Performance of MAX test and degree of dominance index in predicting the mode of inheritance

We evaluate power performance to detect the correct mode of inheritance in gene-disease associations of two different approaches: the MAX test and the degree of dominance index or h-index. The MAX test is a special case of the conditional independence tests that simultaneously test for association and select the most likely genetic model based on a three-dimensional normal distribution. The h-index is based on the philosophy of using orthogonal contrasts to infer the mode of inheritance quantitatively. A population genetic model is developed where the real mode of inheritance is known a priori and power performance can be accurately determined. The simulations showed that none of the two approaches generally outperforms the other, nor each of them provides a panacea to estimate efficiently the mode of inheritance in all parameter space. However, the simultaneous application of both approaches can provide insights in determining the underlying mode of inheritance.     

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

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

A Confidence‐Limit‐Based Approach to the Assessment of Hardy–Weinberg Equilibrium

The classical χ²‐procedure for the assessment of Hardy–Weinberg equilibrium (HWE) is tailored for detecting violations of HWE. However, many applications in genetic epidemiology require approximate compatibility with HWE. In a previous contribution to the field (Wellek, S. (2004). Biometrics, 60 , 694–703), the methodology of statistical equivalence testing was exploited for the construction of tests for problems in which the assumption of approximate compatibility of a given genotype distribution with HWE plays the role of the alternative hypothesis one aims to establish. In this article, we propose a procedure serving the same purpose but relying on confidence limits rather than critical bounds of a significance test. Interval estimation relates to essentially the same parametric function that was previously chosen as the target parameter for constructing an exact conditional UMPU test for equivalence with a HWE conforming genotype distribution. This population parameter is shown to have a direct genetic interpretation as a measure of relative excess heterozygosity. Confidence limits are constructed using both asymptotic and exact methods. The new approach is illustrated by reanalyzing genotype distributions obtained from published genetic association studies, and detailed guidance for choosing the equivalence margin is provided. The methods have been implemented in freely available SAS macros.     

A Robust Genome‐Wide Scan Statistic of the Wellcome Trust Case–Control Consortium

Summary In genome‐wide association (GWA) studies, test statistics that are efficient and robust across various genetic models are preferable, particularly for studying multiple diseases in the Wellcome Trust Case–Control Consortium ( WTCCC, 2007 , Nature 447 , 661–678). A new test statistic, the minimum of the p‐values of the trend test and Pearson's test, was considered by the WTCCC. It is referred to here as MIN2. Because the minimum of two p‐values is no longer a valid p‐value itself, the WTCCC only used it to rank single nucleotide polymorphisms (SNPs) but did not report the p‐values of the associated SNPs when MIN2 was used for ranking. Given its importance in practice, we derive the asymptotic null distribution of MIN2, study some of its analytical properties related to GWA studies, and compare it with existing methods (the trend test, Pearson's test, MAX3, and the constrained likelihood ratio test [CLRT]) by simulations across a wide range of possible genetic models: the recessive (REC), additive (ADD), multiplicative (MUL), dominant (DOM), and overdominant models. The results show that MAX3 and CLRT have greater efficiency robustness than other tests when the REC, ADD/MUL, and DOM models are possible, whereas Pearson's test and MIN2 have greater efficiency robustness if the possible genetic models also include the overdominant model. We conclude that robust tests (MAX3, MIN2, CLRT, and Pearson's test) are preferable to a single trend test for initial GWA studies. The four robust tests are applied to more than 100 SNPs associated with 11 common diseases identified by the two WTCCC GWA studies.     

Nonparametric evaluation of quantitative traits in population-based association studies when the genetic model is unknown

