A multivariate family-based association test using generalized estimating equations: FBAT-GEE

In this paper we propose a multivariate extension of family-based association tests based on generalized estimating equations. The test can be applied to multiple phenotypes and to phenotypic data obtained in longitudinal studies without making any distributional assumptions for the phenotypic observations. Methods for handling missing phenotypic information are discussed. Further, we compare the power of the multivariate test with permutation tests and with using separate tests for each outcome which are adjusted for multiple testing. Application of the proposed test to an asthma study illustrates the power of the approach.     

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.     

Tests for density dependence revisited

We have examined a number of statistical issues associated with methods for evaluating different tests of density dependence. The lack of definitive standards and benchmarks for conducting simulation studies makes it difficult to assess the performance of various tests. The biological researcher has a bewildering choice of statistical tests for testing density dependence and the list is growing. The most recent additions have been based on computationally intensive methods such as permutation tests and boot-strapping. We believe the computational effort and time involved will preclude their widespread adoption until: (1) these methods have been fully explored under a wide range of conditions and shown to be demonstrably superior than other, simpler methods, and (2) general purpose software is made available for performing the calculations. We have advocated the use of Bulmer's (first) test as a de facto standard for comparative studies on the grounds of its simplicity, applicability, and satisfactory performance under a variety of conditions. We show that, in terms of power, Bulmer's test is robust to certain departures from normality although, as noted by other authors, it is affected by temporal trends in the data. We are not convinced that the reported differences in power between Bulmer's test and the randomisation test of Pollard et al. (1987) justifies the adoption of the latter. Nor do we believe a compelling case has been established for the parametric bootstrap likelihood ratio test of Dennis and Taper (1994). Bulmer's test is essentially a test of the serial correlation in the (log) abundance data and is affected by the presence of autocorrelated errors. In such cases the test cannot distinguish between the autoregressive effect in the errors and a true density dependent effect in the time series data. We suspect other tests may be similarly affected, although this is an area for further research. We have also noted that in the presence of autocorrelation, the type I error rates can be substantially different from the assumed level of significance, implying that in such cases the test is based on a faulty significance region. We have indicated both qualitatively and quantitatively how autoregressive error terms can affect the power of Bulmer's test, although we suggest that more work is required in this area. These apparent inadequacies of Bulmer's test should not be interpreted as a failure of the statistical procedure since the test was not intended to be used with autocorrelated error terms.     

Testing Hardy–Weinberg Equilibrium and Homogeneity of Hardy–Weinberg Disequilibrium using Complex Survey Data

Summary For studies on population genetics, the use of representative random samples of the target population can avoid ascertainment bias. Genetic variation data from over a hundred genes were collected in a U.S. nationally representative sample in the Third National Health and Nutrition Examination Survey (NHANES III). Surveys such as the NHANES have complex stratified multistage cluster sample designs with sample weighting that can inflate variances and alter the expectations of test statistics. Thus, classical statistical tests of Hardy–Weinberg equilibrium (HWE) and homogeneity of HW disequilibrium (HHWD) for simple random samples are not suitable for data from complex samples. We propose using Wald tests for HWE and generalized score tests for HHWD that have been modified for complex samples. Monte Carlo simulation studies are used to investigate the finite sample properties of the proposed tests. Rao–Scott corrections applied to the tests were found to improve their type I error properties. Our methods are applied to the NHANES III genetic data for three loci involved in metabolizing lead in the body.     

SIMEX variance component tests in generalized linear mixed measurement error models

In the analysis of clustered data with covariates measured with error, a problem of common interest is to test for correlation within clusters and heterogeneity across clusters. We examined this problem in the framework of generalized linear mixed measurement error models. We propose using the simulation extrapolation (SIMEX) method to construct a score test for the null hypothesis that all variance components are zero. A key feature of this SIMEX score test is that no assumptions need to be made regarding the distributions of the random effects and the unobserved covariates. We illustrate this test by analyzing Framingham heart disease data and evaluate its performance by simulation. We also propose individual SIMEX score tests for testing the variance components separately. Both tests can be easily implemented using existing statistical software.     

Nonparametric tests for homogeneity of species assemblages: a data depth approach

Testing homogeneity of species assemblages has important applications in ecology. Due to the unique structure of abundance data often collected in ecological studies, most classical statistical tests cannot be applied directly. In this article, we propose two novel nonparametric tests for comparing species assemblages based on the concept of data depth. They can be considered as a natural generalization of the Kolmogorov-Smirnov and the Cramér-von Mises tests (KS and CM) in this species assemblage comparison context. Our simulation studies show that the proposed test is more powerful than other existing methods under various settings. A real example is used to demonstrate how the proposed method is applied to compare species assemblages using plant community data from a highly diverse tropical forest at Barro Colorado Island, Panama.     

