Phenome-wide association studies (PheWAS) are a high-throughput approach to evaluate comprehensive associations between genetic variants and a wide range of phenotypic measures. PheWAS has varying sample sizes for quantitative traits, and variable numbers of cases and controls for binary traits across the many phenotypes of interest, which can affect the statistical power to detect associations. The motivation of this study is to investigate the various parameters which affect the estimation of statistical power in PheWAS, including sample size, case-control ratio, minor allele frequency, and disease penetrance.We performed a PheWAS simulation study, where we investigated variations in statistical power based on different parameters, such as overall sample size, number of cases, case-control ratio, minor allele frequency, and disease penetrance. The simulation was performed on both binary and quantitative phenotypic measures. Our simulation on binary traits suggests that the number of cases has more impact on statistical power than the case to control ratio; also, we found that a sample size of 200 cases or more maintains the statistical power to identify associations for common variants. For quantitative traits, a sample size of 1000 or more individuals performed best in the power calculations. We focused on common genetic variants (MAF > 0.01) in this study; however, in future studies, we will be extending this effort to perform similar simulations on rare variants.This study provides a series of PheWAS simulation analyses that can be used to estimate statistical power for some potential scenarios. These results can be used to provide guidelines for appropriate study design for future PheWAS analyses.The online version of this article (10.1186/s12859-018-2135-0) contains supplementary material, which is available to authorized users. Phenome-wide association study (PheWAS) has been implemented in a variety of different studies, like within the eMERGE network –5, usingUnlike genome-wide association studies (GWAS) where one phenotype is investigated in a study population, PheWAS uses a wide range of phenotypes collected for a variety of reasons for each dataset, often with minimal curation. Thus, in PheWAS, the data collected for different measurements can vary considerably in sample size, including the numbers of cases for diagnoses, depending on the rarity of the diagnosis. This makes the estimation of statistical power for PheWAS a challenge. For example, in electronic health record (EHR) data, one of the most commonly used data types to define case-control status is through ICD-9 codes; these codes provide information on disease diagnosis, procedures, and medications in the form of three- to five-digit codes. The longitudinal ICD-9 data collected over many years varies between patients due to multiple factors, such as differences in the frequency of patient visits, differences in length of records due to different start and end dates, and incomplete patient medical history. These factors generate sparseness and missing information in the data and, hence, variability in the number of cases, the case-control ratio, and the overall sample size in case-control study designs. These factors can then affect downstream association testing. Three issues exist for measures with low sample sizes: 1) low statistical power to identify or replicate genetic associations and, 2) potentially biased estimates in analyses with low sample size, and 3) an increase in multiple hypothesis testing burden through including low powered phenotypes that may not provide insights but increase the number of statistical tests.In Table 0 in the regression model and set to 0.1 [To investigate the power of the PheWAS approach and the sample size requirements, we simulated an additive genetic model with a range of effect sizes, case-control ratios, and minor allele frequencies. The simulation design assumes no confounding effects due to the environmental exposure and covariates. Genotypes were simulated as SNPs, which were drawn according to Hardy Weinberg Equilibrium (HWE) probabilities. Since multiple diseases and phenotypes are investigated in an EHR-based PheWAS, the prevalence of the diseases could differ in the study population. In our simulated datasets, we used a constant disease prevalence, which is represented as ßt to 0.1 . We reprt to 0.1  to generFor binary phenotypes, we generated the simulated datasets by varying the following parameter settings: cases, case-control ratios, SNP minor allele frequencies, and disease penetrance. In this simulation study, we only investigated a study population with unbalanced and unmatched cases and controls. For example, we simulated a dataset with a random set of 30 individuals. The parameter settings for this simulated dataset were as follows: case-control ratio = 1:2, cases = 10, controls = 20, disease penetrance =0.15, and SNP MAF = 0.01. The simulated dataset was generated for four SNPs and 10 phenotypes, including one SNP-phenotype model with signal, and other models were simulated as noise. The noise was added to evaluate any systematic bias in the power estimates. The noise SNPs were generated by randomly assigning the genotypes in the study population but keeping the MAF the same as the signal SNP. We randomly assigned the cases for the noise phenotypes. Under each parameter setting, we generated 1000 datasets and then calculated associations using logistic regression. Please refer to Fig. For the continuous or quantitative trait simulations, we investigated the power estimates similarly by varying the sample size, minor allele frequencies, and disease penetrance. The simulated dataset was generated for four SNPs and one phenotype, with one signal SNP-phenotype model, and the rest was noise data. We generated 3 noise SNPs as in the binary phenotype simulations. Again, we generated 1000 datasets for each parameter