Genome-wide dense markers have been used to detect genes and estimate relative genetic values. Among many methods, Bayesian techniques have been widely used and shown to be powerful in genome-wide breeding value estimation and association studies. However, computation is known to be intensive under the Bayesian framework, and specifying a prior distribution for each parameter is always required for Bayesian computation. We propose the use of hierarchical likelihood to solve such problems.Using double hierarchical generalized linear models, we analyzed the simulated dataset provided by the QTLMAS 2010 workshop. Marker-specific variances estimated by double hierarchical generalized linear models identified the QTL with large effects for both the quantitative and binary traits. The QTL positions were detected with very high accuracy. For young individuals without phenotypic records, the true and estimated breeding values had Pearson correlation of 0.60 for the quantitative trait and 0.72 for the binary trait, where the quantitative trait had a more complicated genetic architecture involving imprinting and epistatic QTL.Hierarchical likelihood enables estimation of marker-specific variances under the likelihoodist framework. Double hierarchical generalized linear models are powerful in localizing major QTL and computationally fast. By including genetic marker information, genomic estimated breeding values (GEBV) can be obtained taking into account the information from these markers, and also quantitative trait loci (QTL) can be mapped by associating genotypes at a certain locus to the phenotype observations.Genetic analyses in livestock studies are generally based on information from pedigrees and molecular markers. Traditionally, a kinship matrix can be calculated using the pedigree data, which can be used in a single nucleotide polymorphism (SNP) markers. The dense SNP maps have made genome-wide association (GWA) studies popular for gene detection. Classic GWA methods [hierarchical likelihood (h-likelihood) [Dense marker genotypes along genome can now be affordably obtained due to new and efficient methods for typing  methods , commonl methods -5. Lee a methods . DHGLM e methods . The likelihood) .h-likelihood and model the data using DHGLM. GEBV are calculated from the estimated marker effects, and QTL are mapped by the estimated marker-specific variances.The aim of this paper is to map QTL and report GEBV for the simulated dataset provided by QTLMAS 2010 workshop. We employ a unified analysis via the F0 - F4) was simulated, where F0 contains 5 males and 15 females. Each female was mated once and gave birth to about 30 progeny. Two traits were simulated, where one is quantitative (QT), and the other is binary (BT). Young individuals in F4  had no phenotypic records. The genome was assumed to be about 5 × 108 bp long, consisting of 5 chromosomes, each of which contained about 1 × 108 bp. Each individual was genotyped for 10031 biallelic SNPs in the genome.The dataset used in this paper was simulated for the QTLMAS 2010 workshop . A pedigree consisting of 3226 individuals in 5 generations (y (n × 1 vector) is postulated as a random effect modelDHGLM provides a unified analysis for both QTL mapping and genomic breeding value estimation. Similar to BayesA, the data are modeled on two levels, i.e. both the phenotypic mean and the variance are modeled with random effects. For a quantitative trait, the phenotype y = βX + Zg + e (1)g ~ N) are the SNP effects, λ = ′ are the variances of the SNP effects, and the residuals e ~ N. The fixed effects β included an intercept and the sex effect in our application to reduce the residual errors. The SNP variances λ are modeled aswhere λ = 1a + b (2)log a and normally distributed random effects b. The genomic estimated breeding value (GEBV) for individual i is computed as λ. For a binary trait, the mean of y, is modeled by the same linear predictor βX + Zg through a logit link function.with an intercept b, follow a multivariate normal distribution with a mean of zero and a variance-covariance matrix m is the number of SNPs and k, l are the SNP indices. When ρ = 1, all the SNPs have a constant variance (GLMM); when ρ = 0, the SNPs are assumed to be independent (DHGLM); and for 0 <ρ < 1, the correlation between two SNPs is a monomial function of ρ, which is referred to as the smoothed DHGLM [ρ, regarded as a spatial correlation parameter, was chosen to be 0.9 in this paper, which nicely shrank the SNPs with zero effect.For the marker-specific variances, the correlated random effects, ed DHGLM . We propThe overall phenotypic variance can be expressed aszj. (the j-th column of Z) across individuals. These variance values can be directly calculated from the data. The contribution (heritability) of a particular SNP is expressed by where essed by .g should be drawn through the h-likelihood, fixed effects β through the marginal likelihood, and variance components λ, σ2 and According to the extended likelihood principle, inference of the random SNP effects kelihood . Howeverkelihood ,ĝ,• Solve the following WLS problem for M stands for ‘mean’.Where σ2 by fitting the deviance residuals wM = (1 – qM)/2, where n) and SNPs (n + 1 to n + m), respectively.• Update â and • Solve the following WLS problem for z = log λ + (dM2 – λ)/λ is linearized λ in a gamma GLM with a log link, and L satisfies LL′ = A. The subscript D stands for ‘dispersion’.where wD = (1 – qD)/2, where êD are the last m residuals of (5), and qD are the last m diagonal elements of • Update ρ = 0.9 for both traits  part of the models, marker-specific variances were estimated and used to detect QTL Figure . The ove package  in R [13GEBV were estimated for all the 3226 individuals in the pedigree. Examining out-sample prediction, we compare the GEBV with the true breeding values (TBV) for the young individuals (2327-3226) without phenotypic records , it took about 10-20 minutes, depending on starting values, to obtain our results using our implementation in R. No priors are required for parameters in DHGLM. Main QTL mapped via DHGLM showed very good accuracy though some QTL with small effects were shrunk or smoothed down. An R package h-likelihood: hierarchical likelihood; HGLM: hierarchical generalized linear model; MCMC: Markov chain Monte Carlo; QTL: quantitative trait locus/loci; QTLMAS: quantitative trait loci and marker assisted selection; REML: restricted maximum likelihood; SNP: single nucleotide polymorphism; TBV: true breeding values; WLS: weighted least squares.bp: base pair; DHGLM: double hierarchical generalized linear model; DNA: deoxyribonucleic acid; GEBV: genomic estimated breeding values; GLM: generalized linear model; GLMM: generalized linear mixed model; GWA: Genome-wide association; No competing interest to declare by any of the authors.XS, LR and ÖC initiated the study. XS analyzed the simulated common dataset of the QTLMAS 2010 workshop and drafted the paper. LR initiated the smoothed version of double hierarchical generalized linear models. XS, LR and ÖC worked on the revision together and approved the final manuscript.


Social-communication abilities are heritable traits, and their impairments overlap with the autism continuum. To characterise the genetic architecture of social-communication difficulties developmentally and identify genetic links with the autistic dimension, we conducted a genome-wide screen of social-communication problems at multiple time-points during childhood and adolescence.N ≤ 5,628) using mother-reported Social Communication Disorder Checklist scores. Genome-wide Complex Trait Analysis (GCTA) was conducted for all phenotypes. The time-points with the highest GCTA heritability were subsequently analysed for single SNP association genome-wide. Type I error in the presence of measurement relatedness and the likelihood of observing SNP signals near known autism susceptibility loci (co-location) were assessed via large-scale, genome-wide permutations. Association signals (P ≤ 10−5) were also followed up in Autism Genetic Resource Exchange pedigrees (N = 793) and the Autism Case Control cohort .Social-communication difficulties were ascertained at ages 8, 11, 14 and 17 years in a UK population-based birth cohort  = 0.24) and especially in later adolescence (h2(17 years) = 0.45), with a marked drop during early to middle adolescence (h2(11 years) = 0.16 and h2(14 years) = 0.08). Genome-wide screens at ages 8 and 17 years identified for the latter time-point evidence for association at 3p22.2 near SCN11A  and suggestive evidence at 20p12.3 at PLCB1 . None of these signals contributed to risk for autism. However, the co-location of population-based signals and autism susceptibility loci harbouring rare mutations, such as PLCB1, is unlikely to be due to chance .GCTA heritability was strongest in childhood (Our findings suggest that measurable common genetic effects for social-communication difficulties vary developmentally and that these changes may affect detectable overlaps with the autism spectrum. There was little evidence for a functional role of the associated SNP, although involvement in DNA motif changes is possible. PLCB1 is involved in extracellular signal transduction and catalyses the formation of inositol 1,4,5-trisphosphate and diacylglycerol from phosphatidylinositol 4,5-bisphosphate. The PLCB1 locus has previously been established as a susceptibility locus for ASD, harbouring multiple rare mutations. This includes an ASD-specific, approximately 480 kb inherited deletion [PLCB1 have been linked to early-onset epileptic encephalopathy [The signal at 20p12.3 (rs3761168) was found at deletion . In addideletion , and othalopathy  and schialopathy .PLCB1, a locus where LD is clearly defined, may play a more general functional role in the autism continuum with rare, protein-disrupting PLCB1 mutations contributing to ASD and common PLCB1 variation contributing to subtle changes in social-communication difficulties. Similar relationships, for example, have been demonstrated for low-density lipoprotein receptor (LDLR) gene mutations that cause familial hypercholesterolemia [LDLR variants which are related to smaller elevations in cholesterol [Our search for dimensional ASD QTL found little evidence for the contribution of social-communication related signals to risk for autism, which is consistent with recent views on the role of common genetic variation in ASD . Howevererolemia , as welllesterol . The repPLCB1.The major strength of our study is the exploration of the common genetic architecture of social-communication traits over time, which can be assessed by current chip arrays and imputation efforts, and the exploitation of GCTA to enhance the power of subsequent genome-wide association screens. The identification of specific genetically enriched time-points, however, also limits the pool of available replication samples, as, for example, our main