Genome-wide Association Studies Test Research Articles

Covariate-adjusted multiple testing in genome-wide association studies via factorial hidden Markov models

It is more and more important to consider the dependence structure among multiple testings, especially for the genome-wide association studies (GWAS). The existing procedures, such as local index of significance (LIS) and pooled local index of significance (PLIS), were proposed to test hidden Markov model (HMM)-dependent hypotheses under the framework of compound decision theory, which was successfully applied to GWAS. However, the etiology of complex diseases is not only with respect to the genetic effects, but also the environmental factors. Failure to account for the covariates in multiple testing can produce misleading bias of the association of interest, or suffer from loss of testing efficiency. In this paper, we develop a covariate-adjusted multiple testing procedure, called covariate-adjusted local index of significance (CALIS), to account for the effects of environmental factors via a factorial hidden Markov model. The theoretical results show that our procedure can control the false discovery rate (FDR) at the nominal level and has the smallest false non-discovery rate (FNR) among all valid FDR procedures. We further demonstrate the advantage of our novel procedure over the existing procedures by simulation studies and a real data analysis.

A C6orf10/LOC101929163 locus is associated with age of onset in C9orf72 carriers.

The G4C2-repeat expansion in C9orf72 is the most common known cause of amyotrophic lateral sclerosis and frontotemporal dementia. The high phenotypic heterogeneity of C9orf72 patients includes a wide range in age of onset, modifiers of which are largely unknown. Age of onset could be influenced by environmental and genetic factors both of which may trigger DNA methylation changes at CpG sites. We tested the hypothesis that age of onset in C9orf72 patients is associated with some common single nucleotide polymorphisms causing a gain or loss of CpG sites and thus resulting in DNA methylation alterations. Combined analyses of epigenetic and genetic data have the advantage of detecting functional variants with reduced likelihood of false negative results due to excessive correction for multiple testing in genome-wide association studies. First, we estimated the association between age of onset in C9orf72 patients (n = 46) and the DNA methylation levels at all 7603 CpG sites available on the 450 k BeadChip that are mapped to common single nucleotide polymorphisms. This was followed by a genetic association study of the discovery (n = 144) and replication (n = 187) C9orf72 cohorts. We found that age of onset was reproducibly associated with polymorphisms within a 124.7 kb linkage disequilibrium block tagged by top-significant variation, rs9357140, and containing two overlapping genes (LOC101929163 and C6orf10). A meta-analysis of all 331 C9orf72 carriers revealed that every A-allele of rs9357140 reduced hazard by 30% (P = 0.0002); and the median age of onset in AA-carriers was 6 years later than GG-carriers. In addition, we investigated a cohort of C9orf72 negative patients (n = 2634) affected by frontotemporal dementia and/or amyotrophic lateral sclerosis; and also found that the AA-genotype of rs9357140 was associated with a later age of onset (adjusted P = 0.007 for recessive model). Phenotype analyses detected significant association only in the largest subgroup of patients with frontotemporal dementia (n = 2142, adjusted P = 0.01 for recessive model). Gene expression studies of frontal cortex tissues from 25 autopsy cases affected by amyotrophic lateral sclerosis revealed that the G-allele of rs9357140 is associated with increased brain expression of LOC101929163 (a non-coding RNA) and HLA-DRB1 (involved in initiating immune responses), while the A-allele is associated with their reduced expression. Our findings suggest that carriers of the rs9357140 GG-genotype (linked to an earlier age of onset) might be more prone to be in a pro-inflammatory state (e.g. by microglia) than AA-carriers. Further, investigating the functional links within the C6orf10/LOC101929163/HLA-DRB1 pathway will be critical to better define age-dependent pathogenesis of frontotemporal dementia and amyotrophic lateral sclerosis.

Probabilistic natural mapping of gene-level tests for genome-wide association studies.

