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.     

Effect of Winsorization on Power and Type 1 Error of Variance Components and Related Methods of QTL Detection

Variance components analysis provides an efficient method for performing linkage analysis for quantitative traits. However, power and type 1 error of variance components-based likelihood ratio testing may be affected when phenotypic data are nonnormally distributed (especially with high values of kurtosis) and there is moderate to high correlation among the siblings. Winsorization can reduce the effect of outliers on statistical analyses. Here, we considered the effect of winsorization on variance components-based tests. We considered the likelihood ratio test (LRT), the Wald test, and some robust variance components tests. We compared these tests with Haseman-Elston least squares-based tests. We found that power to detect linkage is significantly increased after winsorization of the nonnormal phenotypes. Winsorization does not greatly diminish the type 1 error for the variance components-based tests for markedly nonnormal data. A robust version of the LRT that adjusts for sample kurtosis showed the best power for nonnormal data. Finally, phenotype winsorization of nonnormal data reduces the bias in estimation of the major gene variance component.     

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.     

A Marginal Likelihood Model for Family-Based Data

This paper presents a marginal likelihood model for family‐based data based upon the transmission of marker alleles from each heterozygous parent to his/her affected children. The proposed model, extending the maximum‐likelihood‐binomial (MLB) method and the disequilibrium maximum‐likelihood‐binomial (DMLB) method ( Abel et al. 1998 ; Abel & Müller‐Myhsok, 1998 ; Huang & Jiang, 1999 ), is adaptive to linkage disequilibrium (LD) and linkage heterogeneity. Compared with other procedures, the likelihood ratio test (LRT) derived from the proposed model enjoys superior qualities. First, simulations indicate that the power of the LRT is greater than that of the TDT or DMLB in all of our studied scenarios. Second, when we applied the LRT and other tests to a Tourette Syndrome data, the result was data favorable to the use of the LRT. Therefore, we recommend the use of the LRT as an additional linkage test wherever applicable, especially when the amount of LD is uncertain.     

Type I and type II error rates for quantitative trait loci (QTL) mapping studies using recombinant inbred mouse strains

Effective mapping strategies for quantitative traits must allow for the detection of the more important quantitative trait loci (QTLs) while minimizing false positives. Type I (false-positive) and Type II (false-negative) error rates were estimated from a computer simulation of QTL mapping in the BXD recombinant inbred (RI) set comprising 26 strains of mice, and comparisons made with theoretical predictions. The results are generally applicable to other RI sets when corrections are made for differing strain numbers and marker densities. Regardless of the number or magnitude of simulated QTLs contributing to the trait variance, thep value necessary to provide genome-wide. 05 Type I error protection was found to be aboutp=.0001. To provide adequate protection against both Type I (α=.0001) and Type II (β=.2) errors, a QTL would have to account for more than half of the between-strain (genetic) variance if the BXD or similar set was used alone. In contrast, a two-step mapping strategy was also considered, where RI strains are used as a preliminary screen for QTLs to be specifically tested (confirmed) in an F₂ (or other) population. In this case, QTLs accounting for ∼16% of the between-strain variance could be detected with an 80% probability in the BXD set when α=0.2. To balance the competing goals of minimizing Type I and II errors, an economical strategy is to adopt a more stringent α initially for the RI screen, since this requires only a limited genome search in the F₂ of the RI-implicated regions (∼10% of the F₂ genome whenp<.01 in the RIs). If confirmed QTLs do not account in the aggregate for a sufficient proportion of the genetic variance, then a more relaxed α value can be used in the RI screen to increase the statistical power. This flexibility in setting RI α values is appropriate only when adequate protection against Type I errors comes from the F₂ (or other) confirmation test(s).     

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.     

Quantitative-trait-locus Mapping in the Presence of Locus Heterogeneity

Locus heterogeneity is a concern for quantitative trait locus mapping where phenotypes are likely to be influenced by more than one gene. We introduce a model which generalizes the locus heterogeneity model of Smith (1961) from dichotomous traits to quantitative traits and consider some test statistics for this model. The type I error rates and the power of these statistics are assessed through simulation studies. These statistics are applied to a linkage study of asthma genes.     