signal at 3p22.2 does not show any association at 11 years, an age at which data on social-communication phenotypes is available within many cohorts. Our study thus emphasises the need to collect social-communication phenotypes across development, including late adolescence and possibly adulthood. Given that genetically backed social-communication measures in near-adult populations are rare, we selected genome-wide permutations to control for type I error. This allowed us to adjust for measurement relatedness without modelling the genetically less enriched phenotypes, which might be affected by increased ‘measurement noise’. Although permutation analysis cannot provide the same robustness as replication in independent samples (and we therefore cannot entirely rule out type I error), at least one identified common signal within our study overlaps with a locus, which has been implicated within ASD through a rare disease mechanism, Together, our findings suggest that social-communication difficulties are developmentally characterised by variation in GCTA heritability, despite some genetic stability, and that these changes may affect detectable overlaps with the autism spectrum. An informed analysis of phenotypes with high GCTA heritability, however, may increase the power of genome-wide association screens.Supplementary information is provided as Additional file ASD: Autism spectrum disorder; ADI–R: Autism Diagnostic Interview–Revised; ACC: Autism Case Control cohort; AGRE: Autism Genetic Resource Exchange; ALSPAC: Avon Longitudinal Study of Parents and Children; GC: Genomic control; GCTA: Genome-wide Complex Trait Analysis; GWAS: Genome-wide association study; HapMapCEU: Utah residents with Northern and Western European ancestry from the Centre d’Etude du Polymorphisme Humain collection; LCL: Lymphoblastoid cell line; LD: Linkage disequilibrium; LDLR: Low-density lipoprotein receptor; MAF: Minor allele frequency; MDS: Multidimensional scaling; QTL: Quantitative trait locus; SCDC: Social Communication Disorder Checklist; SNP: Single-nucleotide polymorphism.The authors declare that they have no competing interests.BSP and KW carried out the statistical analysis. BSP, DME, JPK, SMR and WLM were involved in the preparation of the genotype information. BSP, DHS, WPM and GDS participated in the design of the study. BSP, DHS, WPM, KW, HH, NJT, DMW, JPK, JG and GDS helped to draft the manuscript. All authors read and approved the final manuscript.Additional Note. Genome-wide Complex Trait Analysis. Table S1. Temporal stability of social-communication problems. Table S2. Genetic correlations. Table S3. Genome-wide association signals for social-communication problems at single time-points. Table S4. Longitudinal analysis of the strongest single time-point association signals. Table S5. Functional characterisation of non-coding variation near rs4453791. Table S6. Expression quantitative trait locus analysis. Table S7. Follow-up analysis of social-communication related signals in autism samples. Figure S1. Quantile-quantile plots of genome-wide association signals.Click here for file


Both linkage and association studies have been successfully applied to identify disease susceptibility genes with genetic markers such as microsatellites and Single Nucleotide Polymorphisms (SNPs). As one of the traditional family-based studies, the Transmission/Disequilibrium Test (TDT) measures the over-transmission of an allele in a trio from its heterozygous parents to the affected offspring and can be potentially useful to identify genetic determinants for complex disorders. However, there is reduced information when complete trio information is unavailable. In this study, we developed a novel approach to “infer” the transmission of SNPs by combining both the linkage and association data, which uses microsatellite markers from families informative for linkage together with SNP markers from the offspring who are genotyped for both linkage and a Genome-Wide Association Study (GWAS). We generalized the traditional TDT to process these inferred dosage probabilities, which we name as the dosage-TDT (dTDT). For evaluation purpose, we developed a simulation procedure to assess its operating characteristics. We applied the dTDT to the simulated data and documented the power of the dTDT under a number of different realistic scenarios. Finally, we applied our methods to a family study of alcohol dependence (COGA) and performed individual genotyping on complete families for the top signals. One SNP (rs4903712 on chromosome 14) remained significant after correcting for multiple testing Methods developed in this study can be adapted to other platforms and will have widespread applicability in genomic research when case-control GWAS data are collected in families with existing linkage data. Linkage studies have been successfully used to identify many disease genes such as hyper-cholesterolaemia In general, there are two major types of designs that are commonly used in association research: population-based and family-based studies. As the most common population-based approach, the case-control setup compares an unrelated healthy control group and affected case group. The genotyped SNPs are investigated to identify the allele frequency differences between these two groups. The study then determines whether the SNPs are associated with the genetic trait or disease based on the statistical significance of the differences. The independent samples are typically easier to obtain in a case-control study than family samples. However, many case-control samples select independent cases from existing family data that were originally used in linkage analysis. Because cases can be over-sampled from groups with higher disease prevalence, the differences of allele frequencies in an admixture of ethnic groups may produce spurious associations. Therefore, although case-control studies have shown advantages in identifying association between the disease susceptibility and markers in a candidate gene, the results may reflect type I errors  due to unaccounted confounding factors Unlike the population-based studies, family-based studies are resistant to type I errors arising from population stratification. The family-based Transmission/Disequilibrium Test (TDT) measures the over-transmission of an allele from heterozygous parents to their affected offspring, in which the non-transmitted parental alleles serve, in effect, as a control group. Therefore the TDT is a robust test of association in the presence of geographical or ethnical impact from the population Because neither genotypes nor allele frequencies are required, the TDT is considered robust to the population stratification as mentioned above.A variety of TDT-like tests have been suggested starting with Rubinstein et al Disorders can often have genotype information from only one parent of the affected individuals. As a common practice, these trios are simply discarded Against this background, we note this is a two step procedure. In the first step, the SNP data is used with both parents missing, so that the analysis depends on external allele frequency estimates and is sensitive to population stratification. In the second step, individual genotyping is performed on the parents for the top SNPs from step one, so this step is a traditional TDT and insensitive to the potential biases in step one. Accordingly, having available family DNA is needed to avoid false positive results.Genotype Inference of Familial Individuals for more details). All the trio combinations from the inferred pedigrees are extracted on the condition that the children were affected and at least one parent in the trio was genotyped on microsatellite markers. The dosage-TDT that we have developed in this study is applied on these trio pedigrees using their inferred dosage probabilities. By incorporating the family linkage information into the GWAS data, we can potentially have higher power to detect association between our genotypic markers and the disease susceptibility alleles.As stated above, the traditional TDT requires complete genotypic information from all members of the nuclear families. However, obtaining all genotypes cannot always be feasible for some diseases or families. Therefore the traditional TDT-type studies may not be useful to identify the presence of genetic determinants in data with relatively small amounts of complete trio information. One way to solve this type of issue is to reconstruct the missing parental genotypes under the assumption that they follow the probability distribution of the fully observed cases. However, most studies designed for this purpose do not incorporate the impact from LD and thus may introduce bias to the results. On the other hand, there are data available that have genotype information on both microsatellite and SNP markers for diseases; one example is alcoholism gm denotes the genetic position of a marker, in centi-Morgan (cM) unit; pm denotes the physical position of a marker, in base pair units; lastMS is the last microsatellite marker on a chromosome, st microsatellite marker or larger than the last microsatellite marker, there is only one microsatellite marker that can be referred to compute the genetic position for the SNP. We simply use the convention that 1cM ≈ 106 base pairs to convert a SNP’s physical map to its genetic map. While a SNP is in between two microsatellite markers, we use the ratio relative genetic position of a SNP marker to the microsatellite marker that’s next to it.The common map of both linkage and GWAS data is designed in the following way. With the genetic position of the linkage markers (microsatellite as in here) and physical position of both linkage and GWAS markers (microsatellite and SNPs), the genetic positions of all the GWAS markers (SNPs) are calculated based on Initially, many approaches implicitly imputed missing genotypes based on the potential genotype distribution in a family χ2 statistics, it requires single counts of the transmitted/non-transmitted alleles from the heterozygous parents to the affected offspring. Thus the inferred dosage probabilities cannot be processed through this setup. In this study, we generalize the original TDT by taking all possible allele transmissions in a pedigree into account. trio-dosage set in this work) from the inferred results. ib and ic used in the χ2 calculation of the TDT in each trio i are calculated by summing up the probabilities across all these 11 types of transmissions. Let t denote the probability that allele 1 is transmitted by a heterozygote parent of an affected child. We can then write the dosage probabilities of a child in terms of the dosage probabilities of its parents and t as follows:Because the traditional TDT is a simple representation of the Thus, the frequencies of allele 1 and 2 of a child appearing in a trio are as follows:Based on ib and ic can be written as:i