setting and then used linear regression to calculate associations with the quantitative trait. Please refer to Fig. We calculated the power estimates by counting the number of associations below an alpha value based on total number of tests within each set of 1000 simulated datasets for all parameter settings. For binary trait, we used α = 0.00025 (0.01/40) and for the quantitative trait, we used α = 0.004 (0.01/4).We designed a simulation approach with different combinations of genotype and phenotype parameters and then performed association testing so as to investigate the factors that could influence the power to detect the signal.In Fig. We also performed similar simulation analyses on quantitative traits  to identify a sample size threshold for multi-phenotype -based studies like PheWAS. For quantitative traits, we used different sample sizes for the simulation, ranging from 10 to 25,000, as these are based on sample sizes we observed in previously published EHR or clinical trials based PheWAS analyses  is one of the largest healthcare providers in central Pennsylvania, with an EHR system including ~ 1.2 million unique patients. Using the EHR data in Geisinger for ~ 100,000 consented participants in the MyCode Community Health Initiative , we evalUsing the findings from these simulations, we addressed three issues related to low sample size and its impact on PheWAS approach. First, the impact of low sample size is evident in quantitative trait simulations, which suggests that the sample size of 1000 individuals for each phenotype is important to consider in the study design. However, in binary trait simulations, we observed that overall sample size does not affect the power, but instead the number of cases drives the power estimates. Secondly, low Type 1 error across all parameter settings , but there are now many other statistical methods for phenome-wide association analysis . FurtherPheWAS have become a common tool to explore the genotype-phenotype landscape of large biobanks linked to comprehensive phenotype/trait data collections as in EHRs, clinical trials, or epidemiological cohort studies. This high-throughput analysis approach has been met with much success in recent years , 6, 14. Additional file 1:Figure S1. Binary Trait Type I Errors. The plot shows the Type I errors for different parameter settings. Each panel represents the different case number on the top and case-control ratio on the right which was used for the simulation dataset. The Type I error on the y-axis is calculated based on the number of false positive association below significance level of α = 0.00025. The disease penetrance is represented on the x-axis and each colored point represent different MAF used in the simulations. (PNG 379 kb)Additional file 2:Summary Results: The summary results used to generate the figures. (XLSX 56 kb)


We consider the problem of genetic association testing of a binary trait in a sample that contains related individuals, where we adjust for relevant covariates and allow for missing data. We propose CERAMIC, an estimating equation approach that can be viewed as a hybrid of logistic regression and linear mixed-effects model (LMM) approaches. CERAMIC extends the recently proposed CARAT method to allow samples with related individuals and to incorporate partially missing data. In simulations, we show that CERAMIC outperforms existing LMM and generalized LMM approaches, maintaining high power and correct type 1 error across a wider range of scenarios. CERAMIC results in a particularly large power increase over existing methods when the sample includes related individuals with some missing data , because CERAMIC is able to make use of the relationship information to incorporate partially missing data in the analysis while correcting for dependence. Because CERAMIC is based on a retrospective analysis, it is robust to misspecification of the phenotype model, resulting in better control of type 1 error and higher power than that of prospective methods, such as GMMAT, when the phenotype model is misspecified. CERAMIC is computationally efficient for genomewide analysis in samples of related individuals of almost any configuration, including small families, unrelated individuals and even large, complex pedigrees. We apply CERAMIC to data on type 2 diabetes (T2D) from the Framingham Heart Study. In a genome scan, 9 of the 10 smallest CERAMIC p-values occur in or near either known T2D susceptibility loci or plausible candidates, verifying that CERAMIC is able to home in on the important loci in a genome scan. Case-control association testing has proven to be useful for identification of genetic variants that affect susceptibility to disease. One can expect to gain power for detecting such variants by including relevant covariates in the analysis, by accounting for any relatedness of sampled individuals, and by making use of partial information in the data. For analysis of continuously-varying traits, variations on linear mixed-model (LMM) approaches have proven effective at achieving some of these goals. However, for case-control or binary trait mapping, there remain significant challenges. Direct application of LMM approaches to binary traits suffers from power loss when covariate effects are strong, and existing generalized LMM approaches can perform poorly in the presence of trait model misspecification and partially missing data. We propose CERAMIC, a method for binary trait mapping, which is computationally feasible for large genome-wide studies, and which gains power over previous approaches by improved trait modeling, retrospective assessment of significance, accounting for sample structure, and making use of partially missing data. We illustrate this approach in genome-wide association mapping of type 2 diabetes in data from the Framingham Heart Study. In genetic association analysis of a binary trait, such as presence or absence of