Further Investigation of Linkage Disequilibrium SNPs and their Ability to Identify Associated Susceptibility Loci

There is currently considerable interest in the use of single‐nucleotide polymorphisms (SNPs) to map disease susceptibility genes. The success of this method will depend on a number of factors including the strength of linkage disequilibrium (LD) between marker and disease loci. We used a data set of SNP genotypings in the region of the APOE disease susceptibility locus to investigate the likely usefulness of SNPs in case‐control studies. Using the estimated haplotype structure surrounding and including the APOE locus, and assuming a codominant disease model, we treated each SNP in turn as if it were a disease susceptibility locus and obtained, for each disease locus and markers, the expected likelihood ratio test (LRT) to assess disease association.We were particularly interested in the power to detect association with the susceptibility polymorphism itself, the power of nearby markers to detect association, and the ability to distinguish between the susceptibility polymorphism and marker loci also showing association. We found that the expected LRT depended critically on disease allele frequencies. For disease loci with a reasonably common allele we were usually able to detect association. However, for only a subset of markers in the close neighbourhood of the disease locus was association detectable. In these cases we were usually, but not always, able to distinguish the disease locus from nearby associated marker loci. For some disease loci, no other loci demonstrated detectable association with the disease phenotype. We conclude that one may need to use very dense SNP maps in order to avoid overlooking polymorphisms affecting susceptibility to a common phenotype.     

Likelihood Ratio Tests in Rare Variant Detection for Continuous Phenotypes

It is believed that rare variants play an important role in human phenotypes; however, the detection of rare variants is extremely challenging due to their very low minor allele frequency. In this paper, the likelihood ratio test (LRT) and restricted likelihood ratio test (ReLRT) are proposed to test the association of rare variants based on the linear mixed effects model, where a group of rare variants are treated as random effects. Like the sequence kernel association test (SKAT), a state‐of‐the‐art method for rare variant detection, LRT and ReLRT can effectively overcome the problem of directionality of effect inherent in the burden test in practice. By taking full advantage of the spectral decomposition, exact finite sample null distributions for LRT and ReLRT are obtained by simulation. We perform extensive numerical studies to evaluate the performance of LRT and ReLRT, and compare to the burden test, SKAT and SKAT‐O. The simulations have shown that LRT and ReLRT can correctly control the type I error, and the controls are robust to the weights chosen and the number of rare variants under study. LRT and ReLRT behave similarly to the burden test when all the causal rare variants share the same direction of effect, and outperform SKAT across various situations. When both positive and negative effects exist, LRT and ReLRT suffer from few power reductions compared to the other two competing methods; under this case, an additional finding from our simulations is that SKAT‐O is no longer the optimal test, and its power is even lower than that of SKAT. The exome sequencing SNP data from Genetic Analysis Workshop 17 were employed to illustrate the proposed methods, and interesting results are described.     

Partitioning Phenotypic Variance Due to Parent-of-Origin Effects Using Genomic Relatedness Matrices

We propose a new method, G-REMLadp, to estimate the phenotypic variance explained by parent-of-origin effects (POEs) across the genome. Our method uses restricted maximum likelihood analysis of genome-wide genetic relatedness matrices based on individuals’ phased genotypes. Genome-wide SNP data from parent child duos or trios is required to obtain relatedness matrices indexing the parental origin of offspring alleles, as well as offspring phenotype data to partition the trait variation into variance components. To calibrate the power of G-REMLadp to detect non-null POEs when they are present, we provide an analytic approximation derived from Haseman-Elston regression. We also used simulated data to quantify the power and Type I Error rates of G-REMLadp, as well as the sensitivity of its variance component estimates to violations of underlying assumptions. We subsequently applied G-REMLadp to 36 phenotypes in a sample of individuals from the Avon Longitudinal Study of Parents and Children (ALSPAC). We found that the method does not seem to be inherently biased in estimating variance due to POEs, and that substantial correlation between parental genotypes is necessary to generate biased estimates. Our empirical results, power calculations and simulations indicate that sample sizes over 10000 unrelated parent-offspring duos will be necessary to detect POEs explaining?<?10% of the variance with moderate power. We conclude that POEs tagged by our genetic relationship matrices are unlikely to explain large proportions of the phenotypic variance (i.e. >?15%) for the 36 traits that we have examined.     

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.     

Gene finding strategies

Both linkage and association methods have been used to localise and identify genes related to behaviour and other complex traits. The linkage approach (parametric or non-parametric) can be used for whole genome screens to localise genes of unknown function. The parametric linkage approach is very effective for locating single-gene disorders and is usually based on large family pedigrees. The non-parametric method is useful to detect quantitative trait loci (QTLs) for complex traits and was originally developed for sib pair analyses. Genetic association studies are most often used to test the association of alleles at a candidate gene with a disease or with levels of a quantitative trait. Allelic association between a trait and a marker can be studied in a case-control design, but because of possible problems due to population stratification, within-family designs have been proposed as the optimal test for association.     

