Amongst the new findings provided by GUESS, we revealed a strong association of SORT1 with TG-APOB and LIPC with TG-HDL phenotypic groups, which were overlooked in the larger meta-GWAS and not revealed by competing approaches, associations that we replicated in two independent cohorts. Moreover, we demonstrated the increased power of GUESS over alternative multi-phenotype approaches, both Bayesian and non-Bayesian, in a simulation study that mimics real-case scenarios. We showed that our parallel implementation based on Graphics Processing Units outperforms alternative multi-phenotype methods. Beyond multivariate modelling of multi-phenotypes, our Bayesian model employs a flexible hierarchical prior structure for genetic effects that adapts to any correlation structure of the predictors and increases the power to identify associated variants. This provides a powerful tool for the analysis of diverse genomic features, for instance including gene expression and exome sequencing data, where complex dependencies are present in the predictor space.Genome-wide association studies (GWAS) yielded significant advances in defining the genetic architecture of complex traits and disease. Still, a major hurdle of GWAS is narrowing down multiple genetic associations to a few causal variants for functional studies. This becomes critical in multi-phenotype GWAS where detection and interpretability of complex SNP(s)-trait(s) associations are complicated by complex Linkage Disequilibrium patterns between SNPs and correlation between traits. Here we propose a computationally efficient algorithm (GUESS) to explore complex genetic-association models and maximize genetic variant detection. We integrated our algorithm with a new Bayesian strategy for multi-phenotype analysis to identify the specific contribution of each SNP to different trait combinations and study genetic regulation of lipid metabolism in the Gutenberg Health Study (GHS). Despite the relatively small size of GHS phenotypes in large population cohorts allows multi-trait studies. However, these studies are limited by the lack of flexible models integrated with efficient computational tools for genome-wide multi SNPs-traits analyses. To overcome this problem, we propose a novel Bayesian analysis strategy and a new algorithmic implementation which exploits parallel processing architecture for fully multivariate modeling of groups of correlated phenotypes at the genome-wide scale. In addition to increased power of our algorithm over alternative Bayesian and well-established non-Bayesian multi-phenotype methods, we provide an application to a real case study of several blood lipid traits, and show how our method recovered most of the major associations and is better at refining multi-trait polygenic associations than alternative methods. We reveal and replicate in independent cohorts new associations with two phenotypic groups that were not detected by competing multivariate approaches and not noticed by a large meta-GWAS. We also discuss the applicability of the proposed method to large meta-analyses involving hundreds of thousands of individuals and to diverse genomic datasets where complex dependencies in the predictor space are present. This paper builds upon recent developments in Bayesian Variable Selection (BVS) to propose a novel strategy for studying the association between large sets of predictors  and groups of correlated traits . Such data commonly arise in Genome-Wide Association Studies (GWAS), where a large range of continuous phenotypes are recorded together with hundreds of thousands genetic markers Our strategy exploits the advantages provided by two approaches used in genetic association studies: firstly, the use of BVS to go beyond “single SNP analyses” in GWAS The recently proposed piMASS algorithm implements a BVS strategy for genome-wide association analysis of single continuous phenotypes with a novel prior specification for the variance of the regression coefficients. The implementation of piMASS is based on a single chain Monte Carlo Markov Chain (MCMC) algorithm tuned to analyse a single response, with the aim of demonstrating the feasibility of BVS in a model space with many predictors whilst showing the benefits of considering multivariate SNPs models and model uncertainty. However the specific proposal density used in the MCMC and implemented in piMASS cannot be extended easily in a multi-phenotype setup.http://www.bgx.org.uk/software/guess.html and will soon be available on CRAN.Our algorithm, GUESS, also considers BVS for such a large model space through an Evolutionary Stochastic Search algorithm The SNPTEST package incorporates a Bayesian measure of association through the computation of a BF to quantify the evidence for association between a single explanatory variable and one or several continuous phenotypes. The benefits in terms of interpretability of using BFs rather than frequentist p-values in GWAS have been discussed in a number of papers p, small n” framework. These techniques tend to shrink the regression coefficients towards zero in order to select a sparse subset of covariates and provide better prediction performance. Such methods include, among others: LASSO Penalized regression methods have been proposed to improve Ordinary Least Squares, which often do poorly in both prediction and interpretation, and is not applicable in the “large Recently, the LASSO-type approach has been successfully applied to GWAS Building on well-established dimension reduction techniques, Sparse PLS (SPLS) While both penalized regression and SPLS approaches offer solutions for multivariate GWAS, their use requires a preliminary calibration of the penalty parameters which directly affects the number of selected variables, the value of the regression coefficients and therefore the statistical performances of the models. Calibration procedures usually involve the minimization of the mean square error of prediction through V-fold cross validation. Based on the publicly available implementation of these algorithms, such procedures become computationally expensive when GWAS data are analyzed . Moreover, none of the available implementations of the aforementioned algorithms provide a measure of uncertainty of the SNP(s)-trait(s) associations. While resampling techniques could be employed Beyond the methodological and computational advances of GUESS, one novel aspect of our method is the analysis strategy for groups of correlated phenotypes. This is illustrated in a study of a group of traits linked to lipid metabolism from GHS, where five lipid-related parameters Apolipoproteins A1 (APOA1) and B (APOB), HDL-cholesterol (HDL) and LDL-cholesterol (LDL) and Triglycerides (TG), are measured in 3,175 unrelated individuals As illustrated in 10(RBF) allows to identify the strongest SNP-trait(s) associations and thus better characterise the biological function of the SNP on the associated trait(s).Our strategy is to run GUESS on the phenotypic groups at the top of each tree and on all derived subsets of traits. To compare the results between the different branches of the trees, we propose a new measure for SNP-trait(s) association, the Ratio of Bayes Factors (RBF) , to pinpoint the specific contribution of each SNP to different combinations of traits. For each SNP, by ranking the strength of association with phenotypic groups, the logIn this study, we propose an efficient algorithm that combines the best features of genome-wide multi-SNP analysis with a fast and efficient algorithmic implementation based on Complete Unified Device Architecture (CUDA), which is extended to the analysis of multiple traits. A distinctive benefit provided by GUESS is the ability to perform a fully Bayesian analysis in an ultra-high dimensional model space and to select the best set of SNPs that predict the joint variation of several traits, which can provide direct insights into the polygenic regulation of multiple phenotypes.SORT1 (rs629301), APOB (rs1469513), GCKR (rs780094), LPL (rs336), APOA5 (rs964184), LIPC (rs261333), CETP (rs247617), APOC1 (rs4420638). The only gene not detected by our approach in any combination of phenotypes was LDLR. This is most likely due to the lack of genotype data covering the 5′UTR of the gene where the genetic associations were previously detected (data not shown).Despite the relatively modest sample size of the GHS, we were able to recover eight out of the nine top loci associated with combinations of blood lipid phenotypes that were identified by a large meta-GWAS of blood lipids in >100,000 individuals SORT1 which encodes sortilin, an intra-cellular receptor involved in the processing of APOB-containing particles and modulating hepatic secretion of VLDL, the lipoproteins which have the highest content of TG SORT1, our joint multi-trait analysis (including APOB) enhances the identification of the causal variant in this relatively small sample.The multiple phenotypes approach allowed us to detect SNPs involved in combinations of traits that would have been missed by single trait analysis. For example, LIPC which was detected in the TG-HDL combination  and the descriptions of SNPTEST and piMASS analysis are presented in Material and Methods.10(BF)>5) ZNF259 to SIK3) with TG disappears after including rs964184 in the model, confirming the results obtained with GUESS. Figures S3A–S3B summarise the comparison between GUESS, SNPTEST and piMASS for all the single trait analyses where, for each SNP, the genome-wide BFs of SNPTEST-GUESS and piMASS-GUESS algorithms are plotted. Overall SNPTEST is not able to separate clearly primary/secondary associations from the large bulk of SNPs . There iLPL locus) and rs1469513 (APOB) are not included in the list of SNPs with log10(BF)>5, a conventional threshold adopted for selecting significant SNPs SORT1 with TG-APOB and LIPC with TG-HDL phenotypic groups to illustrate the added value provided by multi-trait analyses to uncover common genetic regulation underlying correlated phenotypes. To replicate both the genetic association and the order of association between the causal SNPs and the phenotypic groups we have used a two-step procedure: (1) identification of the most significant variant associated with TG-APOB and TG-HDL in each independent cohort and (2) investigation of the order of association between the variants detected in (1) and the branches of the two trees in the candidate regions.To demonstrate how GUESS can provide useful insights into new genetic associations with multi-phenotypes, we carried out a replication study in the Copenhagen City Heart Study (CCHS) a priori expected model size (number of true associations) and standard deviation such that the prior model size is likely to range from 0 to 3. LIPC gene, is not present in the CardioMetabochip 2r and D′ level equal to 0.582 and 0.979 between rs261332 and rs8034802 respectively, and 0.838 and 0.982 between rs261332 and rs1077834 respectively, in populations of European ancestry .In the first step, we selected a 2Mb region centred at each identified variant (rs629301 and rs261333) and ran GUESS in each region with an adapted specification of the In the second step, we investigated whether we would find similarities between the order of association obtained previously between the causal SNPs and the phenotypic groups  and thatThe real data analysis