Powerful multi-marker association tests: unifying genomic distance-based regression and logistic regression

To detect genetic association with common and complex diseases, many statistical tests have been proposed for candidate gene or genome-wide association studies with the case-control design. Due to linkage disequilibrium (LD), multi-marker association tests can gain power over single-marker tests with a Bonferroni multiple testing adjustment. Among many existing multi-marker association tests, most target to detect only one of many possible aspects in distributional differences between the genotypes of cases and controls, such as allele frequency differences, while a few new ones aim to target two or three aspects, all of which can be implemented in logistic regression. In contrast to logistic regression, a genomic distance-based regression (GDBR) approach aims to detect some high-order genotypic differences between cases and controls. A recent study has confirmed the high power of GDBR tests. At this moment, the popular logistic regression and the emerging GDBR approaches are completely unrelated; for example, one has to choose between the two. In this article, we reformulate GDBR as logistic regression, opening a venue to constructing other powerful tests while overcoming some limitations of GDBR. For example, asymptotic distributions can replace time-consuming permutations for deriving P-values and covariates, including gene-gene interactions, can be easily incorporated. Importantly, this reformulation facilitates combining GDBR with other existing methods in a unified framework of logistic regression. In particular, we show that Fisher's P-value combining method can boost statistical power by incorporating information from allele frequencies, Hardy-Weinberg disequilibrium, LD patterns, and other higher-order interactions among multi-markers as captured by GDBR.     

Power comparison of admixture mapping and direct association analysis in genome-wide association studies

When dense markers are available, one can interrogate almost every common variant across the genome via imputation and single nucleotide polymorphism (SNP) test, which has become a routine in current genome-wide association studies (GWASs). As a complement, admixture mapping exploits the long-range linkage disequilibrium (LD) generated by admixture between genetically distinct ancestral populations. It is then questionable whether admixture mapping analysis is still necessary in detecting the disease associated variants in admixed populations. We argue that admixture mapping is able to reduce the burden of massive comparisons in GWASs; it therefore can be a powerful tool to locate the disease variants with substantial allele frequency differences between ancestral populations. In this report we studied a two-stage approach, where candidate regions are defined by conducting admixture mapping at stage 1, and single SNP association tests are followed at stage 2 within the candidate regions defined at stage 1. We first established the genome-wide significance levels corresponding to the criteria to define the candidate regions at stage 1 by simulations. We next compared the power of the two-stage approach with direct association analysis. Our simulations suggest that the two-stage approach can be more powerful than the standard genome-wide association analysis when the allele frequency difference of a causal variant in ancestral populations, is larger than 0.4. Our conclusion is consistent with a theoretical prediction by Risch and Tang ([2006] Am J Hum Genet 79:S254). Surprisingly, our study also suggests that power can be improved when we use less strict criteria to define the candidate regions at stage 1.     

Optimal designs for two-stage genome-wide association studies

Genome-wide association (GWA) studies require genotyping hundreds of thousands of markers on thousands of subjects, and are expensive at current genotyping costs. To conserve resources, many GWA studies are adopting a staged design in which a proportion of the available samples are genotyped on all markers in stage 1, and a proportion of these markers are genotyped on the remaining samples in stage 2. We describe a strategy for designing cost-effective two-stage GWA studies. Our strategy preserves much of the power of the corresponding one-stage design and minimizes the genotyping cost of the study while allowing for differences in per genotyping cost between stages 1 and 2. We show that the ratio of stage 2 to stage 1 per genotype cost can strongly influence both the optimal design and the genotyping cost of the study. Increasing the stage 2 per genotype cost shifts more of the genotyping and study cost to stage 1, and increases the cost of the study. This higher cost can be partially mitigated by adopting a design with reduced power while preserving the false positive rate or by increasing the false positive rate while preserving power. For example, reducing the power preserved in the two-stage design from 99 to 95% that of the one-stage design decreases the two-stage study cost by approximately 15%. Alternatively, the same cost savings can be had by relaxing the false positive rate by 2.5-fold, for example from 1/300,000 to 2.5/300,000, while retaining the same power.     