Genome-wide association studies (GWASs) generally focus on a single marker, which limits the elucidation of the genetic architecture of complex traits. Herein, we present a new computational framework, termed probabilistic natural mapping (PALM), for performing gene-level association tests. PALM robustly reveals the inherent genomic structures of genes and generates feature representations that can be seamlessly incorporated into conventional statistic tests. Our approach substantially improves the effectiveness of uncovering associations derived from a subgroup of variants with weak effects, which represents a known challenge associated with existing methods. We applied PALM in a gastric cancer GWAS and identified two additional gastric cancer-associated susceptibility genes, NOC3L and RUNDC2A. The robust susceptibility discoveries of PALM are widely supported by existing studies from other biological perspectives. PALM will be useful for further GWAS analytical strategies that use gene-level analyses.

An enhanced version of Cochran-Armitage trend test for genome-wide association studies

Genome-wide association studies the evaluation of association between candidate gene and disease status is widely carried out using Cochran-Armitage trend test. However, only a small number of research papers have evaluated the distribution of p-values for the Cochran-Armitage trend test. In this paper, an enhanced version of Cochran-Armitage trend test based on bootstrap approach is introduced. The achieved results confirm that the distribution of p-values of the proposed approach fits better to the uniform distribution, and it is thus concluded that the proposed method, which needs less assumptions in comparison with the conventional method, can be successfully used to test the genetic association.

Pleiotropic Genes Affecting Carcass Traits in Bos indicus (Nellore) Cattle Are Modulators of Growth.

Two complementary methods, namely Multi-Trait Meta-Analysis and Versatile Gene-Based Test for Genome-wide Association Studies (VEGAS), were used to identify putative pleiotropic genes affecting carcass traits in Bos indicus (Nellore) cattle. The genotypic data comprised over 777,000 single-nucleotide polymorphism markers scored in 995 bulls, and the phenotypic data included deregressed breeding values (dEBV) for weight measurements at birth, weaning and yearling, as well visual scores taken at weaning and yearling for carcass finishing precocity, conformation and muscling. Both analyses pointed to the pleomorphic adenoma gene 1 (PLAG1) as a major pleiotropic gene. VEGAS analysis revealed 224 additional candidates. From these, 57 participated, together with PLAG1, in a network involved in the modulation of the function and expression of IGF1 (insulin like growth factor 1), IGF2 (insulin like growth factor 2), GH1 (growth hormone 1), IGF1R (insulin like growth factor 1 receptor) and GHR (growth hormone receptor), suggesting that those pleiotropic genes operate as satellite regulators of the growth pathway.

An efficient gene-gene interaction test for genome-wide association studies in trio families.

Several efficient gene-gene interaction tests have been developed for unrelated case-control samples in genome-wide association studies (GWAS), making it possible to test tens of billions of interaction pairs of single-nucleotide polymorphisms (SNPs) in a reasonable timeframe. However, current family-based gene-gene interaction tests are computationally expensive and are not applicable to genome-wide interaction analysis. We developed an efficient family-based gene-gene interaction test, GCORE, for trios (i.e. two parents and one affected sib). The GCORE compares interlocus correlations at two SNPs between the transmitted and non-transmitted alleles. We used simulation studies to compare the statistical properties such as type I error rates and power for the GCORE with several other family-based interaction tests under various scenarios. We applied the GCORE to a family-based GWAS for autism consisting of approximately 2000 trios. Testing a total of 22 471 383 013 interaction pairs in the GWAS can be finished in 36 h by the GCORE without large-scale computing resources, demonstrating that the test is practical for genome-wide gene-gene interaction analysis in trios. GCORE is implemented with C ++ and is available at http://gscore.sourceforge.net rchung@nhri.org.tw Supplementary data are available at Bioinformatics online.

Large-scale multiple testing in genome-wide association studies via region-specific hidden Markov models.

