Differentiating the Cochran‐Armitage Trend Test and Pearson's χ2 Test: Location and Dispersion

In genetic case‐control association studies, a standard practice is to perform the Cochran‐Armitage (CA) trend test with 1 degree‐of‐freedom (d.f.) under the assumption of an additive model. However, when the true genetic model is recessive or near recessive, it is outperformed by Pearson's χ² test with 2 d.f. In this article, we analytically reveal the statistical basis that leads to the phenomenon. First, we show that the CA trend test examines the location shift between the case and control groups, whereas Pearson's χ² test examines both the location and dispersion shifts between the two groups. Second, we show that under the additive model, the effect of location deviation outweighs that of the dispersion deviation and vice versa under a near recessive model. Therefore, Pearson's χ² test is a more robust test than the CA trend test, and it outperforms the latter when the mode of inheritance evolves to the recessive end.     

A Comparison of the Likelihood Ratio Test and the Variance‐Stabilising Transformation‐Based Tests for Detecting Association of Rare Variants

In a recent paper in this journal, the use of variance‐stabilising transformation techniques was proposed to overcome the problem of inadequacy in normality approximation when testing association for a low‐frequency variant in a case‐control study. It was shown that tests based on the variance‐stabilising transformations are more powerful than Fisher's exact test while controlling for type I error rate. Earlier in the journal, another study had shown that the likelihood ratio test (LRT) is superior to Fisher's exact test, Wald's test, and Pearson's χ² test in testing association for low‐frequency variants. Thus, it is of interest to make a direct comparison between the LRT and the tests based on the variance‐stabilising transformations. In this commentary, we show that the LRT and the variance‐stabilising transformation‐based tests have comparable power greater than Fisher's exact test, Wald's test, and Pearson's χ² test.     

A Note on the Asymptotic Properties of Affected-sib-pair Linkage Tests

This article concerns the asymptotic properties of linkage tests for affected‐sib‐pair data under the null hypothesis of no linkage. We consider a popular single‐locus analysis model where the unknown parameters are the disease allele frequency, the three penetrances for the three genotypes at the disease locus, and the recombination fraction between the marker locus and the disease locus. These parameters are completely confounded under the null hypothesis of no linkage. We show that 1) If the total variance of the trait (i.e., the additive variance plus the dominance variance) is “separated” from 0, then the likelihood ratio statistic has an asymptotic 0.5χ²₀+ 0.5χ²₁ distribution; 2) If the prevalence of the trait is “separated” from 0 and the recombination fraction is fixed at 0, then the likelihood ratio statistic has an asymptotic distribution which is a mixture of χ²₀, χ²₁ and χ²₂ . The implications of these results are discussed.     

The Effects of SNP Genotyping Errors on the Power of The Cochran-Armitage Linear Trend Test for Case/Control Association Studies

The questions addressed in this paper are: What single nucleotide polymorphism (SNP) genotyping errors are most costly, in terms of minimum sample size necessary (MSSN) to maintain constant asymptotic power and significance level, when performing case‐control studies of genetic association applying the Cochran‐Armitage trend test? And which trend test or χ² test is more powerful under standard genetic models with genotyping errors? Our strategy is to expand the non‐centrality parameter of the asymptotic distribution of the trend test to approximate the MSSN using a Taylor series linear in the genotyping error rates. We apply our strategy to example scenarios that assume recessive, dominant, additive, or over‐dominant disease models. The most costly errors are recording the more common homozygote as the less common homozygote, and the more common homozygote as the heterozygote, with MSSN that become indefinitely large as the minor SNP allele frequency approaches zero. Misclassifying the heterozygote as the less common homozygote is costly when using the recessive trend test on data from a recessive model. The χ² test has power close to, but less than, the optimal trend test and is never dominated over all genetic models studied by any specific trend test.     