represents the ith trio. By substituting t from ib and ic for all trios, we can compute the χ2 values for each SNP:In summary, the sum of all 11 informative transmissions for itten as:(11)whert can be canceled out with the one in ib and ic thus The beauty of the above equations is the denominator from i. Using form  of the disease is11f, 12f and 22f are the penetrances of the three genotypes 1D1D, 1D2D and 2D2D.Denote the low and high risk alleles at a disease locus f11<f12 = f22), recessive (f11 = f12<f22) and co-dominant (f11<f12 = ½ f22).We have considered three disease models: dominant, recessive and co-dominant. The combinations of the penetrances in these three models are designed as follows: dominant  of linkage disequilibrium (LD) between D and S, we can derive the haplotype frequencies as follows:1q and 2q are the population frequencies of the SNP alleles 1S and 2S. To simplify our model, we will assume ip = iq, where i = 1 or 2. The rationale behind this is that if the SNP marker and disease allele have very different frequencies, then R2 is small and there is little power. Keeping both frequencies equal allow R to vary the full range from −1 to +1.With a predefined correlation coefficient  at locus S to the affected children based on these derived frequencies:By substituting  into 1919, we caiM as the microsatellite marker from the parents. Because of a large number of polymorphisms  for a microsatellite marker, we assume that our microsatellite marker is completely informative  and assign alleles 1M & 2M to the father and 3M & 4M to the mother. Then we randomly select 1M & 3M, 2M & 4M, 1M & 4M or 2M & 3M equally with 0.25 probabilities as the microsatellite genotype for the child.Denote ip ∈, but also with a good coverage of possible natural phenomena, we predefine the following values for the parameters to generate each set of trios:Without losing biological meaning, i.e. with valid N: the number of trios = 1,000;K: prevalence = 0.01, 0.1, or 0.2;R: correlation coefficient of LD between D and S = 0.5, 0.7, 0.9, or 1.0 ;f or g: penetrance of disease genotype iDiD or iDjD, where i or j = 1 or 2 and i ≠ j. To simplify the notation, here we use f to denote f11 and g to denote f12 or f22. As noted, we separate the disease models as dominant , recessive , and co-dominant  where f = 0.0, 0.1K, 0.3K, 0.5K, 0.7K, or 0.9K and g = 1.1K, 0.5, 0.7, 0.9, or 1.0.p, i.e.These values are first permuted to generate all their possible combinations then any combination that produces invalid equation  for SNPsAlcohol dependence is a serious psychiatric disorder in which an individual is characterized as having harmful consequences of repeated or compulsive alcohol use, and (sometimes) physiological dependence on alcohol  The nine-site national Collaborative study On the Genetics of Alcoholism (COGA) funded by National Institute of Alcohol and Alcoholism (NIAAA) aims to identify and characterize genes that affect the susceptibility to develop alcohol dependence and related phenotypes. COGA is applying multiple strategies for genetic research. The most densely affected, multiplex alcoholic families were used in a multi-wave family-based linkage study. 2,283 out of 2,459 individuals from 262 families were genotyped using microsatellite markers in Wave I and Wave II (data denoted as Map03MS) Table 4 : FWA00003624 (SUNY Research Foundation), FWA00007125 (University of Connecticut), FWA00003544 (Indiana University), FWA00003007 (University of Iowa), FWA00004069 , FWA00002284 (Washington University), FWA00003518 , FWA00003913  and FWA00005287 .All participants agreed to share their DNA and phenotypic information for research purposes and provided written informed consent following instructions from institutional review boards at all data collection sites. The study was approved by the institutional review board at each COGA site, with the OHRP Assurance Numbers beingAs described above, the dTDT uses the inferred dosage probabilities of dense SNPs for association detection. The COGA family data provides us such an opportunity to integrate the information from both linkage and association studies.77 is used to conduct a standard case-control study on the CIDRSNP data. With the idea that making use of all the available sample data would increase the power for association detection, we further combine the results from both dTDT and case-control study through the MH test In this study, we first generate the combined pedigrees with both the LinkageMS and CIDRSNP genotype data from COGA. ,  and The Map03MS data have 219 European American (EA), 35 African American (AA) and eight mixed families. 2,283 individuals from these 262 families were genotyped on 328 microsatellite markers. The MarshfieldMS data contain 234 EA, 77 AA and one mixed families, with a total of 1,442 individuals genotyped on 394 microsatellite markers. 1,041,304 SNPs were genotyped for 1,399 EA and 485 AA individuals in the CIDRSNP GWAS. ,  and 6With AA and mixed families excluded, we have 3,016 out of 3,567 EA individuals from 453 families genotyped on microsatellite markers in the LinkageMS data. 471 of these individuals in 398 linkage families were selected for SNP genotyping  Table 6,MS and CIDRSNP data combined together. One combined pedigree has totalind individuals (in rows) with (MSmrk+SNPmrk) markers (in columns). Missing SNPs of  individuals were inferred by MERLIN.  were excluded. In consideration of any possible impact from linkage disequilibrium (LD), we exclude ∼1,500 SNPs that are within 1,000 base pairs flanking each microsatellite marker. We use  MERLIN. Figure 1χ2 test with one degree of freedom. Comparatively, because the dTDT only takes account of values of b and c, the test can be constructed by a 1 × 2 table instead. To maximally benefit from all sample data and multiple studies, we extend the MH test to pool results on each SNP from these two contingency tables in both case-control study and dTDT. Calculation of each term in the MH test is shown in In this study, we apply both the dTDT and Mantel-Haenszel (MH) tests to the COGA data. The MH test was first proposed by Mantel and Haenszel in 1959 K values . In each group there are 100∼120 settings with different R and f values. With each setting we generate 1,000 trios and replicate the inference and dTDT procedures. Because of the large number of these settings , we attach the results as in supplement tables. A plot of the K values . Each blue dot corresponds to a/under that specific setting.We separate the simulation results into nine groups that are combinations of three disease models and three Because there are many factors interacting with each other, we will start with a general comparison of different models then look at the impact from one or two factors while constraining the others constant.K = 0.01) recessive disease model produces higher power compared to common (K = 0.1 or 0.2), co-dominant or dominant disease models. Meanwhile, high R value (0.9 or 1.0) also helps increase dTDT’s ability to detect signals. This is because in a rare recessive case, markers with high LD to the disease allele both parents are heterozygous and both transmit the recessive risk allele to their offspring. Our findings from the simulation validate what we already observed in the biological phenomenon.In general, reading the values of f11 (from 0.9K to 0.0) and increasing f12 & f22 (from 1.1K to 1.0), it shows a noticeable increasing trend as shown on the graphs. Meanwhile, when f22 is small ( = 1.1K), the blue dots are close to the bottom line on each graph. We note that in order to have enough power to detect the signals, we need to have relatively distinguishable penetrances (i.e. f cannot be too close to g) in the model. Indeed, there is no information when all three penetrances are equal to K.In the figure, each graph is broken down into four bins having R = 0.5, 0.7, 0.9 and 1.0. Interestingly, within each bin, when K = 0.01) the dots appear in clear clusters. Each cluster corresponds to a specific f11 value. Taking the top left graph (dominant with K = 0.01) as an example, f11 changes in the order of 0.9K, 0.7K, 0.5K, 0.3K, 0.1K and 0.0. Within each cluster, f12 & f22 increase in the order of 1.1K, 0.5, 0.9 and 1.0. This clustering holds true in the other two disease models (recessive and co-dominant) when K is small (K = 0.01) except some f11 valued clusters are missing because combinations with invalid p were excluded. In summary, the above observation indicates that f11 has a higher impact to the power than f12 & f22 do in a rare disease model. As the prevalence increases (K = 0.1 and 0.2), the clustering effect gradually disappears. In each R valued bin when K is large (0.1 or 0.2), though the penetrances are sorted in the same order as mentioned above, the dots represent certain continuity instead of clustering. This shows that, in a common disease model, f11 is not the only or the most effective factor as it is in a rare disease model. Other factors start to interact with each other. Especially when K = 0.2 and R = 1.0 (the fourth bin in the three graphs on the right), the dots appear in clear fan-shaped sectors. This irregularity can be partially explained by the sensitivity to randomness of the model under such setting, i.e., small changes of the parameters can have high impact on the results.When we generate the trios, we use a roulette wheel algorithm to assign SNPs to the children. This randomness is reflected on the graphs as the dots spread in some irregular patterns. Reading the graphs from left to right, we can see that with low prevalence , the same degree of lift in the penetrances will add more power and move/more quickly to its next level compared to the situation when R is low (such as 0.5). From another point of view, we can imagine these slopes as the (first) derivatives of a convex function in terms of R. On this convex curve, as R moves along to its rightmost end (increases), the derivative of the function increases and the function value (power) improves faster.We can further see how the penetrances differ by looking at the slope of the trend in each bin. Apparently as the value of p-value (P) = 3.4×10−6. A cluster of SNPs was found in a region of chromosome 11 with p-values ranging from 4.8×10−5 to 6.9×10−4. No single SNP showed genome-wide significance (5×10−8). In the following sections, we will compare our results from dTDT on COGA data with these findings from Edenberg et al’s work.In a recent work in a case-control study using GWAS data on the COGA sample, Edenberg et al MS EA group were extracted and used to build the trio-dosage pedigrees. 