a disease, it can be useful to properly control for relevant covariates. Inclusion of environmental risk factors in the analysis has the potential to increase statistical power by reducing phenotypic noise, and adjustment for confounding factors can provide some protection against spurious association , 2. For It is of interest to develop an association mapping method for binary traits that addresses the aforementioned challenges, yet has computational time complexity feasible for current scales of genome-wide association studies (GWASs). The classical transmission disequilibrium test (TDT)  and the We consider a somewhat different setting from that addressed by CARAT and GMMAT. We assume a sample that includes at least some closely-related individuals, for whom the pedigree structure is assumed known. We also want to allow for the possibility of incorporating partially missing data, which is not possible with CARAT or GMMAT. We propose CERAMIC , a mixed-model, binary-trait association testing method, which adjusts for relevant covariates and efficiently incorporates missing data to enhance power. The mean and covariance structure of CERAMIC is tailored for binary traits, and it would therefore be expected to outperform LMM methods such as EMMAX,  GEMMA 1 and MASTMajor features of CERAMIC are summarized as follows: (1) it is applicable to essentially arbitrary combinations of related and unrelated individuals, including small outbred pedigrees and unrelated individuals, as well as large, complex inbred pedigrees; (2) it incorporates information on individuals with partially missing data while correctly accounting for dependence; (3) it corrects binary phenotypes for both covariates and additive polygenic effects using a model that exploits the binary nature of the trait; and (4) it is computationally feasible for current association studies. For comparison, we have developed two other retrospective tests that also have features (1), (2) and (4) above, but which do not model additive polygenic effects: MQLS-LOG, which uses a logistic approach to adjust binary phenotypes only for covariates, not for polygenic effects, and MQLS-LIN, which uses an ordinary least-squares regression approach to do the same. We conduct simulation studies to assess the type 1 error and power of our methods, CERAMIC, MQLS-LOG and MQLS-LIN, and to compare them to previously reported methods, MASTOR, EMMAX, GEMMA, CARAT and GMMAT. Finally, we apply CERAMIC to the analysis of type 2 diabetes (T2D) data from the Framingham Heart Study (FHS).n individuals. Denote the phenotype by a vector, Y, of length n, where the ith element, Yi, is the value of the phenotype for individual i. Suppose we also observe k − 1 ≥ 0 covariates, encoded in a design matrix, X, of dimension n × k, where an intercept  is always included in X in addition to the k − 1 non-trivial covariates. Let Xi denote the column vector of covariate values for individual i (so that ith row of the matrix X). Let G denote the vector of genotypes for the n individuals at a tested biallelic variant, where Gi = 0, 1, or 2 denotes minor allele count of the ith individual at the variant. In the following sub-sections, we first briefly review the CARAT method before presenting our CERAMIC approach.We consider a binary trait measured on a sample of Y given genotype and covariate information are specified. The conditional mean is given byβ is a k-dimensional vector of unknown covariate effects, and γ is the unknown scalar association parameter. The conditional variance is given byΓ is an n × n diagonal matrix with ith diagonal element equal to μi(1 − μi), I is an n × n identity matrix, Φ is a genetic relationship matrix, and 0 ≤ ξ ≤ 1 is an unknown scalar parameter measuring the relative importance of additive polygenic vs. i.i.d. error variance in explaining trait variability. The appearance of the terms Γ1/2 in the formula for Ω in i, the relationship between the conditional mean and variance given by Var = μi(1 − μi) holds, as is necessary for a binary random variable. In CARAT, Φ is taken to be an empirical genetic relationship matrix calculated from genome-wide data. The unknown parameter for CARAT is , where, for association analysis, γ is the main parameter of interest, while  is typically considered a nuisance parameter.CARAT  was deveH0: γ = 0 is tested against H1: γ ≠ 0. To form the CARAT test statistic, we first obtain the null estimate, γ, β, ξ) by iteratively solving a system of estimating equations under the constraint γ = 0. The quasi-score estimating equation for β is given byμ = μ = T. The estimating equation for ξ is given byγ constrained to be 0. The CARAT test is based on the quasi-score statistic,μ, Σ and Γ evaluated at To detect association between the phenotype and the SNP, ferences  and 19]H0: γ = 0p ≤ 1 is the unknown allele frequency of the variant of interest, p. The CARAT test statistic can then be defined byTo assess the significance of the association test, CARAT takes a retrospective approach, in which it is assumed that, under the null hypothesis of no association,γ, β, ξ) by solving a system consisting of Eqs γ:In addition to testing for association, one can also obtain an estimate of the full parameter  Let the kinship matrix derived from the pedigree information be given byϕi,j is the kinship coefficient between individuals i and j, and hi is the inbreeding coefficient of individual i.Consider a sample of Y, we adopt a quasi-likelihood framework similar to that of CARAT, in which the mean structure is given by Y given X and G, shown in Φ. As we will show in the next subsection, we use the pedigree-based kinship matrix as part of our approach to extract additional information for association from partially missing genotype data on related individuals, so it is convenient  to also use the pedigree-based kinship matrix for Φ in To model the binary trait variable X, G, and Y are fully