Power of genome-wide association studies in the presence of interacting loci

Though multiple interacting loci are likely involved in the etiology of complex diseases, early genome-wide association studies (GWAS) have depended on the detection of the marginal effects of each locus. Here, we evaluate the power of GWAS in the presence of two linked and potentially associated causal loci for several models of interaction between them and find that interacting loci may give rise to marginal relative risks that are not generally considered in a one-locus model. To derive power under realistic situations, we use empirical data generated by the HapMap ENCODE project for both allele frequencies and linkage disequilibrium (LD) structure. The power is also evaluated in situations where the causal single nucleotide polymorphisms (SNPs) may not be genotyped, but rather detected by proxy using a SNP in LD. A common simplification for such power computations assumes that the sample size necessary to detect the effect at the tSNP is the sample size necessary to detect the causal locus directly divided by the LD measure r(2) between the two. This assumption, which we call the "proportionality assumption", is a simplification of the many factors that contribute to the strength of association at a marker, and has recently been criticized as unreasonable (Terwilliger and Hiekkalinna [2006] Eur J Hum Genet 14(4):426-437), in particular in the presence of interacting and associated loci. We find that this assumption does not introduce much error in single locus models of disease, but may do so in so in certain two-locus models.     

Asymptotic tests of association with multiple SNPs in linkage disequilibrium

We consider detecting associations between a trait and multiple single nucleotide polymorphisms (SNPs) in linkage disequilibrium (LD). To maximize the use of information contained in multiple SNPs while minimizing the cost of large degrees of freedom (DF) in testing multiple parameters, we first theoretically explore the sum test derived under a working assumption of a common association strength between the trait and each SNP, testing on the corresponding parameter with only one DF. Under the scenarios that the association strengths between the trait and the SNPs are close to each other (and in the same direction), as considered by Wang and Elston [Am. J. Hum. Genet. [2007] 80:353–360], we show with simulated data that the sum test was powerful as compared to several existing tests; otherwise, the sum test might have much reduced power. To overcome the limitation of the sum test, based on our theoretical analysis of the sum test, we propose five new tests that are closely related to each other and are shown to consistently perform similarly well across a wide range of scenarios. We point out the close connection of the proposed tests to the Goeman test. Furthermore, we derive the asymptotic distributions of the proposed tests so that P‐values can be easily calculated, in contrast to the use of computationally demanding permutations or simulations for the Goeman test. A distinguishing feature of the five new tests is their use of a diagonal working covariance matrix, rather than a full covariance matrix as used in the usual Wald or score test. We recommend the routine use of two of the new tests, along with several other tests, to detect disease associations with multiple linked SNPs. Genet. Epidemiol. 33:497–507, 2009. © 2009 Wiley‐Liss, Inc.     

Impact of genotyping errors on statistical power of association tests in genomic analyses: A case study

A key step in genomic studies is to assess high throughput measurements across millions of markers for each participant's DNA, either using microarrays or sequencing techniques. Accurate genotype calling is essential for downstream statistical analysis of genotype‐phenotype associations, and next generation sequencing (NGS) has recently become a more common approach in genomic studies. How the accuracy of variant calling in NGS‐based studies affects downstream association analysis has not, however, been studied using empirical data in which both microarrays and NGS were available. In this article, we investigate the impact of variant calling errors on the statistical power to identify associations between single nucleotides and disease, and on associations between multiple rare variants and disease. Both differential and nondifferential genotyping errors are considered. Our results show that the power of burden tests for rare variants is strongly influenced by the specificity in variant calling, but is rather robust with regard to sensitivity. By using the variant calling accuracies estimated from a substudy of a Cooperative Studies Program project conducted by the Department of Veterans Affairs, we show that the power of association tests is mostly retained with commonly adopted variant calling pipelines. An R package, GWAS.PC, is provided to accommodate power analysis that takes account of genotyping errors ( http://zhaocenter.org/software/ ).     

An improved score test for genetic association studies

Large-scale genome-wide association studies (GWAS) have become feasible recently because of the development of bead and chip technology. However, the success of GWAS partially depends on the statistical methods that are able to manage and analyze this sort of large-scale data. Currently, the commonly used tests for GWAS include the Cochran-Armitage trend test, the allelic χ(2) test, the genotypic χ(2) test, the haplotypic χ(2) test, and the multi-marker genotypic χ(2) test among others. From a methodological point of view, it is a great challenge to improve the power of commonly used tests, since these tests are commonly used precisely because they are already among the most powerful tests. In this article, we propose an improved score test that is uniformly more powerful than the score test based on the generalized linear model. Since the score test based on the generalized linear model includes the aforementioned commonly used tests as its special cases, our proposed improved score test is thus uniformly more powerful than these commonly used tests. We evaluate the performance of the improved score test by simulation studies and application to a real data set. Our results show that the power increases of the improved score test over the score test cannot be neglected in most cases.     