166 SNPs are found with p-values <10−4. This is compared to 93 SNPs at the same level in Edenberg et al’s paper p-values ≤10−5. However, further analysis with MH statistics does not yield consistent results with dTDT. This may be primarily due to the relatively small sample size used in dTDT and accuracy of the “–infer” program.To apply the dTDT on each SNP from COGA, we re-build the inferred pedigrees by extracting all trio combinations in which every child in a trio must be affected and at least one parent was genotyped on microsatellite markers. Because one family can have more than one affected child, the parents can be found in more than one trio. For instance, the family (FAM_ID 20059) as shown in MS and CIDRSNP data respectively, we compute the p-values of MH test based on calculations in 10P–log across all 23 chromosomes from the MH test. The most significant SNP (rs11583322) on chromosome 1 gives a p-value = 1.10×10−8 that meets the GWAS significance level. This SNP lies in the gene Serine/Threonine Kinase 40 (STK40) that connects pluripotency factor Oct4 to the Erk/MAPK pathway controls extraembryonic endoderm differentiation p-values <10−5 and their corresponding case-control study and dTDT p-values. From the table we can see that the p-value of each SNP in the MH test is approximately the product of p-values in the other two tests. However, SNPs with high rankings by p-values in the MH test do not systematically correspond to high rankings in the individual tests. The p-values in the dTDT share the highest variance (2.01×10−3) among the three tests because of the randomness introduced by the inference procedure as well as the difference in sample sizes across the tests. A larger sample size will likely increase the power and generate more robust test results. In total, we have 257 SNPs in 75 genes with p-values <10−4. 14 SNPs at the same level of p-values are found in replication of Edenberg et al’s study CSMD2 on chromosome 1, LZTS2 & PDZD7 on chromosome 10, and Gcom1 on chromosome 15. There are 34 vs. 11 SNPs that have p-values <10−5. Five SNPs across chromosome 1, 3, 9, 12 and 14 show p-values <5.1×10−7 which is more statistically significant than the case-control analysis by Edenberg et al p-values <10−5 (more than five such markers in one cluster) in genes EXOC6B, FTO, NCAM2 and PPEF1 on chromosome 2, 5, 21 and X, respectively.With dTDT and case-control analysis applied to the Linkageth column on Based on results from the MH test  in the simulation.With slight modification of the above technique, we can use our simulation to investigate how the dTDT is affected by population stratification. Simulating different populations to have different prevalences, we can choose two sets of trios using different allele frequencies. Applying the dTDT to this combined set of trios, we can test whether the power is lowered or heightened because of the prevalence difference within the populations. Since there is no information on phase of two markers in a trio, we have not introduced the recombination frequency  were found having p-values against the theoretical model distribution of expected p-values p-values from Case-Control and dTDT are along the diagonal line. We do not observe significant distortion, i.e., type I error, in both tests. On the other hand, the Q-Q plot of MH test lies above the diagonal line. As stated earlier, both Case-Control and the dTDT are not robust to the population stratification because of the dependence of allele frequencies in the populations. MH test is basically a combined statistic of these two tests. Though we have increased the sample size in the combined test, we have reason to believe that such sensitivity to population stratification has been inflated in the MH test. This is a tradeoff that we need to pay attention to in the future studies. However, this issue can be partially addressed by restricting accurate genotypes based on the imputation quality score (IQS) Dealing with errors).The Quantile-Quantile (Q-Q) plots are used in GWAS to assess the inflation of FPRs by comparing the distribution of observed 10  along the diagonal line in the Q-Q plots doesn’t mean these tests agree with each other. To investigate the concordance among these three tests, we rank the -log10  from the dTDT (or MH test) and pick the top 100 signals. Then we plot these values with their corresponding -log10  from the other two tests , SNP rs4903712 on chromosome 14 remained significant. This was the seventh most significant SNP from the MH test. As discussed above, there are several issues affecting the dependability of our test results. As we saw from the Q-Q plots, signals from MH test have been inflated because of double counting of the population stratification factor. On the other hand, the genotyping and imputation accuracy may be taken into account as well. To address these issues, we compute the IQS for the listed top SNP markers on −5 level due to a small sample size (data not shown).After correcting for multiple tests .According to Since the discovery of Mendel’s law, genetic research has been challenged to identify genetic variants that contribute to human diseases. Along with the development of genome sequencing technologies, there have been impressive progresses within the research community over the past decade. Numerous methodologies have been developed and many disease-associated genes have been reported As discussed, traditional case-control studies on GWA often include only unrelated individuals. By including family information in the study, we can expect an increase in power for linkage and association detection. On the other hand, because the traditional TDT requires complete genotypic information from a trio by measuring over-transmission of an allele from heterozygous parents to the affected offspring, it may be less useful in trio data like COGA where there are relatively few complete trios. To overcome these limitations, in this project, we extend the original TDT to the dTDT to accommodate dosage probabilities of a trio. The trio-dosage sets can be inferred through programs like MERLIN. Compared to a recent work from Edenberg et al, the dTDT shows increased power to detect association.Genotype inference allows us to evaluate the evidence for association at the genetic markers that are not directly-genotyped. It helps improve the power of individual scans and is of particular usefulness for combining information from different studies such as linkage and GWAS. However, the accuracy of genotype inference may be impaired for the following reasons. First, because datasets where subjects are genotyped on different platforms may have different genotyping error rates, when we combine these datasets, inference can be problematic. Second, genotype inference for large datasets based on a small amount of shared information may encounter too much uncertainty in the procedure. For a similar reason, SNPs with low MAF may also have a higher chance of being inferred inaccurately.On the other hand, a major advantage of the TDT is that it is not susceptible to population stratification. The dTDT is, however, sensitive in that it depends on the marker allele frequencies in the population. Because of this reason, when we combine results from both Case-Control and dTDT, the MH test potentially inflates errors due to population stratification. This can be noticed in the Q-Q plot as we present in In summary, as we inspect the reasons for having low concordance among the tests as well as poor replication from our test results to the experiment validation, we have the following conclusions:Because the linkage and GWAS data were genotyped on different platforms, they may carry different genotyping errors, which make it difficult to obtain genotype inference accurately; inference across these platforms can generate spurious associations;Due to a great sparsity in the combined dataset, a large number of the markers have to be inferred without sufficient support from the common markers, which introduces too much uncertainty in the inference;Because of possible population stratification, combining both the Case-Control and the dTDT to enhance the sample size may introduce false positive signals.As one solution, when we filter out poorly-inferred SNP markers using IQS, we are able to removes thousands of false positives that can be particularly useful for SNPs with low MAF and when datasets are genotyped on different platforms. However, the tradeoff is we have to exclude a good number of individuals from our database in order to meet such restriction. But this can always be an option when enough samples are available.Despite this area for methodological development, our work posits that the dTDT has considerable utility for linkage and association testing. By exploiting family data with inference and existing case-control information, the dTDT demonstrated here has opened a new window to possible routes for the integration of both population-based and family-based studies.To address the sensitivity issue due to population stratification, it is necessary to validate the SNP genotyping and perform a test such as the PTDT to validate results and use a program such as UNPHASED. This approach minimizes expense when the case/control sample is derived from an existing family study in which relatives have available DNA for typing. Moreover, we may implement additional application to other populations such as African Americans to compare findings with what we have from the European Americans. This will require extending the techniques described above.We may explore using more of the family data instead of only using SNPs. Other sources of information could be captured, such as the copy number variants (CNVs) Finally, the dTDT should not be limited to dosage probabilities from the inference programs only. As a perspective, the next-generation sequencing data will provide a challenge using the method developed in this paper. Similar methods can be used when pedigree members have a SNP chip, and a subset has sequence data. Despite the discordance and poor replication from our test results, we believe that linkage can help identify regions of interest in conjunction with association testing. Computational inference has helped us reduce the experimental cost in that we may only need to do sequencing on a limited number of family members. Keeping this in mind, we need to implement appropriate selection of the most informative families when we do genotyping. All these future directions shall be promising and have potential to inform the field.Table S1Dominant (K  =  0.01).(XLSX)Click here for additional data file.Table S2Dominant (K  =  0.1).(XLSX)Click here for additional data file.Table S3Dominant (K  =  0.2).(XLSX)Click here for additional data file.Table S4Recessive (K  =  0.01).(XLSX)Click here for additional data file.Table S5Recessive (K  =  0.1).(XLSX)Click here for additional data file.Table S6Recessive (K  =  0.2).(XLSX)Click here for additional data file.Table S7Codominant (K  =  0.01).(XLSX)Click here for additional data file.Table S8Codominant (K  =  0.1).(XLSX)Click here for additional data file.Table S9Codominant (K  =  0.2).(XLSX)Click here for additional data file.