observed), to detect association between the binary phenotype and the genetic variant, we first obtain the quasi-score statistic defined as in In the complete data case . Then the null mean assumption allows, e.g., for different sub-populations to have different allele frequencies. The null variance assumption in p is the allele frequency at the variant of interest, which would hold under Hardy-Weinberg equilibrium. Instead we use a more robust variance estimator [P = Φ−1 − Φ−1X(XT Φ−1X)−1XTΦ−1. Note that G on X, with covariance matrix proportional to Φ. The CERAMIC test statistic in the complete data case is then given byZ is defined in c” on CERAMICc stands for “complete data.” Under suitable regularity conditions, significance of association could then be assessed by comparing CERAMICc to a For the retrospective analysis, we make the following modeling assumptions about the distribution of stimator  given byγ, β, ξ) by iteratively solving the system consisting of Eqs Γ and Σ, respectively, evaluated at n, being large. Calculation of the standard error for the estimator, Y, so it is not available in our approach, in which we need only specify the mean and variance structure of the phenotype.In addition to testing for association, one can also obtain an estimate of the full parameter . To be retained in S, individuals with missing genotype are required to have non-missing phenotype and covariate information, and, in addition, to satisfy at least one of the following two conditions: (1) the individual has a genotyped relative; or (2) the individual is in the same pedigree with an individual with non-missing phenotype and covariates who either has non-missing genotype or has a relative with non-missing genotype at the tested variant. Let W = U ∪ S, so W is the set of w = u + s individuals remaining in N who have complete phenotype and covariate information. Notice that the sets N, R, U, V, S, and W can, in principle, vary across tested variants that have different patterns of genotypic missingness. This point is discussed in more detail in the subsection Some computational considerations for CERAMIC.We let R combined with the phenotype and covariate data for the individuals in the set W. As in the case of complete data, we first obtain an estimator of the phenotypic nuisance parameter, , under the null hypothesis, H0: γ = 0. This estimator, which we call γ set to 0, where all vectors and matrices in Eqs W. We then let ZW denote the vector of transformed null phenotypic residuals for the set W, whereΓ, Σ, and μ, respectively, restricted to the individuals in set W and evaluated at YW is the vector Y restricted to the individuals in set W.To form the CERAMIC test statistic in the case of partially missing data, we propose to use the genotype data for the individuals in the set F = MΦRWZW is a vector of length r that incorporates phenotype, covariate, and pedigree information, in which 1r is a vector of length r with every element equal to 1, and ΦRW denotes the submatrix of Φ with rows corresponding to the individuals in R and columns corresponding to the individuals in W, i.e., the th element of ΦRW is 2ϕij, where ϕij is the kinship coefficient between the ith individual in set R and the jth individual in set W. The variance estimator [Q, of individuals in N who have non-missing genotypes and covariates but may or may not have observed phenotypes (U ⊂ Q ⊂ R), q = |Q|, i.e.GQ, XQ, and ΦQ are G, X, and Φ, respectively, restricted to the individuals in set Q. With complete data, i.e., N = R = U = W and S = V = ∅, the CERAMIC statistic of c statistic of We define the CERAMIC statistic with partially missing data to bestimator , σˇG2, iS. More generally, we can letGW, where GW is the vector G restricted to the individuals in set W, 1w is a vector of length w with every element equal to 1, and p, of the variant of interest. Consider ith element of i is in set U, then i, while if individual i is in set S, then i. In other words, if we reorder the individuals in set W so that the individuals in set U come first in the list and the individuals in set S follow, then we can writeGU is the vector G restricted to the individuals in set U, and S. The CERAMIC statistic can then be rewritten , we need to obtain the inverse of the n × n matrix Σ = ξΦ + (1 − ξ)I for different values of ξ. We reduce the computational burden in two ways: (1) Σ is inverted block-wise where each block of Σ corresponds to a pedigree; and (2) a single spectral decomposition, Φ = AJAT , is used to compute the inverse of Σ for different values of ξ, because Σ−1 = A(Jξ + (1 − ξ)I)−1AT, where Jξ + (1 − ξ)I is a diagonal matrix [In order to carry out the iterative solution of the system of estimating equations given by Eqs l matrix .ZW, depends on the set W, which is a function of the genotypic missingness pattern for the genetic variant being tested. In a GWAS, different SNPs often have different genotypic missingness patterns, so this could imply that in the worst case scenario, ZW would need to be computed separately for each SNP in the genome. One possible way to avoid this would be to compute ZW only once per genome scan based on all individuals with non-missing phenotype and covariate information  and use the same ZW for association testing with respect to all SNPs across the genome. Another approach, which is the one we actually take, would be to estimate the variance component (VC) parameter, ξ, once per genome using all individuals with non-missing phenotype and covariate information, and then for each SNP, compute only the regression parameter, β, by solving γ = 0. In this way, we need only solve ZW separately for each SNP, which drastically reduces the computational burden.The calculation of the transformed null phenotypic residual vector, Results section, we compare them, in terms of type 1 error and power, to CERAMIC and to three previously proposed tests, MASTOR, EMMAX, and GLOGS.In