Statistical association between a single nucleotide polymorphism (SNP) genotype and a quantitative trait in genome-wide association studies is usually assessed using a linear regression model, or, in the case of non-normally distributed trait values, using the Kruskal-Wallis test. While linear regression models assume an additive mode of inheritance via equi-distant genotype scores, Kruskal-Wallis test merely tests global differences in trait values associated with the three genotype groups. Both approaches thus exhibit suboptimal power when the underlying inheritance mode is dominant or recessive. Furthermore, these tests do not perform well in the common situations when only a few trait values are available in a rare genotype category (disbalance), or when the values associated with the three genotype categories exhibit unequal variance (variance heterogeneity). We propose a maximum test based on Marcus-type multiple contrast test for relative effect sizes. This test allows model-specific testing of either dominant, additive or recessive mode of inheritance, and it is robust against variance heterogeneity. We show how to obtain mode-specific simultaneous confidence intervals for the relative effect sizes to aid in interpreting the biological relevance of the results. Further, we discuss the use of a related all-pairwise comparisons contrast test with range preserving confidence intervals as an alternative to Kruskal-Wallis heterogeneity test. We applied the proposed maximum test to the Bogalusa Heart Study dataset, and gained a remarkable increase in the power to detect association, particularly for rare genotypes. Our simulation study also demonstrated that the proposed non-parametric tests control family-wise error rate in the presence of non-normality and variance heterogeneity contrary to the standard parametric approaches. We provide a publicly available R library nparcomp that can be used to estimate simultaneous confidence intervals or compatible multiplicity-adjusted p-values associated with the proposed maximum test.     

On Exact Unconditional Test for Linear Trend in Dose‐Response Studies

The one‐degree‐of‐freedom Cochran‐Armitage (C‐A) test statistic for linear trend has been widely applied in various dose‐response studies (e.g., anti‐ulcer medications and short‐term antibiotics, animal carcinogenicity bioassays and occupational toxicant studies). This approximate statistic relies, however, on asymptotic theory that is reliable only when the sample sizes are reasonably large and well balanced across dose levels. For small, sparse, or skewed data, the asymptotic theory is suspect and exact conditional method (based on the C‐A statistic) seems to provide a dependable alternative. Unfortunately, the exact conditional method is only practical for the linear logistic model from which the sufficient statistics for the regression coefficients can be obtained explicitly. In this article, a simple and efficient recursive polynomial multiplication algorithm for exact unconditional test (based on the C‐A statistic) for detecting a linear trend in proportions is derived. The method is applicable for all choices of the model with monotone trend including logistic, probit, arcsine, extreme value and one hit. We also show that this algorithm can be easily extended to exact unconditional power calculation for studies with up to a moderately large sample size. A real example is given to illustrate the applicability of the proposed method.     

Testing for Genetic Association in the Presence of Linkage and Gene–Covariate Interactions

In order to study family‐based association in the presence of linkage, we extend a generalized linear mixed model proposed for genetic linkage analysis (Lebrec and van Houwelingen (2007), Human Heredity 64 , 5–15) by adding a genotypic effect to the mean. The corresponding score test is a weighted family‐based association tests statistic, where the weight depends on the linkage effect and on other genetic and shared environmental effects. For testing of genetic association in the presence of gene–covariate interaction, we propose a linear regression method where the family‐specific score statistic is regressed on family‐specific covariates. Both statistics are straightforward to compute. Simulation results show that adjusting the weight for the within‐family variance structure may be a powerful approach in the presence of environmental effects. The test statistic for genetic association in the presence of gene–covariate interaction improved the power for detecting association. For illustration, we analyze the rheumatoid arthritis data from GAW15. Adjusting for smoking and anti‐cyclic citrullinated peptide increased the significance of the association with the DR locus.     

On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t‐test is optimum. On the other hand, if the differences are not normal, the t‐test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non‐parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro‐Wilk test of normality to decide whether to use the t‐test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t‐tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy‐tailed, than currently used methods.     

Trend tests for case-control studies of genetic markers: power,sample size and robustness

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.     

Choice of Scores in Trend Tests for Case‐Control Studies of Candidate‐Gene Associations

When applying the Cochran‐Armitage (CA) trend test for an association between a candidate allele and a disease in a case‐control study, a set of scores must be assigned to the genotypes. Sasieni (1997, Biometrics 53 , 1253–1261) suggested scores for the recessive, additive, and dominant models but did not examine their statistical properties. Using the criteria of minimizing the required sample size of the CA trend test to achieve prespecified type I and type II errors, we show that the scores given by Sasieni (1997) are optimal for the recessive and dominant models and locally optimal for the additive one. Moreover, the additive scores are shown to be locally optimal for the multiplicative model. The tests are applied to a real dataset.     