shows that SNPTEST has good power to detect the main variants but it includes several additional SNPs, possibly increasing the number of false positives. Using 273,294 SNPs from the GHS study  we carried out two simulation studies for single and multiple traits to quantify the power of SNPTEST and GUESS. In the multiple traits scenario, we also tested the performance of non-Bayesian multiple traits algorithms MANOVA T, T, T, respectively), but we adjusted the error variance of each trait such that the expected proportion of variance explained is not greater than 5%. Given the effects and error variance of each trait, we simulated 20 replicates using a Normal matrix-variate distribution We simulated a group of three traits choosing four chromosomes  and, for each of them and in each replicate, we selected at random without replacement two SNPs. The number of SNPs selected reflects the average number of associations (7.6) found in the multiple traits real data analysis. The effects of the SNPs on the three traits were fixed  curves in For the second simulated scenario, the ROC curves are depicted in The computational time for GUESS for both multiple traits scenarios and 55000 sweeps is on average around 84 hours while MLASSO and SPLS  took about twice and 12 times more CPU time than GUESS, respectively.Similar results are obtained when SNPTEST and GUESS are tested on a single trait. T ) were used to simulate the trait, adjusting the error variance such that the expected proportion of variance explained was not greater than 5%. The single-trait scenario allows us to also compare GUESS with piMASS. Since in As large scale GWAS and meta-analyses of multiple continuous phenotypes are becoming increasingly common, there is a mounting need to develop models and computationally efficient algorithms for joint analysis of multi-SNP and multi-phenotype data. Current state-of-the-art Bayesian approaches have limitations either in the analysis of one SNP at a time n = 3,175) as compared with recent meta-GWAS of blood lipids n>100,000 individuals), we were able to recover eight out of the nine previously identified top associations. In particular, we were able to elucidate the associations between the SORT1 gene and the TG-APOB phenotypic group and uncover the association of LIPC with the TG-HDL group, which would have a low threshold of evidence if an alternative GWAS single SNP Bayesian method was used. By simply contrasting p-values for the four single traits considered and ranking them, Teslovich et al. LIPC and TC as the second associated trait. Our new finding of the association of LIPC with multiple traits rather than with a single phenotype is supported by recent data We integrated the GUESS algorithm with a new strategy for multiple traits analysis and applied this to study lipid metabolism in the Gutenberg Health Study (GHS). Despite the relatively small sample size of the GHS , are influenced by complex LD structures in the SNP data, and as a consequence have increased false positive association rates. The latter complicates, and often masks, the identification of secondary variants that are truly associated with multiple correlated traits. In contrast, the ability of GUESS to separate causal SNPs from correlated SNPs facilitates the application of empirical FDR procedures to declare robustly associated SNPs, which improves the reproducibility of results provided by GUESS.http://www.oxford-man.ox.ac.uk/gpuss/) is fast advancing in this area and our modular algorithm will be able to benefit from these developments. One important factor in the processing speed is the number of subjects involved in the analysis. Large meta-analyses nowadays often involve hundreds of thousands of subjects and running GUESS with such a large number of individuals will be relatively slow even with new GPU implementations in the future. On the other hand, it will be feasible and relatively straightforward to use Bayesian evidence synthesis methods Our implementation of BVS for high dimensional genome-wide data was made possible using the parallel computing power of the GPU interface and accelerated linear algebra libraries. In this paper we demonstrated that, exploiting the power of GPU processing, it is now feasible to run sophisticated Bayesian search algorithms in very high dimensional spaces, opening the path towards more complex model searches that might include interaction terms. On-going work in several bioinformatics and statistical groups .htpp://www.abbottdiagnostics.de). APOB is the primary apolipoprotein of LDL whereas APOA1 is the major protein component of HDL. Genotyping was performed using the Affymetrix Genome-Wide Human SNP Array 6.0 and the Genome-Wide Human SNP NspI/StyI 5.0 Assay kit. Genotypes were called using the Affymetrix Birdseed-V2 calling algorithm. SNPs with a Minor Allele Frequency (MAF)<0.01 or deviating from Hardy-Weinberg equilibrium  were excluded. Only autosomal SNPs were considered for analysis.More details about the GHS study are provided in -T20), 100 iterations of the EM algorithm for each random start (-C100), no haplotype estimation (-H-4), without the determination of the number of clusters (-K1).Missing values for each of the 22 autosomal chromosomes were imputed using FastPhase 2>0.80 level using an in-house method similar to To reduce the number of SNPs prior to analysis, we performed tagging at rn = 8,261) were selected based on the National Danish Civil Registration System to reflect the adult Danish population aged 20–100 years. Data were obtained from a questionnaire, a physical examination, and from blood samples including DNA extraction at the 1991–94 examination. A lipid profile was measured using standard hospital assays and genotyping was performed using customised version of the Illumina CardioMetabochip SORT1) and rs261333 (LIPC) comprising 543 and 204 SNPs, respectively.The Copenhagen City Heart Study We also analyzed 4,663 subjects of European descent from the Data from an Epidemiological Study on the Insulin Resistance syndrome (DESIR) cohort. More details about this study are available in The GUESS implementation extends the original ESS++ code http://developer.nvidia.com/category/zone/cuda-zone). CUDA is a parallel processing architecture that utilizes the processing power of the many processors present on a GPU, allowing significant performance increases for many mathematical operations and algorithms. By rewriting code in CUDA C/C++ parts of the algorithm can be redirected to the GPU rather than the CPU, often greatly speeding up a typical run http://www.culatools.com/), which is freely available to academic users, directly replacing the GNU Scientific Library (http://www.gnu.org/software/gsl) versions of the relevant linear algebra routines present in the ESS++ code, with CUDA C/C++ equivalents from this library.As indicated by its name, the novel implementation involves the use of Graphical Processor Unit (GPU) technologies, specifically using NVIDIA's Complete Unified Device Architecture (CUDA), Beyond the primary extension to ESS++, GUESS also implements a slight difference in the Metropolis-Hastings move type of the underling algorithm see . In partWhilst the change to GPU based linear algebra routines marks a significant performance improvement, even with these changes in place, attempting to evaluate the marginal likelihood for a model with many variables can be prohibitively slow. Because we put a strong penalty on such models through the prior on the number of predictors in the model, they are typically very rarely visited by the unheated chain in the transient phase of the algorithm, when the posterior is being explored. However, in the burn-in phase of the chain, or for the heated chains, such models might be visited more frequently.To prevent inefficiency in the burn-in phase and allowing the successful exploration of the posterior density, we truncate the prior on the number of variables in the model to exclude any models with too many variables. This truncation leads to a re-normalization of the posterior distribution, but as the normalization constant is not required in the acceptance ratio of the affected MCMC moves , in practice, the algorithm rejects any proposed moves to any model with more than the permitted number of variables.T) is set by the user through an additional parameter (F), through the relation T = E+F×S, where E is the expected value and S is the standard deviation of the prior model size pγ. Given the very large number of predictors (SNPs) in a GWAS and the Central Limit Theorem approximation of a binomial distribution already for moderate values of F, for instance F>3, Pr(pγ>T) = 1−Φ(F)≈0, so that the truncation has a negligible effect. The space of candidate models is reduced from 2p to The truncation  and standard deviation of the model size, and F = 7, meaning the prior model size is likely to range from 0 to 56 with a maximum model size of T = 104. In this set-up, given the level of sparsity and the number of predictors , the average prior probability π that a SNP is truly associated with the phenotype is 7.32×10−5 which is well inside the range of the prior probability suggested by g. The analysis was performed on a HPC cluster computer with a 2.8 GHz Dual-Core Xeon processor and an NVidia Tesla C1060 GPU with 8 Gb of RAM. Average computational times for the single- and multi-trait analysis were 252 and 229 hours, respectively , i.e. the most supported multivariate models ranked according to their Model Posterior Probability (MPP). For each multivariate model visited during the MCMC, the log-Posterior  is available and, for each unique model visited, the MPP is equal to the renormalized log-Posterior . See p-values as the proportion of permuted datasets for which the MPPI exceeds the MPPI for the observed data, the permutation strategy allows us to define the MPPI threshold at a specific empirical FDR level. Specifically, for each group of phenotypes, we compute the MPPI for the observed data and, based on the same prior specification and the same parameters for the GUESS algorithm, the MPPI for artificial datasets created by permuting three times the rows (subjects) of the observed traits. Overall for each group of traits 819,882  observations from the null distribution were obtained using this procedure. The MPPI threshold is then defined as the MPPI level for which the ratio between the number of declared associations in the shuffled datasets and the observed dataset is not greater than a specified FDR level. Since the sample size needs to be large to evaluate the tail of the MPPI distribution in the artificial datasets, we combined the MPPI of the null distributions for all the artificial groups of phenotypes with the same dimension . Several alternatives have been proposed in the literature to select significant MPPI either based on prediction considerations jth SNP in the gth group of phenotypes is defined asgth group, it compares two different models (γjg = 1 vs γjg = 0) regardless of the prior probability jth SNP associated with the gth group of phenotypes regardless of the prior probability. It is expected that this threshold varies depending on the number and correlation of the group of phenotypes. The quantityjg≥1 since a SNP is declared associated if jth SNP to be associated with the gth group of phenotypes with respect to