A scan statistic for identifying chromosomal patterns of SNP association

We have developed a single nucleotide polymorphism (SNP) association scan statistic that takes into account the complex distribution of the human genome variation in the identification of chromosomal regions with significant SNP associations. This scan statistic has wide applicability for genetic analysis, whether to identify important chromosomal regions associated with common diseases based on whole-genome SNP association studies or to identify disease susceptibility genes based on dense SNP positional candidate studies. To illustrate this method, we analyzed patterns of SNP associations on chromosome 19 in a large cohort study. Among 2,944 SNPs, we found seven regions that contained clusters of significantly associated SNPs. The average width of these regions was 35 kb with a range of 10-72 kb. We compared the scan statistic results to Fisher's product method using a sliding window approach, and detected 22 regions with significant clusters of SNP associations. The average width of these regions was 131 kb with a range of 10.1-615 kb. Given that the distances between SNPs are not taken into consideration in the sliding window approach, it is likely that a large fraction of these regions represents false positives. However, all seven regions detected by the scan statistic were also detected by the sliding window approach. The linkage disequilibrium (LD) patterns within the seven regions were highly variable indicating that the clusters of SNP associations were not due to LD alone. The scan statistic developed here can be used to make gene-based or region-based SNP inferences about disease association.     

Single‐marker and two‐marker association tests for unphased case‐control genotype data,with a power comparison

In case‐control single nucleotide polymorphism (SNP) data, the allele frequency, Hardy Weinberg Disequilibrium, and linkage disequilibrium (LD) contrast tests are three distinct sources of information about genetic association. While all three tests are typically developed in a retrospective context, we show that prospective logistic regression models may be developed that correspond conceptually to the retrospective tests. This approach provides a flexible framework for conducting a systematic series of association analyses using unphased genotype data and any number of covariates. For a single stage study, two single‐marker tests and four two‐marker tests are discussed. The true association models are derived and they allow us to understand why a model with only a linear term will generally fit well for a SNP in weak LD with a causal SNP, whatever the disease model, but not for a SNP in high LD with a non‐additive disease SNP. We investigate the power of the association tests using real LD parameters from chromosome 11 in the HapMap CEU population data. Among the single‐marker tests, the allelic test has on average the most power in the case of an additive disease, but for dominant, recessive, and heterozygote disadvantage diseases, the genotypic test has the most power. Among the four two‐marker tests, the Allelic‐LD contrast test, which incorporates linear terms for two markers and their interaction term, provides the most reliable power overall for the cases studied. Therefore, our result supports incorporating an interaction term as well as linear terms in multi‐marker tests. Genet. Epidemiol. 34:67–77, 2010. © 2009 Wiley‐Liss, Inc.     

Analyze multivariate phenotypes in genetic association studies by combining univariate association tests

Multivariate phenotypes are frequently encountered in genome‐wide association studies (GWAS). Such phenotypes contain more information than univariate phenotypes, but how to best exploit the information to increase the chance of detecting genetic variant of pleiotropic effect is not always clear. Moreover, when multivariate phenotypes contain a mixture of quantitative and qualitative measures, limited methods are applicable. In this paper, we first evaluated the approach originally proposed by O'Brien and by Wei and Johnson that combines the univariate test statistics and then we proposed two extensions to that approach. The original and proposed approaches are applicable to a multivariate phenotype containing any type of components including continuous, categorical and survival phenotypes, and applicable to samples consisting of families or unrelated samples. Simulation results suggested that all methods had valid type I error rates. Our extensions had a better power than O'Brien's method with heterogeneous means among univariate test statistics, but were less powerful than O'Brien's with homogeneous means among individual test statistics. All approaches have shown considerable increase in power compared to testing each component of a multivariate phenotype individually in some cases. We apply all the methods to GWAS of serum uric acid levels and gout with 550,000 single nucleotide polymorphisms in the Framingham Heart Study. Genet. Epidemiol. 34:444–454, 2010. © 2010 Wiley‐Liss, Inc.     

Adapting the logical basis of tests for Hardy‐Weinberg Equilibrium to the real needs of association studies in human and medical genetics

The standard procedure to assess genetic equilibrium is a χ² test of goodness‐of‐fit. As is the case with any statistical procedure of that type, the null hypothesis is that the distribution underlying the data is in agreement with the model. Thus, a significant result indicates incompatibility of the observed data with the model, which is clearly at variance with the aim in the majority of applications: to exclude the existence of gross violations of the equilibrium condition. In current practice, we try to avoid this basic logical difficulty by increasing the significance bound to the P‐value (e.g. from 5 to 10%) and inferring compatibility of the data with Hardy Weinberg Equilibrium (HWE) from an insignificant result. Unfortunately, such direct inversion of a statistical testing procedure fails to produce a valid test of the hypothesis of interest, namely, that the data are in sufficiently good agreement with the model under which the P‐value is calculated. We present a logically unflawed solution to the problem of establishing (approximate) compatibility of an observed genotype distribution with HWE. The test is available in one‐ and two‐sided versions. For both versions, we provide tools for exact power calculation. We demonstrate the merits of the new approach through comparison with the traditional χ² goodness‐of‐fit test in 2×60 genotype distributions from 43 published genetic studies of complex diseases where departure from HWE was noted in either the case or control sample. In addition, we show that the new test is useful for the analysis of genome‐wide association studies. Genet. Epidemiol. 33:569–580, 2009. © 2009 Wiley‐Liss, Inc.     