Hundreds of new loci have been discovered by genome-wide association studies of human traits. These studies mostly focused on associations between single locus and a trait. Interactions between genes and between genes and environmental factors are of interest as they can improve our understanding of the genetic background underlying complex traits. Genome-wide testing of complex genetic models is a computationally demanding task. Moreover, testing of such models leads to multiple comparison problems that reduce the probability of new findings. Assuming that the genetic model underlying a complex trait can include hundreds of genes and environmental factors, testing of these models in genome-wide association studies represent substantial difficulties.We and Pare with colleagues (2010) developed a method allowing to overcome such difficulties. The method is based on the fact that loci which are involved in interactions can show genotypic variance heterogeneity of a trait. Genome-wide testing of such heterogeneity can be a fast scanning approach which can point to the interacting genetic variants.In this work we present a new method, SVLM, allowing for variance heterogeneity analysis of imputed genetic variation. Type I error and power of this test are investigated and contracted with these of the Levene's test. We also present an R package, VariABEL, implementing existing and newly developed tests.Variance heterogeneity analysis is a promising method for detection of potentially interacting loci. New method and software package developed in this work will facilitate such analysis in genome-wide context. Genome-wide association studies (GWAS) have been instrumental in identifying genetic variants involved in complex diseases. In GWAS, the relation between a trait of interest and genetic variation  is studied by assessing hundreds of thousands of polymorphisms in thousands of individuals. Several hundreds of loci for dozens of complex human diseases and quantitative traits have been discovered using GWAS .11 unique SNP pairs are to be tested. This amount of tests is equivalent to running a standard "direct effects only" GWAS 5 · 105 times.Though GWASs were successful in finding single loci associated with a trait, complex genetic models which include many interacting loci and environmental factors are of interest as they may help finding new loci and improve our understanding of the genetics of complex traits. A search for genetic interactions by direct analysis, in which all possible genetic models are examined, meets substantial computational and methodological difficulties. When millions of SNPs are considered, which nowadays has become routine in GWAS, testing for interaction for all possible pairwise combinations of SNPs becomes cumbersome requiring parallel computations using hundreds or thousands of CPU cores. Also, a large number of models has to be tested, resulting in multiple comparison problem, which weakens the statistical power and the possibility of new findings. For instance, if a simple interaction between two SNPs is considered in analysis of one million SNPs, approximately 5 · 10Thus, already the simple case of interactions between only two SNPs poses serious computational challenges. However, there is no reason why biology should not be more complex, involving more than two interacting variants. In general, a trait can be determined by a complex network of multiple interacting genes and other factors, including environmental ones. Statistical modeling of such complex network of interacting factors in genome-wide context would be a big challenge both methodologically and computationally. Prior information on loci, which are likely to be involved in a trait's control (e.g. genes in pathways implicated for specific trait) can help reducing the space of models to be tested but still does not solve the problem. For example, the protein pathway involved in Alzheimer's disease incorporates hundreds of genes. Each of them may include over 25-50 SNPs.Another approach to dissection of genetic interactions consist of identification of potentially interacting loci, with further search for factors which interact with these loci. For quantitative outcomes interaction of a SNP with an unknown factor can be discovered from the trait's distribution conditional on the genotype: it is expected that trait will have larger variance for an interacting genotype ,3. This Two groups ,3 demonsIn this work we present a method extending variance heterogeneity analysis to imputed genetic data. We also develop VariABEL - an R package implementing variance heterogeneity tests proposed previously and developed in this work.Here we describe existing variance heterogeneity tests and the newly proposed test, which is suitable for the analysis of imputed genetic data (subsection "Variance heterogeneity tests"). Next, we describe the setup of the simulations, which were used to study statistical properties of the new test (subsection "Simulations") and outline the details of implementation of our software (section "The VariABEL package").For measuring variance heterogeneity we have implemented two tests: Levene's test ,7 and thLevene's (the Brown-Forsythe) test is defined as:Zi = |yi - ỹig| is the deviation of the value of the trait of i-th individual, yi, who has genotype gi from the median value of the trait in individuals having that genotype, ỹgi; N is the total sample size, nj is the number of individuals with genotype j, k is the number of possible genotypes, j , and where T2, has an F distribution with df1 = (k - 1) and df2 = (N - k) degrees of freedom. In a case of large N, T2 is approximated well by the k - 1 = 2.Under the null hypothesis of variance homogeneity, the value of the test statistic, Genetic imputations routinely used in GWAS nowadays increase the power in the analysis of individual studies and also allow meta-analysis of the studies using different SNP arrays. In case of imputations the posterior probability of a genotype is estimated for each subject for a given SNP. Because standard variance heterogeneity tests assume that an observation should be known to belong to a certain group , they can not be directly applied to the imputed data.To allow for variance heterogeneity test for imputed SNPs we propose a simple procedure (SVLM) described below. It is known from elementary statistics that by definition the variance is:Y is a random variable, Var(Y) is the variance of Y, E[Y] and E[Y2] are expected values of the variable Y and Y 2 correspondingly. In our case Y is a trait. The variance of Y conditional on the genotype g is V (Y |g) = E[(Y - E[Y |g])2|g]. This means that for each genotype the variance is equal to the mean of the squared residual of the trait conditional on the genotype.where B alleles in the genotype. It is assumed that allele B is interacting with some quantitative factor, hence the variance of the trait is increasing as the number of B alleles, present in an individual's genotype, increases. Figure To explain this idea we provide Figure Technically, the SVLM method consists of two steps. First, a regression analysis is applied where the trait is adjusted for a possible SNP effect and other covariates. Second, a regression analysis is applied to the squared values of residuals obtained from the first stage, using the SNP as the predictor.To study Type I error and power of the SVLM test, we performed a simulations study. Similar to our previous work , we simuyi is the value of the trait for ith individual, μ is the intercept, βg is the direct effect of the SNP, βF is the direct effect of the interacting factor, βgF is the effect of interaction between the SNP and the factor, gi ~ B is a SNP, which is assumed to be binomially distributed with ng = 2  and PB ∈  . μF and variance ϵi is the random error, which follows a normal distribution with a zero mean and a variance of one. We assumed that the distributions of gi, Fi, and ϵi are independent. For our simulations, without loss of generality we can assume that μ = μF = 0, and where βg = 0. We studied four different frequencies of the interacting allele: 5%, 40%, 60% and 95%. Results for 40% are presented in the text below. Results for other frequencies are shown in Additional file Without loss of generality for both type I error and power the SNP effect was set to zero ϵi: one normal distribution and two types of χ2 distributions with degrees of freedom df = 1 and df = 5 respectively. We simulated data for 10000 individuals.Therefore additionally we have studied type I error of the SVLM test for allele frequencies 0.0005, 0.00075, 0.001, 0.002, 0.003, 0.004 and 0.005. As SVLM requires squaring of the trait's residuals, the presence of extreme values can affect the type I error and power of this test. To check this, we studied three types of distribution of the residual error term βF = βgF = 0). Twenty thousands simulations were performed.For studying Type I error effect of the factor and the effect of interaction term were both set to zero  and tested in R of version 2.13.1. The package is distributed under the GNU GPL license (v. 2.0 or later).http://www.r-project.org/). Installation is possible from R directly by running the command "install.packages("VariABEL")". Documentation is available as a part of the distribution and also on-line at the GenABEL project web-site (http://www.genabel.org). Developmental version of the package is available from the GenABEL project development pages (http://genabel.r-forge.r-project.org) located at R-forge [Stable version of the VariABEL package can be downloaded from the Comprehensive R Archive Network, CRAN  . Thus, the VariABEL can be used for regular GWAS as well.ϵi. The SVLM test had acceptable type I error for all of them: αnormal = 0.0471 ± 0.0015, p ≤ 0.05. We did not see any significant deviation from nominal type I error rate of 5% for allele frequencies 5%, 40%, 60% and 95%. To understand the minimum sample size in a genotypic group under which the type I error of SVLM test still stays at a nominal level we measured type I error for allele frequencies 0.0005, 0.00075, 0.001, 0.002, 0.003, 0.004 and 0.005. These allele frequencies correspond to number of heterozygotes in a sample 10, 15, 20, 40, 60, 80 and 100. For those frequencies the type I errors (with its standard errors) were 0.028 ± 0.001, 0.032 ± 0.001, 0.037 ± 0.001, 0.042 ± 0.001, 0.044 ± 0.001, 0.049 ± 0.002 and 0.045 ± 0.002 correspondingly. This suggests that SVLM test has correct type I error rate when sample size in one of a genotypic group is not less then 80, while for smaller values the SVLM test starts being conservative. Levene's test did not show significant deviation from nominal value of 5% under these extremely low sample sizes.As it was mentioned above, we studied three different distributions of ϵi). Figure χ2 distribution with df = 5, the power of the SVLM test and Levene's test are similar (panel B). In case of higher skewness the SVLM test has lower power than the Levene's test (panel C). This can be explained by the fact that Levene's test is known to be robust to the deviations from normality, while the SVLM test is in fact a regression analysis for which the outcome is supposed to follow a normal distribution.Figure Additional file As it is expected the power of both tests decreases when allele frequency decreases. The observation that SVLM's test power is affected by skewness more than the power of Levene's test stays true for all studied allele frequencies.The analysis by the SVLM test of 2543887 SNPs