Statistical Inference for Truncated Dirichlet Distribution and Its Application in Misclassification

This paper is concerned with the statistical inference of a truncated Dirichlet distribution (TDD) arising in the general context of misclassified multinomial models (such as medical screening or diagnostic tests) and experimental design with mixtures. By employing the conditional distribution method, we offer a generating procedure for the TDD. Alternatively, a sampling‐based approach using the Gibbs sampler was provided as a means for developing the posterior moments of interest. Finding the mode of a TDD is equivalent to extracting the constrained maximum likelihood estimate (MLE) of parameter vector in a multinomial model. Based upon a theoretic result, we propose an algorithm to calculate the constrained MLE. Applications in misclassification are presented.     

Analytic power and sample size calculation for the genotypic transmission/disequilibrium test in case‐parent trio studies

Case‐parent trio studies considering genotype data from children affected by a disease and their parents are frequently used to detect single nucleotide polymorphisms (SNPs) associated with disease. The most popular statistical tests for this study design are transmission/disequilibrium tests (TDTs). Several types of these tests have been developed, for example, procedures based on alleles or genotypes. Therefore, it is of great interest to examine which of these tests have the highest statistical power to detect SNPs associated with disease. Comparisons of the allelic and the genotypic TDT for individual SNPs have so far been conducted based on simulation studies, since the test statistic of the genotypic TDT was determined numerically. Recently, however, it has been shown that this test statistic can be presented in closed form. In this article, we employ this analytic solution to derive equations for calculating the statistical power and the required sample size for different types of the genotypic TDT. The power of this test is then compared with the one of the corresponding score test assuming the same mode of inheritance as well as the allelic TDT based on a multiplicative mode of inheritance, which is equivalent to the score test assuming an additive mode of inheritance. This is, thus, the first time the power of these tests are compared based on equations, yielding instant results and omitting the need for time‐consuming simulation studies. This comparison reveals that these tests have almost the same power, with the score test being slightly more powerful.     

A hybrid omnibus test for generalized semiparametric single‐index models with high‐dimensional covariate sets

Numerous statistical methods have been developed for analyzing high‐dimensional data. These methods often focus on variable selection approaches but are limited for the purpose of testing with high‐dimensional data. They are often required to have explicit‐likelihood functions. In this article, we propose a “hybrid omnibus test” for high‐dicmensional data testing purpose with much weaker requirements. Our hybrid omnibus test is developed under a semiparametric framework where a likelihood function is no longer necessary. Our test is a version of a frequentist‐Bayesian hybrid score‐type test for a generalized partially linear single‐index model, which has a link function being a function of a set of variables through a generalized partially linear single index. We propose an efficient score based on estimating equations, define local tests, and then construct our hybrid omnibus test using local tests. We compare our approach with an empirical‐likelihood ratio test and Bayesian inference based on Bayes factors, using simulation studies. Our simulation results suggest that our approach outperforms the others, in terms of type I error, power, and computational cost in both the low‐ and high‐dimensional cases. The advantage of our approach is demonstrated by applying it to genetic pathway data for type II diabetes mellitus.     

Retrospective versus prospective score tests for genetic association with case‐control data

Since the seminal work of Prentice and Pyke, the prospective logistic likelihood has become the standard method of analysis for retrospectively collected case‐control data, in particular for testing the association between a single genetic marker and a disease outcome in genetic case‐control studies. In the study of multiple genetic markers with relatively small effects, especially those with rare variants, various aggregated approaches based on the same prospective likelihood have been developed to integrate subtle association evidence among all the markers considered. Many of the commonly used tests are derived from the prospective likelihood under a common‐random‐effect assumption, which assumes a common random effect for all subjects. We develop the locally most powerful aggregation test based on the retrospective likelihood under an independent‐random‐effect assumption, which allows the genetic effect to vary among subjects. In contrast to the fact that disease prevalence information cannot be used to improve efficiency for the estimation of odds ratio parameters in logistic regression models, we show that it can be utilized to enhance the testing power in genetic association studies. Extensive simulations demonstrate the advantages of the proposed method over the existing ones. A real genome‐wide association study is analyzed for illustration.     