Evaluation of Animal Carcinogenicity Studies: Cochran‐Armitage Trend Test vs. Multiple Contrast Tests

Typical animal carcinogenicity studies involve the comparison of several dose groups to a negative control. The uncorrected asymptotic Cochran‐Armitage trend test with equally spaced dose scores is the most frequently used test in such set‐ups. However, this test based on a weighted linear regression on proportions. It is well known that the Cochran‐Armitage test lacks in power for other shapes than the assumed linear one. Therefore, dichotomous multiple contrast tests are introduced. These build the maximum over several single contrasts, where each of them is chosen appropriately to cover a specific dose‐response shape. An extensive power study has been conducted to compare multiple contrast tests with the approaches used so far. Crucial results will be presented in this paper. Moreover, exact tests and continuity corrected versions are introduced and compared to the traditional uncorrected approaches regarding size and power behaviour. A trend test for any shape of the dose‐response relationship for either crude tumour rates or mortality‐ adjusted rates based on the simple Poly‐3 transformation is proposed for evaluation of carcinogenicity studies.     

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

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

Comparison of independent samples of high-dimensional data by pairwise distance measures

Pairwise distance or association measures of sample elements are often used as a basis for hierarchical cluster analyses. They can also be used in tests for the comparison of pre-defined subgroups of the total sample. Usually this is done with permutation tests In this paper, we compare such a procedure with alternative tests for high-dimensional data based on spherically distributed scores in simulation experiments and with real data. The tests based on the pairwise distance or similarity measures perform quite well in this comparison. As the number of possible permutations is small in very small samples, this might restrict the use of the test. Therefore, we propose an exact parametric small sample version of the test using randomly rotated samples.     

Tests for density dependence

Several statistical tests for density dependence have been proposed in the literature, and so in any practical case the question poses itself which one of these tests to choose. This paper offers a few remarks additional to those made by Fox and Ridsill-Smith (1995) and others. Parametric statistical tested are based on a fully specified mathematical model. Examples of such tests are Bulmer's (1975) first test, and the test of Dennis and Taper (1994). Distribution-free tests are based on far less stringent assumptions. An example of such a test is the one proposed by Pollard et al. (1987). The choice between parametric tests can best be made by considering which one of the underlying mathematical models ist most plausible. If all models are almost equally plausible, considerations of computational requirement and ease of application may be important. Strong doubts concerning the plausibility of mathematical models may lead one to prefer a distribution-free test. An important feature of any test is its power, i.e. the probability of its rejecting the null hypothesis when this hypothesis is not true. Other things being equal, tests are preferable when they have superior powers. But power of a test depends on the true state of nature, and the only way to study power quantitatively is by assuming some mathematical model as approximately representing this true state. As any mathematical model can at best only be an approximation to the situation in nature, a mathematical model and the statistical tests based on it should be robust against small deviations from model assumptions. Solow (1990) showed that Bulmer's test is not robust with respect to the assumption that the residuals in the underlying autoregression model be stochastically independent. Contrary to what was suggested by Fox and Ridsill-Smith (1995), who misinterpreted some statements in Reddingius (1990), the present author thinks this is a serious shortcoming of this test since an ecologist cannot assume a priori that important density-independent ecological factors are not somehow serially correlated. Moreover, he is rather sceptical about the usefulness of statistical tests for density dependence. They have contributed more to misunderstandings than to a significant increase in ecological insight. In any case, statistical tests are designed to test hypotheses that are stated before data are collected, and the question which test to use also has to be answered before the data have been collected. Designing and using statistical tests a posteriori to detect things in data mainly leads to confusion and controversy.     

Non-Parametric Methods for the 2-Period-Cross-Over Design Under Weak Model Assumptions

A nonparametric analysis for the two period cross-over design has first been suggested by Koch (1972) and has been discussed by Hills and Armitage (1979). As known rank tests on sums or differences of the data are applied in this procedure, the results on the one hand are not invariant under monotonous transformations and on the other hand the procedure is only correct for models with additive effects. Therefore, in the present article generalized effects will first be defined in the 2-period cross-over design without the assumption of a linear model and then rank test will be presented which test tese effects without the need of sums or differences of the data. In the appendix the equivalence of the hypothesis for the generalized effects to the known hypotheses for the effects in the linear model will be shown. The application of the procedures will be demonstrated by means of an example in literature.     

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.     

Survival Tests for r Groups

In this paper we give the generalization of the score tests covering the case of ties and we give examples where the expressions in matrix form are completely specified for the weighted tests and the score tests for the case of r groups. It is worth mentioning that although the score tests are not generally included in the commercial software, these tests should be used if it can be assumed that the censoring mechanism is equal in the r groups or if there is no censoring (Lawless , 1982). We establish the equivalence between “numerators” of these families of tests. As result of this equivalence we define four new tests that complete the classification of score and weighted tests. The Kruskal‐Wallis test (1952) appears as a particular case of the score tests for the case of non‐censoring. A simulation study has been done in order to compare the performance of the tests described in this paper. An example is included to make the understanding of the paper easier.     