addition to CERAMIC, we also developed two other alternative approaches to association testing for binary traits in related individuals. These two methods, which we call MQLS-LOG and MQLS-LIN, both have the following features: (1) they incorporate covariates; (2) they are generalizations of the MQLS test ; (3) theZW, by some other type of residual vector that is a function of . To obtain the MQLS-LOG test statistic from the CERAMIC test statistic, we replace ZW by the vector of residuals from the logistic regression model, where this model is given byβ under the model of ϵ be the resulting null phenotypic residual vector, defined to have ith element i ∈ W, where ϵT1w = 0 for logistic regression.The MQLS-LOG and MQLS-LIN test statistics can each be constructed from the CERAMIC test statistic of Eqs ZW by the vector of residuals from the ordinary linear regression model, where this model is given byβ under To obtain the MQLS-LIN test statistic from the CERAMIC test statistic, we replace Under the same assumptions as for CERAMIC, the MQLS-LOG and MQLS-LIN test statistics both have In simulations, we assess the type 1 error and power of the three methods we propose, CERAMIC, MQLS-LOG, and MQLS-LIN, and we compare them to five previously-proposed methods, EMMAX , GEMMA , MASTOR X(2) denote the column vector of age values for a simulated sample. Sampled individuals from three-generation pedigrees have their sex pattern fixed as shown in X(3) denote the column vector of sex values for a simulated sample. Height is simulated as a heritable trait that exhibits correlation among family members and depends on age and sex. Let X(4) denote the column vector of height values for a simulated sample. The model for height is multivariate normal, given byν(X(2), X(3)), has entry 172.5 for a male with age ≥ 65 and entry 176.5 for a male with age < 65. For a female with age ≥ 65, the entry in the mean height vector, ν(X(2), X(3)), is 160.2, while for a female with age < 65, the mean height is set to be 163.2. Let X(5) denote the column vector of the values of the i.i.d. normal covariate for a simulated sample. The entries of X(5) are generated from the N distribution, independently of all other covariates. Let X = , ⋯, X(5)) be the covariate matrix consisting of an intercept and the four covariates described above.We simulate genotype, covariate and phenotype data for a sample that includes some unrelated individuals and some individuals in three-generation pedigrees see . For eaci = 1, …, n, where ith row of X, u = T is a vector of additive polygenic effects, Gi =  represents the genotypes at causal SNPs 1 and 2 for individual i, and f(Gi) is a function of Gi that represents the combined genetic effect, on the phenotype, of causal SNPs 1 and 2. The coefficient vector β is chosen to satisfy two conditions: (1) the mean of the covariate effects on the logit scale, u, independent of covariates, have a multivariate normal distribution with mean 0 and covariance matrix θc = 1 − θa denote the fraction due to covariate effects. We fix θa, θc) takes possible values , , , ,  and , representing a range of the relative importance of additive genetic effects vs. covariate effects. Note that the model also results in Bernoulli error in the phenotype that is conditionally independent across individuals and that accounts for ∼ 20% of total phenotypic variance in our simulation scenarios .Two types of trait model are considered in our simulation studies. One is a mixed-effects logistic regression model, which has the following form:i are encoded in Gi, are unlinked with minor allele frequencies (MAFs) .1 and .2 respectively, and they are generated independently of the covariates and additive polygenic effects. They act on the phenotype epistatically: an individual with at least one copy of the minor allele at causal SNP 1 and at least one copy of the minor allele at causal SNP 2 has mean penetrance E(Yi|G) ≈ .15; an individual with a genotype not satisfying that condition has mean penetrance E(Yi|G) ≈ .05. (Note that a target mean penetrance can be achieved by setting f(Gi) to be an appropriate value, obtained in a simulation-based approach.) The resulting prevalence is E(Yi) ≈ .057.Causal SNPs 1 and 2, whose genotypes for individual Li is the underlying liability for individual i, and λ(Gi) represents the individual’s liability threshold, beyond which the disease is activated, as a function of individual i’s genotypes at causal SNPs 1 and 2. The liability Li consists of three components: ui, the random additive polygenic effects, and ϵi, which represents measurement error or environmental effects assumed to be acting independently across individuals. The Xi’s and the ui’s have the same distributions as described above. The error terms ϵ1,⋯,ϵn are i.i.d. X, u). We fix the total liability variance β and πa, πc) to take on the possible values , , , , and , representing a range of the relative importance of additive genetic vs. covariate effects. Gi, individual i’s genotypes at causal SNPs 1 and 2, has the same distribution as for the mixed-effects logistic regression model, and in each setting, the values of λ(Gi) are chosen so that an individual with at least one copy of the minor allele at causal SNP 1 and at least one copy of the minor allele at causal SNP 2 has mean penetrance E(Yi|Gi) ≈ .15, while an individual with a genotype not satisfying that condition has mean penetrance ≈ .05. The resulting prevalence is again ≈ .057.The other type of model we consider is a liability threshold model,λ(Gi), so that the prevalence is reduced to.01.In addition, we consider two variations on the liability threshold model. In the first variation, we model the effects of shared environment by incorporating a sibship random effect that accounts for 10% of the total liability variance. In that case, we set the error variance to also account for 10% of the total liability variance, and the