BackgroundIdentifying genetic variants associated with complex human diseases is a great challenge in genome-wide association studies (GWAS). Single nucleotide polymorphisms (SNPs) arising from genetic background are often dependent. The existing methods, i.e., local index of significance (LIS) and pooled local index of significance (PLIS), were both proposed for modeling SNP dependence and assumed that the whole chromosome follows a hidden Markov model (HMM). However, the fact that SNP data are often collected from separate heterogeneous regions of a single chromosome encourages different chromosomal regions to follow different HMMs. In this research, we developed a data-driven penalized criterion combined with a dynamic programming algorithm to find change points that divide the whole chromosome into more homogeneous regions. Furthermore, we extended PLIS to analyze the dependent tests obtained from multiple chromosomes with different regions for GWAS.ResultsThe simulation results show that our new criterion can improve the performance of the model selection procedure and that our region-specific PLIS (RSPLIS) method is better than PLIS at detecting disease-associated SNPs when there are multiple change points along a chromosome. Our method has been used to analyze the Daly study, and compared with PLIS, RSPLIS yielded results that more accurately detected disease-associated SNPs.ConclusionsThe genomic rankings based on our method differ from the rankings based on PLIS. Specifically, for the detection of genetic variants with weak effect sizes, the RSPLIS method was able to rank them more efficiently and with greater power.

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.

HYST: A Hybrid Set-Based Test for Genome-wide Association Studies, with Application to Protein-Protein Interaction-Based Association Analysis

The extended Simes' test (known as GATES) and scaled chi-square test were proposed to combine a set of dependent genome-wide association signals at multiple single-nucleotide polymorphisms (SNPs) for assessing the overall significance of association at the gene or pathway levels. The two tests use different strategies to combine association p values and can outperform each other when the number of and linkage disequilibrium between SNPs vary. In this paper, we introduce a hybrid set-based test (HYST) combining the two tests for genome-wide association studies (GWASs). We describe how HYST can be used to evaluate statistical significance for association at the protein-protein interaction (PPI) level in order to increase power for detecting disease-susceptibility genes of moderate effect size. Computer simulations demonstrated that HYST had a reasonable type 1 error rate and was generally more powerful than its parents and other alternative tests to detect a PPI pair where both genes are associated with the disease of interest. We applied the method to three complex disease GWAS data sets in the public domain; the method detected a number of highly connected significant PPI pairs involving multiple confirmed disease-susceptibility genes not found in the SNP- and gene-based association analyses. These results indicate that HYST can be effectively used to examine a collection of predefined SNP sets based on prior biological knowledge for revealing additional disease-predisposing genes of modest effects in GWASs.

PERMORY-MPI: a program for high-speed parallel permutation testing in genome-wide association studies

PERMORY is software for accelerated permutation testing of genome-wide association studies (GWAS). We have parallelized PERMORY using the Message-Passing Interface resulting in a nearly linear speedup. Furthermore, we added accelerated analysis of GWAS using quantitative phenotypes, and an accurate estimation of the effective number of independent tests. Free download from http://permory.org.

A new association test based on Chi-square partition for case-control GWA studies

In case-control genetic association studies, the robust procedure, Pearson's Chi-square test, is commonly used for testing association between disease status and genetic markers. However, this test does not take the possible trend of relative risks, which are due to genotype, into account. On the contrary, although Cochran-Armitage trend test with optimal scores is more powerful; it is usually difficult to assign the correct scores in advance since the true genetic model is rarely known in practice. If the unknown underlying genetic models are misspecified, the trend test may lose power dramatically. Therefore, it is desirable to find a powerful yet robust statistical test for genome-wide association studies. In this paper, we propose a new test based on the partition of Pearson's Chi-square test statistic. The new test utilizes the information of the monotonic (increasing or decreasing) trend of relative risks and therefore in general is more powerful than the Chi-square test; furthermore, it reserves the robustness. Using simulated and real single nucleotide polymorphism data, we compare the performance of the proposed test with existing methods.

Phenosim - A software to simulate phenotypes for testing in genome-wide association studies

BackgroundThere is a great interest in understanding the genetic architecture of complex traits in natural populations. Genome-wide association studies (GWAS) are becoming routine in human, animal and plant genetics to understand the connection between naturally occurring genotypic and phenotypic variation. Coalescent simulations are commonly used in population genetics to simulate genotypes under different parameters and demographic models.ResultsHere, we present phenosim, a software to add a phenotype to genotypes generated in time-efficient coalescent simulations. Both qualitative and quantitative phenotypes can be generated and it is possible to partition phenotypic variation between additive effects and epistatic interactions between causal variants. The output formats of phenosim are directly usable as input for different GWAS tools. The applicability of phenosim is shown by simulating a genome-wide association study in Arabidopsis thaliana.ConclusionsBy using the coalescent approach to generate genotypes and phenosim to add phenotypes, the data sets can be used to assess the influence of various factors such as demography, genetic architecture or selection on the statistical power of association methods to detect causal genetic variants under a wide variety of population genetic scenarios. phenosim is freely available from the authors' website http://evoplant.uni-hohenheim.de