Comparison of three summary statistics for ranking genes in genome‐wide association studies

Problems associated with insufficient power have haunted the analysis of genome‐wide association studies and are likely to be the main challenge for the analysis of next‐generation sequencing data. Ranking genes according to their strength of association with the investigated phenotype is one solution. To obtain rankings for genes, researchers can draw from a wide range of statistics summarizing the relationships between variants mapped to a gene and the phenotype. Hence, it is of interest to explore the performance of these statistics in the context of rankings. To this end, we conducted a simulation study (limited to genes of equal sizes) of three different summary statistics examining the ability to rank genes in a meaningful order. The weighted sum of squared marginal score test (Pan, 2009), RareCover algorithm (Bahtia et al., 2010) and the elastic net regularization (Zou and Hastie, 2005) were chosen, because they can handle common as well as rare variants. The test based on the score statistic outperformed both other methods in almost all investigated scenarios. It was the only measure to consistently detect genes with interacting causal variants. However, the RareCover algorithm proved better at identifying genes including causal variants with small effect sizes and low minor allele frequency than the weighted sum of squared marginal score test. The performance of the elastic net regularization was unimpressive for all but the simplest scenarios. Copyright © 2013 John Wiley & Sons, Ltd.     

Evaluating cost efficiency of SNP chips in genome-wide association studies

Genome-wide association (GWA) studies have recently emerged as a major approach to gene discovery for many complex diseases. Since GWA scans are expensive, cost efficiency is an important factor to consider in study design. However, it often requires extensive and time-consuming computer simulations to compare cost efficiency across different single nucleotide polymorphism (SNP) chips. Here, we propose two simulation-free approaches to cost efficiency comparisons across SNP chips. In the first method, the overall power under a given disease model is calculated for each SNP chip and various sample sizes. Then SNP chips can be compared with respect to the sample sizes required to achieve the same level of power. In the second method, for a desired level of genomic coverage, the effective r(2) threshold values are calculated for each SNP chip. Since r(2) is inversely proportional to the sample size to achieve the same power, the required sample sizes can then be compared among SNP chips. These two methods are complementary to each other. The first approach provides direct power comparisons, but it requires information on disease model and may not be reliable for SNP chips that contain many non-HapMap SNPs. The second approach allows sample size comparisons based on the coverage of SNP chips, and it can be modified for SNP chips that contain non-HapMap SNPs. These methods are particularly relevant for large epidemiological studies in which enough subjects are available for GWA screening and follow-up stages. We illustrate these approaches using five currently available whole genome SNP chips.     

Perspectives on genome-wide multi-stage family-based association studies

With the establishment of large consortiums of researchers, genome-wide association (GWA) studies have become increasingly popular and feasible. Although most of these association studies focus on unrelated individuals, a lot of advantages can be exploited by including families in the analysis as well. To overcome the additional genotyping cost, multi-stage designs are particularly useful. In this article, I offer a perspective view on genome-wide family-based association analyses, both within a model-based and model-free paradigm. I highlight how multi-stage designs and analysis techniques, which are quite popular in clinical epidemiology, can enter GWA settings. I furthermore discuss how they have proven successful in reducing analysis complexity, and in overcoming one of the most cumbersome statistical hurdles in the genome-wide context, namely controlling increased false positives due to multiple testing.     

On multiple-testing correction in genome-wide association studies

The interpretation of the results of large association studies encompassing much or all of the human genome faces the fundamental statistical problem that a correspondingly large number of single nucleotide polymorphisms markers will be spuriously flagged as significant. A common method of dealing with these false positives is to raise the significance level for the individual tests for association of each marker. Any such adjustment for multiple testing is ultimately based on a more or less precise estimate for the actual overall type I error probability. We estimate this probability for association tests for correlated markers and show that it depends in a nonlinear way on the significance level for the individual tests. This dependence of the effective number of tests is not taken into account by existing multiple-testing corrections, leading to widely overestimated results. We demonstrate a simple correction for multiple testing, which can easily be calculated from the pairwise correlation and gives far more realistic estimates for the effective number of tests than previous formulae. The calculation is considerably faster than with other methods and hence applicable on a genome-wide scale. The efficacy of our method is shown on a constructed example with highly correlated markers as well as on real data sets, including a full genome scan where a conservative estimate only 8% above the permutation estimate is obtained in about 1% of computation time. As the calculation is based on pairwise correlations between markers, it can be performed at the stage of study design using public databases.     