Differentiating population stratification from genotyping error using family data

Identifying population stratification and genotyping error are important for candidate gene association studies using the Transmission Disequilibrium Test (TDT). Although the TDT retains the prespecified Type I error in the presence of population stratification, the test may have decreased power in the presence of population stratification. Genotyping error can also cause the TDT to have an elevated Type I error. Differentiating population stratification from genotyping error remains a challenge for geneticists. Both genotyping error and population stratification can result in an increase in the observed homozygosity of a sample relative to that expected assuming Hardy-Weinberg Equilibrium (HWE). We show that when family data are available, even if a limited number of markers are genotyped, evaluating the markers that show statistically significant deviation from HWE with the Mating Type Distortion Test (MTDT)--a test based on the mating type distribution--can reliably differentiate genotyping error from population stratification. We simulate data based on several models of genotyping error in previously published literature, and show how this method could be used in practice to assist in differentiating population stratification from systematic genotyping error.     

Genetic architecture of regulatory variation in Arabidopsis thaliana

Studying the genetic regulation of expression variation is a key method to dissect complex phenotypic traits. To examine the genetic architecture of regulatory variation in Arabidopsis thaliana, we performed genome-wide association (GWA) mapping of gene expression in an F(1) hybrid diversity panel. At a genome-wide false discovery rate (FDR) of 0.2, an associated single nucleotide polymorphism (SNP) explains >38% of trait variation. In comparison with SNPs that are distant from the genes to which they were associated, locally associated SNPs are preferentially found in regions with extended linkage disequilibrium (LD) and have distinct population frequencies of the derived alleles (where Arabidopsis lyrata has the ancestral allele), suggesting that different selective forces are acting. Locally associated SNPs tend to have additive inheritance, whereas distantly associated SNPs are primarily dominant. In contrast to results from mapping of expression quantitative trait loci (eQTL) in linkage studies, we observe extensive allelic heterogeneity for local regulatory loci in our diversity panel. By association mapping of allele-specific expression (ASE), we detect a significant enrichment for cis-acting variation in local regulatory variation. In addition to gene expression variation, association mapping of splicing variation reveals both local and distant genetic regulation for intron and exon level traits. Finally, we identify candidate genes for 59 diverse phenotypic traits that were mapped to eQTL.     

EMK: A Novel Program for Family-Based Allelic and Genotypic Association Tests on Quantitative Traits

The QTDT program is a widely‐used program for analyzing quantitative trait data, but the methods mainly test allelic association. Since the genotype of a marker is a direct observation for an individual, it is of interest to assess association at the genotypic level. In this study, we extended the allele‐based association method developed by Monks and Kaplan (MK method) to genotype‐based association tests for quantitative traits. We implemented a novel extended MK (EMK) program that can perform both allele‐ and genotype‐ based association tests in any pedigree structure. To evaluate the performance of EMK, we utilized simulated pedigree data and real data from our previous report of GSTO1 and GSTO2 genes in Alzheimer disease (AD). Both allele‐ and genotype‐based EMK methods (allele‐EMK and geno‐EMK) showed correct type I error for various pedigree structures and admixture populations. The geno‐EMK method showed comparable power to the allele‐EMK test. By treating age‐at‐onset (AAO) as a quantitative trait, the EMK program was able to detect significant associations for rs4925 in GSTO1 (P= 0.006 for allele‐EMK and P= 0.009 for geno‐EMK), and rs2297235 in GSTO2 (P= 0.005 for allele‐EMK and P= 0.009 for geno‐EMK), which are consistent with our previous findings.     

Allelic heterogeneity and more detailed analyses of known loci explain additional phenotypic variation and reveal complex patterns of association

The identification of multiple signals at individual loci could explain additional phenotypic variance ('missing heritability') of common traits, and help identify causal genes. We examined gene expression levels as a model trait because of the large number of strong genetic effects acting in cis. Using expression profiles from 613 individuals, we performed genome-wide single nucleotide polymorphism (SNP) analyses to identify cis-expression quantitative trait loci (eQTLs), and conditional analysis to identify second signals. We examined patterns of association when accounting for multiple SNPs at a locus and when including additional SNPs from the 1000 Genomes Project. We identified 1298 cis-eQTLs at an approximate false discovery rate 0.01, of which 118 (9%) showed evidence of a second independent signal. For this subset of 118 traits, accounting for two signals resulted in an average 31% increase in phenotypic variance explained (Wilcoxon P< 0.0001). The association of SNPs with cis gene expression could increase, stay similar or decrease in significance when accounting for linkage disequilibrium with second signals at the same locus. Pairs of SNPs increasing in significance tended to have gene expression increasing alleles on opposite haplotypes, whereas pairs of SNPs decreasing in significance tended to have gene expression increasing alleles on the same haplotypes. Adding data from the 1000 Genomes Project showed that apparently independent signals could be potentially explained by a single association signal. Our results show that accounting for multiple variants at a locus will increase the variance explained in a substantial fraction of loci, but that allelic heterogeneity will be difficult to define without resequencing loci and functional work.     