Abstract SY29-01: Gene-based and pathway-based analysis applied to cancer genome-wide association studies

Abstract Genome-wide association studies (GWAS) have proved successful in identifying new loci underlying susceptibility to a wide variety of traits including many cancers. It is becoming clear that in addition to relatively large genetic effects, such as pigmentation-related genetic effects on cancers such as melanoma, there exist a large number of smaller genetic effects which are more difficult to identify (“the polygenic tail” in the distribution of effects). Identifying and characterizing such smaller effects will require approaches which do more than simply focus on the very top hits from GWAS. Gene-based and pathway-based approaches are two approaches which complement single marker based GWAS analysis. We recently developed a versatile gene-based test for genome-wide association studies (GWAS)1. Our approach, called VEGAS (VErsatile Gene-based Association Study), is applicable to all GWAS designs, including family-based GWAS, meta-analyses of GWAS on the basis of summary data, and DNA-pooling-based GWAS, where existing approaches based on permutation are not possible, as well as singleton data, where they are. The test incorporates information from a full set of markers (or a defined subset) within a gene and accounts for linkage disequilibrium between markers by using simulations from the multivariate normal distribution. For an association study using singletons, our approach produces results equivalent to those obtained via permutation in a fraction of the computation time. Applying VEGAS to a DNA-pooling-based GWAS for melanoma (1200 cases with relevant controls), we previously showed that several pigmentation-related genes are clearly associated at the gene-based level. We have subsequently applied VEGAS to an extended melanoma data based on individually genotyped samples (2200 cases with relevant controls), obtaining stronger confirmation of the role of pigmentation related genes (MC1R, ASIP, OCA2, MTAP, TYR, SLC45A2) and suggestive evidence for a number of novel nonpigmentation related genes in melanoma. We also applied VEGAS to two further smaller (500 and 1000 cases, respectively) GWASs of melanoma, again highlighting some, but not all of the pigmentation genes listed above. Work is underway in applying VEGAS to several other cancer types including breast cancer, ovarian cancer, prostate cancer, pancreatic cancer and lung cancer. VEGAS has been extended to use gene-based results in a pathway analysis (VEGAS-pathway). VEGAS-pathway, like VEGAS, uses summary data and is not limited to situations where raw genotypes are available. Currently, this implementation combines gene-based test statistics across genes in a predefined pathway, taking into account situations where genes within the pathway are close to each other in the genome. This implementation has been evaluated using pathways defined by Gene Ontology and Ingenuity. Work is ongoing to extend the approach to further sets of predefined pathways. Applying VEGAS-pathway to the N=2200 melanoma data set using GO defined pathways, several pathways that made good biological sense (e.g., “melanin biosynthetic process,” “pigmentation”) were apparent. VEGAS-pathway results on the smaller (N=500, 1000) melanoma data sets did not clearly identify these GO pathways, indicating that large sample size remains crucial, even for situations such as pigmentation-related pathways in melanoma. Using Ingenuity-defined pathways, it was not possible to identify any statistically significant pathways that made good biological sense in any of the melanoma data sets – at least in this case, the more narrowly defined Ingenuity pathways offered less insight into the trait than the more broadly defined and numerous GO pathways. After removal of the known pigmentation genes, the most significant GO pathway in the N=2200 data set was the small pathway “regulation of growth rate”. This pathway showed nominal replication in a further set of N=1800 melanoma cases (with relevant controls). VEGAS-pathway results for several other cancer types (breast cancer, ovarian cancer, prostate cancer, pancreatic cancer and lung cancer) will be presented at the meeting. 1 Liu JZ, McRae AF, Nyholt DR, Medland SE, Wray NR, Brown KM, Hayward NK, Montgomery GW, Visscher PM, Martin NG, Macgregor S, A versatile gene-based test for genome-wide association studies. Am J Hum Genet 2010, 87(1):139-145. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 102nd Annual Meeting of the American Association for Cancer Research; 2011 Apr 2-6; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2011;71(8 Suppl):Abstract nr SY29-01. doi:10.1158/1538-7445.AM2011-SY29-01