A unifying framework for robust association testing,estimation, and genetic model selection using the generalized linear model

The analysis of genome-wide genetic association studies generally starts with univariate statistical tests of each single-nucleotide polymorphism. The standard approach is the Cochran-Armitage trend test or its logistic regression equivalent although this approach can lose considerable power if the underlying genetic model is not additive. An alternative is the MAX test, which is robust against the three basic modes of inheritance. Here, the asymptotic distribution of the MAX test is derived using the generalized linear model together with the Delta method and multiple contrasts. The approach is applicable to binary, quantitative, and survival traits. It may be used for unrelated individuals, family-based studies, and matched pairs. The approach provides point and interval effect estimates and allows selecting the most plausible genetic model using the minimum P-value. R code is provided. A Monte-Carlo simulation study shows that the asymptotic MAX test framework meets type I error levels well, has good power, and good model selection properties for minor allele frequencies ≥0.3. Pearson''s χ²-test is superior for lower minor allele frequencies with low frequencies for the rare homozygous genotype. In these cases, the model selection procedure should be used with caution. The use of the MAX test is illustrated by reanalyzing findings from seven genome-wide association studies including case–control, matched pairs, and quantitative trait data.     

High Resolution T2 Association Tests of Complex Diseases Based on Family Data

This paper proposes family based Hotelling's T² tests for high resolution linkage disequilibrium (LD) mapping or association studies of complex diseases. Assume that genotype data of multiple markers or haplotype blocks are available for a sample of nuclear families, in which some offspring are affected. Paired Hotelling's T² test statistics are proposed for a high resolution association study using parents as controls for affected offspring, based on two coding methods: haplotype/allele coding and genotype coding. The paired Hotelling's T² tests take not only the correlation between the haplotype blocks or markers into account, but also take the correlation within each parent‐offspring pair into account. The method extends two sample Hotelling's T² test statistics for population case control association studies, which are not valid for family data due to correlation of genetic data among family members. The validity of the proposed method is justified by rigorous mathematical and statistical proof under the large sample theory. The non‐centrality parameter approximations of the test statistics are calculated for power and sample size calculations. From power comparison and type I error calculations, it is shown that the test statistic based on haplotype/allele coding is advantageous over the test statistic of genotype coding. Analysis using multiple markers may provide higher power than single marker analysis. If only one marker is utilized the power of the test statistic based on haplotype/allele coding is nearly identical to that of 1‐TDT. Moreover, a permutation procedure is provided for data analysis. The method is applied to data from a German asthma family study. The results based on the paired Hotelling's T² statistic tests confirm the previous findings. However, the paired Hotelling's T² tests produce much smaller P‐values than those of the previous study. The permutation tests produce similar results to those of the previous study; moreover, additional marker combinations are shown to be significant by permutation tests. The proposed paired Hotelling's T² statistic tests are potentially powerful in mapping complex diseases. A SAS Macro, Hotel_fam.sas, has been written to implement the method for data analysis.     

The Impact of FOXP3 Polymorphism on the Risk of Allergic Rhinitis: A Meta‐Analysis

Polymorphisms of several genes were reported to be associated with the risk of allergic rhinitis. Here, we first conducted a meta‐analysis to evaluate the potential genetic association between the polymorphisms of the FOXP3 (Forkhead Box P3) gene and the susceptibility to allergic rhinitis. A total of 2671 relevant articles were initially retrieved from the databases of PubMed, Web of Science, Embase, WANFANG/CNKI and Scopus, and six eligible case‐control studies were finally enrolled in our meta‐analysis, according to our strict inclusion/exclusion criteria. Based on the extracted data, Mantel–Haenszel statistic, Cochrane's Q statistic, I² test, subgroup meta‐analysis, Begg's test, Egger's test and sensitivity analysis were performed via Stata/SE 12.0 software. The results of the Mantel–Haenszel statistic regarding rs3761548 showed that no significant difference was observed in the allergic rhinitis case and population‐based control group under the genetic models of A versus C, AA versus CC, CA+AA versus CC, AA versus CC+CA and carrier A versus C (all P‐value of Association Test, P_A > 0.05), apart from CA versus CC (P_A = 0.020). The similar results were obtained in the subgroup analysis of Asian. In addition, we did not obtain the positive result in the meta‐analysis of rs2232365 (all P_A > 0.05). We also excluded the presence of large publication bias through Begg's test and Egger's test, and we confirmed the stability of data by sensitivity analysis. In summary, no significant association between rs3761548, rs2232365 polymorphisms of the FOXP3 gene, and an increased susceptibility to allergic rhinitis was identified based on the published data; however, this conclusion should be confirmed by more studies with increased sample sizes.     