Quantitative Variation in Plasma Angiotensin‐I Converting Enzyme Activity Shows Allelic Heterogeneity in the ABO Blood Group Locus

Angiotensin‐I converting enzyme (ACE) occupies a pivotal role in cardiovascular homeostasis. Major loci for plasma ACE have been identified at ACE on Chromosome 17 and at ABO on Chromosome 9. We sought to characterise the genetic architecture of plasma ACE at finer resolution in two populations. We carried out a GWAS in 1810 individuals of Japanese ethnicity; this identified signals at ACE and ABO that together accounted for nearly half of the population variability of the trait. We conducted measured haplotype analysis at the ABO locus in 1425 members of 248 British families using haplotypes of three SNPs, which together tagged the alleles responsible for the principal blood group antigens A1, A2, B and O. Type O alleles were associated with intermediate plasma ACE activity compared to Type A1 alleles (in whom plasma ACE activity was ～36% lower) and Type B alleles (in whom plasma ACE activity was ～36% higher). We demonstrated heterogeneity among A alleles: A2 alleles were associated with plasma ACE activity that was very similar to the O alleles. Variation at ACE accounted for 35% of the trait variance, and variation at ABO accounted for 15%. A further 10% could be ascribed to polygenic effects.     

An Extension of the Regression of Offspring on Mid-Parent to Test for Association and Estimate Locus-Specific Heritability: The Revised ROMP Method

The Regression of Offspring on Mid-Parent (ROMP) method is a test of association between a quantitative trait and a candidate locus. ROMP estimates the trait heritability and the heritability attributable to a locus and requires genotyping the offspring only. In this study, the theory underlying ROMP was revised (ROMP_rev) and extended. Computer simulations were used to determine the type I error and power of the test of association, and the accuracy of the locus-specific heritability estimate. The ROMP_rev test had good power at the 5% significance level with properly controlled type I error. Locus-specific heritability estimates were, on average, close to simulated values. For non-zero locus-specific heritability, the proposed standard error was downwardly biased, yielding reduced coverage of 95% confidence intervals. A bootstrap approach with proper coverage is suggested as a second step for loci of interest.
ROMP_rev was applied to a study of cardiovascular-related traits to illustrate its use. An association between polymorphisms within the fibrinogen gene cluster and plasma fibrinogen was detected (p < 0.005) that accounted for 29% of the estimated fibrinogen heritability. The ROMP_rev method provides a computationally fast and simple way of testing for association and obtaining accurate estimates of locus-specific heritability while minimizing the genotyping required.     

Robustness of Bayesian Multilocus Association Models to Cryptic Relatedness

Population‐based association analyses are more powerful than within‐family analyses in identifying genetic loci associated with a phenotype of interest. However, if the population or sample structure is omitted from the model, population stratification and cryptic relatedness may lead to false positive and negative signals caused by relatedness between individuals, rather than association due to close linkage of the marker and the trait loci. Therefore it is important to correct or account for these confounders in population‐based association analyses. However, there is cumulative evidence that when fitting a multilocus association model, the genetic relationships between the individuals can be captured by the markers themselves, bringing about a possibility to use the models without an additional correction for the population or sample structure. In this work we have further investigated this possibility in the Bayesian multilocus association model context using the extended Bayesian LASSO and the indicator‐based variable selection. In particular, we have studied whether these multilocus models benefit from an insertion of an additional polygenic term representing the genetic variation not captured by the markers and taking account of the residual dependencies between the individuals. We have found that although the models may benefit from the insertion of the polygenic component, omitting the component does not damage the model performance severely.     

TDT statistics for mapping quantitative trait loci

The original transmission disequilibrium test (TDT), was introduced to test for linkage between a marker and a disease-susceptibility locus (Spielman et al . 1993). Allison (1997) extended the TDT procedure to quantitative traits. Allison's test, however, is restrictive in that it requires family trios consisting of one heterozygous parent, one homozygous parent and one child, and considers only the situation where the marker locus is analogous to the quantitative trait locus itself. In this paper, we propose, investigate and apply a general TDT for quantitative traits that permits more than one child per family, does not require only one parent to be heterozygous, and allows for the fact that the various alleles at the marker and trait loci may be at varying degree of linkage disequilibrium. We also show that this TDT for quantitative traits is still a valid test of linkage in the presence of population substructure. To provide guidelines for study design, we develop analytic formulae for calculation of the power of the TDT for mapping quantitative trait loci and investigate the impact of various factors on the power. Power calculations show that the proposed TDT for quantitative traits is more powerful than Allison's basic test statistic and the extreme discordant sib pair linkage method. The proposed TDT statistic for quantitative traits is applied to systolic blood pressure variation in the Rochester Family Heart Study using an extremely discordant sibling pair design.