of 2715 subjects takes 46 minutes on one core of a Sun Fire X4540 Server with Quad-Core AMD Opteron Processor 2356.Genome-wide association analysis is currently a primary tool for identification of loci associated with complex human traits. Testing for association under complex genetic models involving multiple interactions represents methodologically and computationally challenging task.We and others have developed a method allowing testing of SNPs genome-widely for possible involvement into interaction ,3 via teAnalysis of genotypic variances can be of interest to medical research. Assuming that there is a certain genotype associated with high variance of, for instance, blood pressure, the subjects having this genotype can be at risk of having extremely low or extremely high blood pressure.In developing our method for analysis of variances using imputed data we have utilized the fact that the variance is, by definition, the expectation of squared values of the variable in case of zero mathematical expectation of this variable. This allowed us re-formulate the task of estimation and analysis of variances of the trait as a task of regression analysis of transformed trait. In this setting, methodological and computational tools developed for GWAS are applicable for the variance analysis.The most important advantage of the proposed method is the possibility to detect SNPs belonging to a complex genetic network with many interacting factors that is impossible to study with standard tools. These SNPs will show variance heterogeneity and using our method these SNPs can be detected without knowing all the factors involved into this network. To find the factors, which interact with the identified SNP, a follow-up analysis can be applied where interaction between the SNPs found in variance analysis and all other measured SNPs or environmental factors are tested. In a case of interaction with an unknown factor, the SNPs showing significant variance differences still can be used to improve the variance explained as shown in the example below.Consider a scenario in which SNPs, associated with a trait found in regular GWAS's, together explain a certain proportion of total trait's variance:B increases trait's variance: AA, AB, and BB. Assume that allele frequencies and the effects of the SNPs found in GWAS and which contributed into AA, AB, and BB of the interacting SNP. Then the proportions of explained variance for different genotypic groups are:where AA, AB, and BB, respectively, at the SNP identified by the variance analysis. Taking into account that AA compared to AB and BB, and higher in genotypic group AB compared to where  GRAMMAR  implemen GRAMMAR . In GRAMTo increase power of variance analysis by including the data from other studies the same approach as for regular GWAS can be used where the analysis is done for each cohort separately, followed by meta-analysis. The SVLM method can be used for discovering interacting SNPs following any of additive, dominant, recessive, over-dominant (where trait's variance among heterozygotes is increased), or genotypic models. In case of testing the additive variance model only, the SVLM test has maximal power in the case when the SNP follows true additive model and less power in case of dominant, recessive and over-dominant models. It is of interest to note that in case of over dominant model the power to detect interaction by the SVLM test is zero if the minor allele frequency (MAF) is 0.5 and increases with decreasing MAF. In a case when MAF is close to 0.5 Levene's test has higher performance.In this work we present further development of the method for detection of potentially interacting SNPs, extending it to the case of analysis of imputed SNPs. The method is based on testing of heterogeneity of trait's variance conditional on the genotype of locus being tested. We also present an R package, VariABEL, to facilitate for such analysis in genome-wide context. The package implements already existing variance heterogeneity tests, and the SVLM test developed in this work.Project name: VariABEL packageProject home page: http://www.genabel.org/packages/VariABELOperating systems: Linux, Mac OS X, WindowsProgramming language: R, C++Other requirements: R (≥ 2.13.0)License: GNU GPL (≥ 2)Any restrictions to use by non-academics: none except these posed by the licenseGWAS: Genome-Wide Association Study; MAF: Minor Allele Frequency; SNP: Single Nuclear Polymorphism, SVLM: Squared residual Value Linear Modeling.The authors declare that they have no competing interests.MS wrote the software, planned and carried out the simulation study and wrote the manuscript. NA, PE, CvD and YA planned the simulation study and wrote the manuscript. All authors read and approved the final manuscript.Power to detect variance heterogeneity induced by interaction, assuming Normal distribution of residual error. The file contains the figure describing dependency of power to detect variance heterogeneity induced by interaction and the effect of interaction, βgF, using Levene's (circles) and SVLM (triangles) tests. The residual error follows Normal distribution. Scenarios with different frequencies of interacting allele are given in Panel A - 5%, Panel B - 40%, Panel C - 60%, and Panel D - 95%.Click here for filePower to detect variance heterogeneity induced by interaction, assuming distribution of residual error. The file contains the figure describing dependency of power to detect variance heterogeneity induced by interaction and the effect of interaction, βgF, using Levene's (circles) and SVLM (triangles) tests. The residual error follows Click here for filePower to detect variance heterogeneity induced by interaction, assuming . The file contains the figure describing dependency of power to detect variance heterogeneity induced by interaction and the effect of interaction, βgF, using Levene's (circles) and SVLM (triangles) tests. The residual error follows Click here for fileF  as a function of the interaction effect (βgF). The file contains the figure showing the value of optimal effect of interacting factor F, βF, as a function of the effect of interaction, βgF for allele frequencies 5% (black), 40% (red), 60% (green) and 95% (yellow).Click here for file


Genome-wide association studies have been successful at identifying common disease variants associated with complex diseases, but the common variants identified have small effect sizes and account for only a small fraction of the estimated heritability for common diseases. Theoretical and empirical studies suggest that rare variants, which are much less frequent in populations and are poorly captured by single-nucleotide polymorphism chips, could play a significant role in complex diseases. Several new statistical methods have been developed for the analysis of rare variants, for example, the combined multivariate and collapsing method, the weighted-sum method and a replication-based method. Here, we apply and compare these methods to the simulated data sets of Genetic Analysis Workshop 17 and thereby explore the contribution of rare variants to disease risk. In addition, we investigate the usefulness of extreme phenotypes in identifying rare risk variants when dealing with quantitative traits. Finally, we perform a pathway analysis and show the importance of the vascular endothelial growth factor pathway in explaining different phenotypes. In disease association studies, the common disease/common variants (CDCV) model states that common diseases are caused by common variants with minor allele frequencies (MAFs) in the range of 1–5%. Recent studies suggest that although many more common disease susceptibility variants may still exist, they will likely have even smaller effect sizes and thus will be unlikely to explain most of the missing heritability for many of the traits ,3. ThereMany statistical methods have been developed to test association between common SNPs and disease traits, but these methods have low power for identifying rare variants in complex diseases because of the low frequencies and large number of such variants ,6. SeverIn this paper, we apply these methods to the simulated data sets provided by Genetic Analysis Workshop 17 (GAW17) to analyze the effect of rare variants on various phenotypes and evaluate the performance of each method by comparing the results with the true disease model. We also study the effect of trait dichotomization when dealing with quantitative traits. In addition, we investigate the importance of the vascular endothelial growth factor (VEGF) pathway to different phenotypes in the GAW17 data set.The simulated data set from GAW17 consists of 3,205 autosomal genes with 24,487 SNPs genotyped on 697 subjects. Three quantitative phenotypes  and the Affected Status phenotype are generated for each of the unrelated individuals, and 200 simulated replicates are available to us. In particular, each simulation contains 209 case subjects and 488 control subjects for the Affected Status phenotype and three normally distributed quantitative traits . In the disease model, Q1 is caused by SNPs in the VEGF pathway and Q2 is caused by SNPs related to cardiovascular risk and inflammation; the trait Affected Status is affected by SNPs from both biological processes. See Blangero et al.  for moreWe implemented three recently developed strategies for association testing with rare variants: the gene-based combined multivariate and collapsing (CMC) method of Li and Leal , the weiBecause the methods mentioned are designed for case-control studies, we make some adjustments in order to apply them to quantitative traits. First, we dichotomize the quantitative traits by treating the individuals in the upper quartile as case subjects and those in the lower quartile as control subjects and then apply the CMC, WS, and RB methods to these pseudo-case-control data. Another strategy that uses all the available data is to use the modified weighted-sum test suggested by Price et al.  in whichhttp://www.genome.jp/kegg/pathway.html) and in the NCI-pathway interaction database (http://pid.nci.nih.gov/) and found that 53 genes in GAW17 data sets participate in the VEGF pathway. We grouped the 344 rare SNPs  from these 53 genes and evaluated their effects on Affected Status and Q1.The true disease model suggests that Q1 and Affected Status are affected by genes from the VEGF pathway ; therefoA preliminary analysis indicates that the distribution of SNP frequencies is heavily skewed toward low frequencies. In particular, about 87% of SNPs have a MAF less than 0.05, and about 74% have a MAF less than 0.01. Motivated by the large proportion of rare variants, we decided to investigate their effects on different phenotypes even without prior knowledge of the true disease model.p-value and the relative rank of the gene among all genes in each method, using only the first simulated data set. We also obtained the power (or replicability) for each gene, calculated as the percentage of times the resulting gene-based p-value was less than 0.05 across the 200 replicates. After the true disease model was reviewed in GAW17, our results were compared with the true model. In Table We applied the gene-based CMC, WS, and RB methods for the case-control design and dichotomized quantitative traits; we used Price’s method for quantitative traits only. For each gene we computed the In our analyses, we set two different thresholds to define rare variants: MAF < 0.01 (T1) or MAF < 0.05 (T5). We note here that the results from T1 and T5 are consistent but using T5 