additive polygenic and covariate effects are as described in the previous paragraph. In the second variation on the liability threshold model, we modify the threshold values, In simulations, we consider both the case when there is complete genotype, phenotype and covariate data as well as cases with missing data. In the complete data case, we simulate genotype, covariate and phenotype data, according to one of the binary trait models described above, for either a 600-person sample , independently across individuals and families, and his or her genotype at the tested locus is assumed to be non-missing if and only if at least one of the following two conditions holds: (1) the individual has non-missing phenotype and is affected, or (2) at least half of the individual’s first degree relatives who have non-missing phenotypes are affected. Among the unrelated individuals, all their phenotypes and covariates are assumed to be missing, and all their genotypes are assumed to be non-missing, i.e., they are included as controls of unknown phenotype.For scenarios with missing data, we simulate genotype, covariate and phenotype data for either a 1,200-person sample , we use phenotype and covariate information from 27 clinical exams, for cohort 2 , we use information from 7 clinical exams, and for cohort 3  we use information from 1 clinical exam. We determine the T2D phenotype status in a similar way to that in a previous work . For ind−7, the SNP is removed from our study. Under this screening procedure, we exclude an additional 1,032 SNPs, resulting in a final set of 367,770 SNPs to include in the analysis.Among the 9240 study individuals for whom Affymetrix 500K genotype data are available, we exclude individuals who have either (1) completeness  ≤ 96%, or (2) empirical self-kinship coefficient Methods. In each simulation scenario, phenotypes are generated according to either the mixed effects logistic regression model of θa, θc) =   =   individuals with partially missing data are included in the analysis for every statistic . In every scenario, we simulate 1,200 individuals with missing data and use ascertainment setting A. Association is tested with an unlinked, unassociated SNP with MAF .2. We perform additional simulations in which we compare the type 1 error rate of GEMMA to those of CERAMIC, MQLS-LOG and MQLS-LIN. Phenotypes are generated according to the liability threshold model of In an association study, the correct trait model is generally unknown. In particular, it may not be known which covariates should be included in the model. In Methods. GMMAT offers two options to deal with missing genotype data: the missing genotypes can either be removed or replaced by the estimated mean genotype value. In practice, we found that the two options give identical results, so the simulation results we report for GMMAT apply to both options. Panel A in We compare the power of CERAMIC to that of MQLS-LOG, MQLS-LIN, MASTOR, EMMAX, GMMAT and CARAT in various simulated scenarios with missing data and ascertainment, as described in From Panels A and B in θa ≤ .4 or πa ≤ .2), the methods that fit a logistic mean structure outperform the methods that fit a linear mean structure, i.e., CERAMIC and MQLS-LOG outperform MASTOR and MQLS-LIN, while CARAT and GMMAT outperfom EMMAX. For the mixed-effects logistic regression trait model, this is not surprising, because the simulated model also has a logistic mean structure. However, it is notable that, among methods that treat missing data in the same way, the methods that fit a logistic mean structure also outperform those that fit a linear structure in the case of the liability threshold model, in which the simulated model does not have a logistic mean structure. This improvement may be due to the flexibility afforded by the nonlinearity of a logistic mean structure. Through the power comparison of CERAMIC to MQLS-LOG and that of MASTOR to MQLS-LIN, we can see that fitting the additive VC (as in MASTOR and CERAMIC) does not harm power in any scenario, and it improves power when the additive polygenic effects are large relative to covariate effects . When additive polygenic effects are large, GMMAT has the lowest power of all methods. This might reflect known limitations [In Panels A and B of itations  of the pπa, πc) =  with 1,200 individuals in each simulated replicate, under acertainment setting A with missing data. Because the data are simulated, the missing genotype values are actually available, so we can determine what the power would have been for each of the three methods had the genotype data not been missing. This power is represented in the leftmost set of three bars in We perform additional simulations to compare the ability of the three binary trait methods, CERAMIC, GMMAT and CARAT, to recover power from partially missing genotype information when mean values are plugged in for missing genotype values in both GMMAT and CARAT. We simulate under the liability threshold trait model with (From θa < 1 or πa < .8) and does not compromise power in the other cases.Because the trait model typically cannot be known with certainty, it is always possible that relevant covariates may be left out of the fitted model. Panels C and D of θa ≤ .8 in Panel C and πa ≤ .6 in Panel D), likely reflecting the greater robustness to trait model misspecification of the retrospective methods. The power difference between the methods that correct for missing data (CERAMIC and MQLS-LOG) and the others is quite large (up to 6-fold), particularly in the settings in which covariates play an important role.In Panels C and D, it can be seen that CERAMIC has the highest power in all settings. In contrast, GMMAT is severely underpowered, having the lowest power (or power not significantly different from the lowest) for all settings. Among the three methods that do not correct for missing data , CARAT, which is a retrospective method, has