the prior beliefs, while the latter is the “relative measure of risk” of the jth SNP to be associated with the gth group of phenotypes with respect to the MPPI threshold obtained from each group g at a specified FDR level. The denominator in (2) acts as a standardisation factor. For a given SNP j, RBFjg can be compared across groups of traits and provides a formal way to rank them with respect to the strength of association with the SNP.Bayes Factor (BF) for the jg and RBFjh be the RBF defined in (2) for two groups of traits. If RBFjg>RBFjh, thenjg>RBFjh if the ratio of the Bayes Factors of the two groups of traits is larger than the ratio of the Bayes Factors at the FDR baseline level (that can be < or >1). Finally, given a groups of traits, if BFig>BFjg, i≠j, then RBFig>RBFjg, showing that the new measure does not alter the rank of the phenotype(s)-SNP association within each group.Let RBFhttps://mathgen.stats.ox.ac.uk/genetics_software/snptest/snptest.html) automatically performs normal quantile transformation to each trait and adjusts for sex, age and body mass index . We chose the Bayesian analysis (-bayesian 1) with suggested default hyper-parameters for the single trait analysis  and variance 0.02σ2 (-prior_qt_V_b 0.02)) and InverseGamma prior on the error variance σ2 with finite mean 1 (-prior_qt_a 3 and -prior_qt_b 2). For the multiple traits analysis we selected the suggested default values  with covariance matrix 0.02∑ (-prior_qt_V_b 0.02)) and InverseWishart prior on the error variance ∑ (-prior_mqt_c 6 and -prior_mqt_Q 4). The prior probability of association cannot be modified and it is set at π = 10−4. SNPTEST provides the value of the BF automatically. SNPTEST V2.2.0 (www.bcm.edu/cnrc/mcmcmc/pimass) on the residuals from this regression. In order to match the prior on the model size used in GUESS, we set the minimum and the maximum of π (the prior probability that a SNP is truly associated with the phenotype) equal to 1 and 56 out of the total number of SNPs (-pmin 1 -pmax 56), restricting the minimum and maximum number of SNPs in a model to be 1 and 104 (-smin 1 -smax 104) without any constraint on the hyper-parameter h and no cut-off on MAF (-exclude-maf 0). We ran piMASS with 106 warm-up steps followed by 107 sampling steps (-w 1000000 -s 10000000), recording a sample every 10 steps (-num 10). Although we did not match the number of visited models by GUESS with those of piMASS (for piMASS the number of sampling steps coincides with the number of models visited), we are confident that the very large number of sampling steps allows piMASS to explore faithfully the model space. 10(BF) for TG. Since piMASS provides the MPPI through Rao-Blackwellization pγ) = 13.663 which corresponds to E(π) = 5×10−5. Finally Each trait was normal transformed using the R function qqnorm. The effect of sex, age and body mass index was removed by performing standard multiple linear regression software and then running piMASS v0.9 (http://cran.r-project.org/web/packages/rrcov/) to compare the responses' means for each SNP in the simulated experiments. Setting method = rank the classical Wilks' Lambda statistic for testing the equality of the group means for all the responses is modified into a robust version We implemented the frequentist analysis of multiple traits using the function wilks.test from the rrcov R package (http://cran.r-project.org/web/packages/mixOmics/index.html) ncomp = 1) and investigated nine different values of the number of SNPs in this component (KeepX) ranging from 4 to 36. SNPs contributing to the component are defined as those with non-zero loadings coefficient. In this special case where only one component is retained for the regression model, SPLS corresponds to canonical regression We used the spls function from the mixOmics R package  the overall computational time required by SPLS is around 12 times greater than that of GUESS.Building on the known structure of the multiple responses simulated dataset we were able to fix the number of components as well as the number of the SNPs contributing to each component. The analysis of each replicate took approximately 40 minutes. Using the model on real data, these two features have to be assessed by means of a V-fold cross-validation procedure. Using standard 10-fold cross-validation replicated 50 times, each combination of http://cran.r-project.org/web/packages/glmnet/index.html) R package family = mgaussian, and the LASSO penalty was enabled by setting alpha = 1. The penalty λ was calibrated based on the first replicate of each multiple traits simulated dataset such that the number of retained SNPs by MLASSO is consistent with the sequence of values of KeepX defined in the SPLS analysis. The resulting set of nine values for λ was used in all replicates of the two simulated multiple traits scenarios.We fitted a LASSO-type penalized multivariate linear regression model using glmnet Click here for additional data file.Figure S2Heat-map of the squared correlation matrix of the 16 SNPs which were marginally associated with any group of traits using an empirical FDR cut-off of 5%. Squared correlation between rs11902417 and rs13392272 is 0.2711, rs11902417 and rs1469513 is 0.1949 and rs13392272 and rs1469513 is 0.5582. Squared correlation between rs326 and rs17410962 is 0.3305, rs326 and rs17489268 is 0.7901 and rs17410962 and 17489268 is 0.2165.(TIF)Click here for additional data file.Figure S310(BFs) are truncated at 20). (A) Scatterplot of log10(BF) GUESS vs SNPTEST obtained superimposing the scatterplot of each single trait. A horizontal dashed line indicates level of log10(BF) that provides strong evidence of a phenotype-SNP association (log10(BF)>5). Red and black dots highlight significant SNPs found by GUESS while non-significant SNPs are colour coded according to the level of pairwise Pearson correlation with the closest significant GUESS SNP . (B) Scatterplot of log10(BF) GUESS vs piMASS obtained superimposing the scatterplot of each single trait. Colour code used to identify relevant SNPs and horizontal dashed line are the same as defined in (A).Comparison of the marginal phenotype-SNP association provided by GUESS, SNPTEST and piMASS for all single traits of two trees. Click here for additional data file.Figure S4g (blue) and shrinkage factor g/(1+g). In all plots, black vertical dotted line indicates the end of the burn-in phase.GUESS diagnostic plots in the TG-HDL-LDL group analysis. (A) Trace plot of the log-Posterior  of the three chains run in parallel. (B) Trace plot of the size of the models explored by the three chains run in parallel. (C) Trace plot of the selection coefficient (TIF)Click here for additional data file.Figure S5Trace plot of the size of the models explored by the non-heated chain of GUESS in the TG-HDL-LDL group analysis. Letters A-H indicate when GUESS first identifies the top Best Model Visited (A), the second Best Model Visited (B) and etc. with models ranked by the Model Posterior Probability. A black vertical dotted line indicates the end of the burn-in phase.(TIF)Click here for additional data file.Figure S610(BF)” in the single trait TG analysis. Values of log10(BF) are recorded every ten iterations.Trace plot of piMASS “Model log(TIF)Click here for additional data file.Figure S710(BFs) are truncated at 20). Scatterplot of log10(BF) GUESS vs SNPTEST obtained superimposing the scatterplot of each multiple trait group. A horizontal dashed line indicates the level of log10(BF) that provides strong evidence of a phenotype-SNP association (log10(BF)>5). Red and black dots highlight significant SNPs found by GUESS while non-significant SNPs are colour coded according to the level of pairwise Pearson correlation with the closest significant GUESS SNP .Comparison of the marginal phenotype-SNP association provided by GUESS and SNPTEST for all multiple traits of two trees. Click here for additional data file.Figure S810(BF) obtained from GUESS for single trait analysis. (A) LDL, (B) APOB (first tree), (C) HDL and (D) APOA1 (second tree). Significant SNPs found associated at a 5% FDR are depicted by black dots (with the SNP's name) whereas significant SNPs that are also in the top Best Model Visited are represented by red dots (with the SNP's name) (the log10(BF) is truncated at 20).Genome-wide log(TIF)Click here for additional data file.Figure S910(BF) obtained from GUESS in the first tree centred in the LDL pathway. (A) TG-LDL, (B) TG-APOB and (C) LDL-APOB. Significant SNPs found associated at a 5% FDR are depicted by black dots (with the SNP's name) whereas significant SNPs that are also in the top Best Model Visited are represented by red dots (with the SNP's name) (the log10(BF) is truncated at 20).Genome-wide log(TIF)Click here for additional data file.Figure S1010(BF) obtained from GUESS in the second tree centred in the HDL pathway. (A) TG-HDL, (B) TG-APOA1, (C) HDL-APOA1 and (D) TG-HDL-APOA1. Significant SNPs found associated at 5% FDR are depicted by black dots (with the SNP's name) whereas significant SNPs that are also in the top Best Model Visited are represented by red dots (with the SNP's name) (the log10(BF) is truncated at 20).Genome-wide log(TIF)Click here for additional data file.Figure S1110(BF). The number of SNPs retained by SPLS under different levels of penalization is indicated. For MANOVA Wilks, the ROC curve is derived using SNPs declared significant over a range of FDR levels. The number of false positives (x-axis) is indicated at the top of the figure while the proportion of false positives is presented at the bottom. Given the large number of predictors , false positives are truncated at 10−4 at which level a large number already occurs (27.5).Receiver Operating Characteristic (ROC) curves comparison. ROC curves of SNPTEST (black), SPLS (blue), MANOVA (purple), and GUESS (red) for the first (A–C) and second (B–D) multiple-trait simulated datasets when the definition of positive associations is relaxed, i.e. considering a single association in an interval centred at each top hit and spanning 25 kb, 50 kb and 100 kb on both sides. For GUESS, separate ROC curves are obtained using the top Best Model Visited (red star) and the Marginal Posterior Probability of Inclusion (solid red line). For SNPTEST, the ROC curve is calculated using the log(TIF)Click here for additional data file.Figure S1210(BF). For ANOVA Kruskal, the ROC curve is derived using SNPs declared significant over a range of FDR levels. The number of false positives (x-axis) is indicated at the top of the figure while the proportion of false positives is presented at the bottom. Given the large number of predictors , false positives are truncated at 10−4 at which level a large number already occurs (27.5).Receiver Operating Characteristic (ROC) curves comparison. ROC curves of SNPTEST (black), ANOVA (purple) and GUESS (red) for the first (A) and second (B) single-trait simulated datasets when the definition of positive associations is relaxed, i.e. considering a single association in an interval centred at each top hit and spanning 25 kb, 50 kb and 100 kb on both sides. For GUESS, separate ROC curves were obtained using the top Best Model Visited (red star) and the Marginal Posterior Probability of Inclusion (MPPI) (solid red line). For SNPTEST, the ROC curve is calculated using the log(TIF)Click here for additional data file.Figure