Efficient Calculation of P‐value and Power for Quadratic Form Statistics in Multilocus Association Testing

We address the asymptotic and approximate distributions of a large class of test statistics with quadratic forms used in association studies. The statistics of interest take the general form D=X^TA X , where A is a general similarity matrix which may or may not be positive semi‐definite, and X follows the multivariate normal distribution with mean μ and variance matrix Σ, where Σ may or may not be singular. We show that D can be written as a linear combination of independent χ² random variables with a shift. Furthermore, its distribution can be approximated by a χ² or the difference of two χ² distributions. In the setting of association testing, our methods are especially useful in two situations. First, when the required significance level is much smaller than 0.05 such as in a genome scan, the estimation of p‐values using permutation procedures can be challenging. Second, when an EM algorithm is required to infer haplotype frequencies from un‐phased genotype data, the computation can be intensive for a permutation procedure. In either situation, an efficient and accurate estimation procedure would be useful. Our method can be applied to any quadratic form statistic and therefore should be of general interest.     

An alternative linear trend analysis for assessing the results of the mouse lymphoma assay

As recommended by the mouse lymphoma assay (MLA) Workgroup of the International Workshop on Genotoxicity Testing (Aberdeen, 2003), a trend test is critical if an induced mutant frequency (MF) of at least 126 × 10(-6) (global evaluation factor, GEF) is achieved at one or more test concentrations. Only those responses that both achieve the GEF and a significant trend are biologically relevant. While no specific trend test was recommended by the Workshop, a trend test was recommended by the UK Environmental Mutagen Society (1989). The test uses MF (untransformed) averaged over replicate cultures following a consistency test (against a historical heterogeneity factor) in a weighted linear regression with chi-square (χ(2)) test for slope and returns significant results in virtually all cases that are positive for the GEF, including those with no apparent dose-response. We have explored an alternative method where the natural logarithm of MF and its variance are estimated for each replicate culture separately and used in a weighted ordinary linear regression with t-test for slope. Using test cases positive for the GEF, the P-value from this model is shown to be sensitive to changes in the number of replicates, the shape and magnitude of mutant induction, in contrast to the χ(2) model. Cases with no apparent dose-response and thereby questionable biological significance are tested negative by our method but positive by the χ(2) model. Our method is thus straight-forward and provides a meaningful complement to the GEF in assessing the biological significance of the MLA results.     

Can the Allelic Test be Retired from Analysis of Case‐Control Association Studies?

It has been stated that, when Hardy‐Weinberg equilibrium (HWE) holds in the combined case‐control samples, the allelic test is asymptotically equivalent to the trend test (for the additive model) for testing genetic association, and hence the allelic test should not be used. A recent publication shows that the allelic test and the trend test are asymptotically equivalent when HWE holds in the population. It is known that, when HWE does not hold, the trend test can still be used while the allelic test is no longer valid. Therefore, the allelic test is either not valid or is asymptotically equivalent to the trend test. It appears that the allelic test is a nuisance test. Can it be retired from the analysis of case‐control association studies? It all depends on data and model assumptions. We give conditions under which the allelic test and the trend test are asymptotically equivalent under both null and alternative hypotheses.     

