, each subset containing about half the original data set. This division was repeated 400 times, thus generating 400 different pairs of subsets. Each statistical test was used separately on the pairs of subsets to test the null hypothesis that the two samples of each pair came from the same population;; this procedure generated 400 statistics for each test, one for each pair of subsets. An estimate of the reliability of each statistical test was obtained by comparing the number of times the test actually indicated a significant difference between subsets to the number of times it might be expected to do so out (20 out of 400 when tested at the 5% level of significance). This procedure was repeated on ten different sets of data. The outcome of the evaluation suggested that, from an empirical point of view, Student's t, the Mann Whitney, the Wallraff and the Watson U² tests may be useful in assessing differences among the data we analyzed. The variance-ratio F test and the Watson Williams test for the concentration parameter k were clearly not usable. The Watson Williams test for means might be useful in some circumstances. Performing an arcsine transformation of the data did not significantly alter these results. Possible causes of the inapplicability of some of these tests to phase data are discussed.     

Linkage analysis for diseases with variable age of onset

We present a method for the multivariate linkage analysis of the age of onset of a disease. The approach allows the incorporation of covariates for the study of gene by environment interactions. It is applicable to general pedigrees. The likelihood of the data is expressed as a function of the number of alleles identical by descent at a marker, the censored ages of onset and disease status, and environmental exposures. In a simulation study, we compare the power to detect linkage under different sampling schemes for either a dominant or recessive trait when approximately 10% of individuals are gene carriers. The majority of the linkage information from a sample of randomly selected sib pairs was retained when the analyses were limited to sibships with one sibling having early-onset disease (<59 years old). Incorporating parental phenotypes could improve the power to detect the gene. When the sample consists of affected sib pairs (ASPs) having variable age of onset, the likelihood ratio (LR) test had higher power than the means (t(2)) test for detecting a locus with a large genetic relative risk (R(g) = 20). However, the power of the two tests was similar when ASPs are selected so that the proband has an early onset of disease. Lastly, the LR test had more power than the t(2) test to detect linkage in the presence of gene by environment interactions.     

A Study of Linkage Disequilibrium in Populations of DROSOPHILA MELANOGASTER

This paper presents the results of a study of linkage disequilibrium between five polymorphic enzyme genes located on chromosome 3 of D. melanogaster. Three sets of chromosomes were examined: two represented samples from successive years of the same natural population, and one came from a large laboratory population. Out of the thirty possible tests for linkage disequilibrium between pairs of loci, two were significant at the 5% level and two at the 1% level. This result cannot reasonably be ascribed to chance alone. The pairs of loci that had a significant correlation in one sample had higher than average correlations in the other samples (though not necessarily in the same direction); this effect was highly significant statistically. There was no tendency for the high correlations to be associated with tightness of linkage between the loci concerned. All five loci were involved in at least one significant effect. It was concluded that these results are difficult to explain on the neutral allele theory of protein polymorphism, but are consistent with the concept of selective control of allele frequencies.     

Testing and estimation of proportion (or risk) ratio under the matched‐pair design with multiple binary endpoints

The proportion ratio (PR) of responses between an experimental treatment and a control treatment is one of the most commonly used indices to measure the relative treatment effect in a randomized clinical trial. We develop asymptotic and permutation‐based procedures for testing equality of treatment effects as well as derive confidence intervals of PRs for multivariate binary matched‐pair data under a mixed‐effects exponential risk model. To evaluate and compare the performance of these test procedures and interval estimators, we employ Monte Carlo simulation. When the number of matched pairs is large, we find that all test procedures presented here can perform well with respect to Type I error. When the number of matched pairs is small, the permutation‐based test procedures developed in this paper is of use. Furthermore, using test procedures (or interval estimators) based on a weighted linear average estimator of treatment effects can improve power (or gain precision) when the treatment effects on all response variables of interest are known to fall in the same direction. Finally, we apply the data taken from a crossover clinical trial that monitored several adverse events of an antidepressive drug to illustrate the practical use of test procedures and interval estimators considered here.     

Isolation of Drosophila flightless mutants which affect myofibrillar proteins of indirect flight muscle

Summary A large number of dominant flightless mutants of Drosophila were chemically induced, and their thorax proteins were examined by means of two-dimensional gel electrophoresis (O'Farrell 1975). Among them, 26 lines were found to have deficiency or reduction of some of myofibrillar proteins in indirect flight muscle (IFM). The gel patterns of the mutants could be classified into eleven groups. In general, more than a few polypeptides were either absent or reduced in each mutant line. Although the mutations affect myofibrillar proteins in apparently complex and diverse ways, logical correlations were found among the changes. There are pairs of proteins which always change together when a number of mutants are compared. There are also many pairs in which presence of one protein is necessary, but not sufficient for presence of the other. This suggests that absence of one component leads to disappearance or reduction of others which are either spatially or functionally related to the former. The correlation is possibly due to a hierarchy of the proteins in the myofibrillar assembly processes.Chromosomal loci of eleven typical mutants were examined, and it was found that most of them are located in two small regions of the second and the third chromosomes. IFM myofibrils of these mutants are either abnormal or absent in homozygotes as well as in heterozygotes.     