S13a priori expected model size, i.e E = 10 (blue), E = 20 (red) and E = 40 (green) for five replicates of the first (A) and second (B) multi-trait simulated dataset are depicted. Separate ROC curves were obtained using the top Best Model Visited (star) and the Marginal Posterior Probability of Inclusion (solid line). Given the large number of predictors , false positives are truncated at 10−4 at which level a large number already occurs (27.5).Receiver Operating Characteristic (ROC) curves of GUESS under different parameterization. ROC curve of the (TIF)Click here for additional data file.Table S110(RBF), i.e. rescaled marginal phenotype-SNP association, are included with significant SNPs depicted in black and significant SNPs that are also in the top BMV indicated in red (the log10(RBF) is truncated at 20). The Ratio of Bayes Factors (RBF) is a rescaled measure of SNP-trait(s) association and it is defined as the ratio between the BF to test the SNP-trait(s) association hypothesis and the “baseline” BF level obtained through permutations.Post-processed output obtained from GUESS for all the elements of the two trees (green and blue colour coded) and TG-HDL-LDL. Horizontal lines separating groups of traits with the same cardinality . Model Posterior Probability (MPP) of the top Best Model Visited (BMV) and the cumulative MPP of the five top BMV are indicated in the first two columns of the table, respectively. The unique set of significant SNPs (FDR<0.05) which predict a group of phenotypes is indicated on the top of the table as well as the associated locus. Based on Ensembl R66 annotation, each locus is classified as: (1) intronic, (2) 3′UTR, (3) downstream, (4) previously associated and (5) a tagSNP of a previously associated SNP. In the centre of the table log(PDF)Click here for additional data file.Table S2Comparison of prior setup between GUESS, SNPTEST and piMASS. In SNPTEST the hyper-priors on π and on the variance of the regression coefficients are not specified. piMASS differs from GUESS by a different specification of the priors on the regression coefficients and on their variance.(PDF)Click here for additional data file.Table S37 iterations with 106 as burn-in. GUESS analysis was performed on an HPC cluster computer with a 2.8 GHz Dual-Core Xeon processor and a NVidia Tesla C1060 GPU with 8 Gb of RAM, while piMASS was run on a 3 GHz computer with a 1024 KB cache size Dual-Core AMD Opteron processor and 16 Gb of RAM. “Computational time” is reported in hours (rounded to the nearest integer). “Number of models evaluated” includes the burn-in phase, while “Number of unique model visited” and “Number of model visited before (visiting) top Best Model Visited” are calculated after the burn-in phase. “Average model size” is the average dimension (standard deviation in brackets) of the model recorded in GUESS (from the non-heated chain) and piMASS (every 10 iterations). For piMASS the number of models evaluated corresponds to the number of iterations and is roughly equal to a third of the models evaluated by GUESS.Comparison of the MCMC efficiency between GUESS and piMASS. GUESS was run for 100,000 sweeps with 10,000 as burn-in and with 3 chains. piMASS was run for 1.1×10(PDF)Click here for additional data file.Table S410(Bayes Factor) for significant SNPs found associated by GUESS is included (the log10(BF) is truncated at 20).Output obtained from SNPTEST for all elements of the two trees (green and blue colour coded) and TG-HDL-LDL. Horizontal lines separating groups of traits with the same cardinality . The unique set of significant SNPs (FDR<0.05) found by GUESS which predict a group of phenotypes is indicated on the top of the table as well as the associated locus. Based on Ensembl R66 annotation, each locus is classified as: (1) intronic, (2) 3′UTR, (3) downstream, (4) previously associated and (5) tagSNP of previously associated SNP. In the centre of the table the SNPTEST log(PDF)Click here for additional data file.Table S5+), 6 pairs (++) and 3 triplets (+++), were used to calculate the empirical FDR), the MPPI threshold at an FDR of 5% and the number of significant SNPs associated with each group of phenotypes.Results of the empirical FDR procedure. For each element of the two trees centred on the LDL and HDL pathways and TG-HDL-LDL, we report the sample size of the null distribution used in the empirical FDR procedure that we obtained combining the Marginal Posterior Probability of Inclusion (MPPI) for all the artificial groups of phenotypes with the same dimension Click here for additional data file.Table S62r and D′ are 0.582 and 0.979 between rs261332 and rs8034802 in (A) and 0.838 and 0.982 between rs261332 and rs1077834 in (B), respectively.Genetic associations for selected phenotypic groups  detected by GUESS. Two independent replication datasets were used (a) Copenhagen City Heart Study (CCHS) and (b) Data from an Epidemiological Study on the Insulin Resistance syndrome (DESIR). Each region centred at the identified causal variant in the discovery dataset  and spanning 2 Mb is regressed against the phenotypic groups previously associated with the variant. Genetic markers with the largest significant BF obtained with SNPTEST in each region are reported in each table as well as their position in the genome. SNP rs261332 is not present in the CardioMetabochip. Using 381 Caucasian individuals from the 1000 Genomes project, (PDF)Click here for additional data file.Table S7Strength of the genetic association provided by the RBF between genetic variants identified in (PDF)Click here for additional data file.Table S8Post-processed output obtained from GUESS with different prior specification. For selected elements of the two trees (green and blue colour coded) and TG-HDL-LDL GUESS was run using a conjugate hierarchical independent prior for the genetic effects with a diffuse exponential hyper-prior for the variance of the regression coefficients. Model Posterior Probability (MPP) of the top Best Model Visited (BMV) and the cumulative MPP of the five top BMV are indicated in the first two columns of the table, respectively. The previously identified unique sets of significant SNPs (FDR<0.05) which predict a group of phenotypes is indicated on the top of the table as well as the associated locus. Based on Ensembl R66 annotation, each locus previously identified is classified as: (1) intronic, (2) 3′UTR, (3) downstream, (4) previously associated and (5) a tagSNP of a previously associated SNP. SNP-trait(s) association identified in the BMV by GUESS with the new prior specification are presented in the centre of the table. The BMV for the selected elements of the two trees are as depicted in (PDF)Click here for additional data file.


LIN28B, approached a significant association with pubertal growth. Second, we show how using the twin data to control statistically for environmental variance can provide insight into the ultimate magnitude of SNP effects and consequently the genetic architecture of a phenotype. Specifically, we computed a genetic score by weighting SNPs according to their effects as assessed via meta-analysis. This weighted score accounted for 9.2% of the phenotypic variance in height, but 14.3% of the corresponding genetic variance. Longitudinal samples will be needed to understand the developmental context of common genetic variants identified through GWAS, while genetically informative designs will be helpful in accurately characterizing the extent to which these variants account for genetic, and not just phenotypic, variance.In a sample of 3,187 twins and 3,294 of their parents, we sought to investigate association of both individual variants and a genotype-based height score involving 176 of the 180 common genetic variants with adult height identified recently by the GIANT consortium. First, longitudinal observations on height spanning pre-adolescence through adulthood in the twin sample allowed us to investigate the separate effects of the previously identified SNPs on pre-pubertal height and pubertal growth spurt. We show that the effect of SNPs identified by the GIANT consortium is primarily on prepubertal height. Only one SNP, rs7759938 in  LIN28B, approached a significant association with pubertal growth. Second, using standard twin heritability models, we investigated the extent to which the collective effect of these SNPs explained genetic variance in height—as opposed to phenotypic variance, as other studies have done. We computed a genetic score by weighting SNPs according to their effects as assessed via meta-analysis. We show that, while the score accounts for ∼9% of the phenotypic variance in height , it accounts for ∼14% of the corresponding genetic variance. Longitudinal samples are necessary to understand the developmental context of common genetic variants identified through GWAS, while twin samples will be helpful in accurately characterizing the extent to which these variants account for genetic, and not just phenotypic, variance.We evaluated the developmental specificity of 176 SNPs known to affect adult height based on meta-analysis from the GIANT consortium. First, longitudinal observations on height spanning pre-adolescence through adulthood in a twin sample allowed us to investigate the individual effects of the previously identified SNPs on both pre-pubertal height and pubertal growth spurt. We show that the effect of the SNPs identified by the GIANT consortium is primarily on prepubertal height. Only one SNP, rs7759938 in  Adult height is a model multigenic phenotype for genetic association studies. Twin and adoption studies suggest that height is highly heritable (∼80%) For any person adult height reflects roughly two decades of growth. Change in height is relatively rapid throughout infancy, slows down in early childhood, and increases again during puberty when a notable growth spurt occurs The sample (N = 6481) was drawn from Caucasian participants in the Minnesota Twin Family Study (MTFS). The MTFS a 20-year, longitudinal, community-representative study conducted at the University of Minnesota and approved by the University of Minnesota Institutional Review Board continuously since inception. The study has been extensively described previously The twin sample was used to develop a genetically mediated, longitudinal growth model for height. There are many approaches to analysis of longitudinal designs, and choosing the best approach often requires assumptions about, or explicit knowledge of, the (unknown) true growth trajectories generating the data The growth model was constructed on the twins to evaluate SNP effects on age-11 height and pubertal growth after age 11. In the growth model, age-11 height corresponds to the intercept and pubertal growth corresponds to the slope. A diagram of the model is depicted in In the path diagram of the phenotypic model  the obseAn important caveat with the proposed growth model as described in As portrayed in Analogous to the phenotypic growth model is the twin growth model, partially displayed in the box inset in With the proposed growth model, we could estimate the effect of SNPs on the intercept  and slope . Using twins, we could further estimate the influence of SNP effects on genetic variance—as opposed to overall phenotypic variance—thus providing a direct estimate of the genetic variance accounted for by