Influence of population stratification on population‐based marker‐disease association analysis

Population‐based genetic association analysis may suffer from the failure to control for confounders such as population stratification (PS). There has been extensive study on the influence of PS on candidate gene‐disease association analysis, but much less attention has been paid to its influence on marker‐disease association analysis. In this paper, we focus on the Pearson χ² test and the trend test for marker‐disease association analysis. The mean and variance of the test statistics are derived under presence of PS, so that the power and inflated type I error rate can be evaluated. It is shown that the bias and the variance distortion are not zero in the presence of both PS and penetrance heterogeneity (PH). Unlike candidate gene‐disease association analysis, when PS is present, the bias is not zero no matter whether PH is present or not. This work generalises the published results, where only the fully recessive penetrance model is considered and only the bias is calculated. It is shown that candidate gene‐disease association analysis can be treated as a special case of marker‐disease association analysis. Consequently, our results extend previous studies on candidate gene‐disease association analysis. A simulation study confirms the theoretical findings.     

On Sample Size and Power Calculation for Variant Set‐Based Association Tests

Sample size and power calculations are an important part of designing new sequence‐based association studies. The recently developed SEQPower and SPS programs adopted computationally intensive Monte Carlo simulations to empirically estimate power for a series of variant set association (VSA) test methods including the sequence kernel association test (SKAT). It is desirable to develop methods that can quickly and accurately compute power without intensive Monte Carlo simulations. We will show that the computed power for SKAT based on the existing analytical approach could be inflated especially for small significance levels, which are often of primary interest for large‐scale whole genome and exome sequencing projects. We propose a new χ²‐approximation‐based approach to accurately and efficiently compute sample size and power. In addition, we propose and implement a more accurate “exact” method to compute power, which is more efficient than the Monte Carlo approach though generally involves more computations than the χ² approximation method. The exact approach could produce very accurate results and be used to verify alternative approximation approaches. We implement the proposed methods in publicly available R programs that can be readily adapted when planning sequencing projects.     

A Likelihood Ratio Test for Genome‐Wide Association under Genetic Heterogeneity

Most existing association tests for genome‐wide association studies (GWASs) fail to account for genetic heterogeneity. Zhou and Pan proposed a binomial‐mixture‐model‐based association test to account for the possible genetic heterogeneity in case‐control studies. The idea is elegant, however, the proposed test requires an expectation‐maximization (EM)‐type iterative algorithm to identify the penalised maximum likelihood estimates and a permutation method to assess p‐values. The intensive computational burden induced by the EM‐algorithm and the permutation becomes prohibitive for direct applications to GWASs. This paper develops a likelihood ratio test (LRT) for GWASs under genetic heterogeneity based on a more general alternative mixture model. In particular, a closed‐form formula for the LRT statistic is derived to avoid the EM‐type iterative numerical evaluation. Moreover, an explicit asymptotic null distribution is also obtained, which avoids using the permutation to obtain p‐values. Thus, the proposed LRT is easy to implement for GWASs. Furthermore, numerical studies demonstrate that the LRT has power advantages over the commonly used Armitage trend test and other existing association tests under genetic heterogeneity. A breast cancer GWAS dataset is used to illustrate the newly proposed LRT.     

Tests for Candidate-Gene Association Using Case-Parents Design

In the case‐parents design for testing candidate‐gene association, the conditional likelihood method based on genotype relative risks has been developed recently. A specific relation of the genotype relative risks is referred to as a genetic model. The efficient score tests have been used when the genetic model is correctly specified under the alternative hypothesis. In practice, however, it is usually not able to specify the genetic model correctly. In the latter situation, tests such as the likelihood ratio test (LRT) and the MAX3 (the maximum of the three score statistics for dominant, additive, and recessive models) have been used. In this paper, we consider the restricted likelihood ratio test (RLRT). For a specific genetic model, simulation results demonstrate that RLRT is asymptotically equivalent to the score test, and both are more powerful than the LRT. When the genetic model cannot be correctly specified, the simulation results show that RLRT is most robust and powerful in the situations we studied. MAX3 is the next most robust and powerful test. The TDT is the easiest statistic to compute, compared to MAX3 and RLRT. When the recessive model can be eliminated, it is also as robust and powerful as RLRT for other genetic models.     