Usefulness of non-invasive tests for diagnosing Helicobacter pylori infection in patients undergoing dialysis for chronic renal failure

BACKGROUND: Helicobacter pylori infection in chronic renal failure patients has been linked to peptic ulcer and gastric neoplasia after kidney transplantation. It may also contribute to the accelerated arteriosclerosis that is usual in this population. Few data are available on the usefulness of noninvasive diagnostic tests for H. pylori infection in dialyzed patients, especially regarding the new fecal antigen detection tests. The objective of this study was to determine the efficacy of a noninvasive test for H. pylori infection in patients with chronic renal failure. METHODS: Eighty-six patients were included in a cross-sectional study. Urea breath test, serology and three fecal tests--FemtoLab H. pylori (Connex, Germany), Premier Platinum HpSA (Meridian, USA) and Simple H. pylori (Operon SA, Spain) were performed. Helicobacter pylori status was determined by concordance of the tests. Sensitivity, specificity and positive and negative predictive values were calculated for each test. RESULTS: Sensitivity, specificity, positive and negative predictive values were 94%, 96%, 94% and 96% for the urea breath test; 97%, 64%, 66% and 97% for serology; 86%, 100%, 100% and 91%, for FemtoLab H. pylori; 58%, 96%, 91% and 76% for Premier Platinum HpSA and 61%, 78%, 74% and 67% for Simple H. pylori. CONCLUSIONS: The urea breath test seems to be the most reliable diagnostic method for H. pylori infection in patients with chronic renal failure. Serology has a low specificity, and the results of the fecal tests vary widely.     

Distance-based tests for homogeneity of multivariate dispersions

Summary The traditional likelihood‐based test for differences in multivariate dispersions is known to be sensitive to nonnormality. It is also impossible to use when the number of variables exceeds the number of observations. Many biological and ecological data sets have many variables, are highly skewed, and are zero‐inflated. The traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non‐Euclidean dissimilarity would be more appropriate. Distance‐based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed here. They rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes. The tests are straightforward multivariate extensions of Levene's test, with P‐values obtained either using the traditional F‐distribution or using permutation of either least‐squares or LAD residuals. Examples illustrate the utility of the approach, including the analysis of stabilizing selection in sparrows, biodiversity of New Zealand fish assemblages, and the response of Indonesian reef corals to an El Niño. Monte Carlo simulations from the real data sets show that the distance‐based tests are robust and powerful for relevant alternative hypotheses of real differences in spread.     

A bootstrap test for the analysis of microarray experiments with a very small number of replications

In microarray studies it is common that the number of replications (i.e. the sample size) is small and that the distribution of expression values differs from normality. In this situation, permutation and bootstrap tests may be appropriate for the identification of differentially expressed genes. However, unlike bootstrap tests, permutation tests are not suitable for very small sample sizes, such as three per group. A variety of different bootstrap tests exists. For example, it is possible to adjust the data to have a common mean before the bootstrap samples are drawn. For small significance levels, which can occur when a large number of genes is investigated, the original bootstrap test, as well as a bootstrap test suggested for the Behrens-Fisher problem, have no power in cases of very small sample sizes. In contrast, the modified test based on adjusted data is powerful. Using a Monte Carlo simulation study, we demonstrate that the difference in power can be huge. In addition, the different tests are illustrated using microarray data.     

Most Powerful Permutation Invariant Tests for Relatedness Hypotheses Using Genotypic Data

The problem of inferring kinship structure among a sample of individuals using genetic markers is considered with the objective of developing hypothesis tests for genetic relatedness with nearly optimal properties. The class of tests considered are those that are constrained to be permutation invariant, which in this context defines tests whose properties do not depend on the labeling of the individuals. This is appropriate when all individuals are to be treated identically from a statistical point of view. The approach taken is to derive tests that are probably most powerful for a permutation invariant alternative hypothesis that is, in some sense, close to a null hypothesis of mutual independence. This is analagous to the locally most powerful test commonly used in parametric inference. Although the resulting test statistic is a U-statistic, normal approximation theory is found to be inapplicable because of high skewness. As an alternative it is found that a conditional procedure based on the most powerful test statistic can calculate accurate significance levels without much loss in power. Examples are given in which this type of test proves to be more powerful than a number of alternatives considered in the literature, including Queller and Goodknight's (1989) estimate of genetic relatedness, the average number of shared alleles (Blouin, 1996), and the number of feasible sibling triples (Almudevar and Field, 1999).     

Improved Logrank-Type Tests for Survival Data Using Adaptive Weights

Summary .  For testing for treatment effects with time-to-event data, the logrank test is the most popular choice and has some optimality properties under proportional hazards alternatives. It may also be combined with other tests when a range of nonproportional alternatives are entertained. We introduce some versatile tests that use adaptively weighted logrank statistics. The adaptive weights utilize the hazard ratio obtained by fitting the model of Yang and Prentice (2005,  Biometrika   92 , 1–17). Extensive numerical studies have been performed under proportional and nonproportional alternatives, with a wide range of hazard ratios patterns. These studies show that these new tests typically improve the tests they are designed to modify. In particular, the adaptively weighted logrank test maintains optimality at the proportional alternatives, while improving the power over a wide range of nonproportional alternatives. The new tests are illustrated in several real data examples.     

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.