Joint testing of genotype and ancestry association in admixed families

Current genome-wide association studies (GWAS) often involve populations that have experienced recent genetic admixture. Genotype data generated from these studies can be used to test for association directly, as in a non-admixed population. As an alternative, these data can be used to infer chromosomal ancestry, and thus allow for admixture mapping. We quantify the contribution of allele-based and ancestry-based association testing under a family-design, and demonstrate that the two tests can provide non-redundant information. We propose a joint testing procedure, which efficiently integrates the two sources information. The efficiencies of the allele, ancestry and combined tests are compared in the context of a GWAS. We discuss the impact of population history and provide guidelines for future design and analysis of GWAS in admixed populations.     

An evaluation of statistical approaches to rare variant analysis in genetic association studies

Genome‐wide association (GWA) studies have proved to be extremely successful in identifying novel common polymorphisms contributing effects to the genetic component underlying complex traits. Nevertheless, one source of, as yet, undiscovered genetic determinants of complex traits are those mediated through the effects of rare variants. With the increasing availability of large‐scale re‐sequencing data for rare variant discovery, we have developed a novel statistical method for the detection of complex trait associations with these loci, based on searching for accumulations of minor alleles within the same functional unit. We have undertaken simulations to evaluate strategies for the identification of rare variant associations in population‐based genetic studies when data are available from re‐sequencing discovery efforts or from commercially available GWA chips. Our results demonstrate that methods based on accumulations of rare variants discovered through re‐sequencing offer substantially greater power than conventional analysis of GWA data, and thus provide an exciting opportunity for future discovery of genetic determinants of complex traits. Genet. Epidemiol. 34: 188–193, 2010. © 2009 Wiley‐Liss, Inc.     

Comparisons of multi‐marker association methods to detect association between a candidate region and disease

The joint use of information from multiple markers may be more effective to reveal association between a genomic region and a trait than single marker analysis. In this article, we compare the performance of seven multi‐marker methods. These methods include (1) single marker analysis (either the best‐scoring single nucleotide polymorphism in a candidate region or a combined test based on Fisher's method); (2) fixed effects regression models where the predictors are either the observed genotypes in the region, principal components that explain a proportion of the genetic variation, or predictors based on Fourier transformation for the genotypes; and (3) variance components analysis. In our simulation studies, we consider genetic models where the association is due to one, two, or three markers, and the disease‐causing markers have varying allele frequencies. We use information from either all the markers in a region or information only from tagging markers. Our simulation results suggest that when there is one disease‐causing variant, the best‐scoring marker method is preferred whereas the variance components method and the principal components method work well for more common disease‐causing variants. When there is more than one disease‐causing variant, the principal components method seems to perform well over all the scenarios studied. When these methods are applied to analyze associations between all the markers in or near a gene and disease status for an inflammatory bowel disease data set, the analysis based on the principal components method leads to biologically more consistent discoveries than other methods. Genet. Epidemiol. 34: 201–124, 2010. © 2009 Wiley‐Liss, Inc.     

To stratify or not to stratify: power considerations for population-based genome-wide association studies of quantitative traits

Meta-analyses of genome-wide association studies require numerous study partners to conduct pre-defined analyses and thus simple but efficient analyses plans. Potential differences between strata (e.g. men and women) are usually ignored, but often the question arises whether stratified analyses help to unravel the genetics of a phenotype or if they unnecessarily increase the burden of analyses. To decide whether to stratify or not to stratify, we compare general analytical power computations for the overall analysis with those of stratified analyses considering quantitative trait analyses and two strata. We also relate the stratification problem to interaction modeling and exemplify theoretical considerations on obesity and renal function genetics. We demonstrate that the overall analyses have better power compared to stratified analyses as long as the signals are pronounced in both strata with consistent effect direction. Stratified analyses are advantageous in the case of signals with zero (or very small) effect in one stratum and for signals with opposite effect direction in the two strata. Applying the joint test for a main SNP effect and SNP-stratum interaction beats both overall and stratified analyses regarding power, but involves more complex models. In summary, we recommend to employ stratified analyses or the joint test to better understand the potential of strata-specific signals with opposite effect direction. Only after systematic genome-wide searches for opposite effect direction loci have been conducted, we will know if such signals exist and to what extent stratified analyses can depict loci that otherwise are missed.