Rank‐Based Tests for Identifying Multiple Genetic Variants Associated with Quantitative Traits

We consider the analysis of multiple genetic variants within a gene or a region that are expected to confer risks to human complex diseases with quantitative traits, where the trait values do not follow the normal distribution even after some transformations. We rank the phenotypic values, calculate a score to measure the trend effect of a particular allele for each marker, and then construct three statistics based on the quadratic frameworks of methods Hotelling T², the summation of squared univariate statistic and the inverse of the square root weighted statistics to combine the scores for different marker loci. Simulation results show that the above three test statistics can control the type I error rate well and are more robust than standard tests constructed based on linear regression. Application to GAW16 data for rheumatoid arthritis successfully detects the association between the HLA‐DRB1 gene and anticyclic citrullinated protein measure, while the standard methods based on normal assumption cannot detect this association.     

A Comparison of Case-Control and Family-Based Association Methods: The Example of Sickle-Cell and Malaria

There has been much debate about the relative merits of population‐ and family‐based strategies for testing genetic association, yet there is little empirical data that directly compare the two approaches. Here we compare case‐control and transmission/disequilibrium test (TDT) study designs using a well‐established genetic association, the protective effect of the sickle‐cell trait against severe malaria. We find that the two methods give similar estimates of the level of protection (case‐control odds ratio = 0.10, 95% confidence interval 0.03–0.23; family‐based estimate of the odds ratio = 0.11, 95% confidence interval 0.04–0.25) and similar statistical significance of the result (case‐control: χ²= 41.26, p= 10^?10 , TDT: χ²= 39.06, p= 10^?10 ) when 315 TDT cases are compared to 583 controls. We propose a family plus population control study design, which allows both case‐control and TDT analysis of the cases. This combination is robust against the respective weaknesses of the case‐control and TDT study designs, namely population structure and segregation distortion. The combined study design is especially cost‐effective when cases are difficult to ascertain and, when the case‐control and TDT results agree, offers greater confidence in the result.     

A Family‐Based Joint Test for Mean and Variance Heterogeneity for Quantitative Traits

Traditional quantitative trait locus (QTL) analysis focuses on identifying loci associated with mean heterogeneity. Recent research has discovered loci associated with phenotype variance heterogeneity (vQTL), which is important in studying genetic association with complex traits, especially for identifying gene–gene and gene–environment interactions. While several tests have been proposed to detect vQTL for unrelated individuals, there are no tests for related individuals, commonly seen in family‐based genetic studies. Here we introduce a likelihood ratio test (LRT) for identifying mean and variance heterogeneity simultaneously or for either effect alone, adjusting for covariates and family relatedness using a linear mixed effect model approach. The LRT test statistic for normally distributed quantitative traits approximately follows χ²‐distributions. To correct for inflated Type I error for non‐normally distributed quantitative traits, we propose a parametric bootstrap‐based LRT that removes the best linear unbiased prediction (BLUP) of family random effect. Simulation studies show that our family‐based test controls Type I error and has good power, while Type I error inflation is observed when family relatedness is ignored. We demonstrate the utility and efficiency gains of the proposed method using data from the Framingham Heart Study to detect loci associated with body mass index (BMI) variability.     