the SNPs. Studies on height heretofore have been unable to accomplish this, and have been restricted to comparing estimates of the percent of variance accounted for by SNPs in their sample to estimates of heritability obtained in separate studies of twins  we used the Pubertal Development Scale 2 = .17, p<2e−16) and 14 . In females at age 11 age, puberty score, and menarche significantly predicted height . At age 14 only the puberty score was a significant predictor . We computed the sum of age and puberty score  weighted by their corresponding regression weight and scaled the result to have the mean and variance of the original chronological age at 11 and 14. This gives an estimate of each subject's pubertal age—as opposed to chronological age—as it relates to growth in height. In supplementary analyses these pubertal ages were used in place of chronological age for the age-11 and age-14 assessments in the growth model, in an attempt to more precisely gauge the developmental specificity of each SNP.We incorporated pubertal status into the growth model in the following way. For each age of assessment we regressed height on age and puberty score and, for females, an indicator variable measuring whether menarche had occurred by that assessment. For males both age and puberty score were highly significant in predicting height at age 11  and one duplicate sample. Markers were excluded if . One SNP (rs5017948) was not contained in the 1000 Genomes 2010-06 or 2010-08 reference datasets and so was discarded. The average r2 of the remaining 124 imputed SNPs was .96 . In total, 176 of the 180 SNPs from the GIANT Consortium meta-analysis Of the 180 SNPs described by the GIANT consortium as associated with height, 52 existed on the Illumina 660quad. The remaining were imputed with best-guess genotypes using MaCH −102) and accounted for 9.2%  of the phenotypic variance in adult height (in the full sample N = 5706). This result is not significantly different from the r2 of 10.5% found by the GIANT Consortium in First, we ran association tests for each of the 176 SNPs on adult height in the full sample. A subset of twins (N = 775) had not yet reached adult height and were excluded from this analysis . To account statistically for within-family clustering a generalized least squares method was used Fitting the growth model, we found the intercept was 86% heritable  and the slope 84% heritable . Shared environmental effects accounted for 9%  and 11%  of the variance in the intercept and slope, respectively. Unshared environment accounted for 5%  in the intercept and 5%  in the slope. Clearly, both age-11 height and pubertal growth  were highly heritable, consistent with previous reports p = 2e−9), but was chance-level for the slope (p = .06). The QQ plot for age-11 height clearly showed higher −log10(p) values than expected by chance, and the genetic score was highly significant . Alternatively, the QQ plot for pubertal growth consistently showed chance-level effects. The genetic score effect here was nominally significant .Individual SNP and genetic score effects on the intercept and slope were computed simultaneously. Log-transformed p-values are displayed in p-values for the SNP effects on the growth model intercept, the growth model slope, and adult height are listed in All regression coefficients, standard errors, and h) is a quadratic function of the intercept (i), slope (s), and age (a),To summarize findings from the growth model, we computed the variance in height accounted for by the genetic risk score's effect on the intercept and slope. A linear growth model is linear in the mean function but quadratic in the variance. In the present model, phenotypic height (2 with the genetic score. We plotted phenotypic r2 in 2). The model with a genetic score effect only on the slope did account for variance in height at all ages, but not as much as the model with a score effect only on the intercept. More to the point, the model with score effect simultaneously on the slope and intercept negligibly improved over the model with score effect only on the intercept. This indicated that the score has only a slight relationship with the unique SNP variance in pubertal growth. However, because the score reduces variance in height even when it was only allowed to load onto the slope, it did appear that the score is related to genetic variance overlapping between the intercept and slope .This parabola was computed for the base model, with no genetic score effect, and used to compute r2, or the genetic variance in height accounted for by the genetic score. This is plotted in red in 2 was greater than the phenotypic r2 for the entire age range under investigation. Comparing the maximum phenotypic r2 to the maximum genetic r2 for the model with a score effect on both the intercept and slope, one notices a jump from 9.2%  to 14.3% . That is, by using the twins to partition environmental variance, we were able to increase the magnitude of the SNP effect and consequently the sensitivity of the analysis.Using the twins, we also computed the genetic r2 = 78.04, df = 9, p = 4e−13). While heritability of the intercept was similar  the heritability of the slope was different . Females had a larger shared environmental contribution to their slope variance . Males and females had similar phenotypic correlations between the intercept and slope . The genetic and environmental contributions to this correlation were different between the sexes. The genetic correlation between intercept and slope was −.60 for males and −.47 for females. The shared environment correlation was −.04 for males and −.14 for females. The unshared environmental correlation was −.02 in males and −.03 in females.Growth models were also fit separately to the male and female subsamples. After scaling male heights for each age of assessment to have the female's mean and variance, the growth model variance component parameters were different between the sexes . All SNP and score statistics are included in Growth model parameters did not change dramatically after correcting chronological ages at 11 and 14 for pubertal status. The negative correlation between the intercept and slope was unchanged (−.62) with a larger genetic contribution (−.58) and smaller contributions by shared environment (−.02) and unshared environment (−.03). The intercept was 88% heritable with contributions of 7% and 5% from shared and unshared environment, respectively. The slope was 84% heritable with contributions of 10% and 6% from shared and unshared environment, respectively.r = .99) and the slope (r = .99). The mean regression weight onto the intercept in the uncorrected model was .06 (SD = .43) versus .05 (SD = .43). The mean weight onto the slope in the uncorrected model was −.005 (SD = .06) versus −.003 (SD = .06). Correlations between standard errors and p-values were equally similar between the corrected and uncorrected models.SNP associations also remained largely unchanged after correcting for pubertal status. FTO, where the minor allele of rs9939609 is a well established risk-factor for adult obesity, illustrates the importance of a developmental perspective. Specifically, the minor allele of rs9939609 is negatively associated with body mass index (BMI) until the age of 2.5, but, because it is associated with an earlier onset of the adiposity rebound that occurs in childhood, positively associated with BMI after age 5.5 years The first wave of GWAS research has been successful in identifying numerous common variants associated with various adult disorders and traits The present study extended genetic analysis of developmental phenotypes by implementing a growth model to partition observed measures into two biologically meaningful constructs: pre-pubertal height and pubertal growth. We focused here on an established literature of SNP effects on height. This is necessary because individual genetic effects are too small to be detected at genome-wide levels by most individual studies, and combining longitudinal studies with commensurate phenotypes can be prohibitively difficult . It may be that consortia of cross-sectional data will largely be necessary to discover replicable genetic variants while smaller, methodologically-unique individual studies will be left to understand those effects within a developmental context.The vast majority of SNPs identified by Allen et al. LIN28B has previously been identified as relevant for adult height Lin28a were found to have accelerated growth during the first 60 weeks of life in addition to later onset of puberty While most SNPs were unrelated to pubertal growth, one was. rs7759938 in 2, but so-called “genetically informative” samples such as twins or adoptive families are. An array of statistical techniques have been developed for such samples We also evaluated the effect of a genetic score on zygosity-derived genetic variance, as opposed to phenotypic variance, using a sample of twins. The score accounted for 14.3% of genetic variance in adult height, but only 9.2% of phenotypic variance, illustrating the possible advantages of using a twin sample. The use of twins provides concrete advantages over analyses that estimate the fraction of heritable variance attributable to multiple loci indirectly either based on previously reported heritability estimates or genome-wide markers in unrelated individuals. Admittedly, the advantage may not be extremely powerful in the present context, given height's high heritability, where the genetic variance is 80% or more of the total phenotypic variance. However, for less-heritable phenotypes, or where heritability is less well known, the approach will provide improved information about the magnitude of a SNP's  relationship to the phenotype. A growth model is not necessary to evaluate genetic rIn summary, genomic findings from consortia may be fruitfully characterized within a developmental framework. Many analytic approaches exist, and the best may depend on the data structure at hand. Genetically informative samples such as twins remain important and viable tools in investigating genomic variation, even as genotyping or sequencing becomes routine.Figure S1Scatterplot Matrix of Regression Coefficients. “Meta-Analysis” refers to the SNPs reported in Allen et al. (TIF)Click here for additional data file.Figure S2(a) Refers to males; (b) to females. All other aspects are described in the caption to (TIF)Click here for additional data file.Figure S3SNP and Score Effects on the Puberty-Corrected Growth Model Intercept and Slope. Univariate plot (a) and QQ plot (b) of SNP effects on intercept. Univariate plot (c) and QQ plot (d) of SNP effects on the slope. All symbols are described in the caption for (TIF)Click here for additional data file.Table S1Gene/allele information, regression coefficients, standard errors, and p-values for all tests in the growth model and the adult height analysis.(XLS)Click here for additional data file.Table S2Regression Correlation Coefficient Matrix. These are Spearman rank order correlations between all 176 regression coefficients computed for each height phenotype under study. Confidence intervals were computed by bootstrap with 2000 pseudo-replications. “Meta-Analysis” refers to sex-combined regression coefficients from the GIANT Consortium meta-analysis (DOC)Click here for additional data file.