Statistical analysis of microarray data: a Bayesian approach

The potential of microarray data is enormous. It allows us to monitor the expression of thousands of genes simultaneously. A common task with microarray is to determine which genes are differentially expressed between two samples obtained under two different conditions. Recently, several statistical methods have been proposed to perform such a task when there are replicate samples under each condition. Two major problems arise with microarray data. The first one is that the number of replicates is very small (usually 2-10), leading to noisy point estimates. As a consequence, traditional statistics that are based on the means and standard deviations, e.g. t-statistic, are not suitable. The second problem is that the number of genes is usually very large (approximately 10,000), and one is faced with an extreme multiple testing problem. Most multiple testing adjustments are relatively conservative, especially when the number of replicates is small. In this paper we present an empirical Bayes analysis that handles both problems very well. Using different parametrizations, we develop four statistics that can be used to test hypotheses about the means and/or variances of the gene expression levels in both one- and two-sample problems. The methods are illustrated using experimental data with prior knowledge. In addition, we present the result of a simulation comparing our methods to well-known statistics and multiple testing adjustments.     

Chromosomes of the liver fluke, Clonorchis sinensis

A karyological study was carried out in order to compared the chromosome numbers, chromosome morphologies and karyotypes of the oriental liver fluke, Clonorchis sinensis (Trematoda: Opisthorchiidae), collected from Korea and China. Chromosome preparations were made by means of air-drying method. The chromosome number was 2n = 56 in both Korean and Chinese flukes, and chromosomes were divided into two groups based on this size; consisting of 8 pairs of large and 20 pairs of small chromosomes. However, the karyotypes showed some differences between Korean and Chinese flukes. The karyotype of liver flukes from Korea consisted of three metacentric pairs, one meta-/submetacentric pair, 16 submetacentric pairs and eight subtelocentric pairs of chromosomes. On the other hand, liver flukes from China consisted of two metacentric pairs, two meta-/submetacentric pairs, 16 submetacentric pairs and eight subtelocentric pairs of chromosomes.     

Statistical test for detecting overdispersion in offspring number based on kinship information

In this paper, we develop a theory of a new statistic that tests overdispersion in offspring number on the basis of exactly known kinship relationships. The statistic utilizes the sample size and the number of kinship pairs found in a sample, specially the number of mother–offspring (MO) and maternal–half-sibling (MHS) pairs. Given a sufficiently large sample size, the statistic proposed in this paper approximately follows a standard-normal distribution under non-overdispersed conditions (Poisson’s variance). We found that (1) the value of the statistic (\(\ge 2\) or \(<2\)) reasonably indicates whether reproduction is overdispersed at the 5% significance level; (2) the power of the statistic is determined primarily by the balance between the degree of overdispersion and the sample size; (3) in many cases, if the number of kinship pairs can be approximated by a normal distribution, false-positive and false-negative situations can be avoided. The proposed method can detect moderate-weak levels of overdispersion that produce few MHS pairs in a sample because the effect of the population size (which determines the number of detected MHS pairs) is canceled by the detection of the number of MO pairs. Once the kinship determination procedure is established, this indirect measurement will be readily applicable to species even with weak overdispersion, expanding the available opportunities for understanding how overdispersion in offspring number affects ecological processes.     

Simultaneous Non‐inferiority Test of Sensitivity and Specificity for Two Diagnostic Procedures in the Presence of a Gold Standard

Sensitivity and specificity have traditionally been used to assess the performance of a diagnostic procedure. Diagnostic procedures with both high sensitivity and high specificity are desirable, but these procedures are frequently too expensive, hazardous, and/or difficult to operate. A less sophisticated procedure may be preferred, if the loss of the sensitivity or specificity is determined to be clinically acceptable. This paper addresses the problem of simultaneous testing of sensitivity and specificity for an alternative test procedure with a reference test procedure when a gold standard is present. The hypothesis is formulated as a compound hypothesis of two non‐inferiority (one‐sided equivalence) tests. We present an asymptotic test statistic based on the restricted maximum likelihood estimate in the framework of comparing two correlated proportions under the prospective and retrospective sampling designs. The sample size and power of an asymptotic test statistic are derived. The actual type I error and power are calculated by enumerating the exact probabilities in the rejection region. For applications that require high sensitivity as well as high specificity, a large number of positive subjects and a large number of negative subjects are needed. We also propose a weighted sum statistic as an alternative test by comparing a combined measure of sensitivity and specificity of the two procedures. The sample size determination is independent of the sampling plan for the two tests.     