On the follow-up of genome-wide association studies: an overall test for the most promising SNPs

Even in large-scale genome-wide association studies (GWASs), only a fraction of the true associations are detected at the genome-wide significance level. When few or no associations reach the significance threshold, one strategy is to follow up on the most promising candidates, i.e. the single nucleotide polymorphisms (SNPs) with the smallest association-test P-values, by genotyping them in additional studies. In this communication, we propose an overall test for GWASs that analyzes the SNPs with the most promising P-values simultaneously and therefore allows an early assessment of whether the follow-up of the selected SNPs is likely promising. We theoretically derive the properties of the proposed overall test under the null hypothesis and assess its power based on simulation studies. An application to a GWAS for chronic obstructive pulmonary disease suggests that there are true association signals among the top SNPs and that an additional follow-up study is promising.

A Versatile Gene-Based Test for Genome-wide Association Studies

We have derived a versatile gene-based test for genome-wide association studies (GWAS). Our approach, called VEGAS (versatile gene-based association study), is applicable to all GWAS designs, including family-based GWAS, meta-analyses of GWAS on the basis of summary data, and DNA-pooling-based GWAS, where existing approaches based on permutation are not possible, as well as singleton data, where they are. The test incorporates information from a full set of markers (or a defined subset) within a gene and accounts for linkage disequilibrium between markers by using simulations from the multivariate normal distribution. We show that for an association study using singletons, our approach produces results equivalent to those obtained via permutation in a fraction of the computation time. We demonstrate proof-of-principle by using the gene-based test to replicate several genes known to be associated on the basis of results from a family-based GWAS for height in 11,536 individuals and a DNA-pooling-based GWAS for melanoma in approximately 1300 cases and controls. Our method has the potential to identify novel associated genes; provide a basis for selecting SNPs for replication; and be directly used in network (pathway) approaches that require per-gene association test statistics. We have implemented the approach in both an easy-to-use web interface, which only requires the uploading of markers with their association p-values, and a separate downloadable application.

A new gene-based association test for genome-wide association studies

Genome-wide association studies are widely used today to discover genetic factors that modify the risk of complex diseases. Usually, these methods work in a SNP-by-SNP fashion. We present a gene-based test that can be applied in the context of genome-wide association studies. We compare both strategies, SNP-based and gene-based, in a sample of cases and controls for rheumatoid arthritis.We obtained different results using each strategy. The SNP-based test found the PTPN22 gene while the gene-based test found the PHF19-TRAF1-C5 region. That suggests that no single strategy performs better than another in all cases and that a certain underlying genetic architecture can be delineated more easily with one strategy rather than with another.

Multiple testing in genome-wide association studies via hidden Markov models

Genome-wide association studies (GWAS) interrogate common genetic variation across the entire human genome in an unbiased manner and hold promise in identifying genetic variants with moderate or weak effect sizes. However, conventional testing procedures, which are mostly P-value based, ignore the dependency and therefore suffer from loss of efficiency. The goal of this article is to exploit the dependency information among adjacent single nucleotide polymorphisms (SNPs) to improve the screening efficiency in GWAS. We propose to model the linear block dependency in the SNP data using hidden Markov models (HMMs). A compound decision-theoretic framework for testing HMM-dependent hypotheses is developed. We propose a powerful data-driven procedure [pooled local index of significance (PLIS)] that controls the false discovery rate (FDR) at the nominal level. PLIS is shown to be optimal in the sense that it has the smallest false negative rate (FNR) among all valid FDR procedures. By re-ranking significance for all SNPs with dependency considered, PLIS gains higher power than conventional P-value based methods. Simulation results demonstrate that PLIS dominates conventional FDR procedures in detecting disease-associated SNPs. Our method is applied to analysis of the SNP data from a GWAS of type 1 diabetes. Compared with the Benjamini-Hochberg (BH) procedure, PLIS yields more accurate results and has better reproducibility of findings. The genomic rankings based on our procedure are substantially different from the rankings based on the P-values. By integrating information from adjacent locations, the PLIS rankings benefit from the increased signal-to-noise ratio, hence our procedure often has higher statistical power and better reproducibility. It provides a promising direction in large-scale GWAS. An R package PLIS has been developed to implement the PLIS procedure. Source codes are available upon request and will be available on CRAN (http://cran.r-project.org/). zhiwei@njit.edu Supplementary data are available at Bioinformatics online.