the highest power for all the settings in which the model is misspecified (θa ≤ .2 or πa ≤ .2). For EMMAX, in particular, this could be explained in more detail by the fact that EMMAX assesses the variation of the test statistic based on phenotypic variance , and failure to adjust for covariates would lead to inflation of the estimated phenotypic variance, and hence, to a reduction in power, whereas the retrospective methods such as MASTOR and CERAMIC assess variation based on genotypic variance , so are robust to power loss arising from misspecification of the phenotype model. Regardless of whether or not covariates are adjusted for, CERAMIC has higher power than EMMAX when covariate effects are large relative to additive polygenic effects (θa ≤ .2 or πa ≤ .2), and maintains similar power to EMMAX in other scenarios.With complete data, when all relevant covariates are included in the model, the three binary trait methods have approximately equal power in all scenarios. However, when relevant covariates are excluded from the model, the power of the prospective method GMMAT is lower than that of the retrospective methods CERAMIC and CARAT. The MQLS-LOG method, which has a logistic mean structure but does not include an additive polygenic VC has power approximately equal to that of the most powerful binary trait methods (CERAMIC and CARAT) when the additive polygenic variance is low, but loses power when the additive polygenic variance accounts for a high proportion of the trait variance. The EMMAX and MASTOR methods, which have an additive polygenic VC but linear instead of logistic mean structure, have power approximately equal to that of the most powerful methods  when covariates do not play an imporant role in the trait model and additive polygenic variance does. Compared to the retrospective method MASTOR, the prospective method EMMAX loses power when relevant covariates are omitted from the fitted model in the settings in which covariates play an important role in the trait model , we restrict consideration to methods that did not experience type 1 error problems in our simulations. We compare CERAMIC, MASTOR, EMMAX, MQLS-LOG and MQLS-LIN. Tables −5. The two SNPs with the smallest p-values, rs4506565 and rs7901695, are in an intron of TCF7L2 (MIM 602228), which has been extensively reported to have strong association with T2D [TLE1 (MIM 600189) has previously been reported and replicated as a T2D susceptibility locus [GALNT9 (MIM 606251) is the left flanking gene of SNP rs10747083 previously found to be significantly associated with fasting glucose [DLGAP1 has previously been associated with serum insulin-like growth factor-binding protein 3 (IGFBP-3) levels [DLGAP1 has also been previously associated [PALLD has previously been associated [with T2D –28. Amonty locus , 30, and glucose . DLGAP1 ) levels . DLGAP1 sociated  with levsociated  to be associated  with aspsociated .From http://www.stat.uchicago.edu/~mcpeek/software/index.html. We report run times for CERAMIC in simulations and in analysis of the FHS data set. All runs are completed using only one core (at 3.5GHz) of Intel Xeon CPU E5-2637 v3. In CERAMIC, the time-limiting step is the incorporation of missing data, which depends strongly on the sizes of the individual pedigrees making up the sample because the missing data incorporation is based only on genotype and phenotype information from the close relatives of the individuals with missing genotype. Therefore, because incorporation of missing data is the slowest step, for fixed size of the families making up the sample, the computation could be expected to scale approximately linearly in sample size. We report run times on simulated data sets with varying sample sizes, where each sample consists of 20% unrelated individuals and 80% related individuals in families of the type in 3, 2 × 103, 4 × 103, 6 × 103, 8 × 103 and 1 × 104, we obtain run times of 4.9, 7.8, 16.2, 23.6, 32.8 and 39.0 minutes, respectively. These run times are plotted in CERAMIC as well as MQLS-LOG and MQLS-LIN are implemented in the CERAMIC softare, which will be made freely available at For the FHS data set, which contains 8080 individuals, it takes approximately 4.2 hours to perform a scan of 367,770 SNPs with phenotypic residuals computed once per genome screen, and approximately 6.1 hours if phenotypic residuals are computed separately for each SNP. This time is greater than would be needed for a sample of 8080 individuals of the type in our simulations because Framingham includes several families that each have hundreds of individuals, for whom the missing data step is much more time consuming. However, even in this case, the computation time would be expected to scale approximately linearly as the sample size increased, for a fixed family complexity. With complete data, the computations could be substantially sped up by taking a different algorithmic approach, similar to those used in LMM methods. However, such an approach is not optimal for partially missing data. In all cases, the computations are easily parallelized across tested variants.For genetic association mapping of binary traits in samples with related individuals, we have developed a new method, CERAMIC, which incorporates pedigree and covariate information and effectively handles partially missing data. CERAMIC is applicable to samples that contain essentially arbitrary combinations of related and unrelated individuals. CERAMIC can be viewed as a hybrid of logistic regression and LMM approaches. Like LMM methods, CERAMIC incorporates an additive component of variance and can accommodate related individuals in a computationally feasible way. Like logistic regression methods, CERAMIC uses a logistic function to model the effects of covariates on a binary trait, and it accounts for the dependence of the variance on the mean (i.e. Bernoulli variance). As a result, CERAMIC is able to gain