Inheritance of longevity evinces no secular trend among members of six New England families born 1650–1874

This study investigated the historical trend in resemblance between first-degree relatives for age at death. Data from genealogies of six New England families (N = 13,656) were divided into nine 25 year birth cohorts, 1650–1874, to test the hypothesis that familial influence on human longevity has changed during the past 300 years. Heritability (h²) for longevity demonstrated no historical trend, whether calculated by regression of offspring's longevity on paternal, maternal, or mid-parental longevity or by intraclass correlations (t) among sibships. Ninety-five percent confidence intervals (C.I.) for h² (additive genetic variance) were in the range 0.10–0.33 for parent–offspring regressions and 0.16–0.22 based on mean of sibship regressed on mean of parents. Based on sibship t, the 95% C.I. for the upper limit to h² (which includes variance contributions caused by dominance interactions and common developmental environment as well as additive genetic effects) was 0.33–0.41. In this socially elite sample, the statistical contribution of first-degree relatives to age at death has varied within a historically consistent range over the past 300 years, directly implying a persistent genetic influence on longevity. The magnitude of this influence with respect to additive genetic variance, about 10–30%, may overestimate h² because of the elite nature of the sample. Nevertheless, these results support a genetic component to lifespan even though the majority of variation in human longevity is not explained by genetic factors.     

The mean measure of divergence: Its utility in model‐free and model‐bound analyses relative to the Mahalanobis D2 distance for nonmetric traits

The mean measure of divergence (MMD) distance statistic has been used by researchers for nearly 50 years to assess inter‐sample phenetic affinity. Its widespread and often successful use is well documented, especially in the study of cranial and dental nonmetric traits. However, the statistic has accumulated some undesired mathematical baggage through the years from various workers in their attempts to improve or alter its performance. Others may not fully understand how to apply the MMD or interpret its output, whereas some described a number of perceived shortcomings. As a result, the statistic and its sometimes flawed application(s) have taken several well‐aimed hits; a few researchers even argued that it should no longer be utilized or, at least, that its use be reevaluated. The objective of this report is to support the MMD, and in the process: (1) provide a brief history of the statistic, (2) review its attributes and applicability relative to the often‐used Mahalanobis D² statistic for nonmetric traits, (3) compare results from MMD and D² model‐free analyses of previously‐recorded sub‐Saharan African dental samples, and (4) investigate its utility for model‐bound analyses. In the latter instance, the ability of the D² and other squared Euclidean‐based statistics to approximate a genetic relationship matrix and Sewall Wright's fixation index using phenotypic data, and the inability of the MMD to do so, is addressed. Three methods for obtaining such results with nonlinear MMD distances, as well as an assessment of the fit of the isolation‐by‐distance model, are presented. Am. J. Hum. Biol., 2010. © 2009 Wiley‐Liss, Inc.     

Comparison of wheat and rye flour solutions for skin prick testing: a multi‐centre study (Stad 1)

Background Skin prick testing (SPT) is the basic method for diagnosing IgE‐mediated allergies. However, skin reactivity is related to the quality of allergen extracts, which are often poorly defined for occupational allergens. Objective To compare wheat and rye flour SPT solutions from different producers. Materials and methods Standardized SPTs were performed in seven allergy centres with wheat and rye flour solutions from four producers in 125 symptomatic bakers. Optimal cut‐off levels for weal sizes were assessed with the Youden Index. Comparisons between SPT results of different solutions were made with flour‐specific IgE (sIgE) as the gold standard. Sensitivities, specificities, positive and negative predictive values, and test efficiencies were calculated and compared with McNemar and χ²‐tests. The influence of the choice of the gold standard (sIgE or challenge) test was examined for 95 subjects. Additionally, SPT solutions were analysed for protein and antigen content. Results The optimal cut‐off level for all SPT solutions was a weal size of 1.5 mm. While differences between wheat and rye flours were small, differences between producers were important. Variability of sensitivities (0.31–0.96), negative predictive values (0.42–0.91), and test efficiencies (0.54–0.90) were higher than variations of specificities (0.74–1.00) and positive predictive values (0.88–1.00). Similar results were obtained when using challenge test results as the gold standard. Variability could be explained by the different antigen contents of the SPT solutions. Conclusion There is a wide variability of SPT solutions for wheat and rye flour from different producers, mainly with respect to sensitivities, negative predictive values, and test efficiencies. Improvement and standardization of SPT solutions used for the diagnosis of baker's asthma are highly recommended.