Differences in transcription between the two alleles at an autosomal locus can be used to test the association between candidate polymorphisms and the modulation of gene expression in cis. This type of approach requires at least one transcribed polymorphism and one candidate polymorphism. In the past five years, different methods have been proposed to analyse such data. Here we use simulations and real data sets to compare the power of some of these methods. The results show that when it is not possible to determine the phase between the transcribed and potentially cis acting allele there is some advantage in using methods that estimate phased genotype and effect on expression simultaneously. However when the phase can be determined, simple regression models seem preferable because of their simplicity and flexibility. The simulations and the analysis of experimental data suggest that in the majority of situations, methods that assume a lognormal distribution of the allelic expression ratios are both robust to deviations from this assumption and more powerful than alternatives that do not make these assumptions.Genome wide association studies frequently reveal associations between disease susceptibility and polymorphisms outside coding regions. Such associations cannot always be explained by linkage disequilibrium with changes affecting the transcription products. This has stimulated the interest in characterising sequence variation influencing gene expression levels, in particular in changes acting in  Changes acting in trans affect both alleles. In individuals heterozygous for one or more transcribed polymorphisms, the contribution of each of the two alleles is assessed by quantifying the relative amount of transcripts from each. Unequal expression designated here as allelic expression imbalance , in individuals heterozygous for a putative cis acting polymorphism is seen as evidence for cis acting effects. The principle is depicted in cis acting polymorphism with alleles T and C; and a transcribed polymorphism with alleles A and G. This second polymorphism allows us to ascertain the origin of each transcript. The figure shows that transcripts carrying the A allele are more abundant than those carrying the allele G. This is consistent with the T allele of the cis acting polymorphism being associated with overexpression . The rationale behind the use of allelic expression as a tool for mapping cis acting polymorphisms is that it should be relatively insensitive to influences affecting both alleles such as sample degradation or trans acting effects, compared to methods that analyse expression from both alleles as a pool. Indeed, several recent reports have found that allelic expression analysis can be more powerful in detecting cis acting variants than traditional expression quantitative trait locus (eQTL) analysis In recent years, analysis of allelic expression has increasingly been used to ascertain in vivoAllelic expression is often assessed by applying established genotyping methods to cDNA instead of genomic DNA. In a typical experiment DNA and RNA samples are collected from a panel of individuals. We will use here the term sample size for the number of individuals in the panel. The DNA is genotyped for a set of markers that includes at least one marker that is located within the transcript of interest. For individuals heterozygous for the transcribed polymorphism, the RNA, usually after reverse transcription, is used to quantify the relative amounts of transcripts originating from each of the two alleles. The outcomes of such an experiment are the genotype frequencies in the panel of individuals and allelic expression ratios for the individuals that are heterozygous for the transcribed marker. Allelic expression can be treated as a qualitative trait, describing the presence or absence of imbalance and perhaps which allele is overexpressed. It can also be considered as a quantitative trait. We will use here the term allelic expression ratio (AER) for the ratio of the signal intensity emanating from one allele, as defined by the transcribed polymorphism, divided by that from the other e.g. . Alternacis acting effect for each polymorphism from a set of SNPs. Here we will concentrate on the simplest case where only one candidate polymorphism is tested and only one transcribed polymorphism is used. For an individual that is heterozygous at both the transcribed and the cis acting sites, the transcribed allele that is overexpressed will be the one that is on the same chromosome as the as the cis acting allele causing overexpression. The phase between alleles at the two sites can vary from individual to individual. Therefore assessing the effect of a putative cis acting polymorphism may require determining the phase of the alleles at the transcribed and cis acting sites. This is particularly simple when the polymorphism of interest is the transcribed polymorphism itself, resulting in the systematic overexpression of the same allele in heterozygotes, or when both polymorphisms are in complete linkage disequilibrium, where little imbalance will be expected for those homozygous at the cis acting site, while those that are heterozygous should show systematic overexpression of the same transcribed allele cis-acting effect can then be assessed. This analysis can be carried out either using the most likely phased genotype cis-acting effect simultaneously . The distinct approaches arise due to the extent of linkage disequilibrium present. Panel A represents the relationship between the genotype and allelic expression ratios in the general situation. Panels B to D illustrate the reasoning underlying different tests that have been used. In the simplest case where both polymorphisms are in perfect disequilibrium or the cis acting and the transcribed polymorphism are one and the same, testing the effect is consistent with assessing a systematic deviation from balanced expression in one direction (Panel B). Panel C depicts the situation where there is complete disequilibrium. Here the effect of the putative functional polymorphism should result in the systematic overexpression of one and the same transcribed allele in heterozygotes that is not observed among homozygotes. This suggests using a test to assess differences in allelic expression ratios from both groups  is strong (D′ = 1), or Panel D when phase can be confidently inferred.cis and trans acting factors, therefore although a polymorphism may act in cis in a certain context this effect is mediated by trans acting factors that may themselves by the subject of variation caused either by environmental In order to test the power we simulate the allelic expression. The simplest assumption is to presume that expression from each allele is lognormally distributed. However in practice the patterns observed are more complicated. Several elements contribute to this. Detailed studies routinely uncover that transcription is influenced by more than one polymorphism cis. We concentrate on the association between allelic expression and particular biallelic polymorphisms and we compare the power using simulated and published data sets.Here we examine several different but commonly used approaches to the analysis of allelic expression. We focus on the power of different methods to identify sites associated with expression differences in This section has of two parts. In the first part we present the different statistical methods to be compared and in the second we describe and discuss the models used in the simulations. The simulations are used to test the power to detect the effect of a single polymorphism on the AER measured using a single transcribed polymorphism. We consider only biallellic polymorphisms. A significance level threshold of 0.05 is assumed throughout.Statistical Tests: We limit our consideration to previously published approaches or existing methods. These tests assume that the data consist of a set of individuals who have been typed for a putative cis acting polymorphism and transcribed marker, and that allelic expression has been measured in those individuals that are heterozygous for the transcribed marker. We use the term sample size to describe the number of individuals genotyped, irrespective of the transcribed marker genotype. The tests can be divided in two groups. The first group relies on a model of the process generating allelic expression. The second group consists of simple statistical tests appropriate for one or more of the instances depicted in i-th individual we designate with iT the genotyping results for both loci and with iI the log of the allelic expression ratio. The first set of tests  are likelihood ratio tests that rely on the assumption that the allelic expression ratios are lognormally distributed with a genotype dependent mean cis acting polymorphism and M and m the alleles at the transcribed site and with C and c the alleles of the cis acting polymorphism thenLRT.j jointly maximises the parameters describing both expression and haplotype frequencies. The likelihood L can be decomposed in two components: g given the haplotype frequencies, the index j runs through all individuals in the sample for which no AER was measured and g through all phased genotypes defined by the two polymorphisms. The second component describes the contribution of samples for which AER was measured i runs through all individuals. This test (LRT.j) represents an extension of the procedure described by Teare et al.We use the following notation: For the LRT.p which maximises the probability of the log expression ratio given the genotyping results :j in that the haplotype frequencies are inferred (or ‘prephased’) from the genotyping results through a preliminary step. For the results presented here this was done here using an Expectation Maximisation (EM) algorithm.The second test we consider is LRT.b uses instead of the distribution of haplotypes only the most likely haplotype. Thus the likelihood of interest can be described as i. This specific application is equivalent to fitting a simple linear regression.Test LRT.k uses the true or known genotypes instead of the most likely ones. This final LRT test is examined as a gold standard comparison, but in practice the true haplotypes for double heterozygotes are frequently not known.Test cis acting SNP. The second approach uses the Mann-Whitney or Wilcoxon test (W) to assess whether there is a difference in AER between individuals that are homozygous or heterozygous at the cis acting site. In perfect LD (only the double heterozygote group is observed) the test is assumed to fail and a nonsignificant result is returned. These two tests do not assume lognormality of the ratios.The second set of procedures we include, rely on some widely used tests, whose application to the analysis of allelic expression is motivated by the considerations discussed in relation to A third procedure investigates whether there is a correlation between AER and the phased genotype . This requires assigning one phased genotype to each individual. To apply the test in the case of haplotype phase uncertainty (C) the heterozygous individuals are assigned the most likely phase, resulting in only one heterozygote group. We compare the results using the same test but using the true or known simulated genotypes (C.k).cis acting site using an F-test (V). This test would appear most suitable under linkage equilibrium , that were also typed for a polymorphism in the promoter in vitroin vitroata sets . In thesThe results are summarised and presented in five figures. Each investigates the power of the different tests when conditions such as sample size, extent of LD or allele frequency are permitted to vary. The first four figures use simulated data. First we assess the power of the different tests when AER simply follows a log normal distribution , then wecis acting polymorphism and a lognormal distribution for the allelic expression ratios. In panel A the transcribed and cis acting polymorphisms are in linkage equilibrium (D′ = 0). The effect of sample size on the power to detect an association with the seven different methods can be seen. The joint phase and effect estimation (LRT.j) performs better than methods where the haplotypes are estimated first and the effect assessed in a separate step. However, its performance is strongly affected by the extent of disequilibrium between both markers. Panel B shows for a sample size of 100, that although there is a difference between both types of methods (LRT.j and LRT.p) the advantage diminishes quickly with increasing disequilibrium. It should also be noted that the most likely phase test (LRT.b) achieves a comparable power to LRT.j for modest disequilibrium (D′ = 0.5). The variance test works very well when D′ = 0 however as D′ increases this test only reduces in power whereas all the other tests improve.cis acting in panel G and H. The power increases with the proportion of heterozygotes for the transcribed markers that are also heterozygotes for the cis acting variant. Therefore in the absence of disequilibrium (D′ = 0) power increases with heterozygosity (panels E and G), while for D′ = 1 a maximum is achieved in the presence of matching allele frequencies for the model based tests . In panels F and H we see the drop in power of the two tests C.k and W as the system approaches perfect disequilibrium, reflecting the fact that when there is perfect disequilibrium all heterozygotes have the same genotype.Changes in effect size are explored in panels C (for D′ = 0) and D (D′ = 1). Where there is no effect (i.e. β = 0) the type I error seems adequately controlled. Under linkage equilibrium the LRT.j is slightly more powerful than LRT.p. While the test using Pearson's correlation coefficient (test C) works very well when phase is known, the test is ineffective when haplotype uncertainty exists. The effects of changes in the allele frequency for the transcribed polymorphism are explored in panels E and F, and of the putative 0, LRT.j.