The presence of a cryptic barrier in the West Pacific Ocean suggests the effect of glacial climate changes on a widespread sea‐dispersed plant,Vigna marina (Fabaceae)

Ocean currents are an important driver of evolution for sea‐dispersed plants, enabling them to maintain reciprocal gene flow via sea‐dispersed diaspores and obtain wide distribution ranges. Although geographic barriers are known to be the primary factors shaping present genetic structure of sea‐dispersed plants, cryptic barriers which form clear genetic structure within oceanic regions are poorly understood. To test the presence of a cryptic barrier, we conducted a phylogeographic study together with past demographic inference for a widespread sea‐dispersed plant, Vigna marina, using 308 individuals collected from the entire Indo‐West Pacific (IWP) region. Chloroplast DNA variation showed strong genetic structure that separated populations into three groups: North Pacific (NP), South Pacific (SP) and Indian Ocean (IN) (F′_CT among groups = 0.954–1.000). According to the Approximate Bayesian computation inference, splitting time between NP and SP was approximately 20,200 years (95%HPD, 4,530–95,400) before present. Moreover, a signal of recent population expansion was detected in the NP group. This study clearly showed the presence of a cryptic barrier in the West Pacific region of the distributional range of V. marina. The locations of the cryptic barrier observed in V. marina corresponded to the genetic breaks found in other plants, suggesting the presence of a common cryptic barrier for sea‐dispersed plants. Demographic inference suggested that genetic structure related to this cryptic barrier has been present since the last glacial maximum and may reflect patterns of past population expansion from refugia.     

Robust trend tests for genetic association in case-control studies using family data

We studied a trend test for genetic association between disease and the number of risk alleles using case-control data. When the data are sampled from families, this trend test can be adjusted to take into account the correlations among family members in complex pedigrees. However, the test depends on the scores based on the underlying genetic model and thus it may have substantial loss of power when the model is misspecified. Since the mode of inheritance will be unknown for complex diseases, we have developed two robust trend tests for case-control studies using family data. These robust tests have relatively good power for a class of possible genetic models. The trend tests and robust trend tests were applied to a dataset of Genetic Analysis Workshop 14 from the Collaborative Study on the Genetics of Alcoholism.     

Robust genomic control for association studies

Population-based case-control studies are a useful method to test for a genetic association between a trait and a marker. However, the analysis of the resulting data can be affected by population stratification or cryptic relatedness, which may inflate the variance of the usual statistics, resulting in a higher-than-nominal rate of false-positive results. One approach to preserving the nominal type I error is to apply genomic control, which adjusts the variance of the Cochran-Armitage trend test by calculating the statistic on data from null loci. This enables one to estimate any additional variance in the null distribution of statistics. When the underlying genetic model (e.g., recessive, additive, or dominant) is known, genomic control can be applied to the corresponding optimal trend tests. In practice, however, the mode of inheritance is unknown. The genotype-based chi (2) test for a general association between the trait and the marker does not depend on the underlying genetic model. Since this general association test has 2 degrees of freedom (df), the existing formulas for estimating the variance factor by use of genomic control are not directly applicable. By expressing the general association test in terms of two Cochran-Armitage trend tests, one can apply genomic control to each of the two trend tests separately, thereby adjusting the chi (2) statistic. The properties of this robust genomic control test with 2 df are examined by simulation. This genomic control-adjusted 2-df test has control of type I error and achieves reasonable power, relative to the optimal tests for each model.     

An Investigation on the Allelic Chi‐Square Test Used in Genetic Association Studies

Case‐control studies are primary study designs used in genetic association studies. Sasieni (Biometrics 1997, 53, 1253–1261) pointed out that the allelic chi‐square test used in genetic association studies is invalid when Hardy‐Weinberg equilibrium (HWE) is violated in a combined population. It is important to know how much type I error rate is deviated from the nominal level under violated HWE. We examine bounds of type I error rate of the allelic chi‐square test. We also investigate power of the goodness‐of‐fit test for HWE which can be used as a guideline for selecting an appropriate test between the allelic chi‐square test and the modified allelic chi‐square test, the latter of which was proposed for cases of violated HWE. In small samples, power is not large enough to detect the Wright's inbreeding model of small values of inbreeding coefficient. Therefore, when the null hypothesis of HWE is barely accepted, the modified test should be considered as an alternative method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)