Asymmetrical correlated responses to selection under an infinitesimal genetic model

Summary Asymmetry in correlated responses to selection is expected when more than one cycle of selection is practised due to changes in genetic parameters produced by selection. In large populations, under the infinitesimal model these changes are due to linkage disequilibrium generated by selection and not to gene frequency changes. This study examines the conditions under which asymmetrical correlated responses are to be expected when an infinitesimal model is considered. Asymmetrical correlated responses in two traits in respect to which trait is selected are expected if the two traits have different heritabilities. Predicted asymmetry increases with the absolute value of the genetic correlation between the two traits, the difference between the two heritabilities, the intensity of selection and the number of generations of selection. Linkage disequilibrium generated by selection should be taken into account in explaining asymmetrical correlated responses observed in selection experiments.     

The design of diagnostic tests for defective colour vision.

Pseudoisochromatic plates are among the most popular tests for defective colour vision. They are particularly good for screening but are less good in assessing the degree and type of the colour vision defect. To select colours for use in diagnostic plates a large number of colour defective subjects have made colour matches with the Lovibond Tintometer and the isochromatic data collected. Pseudoisochromatic plates have been printed using pairs of colours only and incorporating both a random dot and a regular dot format. These plates have proved effective in a clinical trial. Not only must pairs of inks be carefully selected to lie upon appropriate isochromatic lines but the luminance contrast between the two colours must be kept within 5%. Failure to control luminance contrast is as much a source of error in currently available pseudoischromatic tests as the inappropriate use of colour.     

Sequence-similar, structure-dissimilar protein pairs in the PDB

It is often assumed that in the Protein Data Bank (PDB), two proteins with similar sequences will also have similar structures. Accordingly, it has proved useful to develop subsets of the PDB from which "redundant" structures have been removed, based on a sequence-based criterion for similarity. Similarly, when predicting protein structure using homology modeling, if a template structure for modeling a target sequence is selected by sequence alone, this implicitly assumes that all sequence-similar templates are equivalent. Here, we show that this assumption is often not correct and that standard approaches to create subsets of the PDB can lead to the loss of structurally and functionally important information. We have carried out sequence-based structural superpositions and geometry-based structural alignments of a large number of protein pairs to determine the extent to which sequence similarity ensures structural similarity. We find many examples where two proteins that are similar in sequence have structures that differ significantly from one another. The source of the structural differences usually has a functional basis. The number of such proteins pairs that are identified and the magnitude of the dissimilarity depend on the approach that is used to calculate the differences; in particular sequence-based structure superpositioning will identify a larger number of structurally dissimilar pairs than geometry-based structural alignments. When two sequences can be aligned in a statistically meaningful way, sequence-based structural superpositioning provides a meaningful measure of structural differences. This approach and geometry-based structure alignments reveal somewhat different information and one or the other might be preferable in a given application. Our results suggest that in some cases, notably homology modeling, the common use of nonredundant datasets, culled from the PDB based on sequence, may mask important structural and functional information. We have established a data base of sequence-similar, structurally dissimilar protein pairs that will help address this problem (http://luna.bioc.columbia.edu/rachel/seqsimstrdiff.htm).     

Serological and immunoelectrophoretic relationships among viruses in the tombusvirus group

Serological cross-reactions among eighteen virus isolates of the tombusvirus group were compared in precipitin tube and immunodiffusion serological tests. The isolates were also compared by immunoelectrophoresis in agar gel. Although precipitin tube tests showed considerable and reproducible differences between the various isolates, the results were too greatly affected by other factors to be of value in assessing strain relationships. When pairs of isolates were compared for spur formation in gel-diffusion tests, the results suggested that most isolates could be placed in one of two groups; one group comprised isolates from pelargonium (leaf curl), the other consisted of petunia asteroid mosaic virus and artichoke mottled crinkle virus isolates from Italy and tomato bushy stunt isolates from soil around this Institute and from cherry. Four isolates did not fall into either of these groups; they nearly always formed spurs when compared among themselves, or with viruses in either of the two groups. Pairs of isolates that could be distinguished from each other in spur-formation tests using antiserum homologous to one of them could not always be differentiated when antiserum heterologous to both isolates was used. Immunoelectrophoresis gave consistent results with several methods of virus preparation; it indicated grouping and separation of the isolates in general agreement with the results of gel-diffusion tests: all pelargonium leaf curl isolates were grouped together with slow migration towards the cathode. The petunia asteroid mosaic isolate and the isolates from cherry and from soil from this Institute (GCRI) moved slowly towards the anode. Tomato bushy stunt virus type strain migrated rapidly to the cathode, differing greatly from all other isolates. The method offers a relatively simple means of typing isolates of the tombusvirus group.