μ0, LRT.k.μ0 and LRT.b.μ0). The lines depicted in grey present analyses in which the type I error rate is not adequately controlled. The sign test S uses only the AER observed on heterozygotes so this test will be affected by the transcribed SNP effect. If analysis is conducted using the LRT approach but the β0 parameter is neglected then these methods are compromised. Therefore we used a set of tests , that allow for an effect of the transcribed polymorphism itself. Here the expected log AER is described as 0 represents the effect of the transcribed polymorphism. The LRT.j and LRT.p both perform well and much better than the LRT.b and the correlation test (C).cis acting loci, though there is now no effect of the transcribed marker itself. The simulations constrain the effect of the candidate SNP to be the strongest effect. As should be anticipated power reduces with increasing additional causative loci. This figure assumes that D′ = 0.cis acting polymorphisms are affected to a lesser degree. With increasing outlier frequency the performance of the LRT.p and that of the LRT.j method becomes similar. The presence of outliers does not affect the ability to determine haplotype frequencies in the former. The power of the variance test appears to be higher than the LRT methods. However, this is accompanied by an inflated type I error as can be seen in panel B when the true effect size is zero. Panels C and D show that when the log of the expression of each allele follows a t-distribution with 2 degrees of freedom, there is a substantial loss of power in particular for the model based methods where the phase needs to be estimated.cis acting effect of the nontranscribed SNPs. The Figure explores the effect of varying sample size. These two examples differ in two aspects. The first is that in MMP1 (panel A) the cis acting and transcribed markers are close to linkage equilibrium  while in MLH1 (panel B) they are in strong disequilibrium . Therefore we see differences in the power attained by the LRT.j, LRT.p and LRT.b tests for MMP1 (panel A), while for MLH1 the curves for these same tests are indistinguishable. The second difference is that while there is no evidence for a transcribed marker associated effect for MMP1MLH1cis acting polymorphism is predominantly occurring in phase with the underexpressed transcribed marker allele. So those tests that do not allow for a β0 parameter will perform very poorly in this situation.2 = D′ = 0), power increases with increasing heterozygosity at both the transcribed and the cis-acting sites (2 = 1).The results demonstrate that phase uncertainty is the main factor determining the power of the tests. However, closer inspection of the Figures shows that this effect depends on the allele frequencies at both sites. In the absence of disequilibrium, i.e. when the two loci are not associated , have an inflated type I error, wrongly attributing an effect where there is none. This can be circumvented by using a test that allows for an effect associated with the transcribed polymorphism. However the evidence for the second effect becomes difficult to obtain when the polymorphisms are closely associated.We explored the situation where the transcribed polymorphism influences transcription and we wish to assess an additional effect associated with a second polymorphism. In this case, likelihood ratio tests that do not include a baseline term (μcis acting sites. Since outliers distort the distribution of allelic expression ratios, it is not surprising to observe that a nonparametric method such as that relying on the Wilcoxon test, is less affected. With increasing outlier frequency the performance of the LRT.p and that of the LRT.j method become similar. This is consistent with the fact that the presence of outliers does not affect the ability to determine haplotype frequencies in the former, while a misspecified model will impair estimation of haplotype frequencies in the latter. However, the performance of the joint estimation method does not drop below that of the method where allele frequencies are estimated without using expression information. Similar observations can be made when there are several cis acting loci and where the distribution of expression is heavy tailed. Also in these cases the allelic expression ratios given the genotype at the cis acting and transcribed polymorphism do not follow a lognormal distribution.The presence of outliers quickly degrades the ability to detect MMP1 data where for a sample size of 105 the first test has 85% power, while the second 70% and the third 65% (MLH1 data. Joint determination of phase and effect is more cumbersome than using the predetermined haplotype frequencies, a task for which a wide range of tools has been developed over the past fifteen years. Indeed, a dense enough panel of typed markers may eliminate most of the haplotype uncertainty, leading to situations where testing can be done using linear regression (see cis acting polymorphisms and other co-variates. In our simulations and analysis the model LRT.b is equivalent to fitting simple linear regression.Throughout our simulations the test relying on joint maximisation of effect and haplotype frequencies is more powerful than the test where the genotypes are estimated separately in a first step and this is in turn more powerful than the a test were only the most likely phase is used. This difference is substantial for the hird 65%  panel A.sion see  panel D.cis. The variance test that groups the homozygotes and heterozygotes together is a powerful test when linkage equilibrium exists, though power reduces as disequilibrium increases. However, this test gives an inflated type I error in the presence of outliers and extreme values.Another conclusion from our data is that for the range of situations studied the advantage gained by using tests that dispense from the lognormality assumption is at best slight and led, in the majority of the simulations scenarios and in the real data sets used, to a substantial loss of power in the ability to detect experimentally supported polymorphisms acting in cis acting allele there is generally some advantage in using methods that estimate genotype and effect on expression simultaneously. However when the phase can be determined, simple regression models seem preferable. The scenarios explored here by simulation and through experimental data show that methods assuming lognormal distributions are the most powerful and are generally robust with respect to presence of outliers and other deviations from lognormality.In summary our investigation shows that when it is not possible to determine the phase between the transcribed and potentially


The theoretical basis of genome-wide association studies (GWAS) is statistical inference of linkage disequilibrium (LD) between any polymorphic marker and a putative disease locus. Most methods widely implemented for such analyses are vulnerable to several key demographic factors and deliver a poor statistical power for detecting genuine associations and also a high false positive rate. Here, we present a likelihood-based statistical approach that accounts properly for non-random nature of case–control samples in regard of genotypic distribution at the loci in populations under study and confers flexibility to test for genetic association in presence of different confounding factors such as population structure, non-randomness of samples etc.FGF20 and PARK8, without invoking false positive risk.We implemented this novel method together with several popular methods in the literature of GWAS, to re-analyze recently published Parkinson’s disease (PD) case–control samples. The real data analysis and computer simulation show that the new method confers not only significantly improved statistical power for detecting the associations but also robustness to the difficulties stemmed from non-randomly sampling and genetic structures when compared to its rivals. In particular, the new method detected 44 significant SNPs within 25 chromosomal regions of size < 1 Mb but only 6 SNPs in two of these regions were previously detected by the trend test based methods. It discovered two SNPs located 1.18 Mb and 0.18 Mb from the PD candidates, We developed a novel likelihood-based method which provides adequate estimation of LD and other population model parameters by using case and control samples, the ease in integration of these samples from multiple genetically divergent populations and thus confers statistically robust and powerful analyses of GWAS. On basis of simulation studies and analysis of real datasets, we demonstrated significant improvement of the new method over the non-parametric trend test, which is the most popularly implemented in the literature of GWAS. Rapid advancement in high-throughput sequencing techniques has greatly inspired the wave of genome-wide association studies (GWAS) to unravel the genetic basis underlying complex traits in plants, animals and humans -3. The t2 or trend tests etc. and rarely account for the biases described above  + (n12 + n22)Log[2p(1 - p)] + (n13 + n23)Log[(1 - p)2], which is a function of the vector N and marker allele frequency p only and is independent of the other parameters, q and D. Thus, the likelihood ratio test statistic can be approximated by a χ2 distribution with 2 degrees of freedom (d.f.). Under D = 0, the MLE of p is given byIt is important to note that the likelihood function under the null hypothesis can be simplified to be k subpopulations or genetic cohorts, we formulate the likelihood of the congregate case and control sample as the sum of the likelihoods for the case and control sample from each of these cohorts as given byWhen the cases and controls are collected independently from qi)( and D(i) from the case and control sample of the ith subpopulation separately using the method described above, and then to sum up the likelihoods for all the case and control samples. Although the likelihood ratio statistic confers the flexibility to test for significance of LD in any subpopulations, we are interested here in testing for a conservative null hypothesis that there is no LD between marker and disease loci in all the subpopulations based on the ratio of the congregate likelihood withDi). The Armitage’s method basically tests for association of a polymorphic genetic marker with a disease phenotype through testing for significance of the difference in allele frequency at the marker between cases and controls. In the presence of genetic structure, the difference in frequency of the marker allele M between the cases and controls contains a term ri and si stand for proportion of cases and controls collected from the ith population and M in the population as shown in equation (1). We removed this term from the numerator of Armitage’s trend test statistic, adjusted the corresponding sampling variance of the difference term and proposedend test  for geneni)( are the number of cases (or controls) and size of cases and controls from the ith population or cohort.to be the test statistic which follows the chi-square distribution with 1 d.f.. In equation (8), the denominator was the sampling variance of the numerator under the null hypothesis, i.e. there is no LD in either subpopulation. Method 3 is virtually the Armitage’s trend test, which is the most commonly implemented approach in the literature of GWAS with a case and control design. The test statistic is built upon the number of genotypes, nij with i =1, 2 corresponding to case and control and j = 1, 2, 3 to three genotypes at a tested marker, and has a form ofχ2 distribution with 1 d.f., where xj  are the weights of effects of different marker genotypes on the disease trait [x1-3 take a form of ,  or , the test statistic corresponds to testing for genetic association of the tested marker with a putative disease trait showing dominant, recessive, and additive genetic inheritance respectively. As demonstrated by Jackson et al. [Under the null hypothesis of no association between the marker and the disease locus, this follows a se trait . When x1under an additive genetic inheritance model ,13, wherMethod 3 here) and the genome-wide significance level was determined by the Bonferroni correction for the probability of an overall type I error at 5%.We implemented the three methods described above to re-analyze the PD dataset which was recently published by Simon-Sanchez et al. . The stuf1-3 for genotypes at the disease locus to be , while in the third sampling scheme, scheme C, mild penetrance parameters (i.e. f1-3 < 1) were used. Sampling schemes A and C collected cases and controls from a single population, while scheme B sampled individuals from two genetically divergent populations with regard to a tested marker and a putative disease locus. A simulated population in the present study was fully characterized by a set of population genetic parameters, p, q and D , and quantitative genetic parameters, f1, f2 and f3 (penetrance of three genotypes at the disease locus). For a given set of these parameters, genotype of a case