tends to give higher power compared to T1 because the sample size (697) is relatively small and the disease model involves variants with MAF greater than 0.01. Therefore, for the sake of clarity, we report here and in Table p-values and their ranks from the WS, CMC, and RB methods highly agree with each other; this makes us confident about the true effect if we observe strong signals across different methods. Our analyses included either all the genes  harboring rare variants or only those genes  containing nonsynonymous rare variants. Among these genes, FLT1 is the easiest to identify. For Q1, FLT1 is statistically significant with p-values less than the Bonferroni-corrected significance level (~1.67 × 10−5) and ranks first among all genes in the first simulation. For Affected Status, FLT1 has a very low p-value, close to the Bonferroni-corrected threshold, especially when only nonsynonymous rare mutations are used. For Q2, disease gene VNN3 is top-ranked with all three methods.We observe that for case-control or dichotomized quantitative trait designs, FLT1 and KDR for Q1 and VNN3 for Q2, which may suggest that they are true positives.For quantitative traits, the two strategies (dichotomization or not) result in different risk genes being detected; in particular, dichotomization identifies more genes with higher power compared with Price’s method. This is partly because dichotomization likely eliminates rare disease variants contained only in individuals with modest phenotype values but it amplifies the signal for rare variants with strong effect that show up in extreme phenotypes. Genes identified by dichotomization tend to have high power because the statistical signal can be consistently detected after removing unidentifiable noise. Therefore selecting extreme phenotypes can sometimes be an alternative strategy in dealing with quantitative traits. On the other hand, these two strategies both detect the same disease genes: Overall, our results are encouraging on these simulated data. In particular, we successfully detected six true disease genes for Q1, five for Q2, and three for Affected Status in the top 50 most significant genes using only the first simulated data set.GOLGA1 for Affected Status, PPP1R14BP1 for Q1, and MAP3K8 for Q2, all have power greater than 50% (data not shown). This can happen for various reasons. Both sequencing errors and population stratification can create these replicable false signals, especially in these data sets, where the genotype data are fixed across simulations. Another reason might be long-range correlation between markers. Our investigation showed that certain consistent false positives are highly correlated with true causal SNPs and that it is difficult to eliminate this artificial linkage effect because of the fixed-genotype design across all simulations.Many of the disease genes are highly replicable in 200 simulations with power greater than 50% Table , indicatWe compared the results using only nonsynonymous rare variants with those using all rare variants and observed that nonsynonymous rare variants improve the rank of true disease genes and the power for most disease genes, especially for trait Q2 Table . This isp-value than using all variants does; this is consistent with previous gene-based analyses and emphasizes again that filtering nonsynonymous mutations is important for identifying influential rare SNPs.We collapsed the rare SNPs (MAF < 0.05) in the VEGF pathway and evaluated their effects on traits Affected Status and Q1 using different statistical strategies Table . The WS In this study, we applied three statistical methods to detect rare variants and successfully pinpointed several true disease genes containing rare SNPs in the GAW17 data sets. We started with the first data set and then made use of the remaining 199 replicates to evaluate the replicability of the discoveries in the first data set. This strategy agrees with the natural procedure when dealing with real data; namely, a first data set is used for discovery purposes, and additional independent data sets are used for replication purposes. We showed that using 5% as the MAF cutoff is better than using 1% in the GAW17 data set because the current sample size is small and disease variants occur at higher frequencies as well. Also, including only nonsynonymous mutations can substantially increase the signal. We also showed that selection of extreme phenotypic individuals can be a useful strategy for rare variant analysis with quantitative traits. Our analyses encountered the difficulty of numerous false positives, some probably resulting from sequencing errors or population stratification. Nevertheless, in real life without knowledge of the true model, the methods we investigated here can help us to detect rare disease SNPs. The consistency of results from different methods can be an indicator of true signal. The three methods performed similarly for the disease genes in these simulations. However, we did observe that the replication-based approach was the most powerful approach for large genetic regions such as pathways, followed by the weighted-sum and collapsing approaches.The authors declare that they have no competing interests.II and RF designed the study. II, RF, SHL and TZ performed the research. II, RF and CHH analyzed data. II and RF wrote the paper. All authors read and approved the final manuscript.


Applying a statistical method implies identifying underlying (model) assumptions and checking their validity in the particular context. One of these contexts is association modeling for epistasis detection. Here, depending on the technique used, violation of model assumptions may result in increased type I error, power loss, or biased parameter estimates. Remedial measures for violated underlying conditions or assumptions include data transformation or selecting a more relaxed modeling or testing strategy. Model-Based Multifactor Dimensionality Reduction (MB-MDR) for epistasis detection relies on association testing between a trait and a factor consisting of multilocus genotype information. For quantitative traits, the framework is essentially Analysis of Variance (ANOVA) that decomposes the variability in the trait amongst the different factors. In this study, we assess through simulations, the cumulative effect of deviations from normality and homoscedasticity on the overall performance of quantitative Model-Based Multifactor Dimensionality Reduction (MB-MDR) to detect 2-locus epistasis signals in the absence of main effects.t with constant or non-constant phenotypic variances. All data are analyzed with MB-MDR using the built-in Student’s t-test for association, as well as a novel MB-MDR implementation based on Welch’s t-test. Traits are either left untransformed or are transformed into new traits via logarithmic, standardization or rank-based transformations, prior to MB-MDR modeling.Our simulation study focuses on pure epistasis models with varying degrees of genetic influence on a quantitative trait. Conditional on a multilocus genotype, we consider quantitative trait distributions that are normal, chi-square or Student’s t-tests are generally lower than those for MB-MDR with Student’s t-tests. Trait transformations involving ranks tend to lead to increased power compared to the other considered data transformations.Our simulation results show that MB-MDR controls type I error and false positive rates irrespective of the association test considered. Empirically-based MB-MDR power estimates for MB-MDR with Welch’s t-tests as internal tests for association.When performing MB-MDR screening for gene-gene interactions with quantitative traits, we recommend to first rank-transform traits to normality and then to apply MB-MDR modeling with Student’s  The search for epistasis or gene-gene interaction effects on traits of interest is marked by an exponential growth. From an application point of view, these searches often focus on candidate genes or pathways. The reasons for this are often logistic ones: First, genome-wide screening for epistasis requires large sample sizes to ensure sufficient power detection, which may only become available when having access to consortia data. Second, exhaustive genome-wide epistasis screenings assumes the availability of sufficient computer power and an adequate infrastructure to store and analyze the data, as well as to store and process the analysis results. From a methodological point of view, searches for epistasis effects are performed with the goal in mind to develop methods that can narrow the gap between statistical and biological epistasis. To date, several epistasis detection approaches exist, each addressing differential aspects of the underlying theoretical epistasis model, and with different performances in terms of Type I error control or power detection . AlthougOne of the pioneer methods used in the context of dimensionality reduction and gene-gene interaction detection is the Multifactor Dimensionality Reduction (MDR) method, initially developed by Ritchie et al. . MDR offt-test is identical to that of an ANOVA computed for two groups; the t-statistic is the square root of the F-statistic used in ANOVA. Hence, in principle, the validity of MB-MDR epistasis results may depend on whether or not ANOVA assumptions are met, which warrants further investigation.The ‘modeling’ part in MB-MDR arises from the need to embrace parametrics when adjusting for lower-order (main) effects within a regression framework. The necessity of lower-order effects corrections in quantitative MB-MDR analyses has been discussed elsewhere . In pureMany authors have studied the effects of model violations in regression settings in general and have suggested alternative strategies when violations cannot be remediated ,8.t-tests are scarcely affected by non-normality of population distributions , as was shown before [t-test performed on the ranks of the original outcome measurements and its p-values are mathematically identical to Kruskal–Wallis one-way analysis of variance by ranks [Due to the aforementioned link between MB-MDR and ANOVA, we are particularly interested in violations regarding the latter. The three main ANOVA assumptions are: 1) the observations are independent, 2) the sample data have a normal distribution within factor levels  and 3) the dependent variable’s variances within each factor level are homogeneous (homoscedasticity) . Generalns e.g, ,11). Nev. Nevt-tens  affects the choice for a particular test as well. Gibbons and Chakraborti  emphasiznd Zumbo  suggest  in Danh  that theSzymczak  and RupaSzymczak  suggest Szymczak  has arguSzymczak .t-test/ANOVA models . ,28. t-te2) between individuals is varied as 0, 5 and 10%.We simulate 18 two-locus settings of an epistasis model following , each set; the same distribution is assumed for each of the 9 levels of the two-locus genotypes derived from SNP1 and SNP2 combined. To investigate the MB-MDR cumulative effects of heteroscedasticity, we consider for every aforementioned setting, constant and non-constant phenotypic variances according to the following scenarios.To assess the effect of violated normal trait distributions, we consider trait distributions that are, apart from normal, also chi-squared or Student’s We simulate 9 variances from U ,10, one Quantitative traits are generated from a central chi-square distribution with 2 degrees of freedom (df), inducing a constant trait variance for every two-locus genotype combination. To simulate settings with heteroscedasticity, non-central