Rapid and Accurate Multiple Testing Correction and Power Estimation for Millions of Correlated Markers

With the development of high-throughput sequencing and genotyping technologies, the number of markers collected in genetic association studies is growing rapidly, increasing the importance of methods for correcting for multiple hypothesis testing. The permutation test is widely considered the gold standard for accurate multiple testing correction, but it is often computationally impractical for these large datasets. Recently, several studies proposed efficient alternative approaches to the permutation test based on the multivariate normal distribution (MVN). However, they cannot accurately correct for multiple testing in genome-wide association studies for two reasons. First, these methods require partitioning of the genome into many disjoint blocks and ignore all correlations between markers from different blocks. Second, the true null distribution of the test statistic often fails to follow the asymptotic distribution at the tails of the distribution. We propose an accurate and efficient method for multiple testing correction in genome-wide association studies—SLIDE. Our method accounts for all correlation within a sliding window and corrects for the departure of the true null distribution of the statistic from the asymptotic distribution. In simulations using the Wellcome Trust Case Control Consortium data, the error rate of SLIDE's corrected p-values is more than 20 times smaller than the error rate of the previous MVN-based methods' corrected p-values, while SLIDE is orders of magnitude faster than the permutation test and other competing methods. We also extend the MVN framework to the problem of estimating the statistical power of an association study with correlated markers and propose an efficient and accurate power estimation method SLIP. SLIP and SLIDE are available at http://slide.cs.ucla.edu.

Two-stage association tests for genome-wide association studies based on family data with arbitrary family structure

Recently, Steen et al proposed a two-stage approach for genome-wide family-based association studies. In the first stage, a screening test is used to select markers, and in the second stage, a family-based association test is performed on a much smaller set of the selected markers. The two-stage method can be much more powerful than the traditional family-based association tests. In this article, we extend the approach so that it can incorporate parental information and can be applied to an arbitrary pedigree structure. We use simulation studies to evaluate the type I error rates and the power of the proposed methods. Our results show that the two-stage approach that incorporates founders' phenotypes has the correct type I error rates, and is much more powerful than the two-stage approach that uses children's phenotypes only. Also, by carefully choosing the number of markers retained in the first stage, the power of a two-stage approach can be much more than that of the corresponding one-stage approach.

Nonlinear Tests for Genomewide Association Studies

As millions of single-nucleotide polymorphisms (SNPs) have been identified and high-throughput genotyping technologies have been rapidly developed, large-scale genomewide association studies are soon within reach. However, since a genomewide association study involves a large number of SNPs it is therefore nearly impossible to ensure a genomewide significance level of 0.05 using the available statistics, although the multiple-test problems can be alleviated, but not sufficiently, by the use of tagging SNPs. One strategy to circumvent the multiple-test problem associated with genome-wide association tests is to develop novel test statistics with high power. In this report, we introduce several nonlinear tests, which are based on nonlinear transformation of allele or haplotype frequencies. We investigate the power of the nonlinear test statistics and demonstrate that under certain conditions, some nonlinear test statistics have much higher power than the standard chi2-test statistic. Type I error rates of the nonlinear tests are validated using simulation studies. We also show that a class of similarity measure-based test statistics is based on the quadratic function of allele or haplotype frequencies, and thus they belong to nonlinear tests. To evaluate their performance, the nonlinear test statistics are also applied to three real data sets. Our study shows that nonlinear test statistics have great potential in association studies of complex diseases.