or control subject at both the marker and disease loci was generated by randomly sampling from the conditional probability distribution given in Table To investigate statistical properties and limitations of the method developed in the present study, we considered three schemes for sampling cases and controls from computer simulated randomly mating populations. In the first two sampling schemes, scheme A and B, we fixed the penetrance parameters P) of genetic association tests across the 23 human chromosomes using the three PD case and control SNP datasets from the stage I, stage II and stage I and II combined, respectively. We analyzed each of the datasets using the three methods described above. For Method 1, the associations were detected under additive genetic inheritance mode with f1-3 = 1, ½, 0. It can be seen from the stage I data analysis  detected 44 significant SNPs, which are distributed across 25 chromosomal regions of size < 1 Mb (Table Methods 2 and 3 detected significant SNPs in only two (4q21 and 17q21) of the 25 regions at the same Bonferroni corrected significance threshold (P ≤ 1.1 × 10-7). No extra significant association was detected by Method 2 or 3 outside the 25 regions screened by Method 1. To explore genetic dependence among the 44 significant SNPs detected using Method 1, we calculated the coefficient of LD between all SNP pairs using an approach that accounts properly for the case and control sampling scheme [r2 is 0.002 between different chromosome regions), removing the concern that the detected SNP-disease associations might be due to autocorrelation in genotypic distribution among the SNPs between these regions. In particular, Method 1 uniquely detected six SNPs on chromosome region 10p11.21, of which, rs7923172 and rs4934704 locate at the introns of gene CUL2. The CUL2 gene encodes a protein of the E3 ubiquitin ligase complex [PARK2 also encodes a parkin protein in the same complex [CUL2 could be a newly detected PD candidate gene. Moreover, Method 1 detected three SNPs on chromosome region 8p22 (the most significant P = 9.9 × 10-10 at rs2736050), which were only 1.2 Mb apart from a previously reported PD susceptible gene FGF20. FGF20 and SNCA have previously been reported to be synergistically associated with PD [To assess the population structure in the stage I dataset, PCA was carried out using whole-genome genotype data and illustrated in Additional file g scheme . The dis complex . The fac4q21 and q21 of th with PD .P values) and the bootstrap posterior probabilities (BPP) calculated for each of the three methods. BPP was calculated as the proportion of bootstraps in which the SNP of interest was detected given the empirical Bonferroni P value threshold of 1.1 × 10-7. Of the three methods tested on the stage I dataset, we find that Method 1 confers the most powerful test for the genetic association. The BPP values calculated from repeated bootstrap samples by analysis using Method 1 were consistently higher , suggesting the method is more robust to variation caused from sampling than the other two methods tested in this study. Method 2 and 3 had comparable BPP values  and hence similar robustness to sampling variation.To assess variation of the predicted genetic associations, we carried out bootstrap sampling with replacement from the stage I dataset  and calculated the empirical posterior probability at each of the 44 significant SNPs. Table Method 3[Method 1 in our analysis of the stage I data. Using a Bonferroni corrected genome-wide P = 0.05 significant threshold (1.5 × 10-4), we found the SNPs located within 4q21 and 17q21 to be repeatedly detected by all the three methods in the stage II dataset, while an additional six SNPs were detected by Method 1 to be in significant association with the disease trait at the Bonferroni genome-wide threshold  within chromosome 12q12, (P = 2.2 × 10-5), located just 176 Kb from the previously identified PD candidate gene PARK8[Method 2 nor 3 detected this significant SNP. In addition, a significant SNP (rs2878172) within chromosome 14q22.2 detected by Method 1 is only 4 Kb from the gene GCH1, which was recently found to be associated with PD through meta-analysis of multiple PD GWAS datasets and curated in the PDGene database [Before reporting our analysis of the stage II dataset, it is worth stressing that the 345 SNPs originally genotyped were selected only from the previous analysis using Method 3. The datene PARK8; neitherdatabase . A full P = 0.05 threshold (1.5 × 10-4) using Method 1, including all the twenty seven significant SNPs detected using the same method in stage II analysis and eight significant SNPs detected by the same method in stage I data analysis . The SNP marking the PD candidate gene, PARK8, detected in the analysis of the stage II dataset, was also repeated in analysis with the combined dataset. Query against the PDGene database [When the two datasets (stage I and stage II) were combined, 90 SNPs were detected significant at the Bonferroni corrected hreshold .5 × 10-4s Figure . The 55 database , we founSNCA[MAPT[Method 1 discovered two additional SNPs located 1.18 Mb and 0.18 Mb from the PD candidates, FGF20[PARK8[There have been a total of twenty five candidate genes discovered so far to predispose individuals to Parkinson’s disease (the OMIM database with entry 168600). We explored the extent to which these candidate genes can be revealed in the present genetic association study. Listed in Figure SNCA on humanSNCA[MAPT on chromF20[PARK8 respectiMethods 1 and 3 were only implemented to analyze the simulation data because Methods 2 and 3 are effectively interchangeable when the case and control samples were known to come from a single randomly mating population. When the marker and disease genes were in LE (i.e. D = 0 in simulated populations 1–3), marker genotype provides no information on the unobservable genotype at the disease locus and thus no estimate of disease allele frequency q was attempted under the circumstances. Means of the test statistic in these populations were approximately equal to 2.0 or 1.0 for Method 1 or 3 respectively, corresponding to means of the chi-square variable with 2 or 1 degree of freedom as expected for the likelihood ratio test in Method 1 and the chi-square test in Method 3. This demonstrates the adequacy of proposed distribution of the test statistic constructed in these methods under the null hypothesis and, in turn, the appropriate control of the type I error of the statistical tests. When LD was actually present (populations 4–10), Method 1 estimated the model parameters, q and D, adequately (coefficient of variation of the square root of the mean square error is less than 0.5 for parameter estimates) and, provided a consistently higher statistical power (ρ) to test for significance of the disequilibrium than Method 3. We did also investigate the performance of Method 1 when the disease allele showed either dominant or recessive inheritance and the simulation study demonstrates that the method predicted the model parameters well under these different genetic models , Method 1 was able to detect it with a remarkably higher statistical power than the other two methods. In particular, when the disequilibrium had opposite signs in the two contributing populations , the highest detecting power was observed for Method 1 no matter whether the case and control sample was collected from the contributing populations separately or as admixture of the populations. In contrast, the χ2 test based methods failed to detect the disequilibrium under such circumstances. These findings strongly support the improved statistical efficiency of the likelihood-based method presented here and its robustness to inherent genetic structure in the case and control samples.We explored the influence of using case and control samples collected from genetically divergent populations (or cohorts) on performance of the three methods. Table Method 1 when true values of the penetrance parameters were used and when the penetrance parameters were set to be constant, i.e. f1-3 =  as those implemented in the above data analysis. It can be seen that mis-specification of the penetrance parameters in analysis of the simulation data with the method has not caused a marked loss of statistical power in detecting the genetic association nor led to increase in false positive inference of association when the marker and disease loci are truly in linkage equilibrium (D = 0). This thus removes the concern about appropriateness to use the predefined setting of disease penetrance in the method developed here.While the true penetrance parameters at a disease locus are indeed unknown in practice, we proposed incorporation of the penetrance parameters with predefined values of ,  or  into the analysis. This is mainly to ease the problem of over-parameterization and to set the penetrance differ among the different disease genotypes whereas the true but unknown penetrance parameters could be far less than 1 for any single locus genotype that contributes to genetic variation of common polygenic disease traits. We investigated how the use of mis-specified penetrance values would influence performance of the association tests through computer simulation. The simulation considered the scenario where the disease genotypes had very low levels of penetrance. Table We have shown that Armitage’s trend test , the mosWe have solved three major problems in the methodology development. Firstly, genotype at the disease locus is not observable. This has led formulation of the model parameter estimation to be built on the principles of statistical analysis with missing data . Secondlwhich follows a chi-square distribution with 1 d.f.. Comparison of equation (11) to equation (8) shows that the two test statistics share the same numerator, and the denominator of p, from control samples. This is largely because the likelihood function (2) is formulated in terms of two intermediate variables Q and R as described in Table p, q and D. It is thus impractical to work out estimates of these three parameters independently and simultaneously. We explored how the proposed method to estimate p would affect estimation of the parameters q and D and in turn statistical power of the association test. In fact, the marker allele frequency, p, in the controls can be expressed in the present notation asIt needs to be pointed out that a full model involves a total of six unknown parameters and thus presents an over-parameterization problem to statistical analysis under the model. To ease the problem, we have firstly proposed to estimate the marker allele frequency, π = f1q2 + 2f2q(1 - q) + f3(1 - q)2 is the population prevalence of the disease attributed to the disease locus. The second term, i.e. the bias, will be negligible when π is low. To illustrate magnitude of the bias, we worked out the absolute difference |p’-p| and illustrated for a wide ranges of the population settings in Additional file π is less than 10%. It should be stressed that the bias presented here is its largest possible value because it was calculated at the maximum value of the disequilibrium parameter D. In addition, we compared the rate of false positive and statistical power of Method 1 when the true and biased values of marker allele frequency were used in analysis of simulation data under a wide range of settings. The results of the analysis summarized in Additional file where D as defined in Table D’, r2 and some others [D will be inherent to that of the transformed versions of the parameter [In spite that the population genetic model has been focused on the most prominent LD measure, e others , which aarameter . AlthougWe have developed a novel likelihood based statistical approach to model linkage disequilibrium between any genetic marker locus and a putative disease locus in a randomly matting population and to infer the disequilibrium parameter and other population genetic parameters from case and control samples from the population under a likelihood based framework. The model and likelihood based approach are implemented to re-analyze large SNP datasets of the Parkinson disease case and control samples collected from multiple human cohorts. Statistical properties and utility limitations are investigated through simulation studies. Based on the simulation data analysis and analysis with the Parkinson disease case and control sample, we demonstrate that the likelihood based approach outperforms the trend test and logistic regression methods for an increased statistical power and reduced false positive inference, which are popularly implemented in the GWAS literature.The authors declare that they have no competing interests.ZL conceived and designed the study. ZL and MW developed the theoretical analysis. MW, LW, NJ and TJ implemented the simulation and analyzed the PD datasets. ZL and MW wrote the paper. All authors read and approved the final manuscript.Mathematical forms of the coefficients in normal equations (4) and (5).Click here for fileTop two principal components from principal component analysis (PCA) of the stage I dataset.Click here for fileAssociation scans from stage II and two-stage combined samples.Click here for fileScheme A simulation under dominant and recessive genetic models.Click here for fileStructured association using logistic regression.Click here for filePredicting marker allele frequencies from control samples.Click here for fileSimulation results from using biased estimates of maker allele frequency.Click here for file