chi-square distributions are used, df randomly selected from the uniform distribution U . The nont-distribution with 3 degrees of freedom. Non-equal phenotypic variances are introduced by generating data for the 9 multilocus genotype combinations from the uniform distribution U [We consider quantitative traits from a bution U .t-test to make such group comparisons for quantitative traits. This implementation is based on simulation studies that assumed traits to be normally distributed with equal genotypic variances for each of the multi-locus genotype combinations corresponding to a bi-allelic functional SNP pair [t-tests are always pooled variance t-tests. A novel implementation allowing unequal group variances based on the Welch’s t-test (WT) for two-group comparisons is included in the MB-MDR software version 2.7.5. For a graphical representation of the quantitative MB-MDR method, we refer to Figure Model-Based Multifactor Dimensionality Reduction (MB-MDR) is a data mining technique that enables the fast identification of gene-gene interactions among thousands of SNPs, without the need to make restrictive assumptions about the genetic modes of inheritance. The most commonly used implementation of MB-MDR involves testing one multi-locus genotype cell versus the remaining multi-locus cells . By construction, this procedure creates two (possibly highly) imbalanced genetic groups that subsequently need to be compared in terms of mean or median trait differences. To date, MB-MDR has used Student’s SNP pair ,29. The t-test (ST) as well as the novel Welch’s t-test (WT) implementation to assess power and type I error. Prior to MB-MDR submission, original traits are either left untransformed or transformed into new traits via logarithm transformations (Log), standardization transformation (Stz) or via rank-based transformations. The latter transformations involve the assignment of absolute ranks to all available trait measurements in a serially increasing order (Rank), after which the ranks are transformed to normality (Rtn). Data transformations are conducted in R.2.15.0 [t- or non-parametric Mann-Whitney U- tests of association. Overall significance is assessed at 5% level of significance after correction for multiple testing via the permutation-based step-down maxT multiple testing correction of [All simulated data are analyzed with MB-MDR, with Student’s R.2.15.0 . We are ction of  . PermutFigure t) comparing a single multi-locus genotype  with the 8 remaining ones, Figure H-cells versus {L,O}-cells, and one involving L-cells versus {H,O}-cells). Comparison of Figure For the same scenarios as before, yet using all SNP pairs, and the 999 permutations F-statistics data, we create quantile-quantile plots (qq-plots) for a theoretical F distribution with  degrees of freedom. Here, n=500 is the number of individuals in a dataset and g=2 is the number of groups (i.e. 1 cell versus 8 remaining cells). Note that since no missing data were considered, all theoretical distributions for association tests within MB-MDR, whatever SNP pair is considered, should be F1,498). Whereas Figure 98. Where2=0  and false positive rates corresponding to g2=0.05 and 0.1 (scenarios of epistasis in the absence of main effects). We observe that, irrespective of the original trait distribution and whether or not a data transformation preceded MB-MDR analysis, the estimated rates satisfy Bradley’s [Table radley’s  liberal 2, i.e., with increasing proportion of epistatic variance to total trait variance. MB-MDR analysis with Welch’s t-test has generally lower power than MB-MDR with the Student’s t-test. This power loss is most severe for normal data. A (moderate) power gain is observed for settings where traits are t-distributed, variance homogeneity is present, epistatic variance is 10% and data are either left untransformed or are log-transformed prior to MB-MDR analysis. Parametric Student’s t-tests with the original trait measurements lead to reduced overall MB-MDR power when trait distributions deviate from normality. For non-normally distributed traits, there is a tendency for MB-MDR with Student’s t applied to rank-transformed data to outperform other MB-MDR analysis approaches . A worthy competitor is MB-MDR with Student’s t after rank-transforming original traits to normality. The differences in power performance between MB-MDR using untransformed traits or transformed traits are the largest for rank-based transformations compared to logarithmic transformations. No significant differences are observed between empirical power estimates derived from MB-MDR analysis on untransformed traits compared to those analyses based on trait standardization transformations (results not shown).MB-MDR empirical power estimates for correctly identifying the causal epistatic SNP are given in Table p-values, multiple testing corrected, as output by the MB-MDR software), one for each data set generated under a particular simulation setting , is given in Figure t is considered. Results are depicted for scenarios where the original trait data are derived from a normal (symmetric) or from a chi-squared (non-symmetric) distribution, and then subjected to different data transformation strategies. The scatter plot matrices of Figure p-values to be generated after rank-based data transformations compared to other type of transformations, including the identity transformation . This tendency becomes more extreme for chi-square distributed traits with non-equal variance (Panel D). Here, it becomes apparent that rank-transformation generally leads to larger p-values as compared to rank-transformations to normality. For settings where traits are chi-squared distributed and variance homogeneity is present, the scatter plots of Figure A graphical representation of the 1000 MB-MDR epistasis test results for the causal SNP pair (Proposed data mining methods for epistasis detection are seldom thoroughly discussed in terms of their underlying (model) assumptions and their effects on overall power or type I error control. For instance, another well-known data dimensionality reduction method for quantitative traits   is basedt-tests subsequently creates two groups, where one group refers to one multilocus genotype and the other to the remaining multilocus genotype combinations. Internally, 2-group comparison tests are performed so as to assign a “label” to each multilocus genotype. This procedure naturally creates highly imbalanced groups, with potentially extreme cases of heteroscedasticity. Although Welch’s test is designed to give a valid t-test in the presence of different population variances, Welch’s t-test combined with MB-MDR shows no improved power over the Student’s t-test for scenarios with unequal variances, even for normally distributed traits , MB-MDR with association It is well-known that parametric methods have improved statistical power over non-parametric methods when all parametric model assumptions are valid ,38. Whent association testing. This can be understood by noting that the ranks, which are computed for the pooled set of all available quantitative trait measurements, in general reduces the influence of skewness and deviations from normality in the population distribution [t-test on original traits, a small power loss is observed. Goh and Yap [t-testing, deviations from normality seem to be more influential to the power of an MB-MDR analysis with Student’s t than deviations from homoscedasticity  to avoid potential false positives (or redundant interactions) due to highly correlated SNPs. No such observation was previously made for dichotomous traits. For dichotomous traits, MB-MDR uses a score test, in particular, the Pearson’s chi-squared test. This test is known to be affected by unbalanced data, or sparse data, as is the case for rare variants [In many of our practical applications though, we observed a tendency of increased numbers of significant epistasis results with MB-MDR applied to quantitative traits, even after SNP pruning  show that subset pivotality is likely to be violated in real-life settings, giving rise to inflated error rates in the presence of multiple epistasis signals.(work in progress). Note that the standard bootstrap method provides the asymptotically correct null distribution for multiple testing and does not require the subset pivotality condition given in Westfall and Young [In real-life applications, we do not know nd Young . The invScale transformations are quite common as remedial strategies to meet statistical testing assumptions. However, since the optimal scale transformation is often based on theoretical motivations or statistical convenience, it often leads to new constructs that are hard to interpret or are biologically meaningless. Another concern related to implementing scale transformations is that non-additive signals may be removed as a direct consequence of such transformations prior to analysis .Our results confirmed that rank-based transformations are generally most powerful when quantitative traits are non-normally distributed. Rank transformations serve as a bridge between non-parametrics and parametrics . They nat-tests for association, and that this influence is primarily driven by the sparser multilocus genotype combinations. Improved power can be obtained by pre-analysis data transformations. MB-MDR permutation-based maxT correction for multiple testing keeps type I error and false positive rates under control, since in all considered simulation scenarios, the assumption of subset pivotality of the maxT permutation strategy was plausible.In this study, we assessed the performance of MB-MDR in terms of power and familywise error rate, with different choices of parametric and non-parametric association tests, in the absence or presence of trait transformations. We observed that non-normally distributed traits can affect the final test statistics of MB-MDR with classical t test as preferred association test. This practice will not only guarantee adequate type I error control, but will also offer an optimal power performance under a wide variety of data applications.When performing MB-MDR screening for gene-gene interactions with quantitative traits, we recommend to rank-transform traits to normality prior to MB-MDR analysis with Student’s All MB-MDR association tests discussed in this study are implemented in the MB-MDR software (version 2.7.5), which is available upon request.The authors have declared that no competing interests exist.JMMJ and KVS designed the analysis, simulated data, performed the analysis and drafted the manuscript. FVL contributed to software related issues. ESG contributed to the design of the analysis. All authors read and approved the final manuscript.Group comparison test maintaining adequate Type 1 error control, when group sizes are equal. Legend: When several tests are listed, they are listed from most (top) to least (bottom) powerful. The tests in a square box and blue font should be avoided in MB-MDR due to reasons mentioned next to them.Click here for fileQq-plots of observed squared Student’s t- test values for association between the multi-locus genotype combination cell 2-2 versus the remaining pooled multi-locus genotypes, for normal and chi-squared trait distributions or non-transformed and rank-transformed to normal data. For each setting, one replicate with epistatic variance 10% is considered and F-statistics are pooled for all SNP pairs over the 999 permutations. A generated F-distribution according to F is taken as the reference.Click here for file