power, over LMM methods, for association mapping of binary traits. In addition to adjusting for covariates, CERAMIC can increase power by incorporating partially missing data. CERAMIC is based on a set of estimating equations, and we take a retrospective approach to assessment of significance of the test statistic, which provides a way to more easily incorporate partially missing data and also leads to robustness of the method to misspecification of the phenotype model. CERAMIC is implemented in freely-available software and is computationally feasible for current genome-wide association studies.In simulations, we demonstrate that CERAMIC outperforms previously-proposed binary-trait methods GMMAT and CARAT in scenarios with partially missing data, with CERAMIC giving large increases in power over the other two methods in many scenarios. CERAMIC also outperforms GMMAT when the trait model is misspecified, with both large increases in power and also improved type 1 error control over GMMAT. When there are no missing data and the correct set of covariates is included in the fitted model, the three methods have approximately equal power. We show that the sophisticated handling of partially missing data in CERAMIC can recover a large portion of the power of complete data. In contrast, imputation of the mean genotype value for missing genotype data in GMMAT or CARAT does a poor job, recovering almost no power.In a range of simulated scenarios with different types of trait models, various levels of relative importance of covariate effects and additive polygenic effects within a trait model, and either complete data or partially missing data, CERAMIC outperforms or performs as well as the best-performing of the other methods considered, MQLS-LOG, MQLS-LIN, MASTOR, EMMAX, GMMAT and CARAT. In addition, we have demonstrated that, when covariates play a major role in the trait model and relevant covariates are included in the fitted model, the methods that incorporate a logistic mean structure tend to perform better than those that incorporate a linear mean structure, even when the underlying trait model is not logistic, but instead follows a liability threshhold model. When additive polygenic effects play a major role in the trait model, the methods that include an additive polygenic VC tend to have higher power than the other methods. When data are partially missing among related individuals, the retrospective methods that incorporate sophisticated missing data handling  boost power by exploiting information contained in partially missing data.TCF7L2 [TLE1 [We apply our methods to analysis of T2D in the FHS data, where we replicate association with two previously-identified T2D susceptibility loci TCF7L2 –28, and L2 [TLE1 , 30. In γ, for example, as a way to quantify the strength and direction of association and/or to build a predictive phenotype model. With complete data, γ can be estimated by γ, rather than to test for association, then we first need to make a careful choice of the set of individuals to be included in the estimation. We would naturally include all the individuals in U, the set of individuals with complete data, and we can also include a subset S′ ⊂ S, where S′ is a set of individuals who have non-missing phenotype and covariate information, and whose genotypes can be informatively estimated from genotyped relatives. We can then set W′ = U ∪ S′ and use W′ and S′ substituted for W and S, to obtain the BLUP vector γ can be estimated by solving the system of Eqs W′ and where G. In the presence of substantial amounts of missing data, the choice of the set S′ could potentially impact the properties of the resulting estimator of γ. While including in the estimation some ungenotyped individuals whose genotypes can be informatively estimated could improve the precision of the estimator by drawing information from genotyped relatives, inclusion of ungenotyped individuals on whose genotypes the data provide relatively low information could bias the estimate of γ. This is in contrast to the testing problem, in which including such individuals would simply provide a relatively low amount of additional power.In genetic association studies, it can be of interest to estimate the association parameter g(x) and V(μ) are the link and variance functions, respectively, chosen for the given exponential-family distribution [g(x) and V(μ) are the logit function (i.e. log(x/(1 − x))) and Bernoulli variance μ(1 − μ), while for Poisson traits, typical choices would be g(x) = log(x) and V(μ) = μ. Furthermore, the correlation matrix Σ can be extended to include more VCs  by assuming In deriving phenotypic residuals, we have used an estimating equation framework to estimate the regression coefficients and VCs. This framework, although built specifically for binary traits, can be generalized to traits with a general exponential family distribution, e.g., count phenotypes distributed as Poisson. For such traits, we propose a general mean and variance structure for testing the null hypothesis,ribution . For binS1 Text(PDF)Click here for additional data file.S2 Text(PDF)Click here for additional data file.S1 Table(PDF)Click here for additional data file.S2 Table(PDF)Click here for additional data file.S3 Table(PDF)Click here for additional data file.S4 Table(PDF)Click here for additional data file.S5 Table(PDF)Click here for additional data file.S6 Table(PDF)Click here for additional data file.S7 Table(PDF)Click here for additional data file.S8 Table(PDF)Click here for additional data file.S9 Table(PDF)Click here for additional data file.S10 Table(PDF)Click here for additional data file.S11 Table(PDF)Click here for additional data file.S12 Table(PDF)Click here for additional data file.S13 Table(PDF)Click here for additional data file.S1 Fig(TIF)Click here for additional data file.


Gene-gene interactions (GGIs) are a known cause of missing heritability. Multifactor dimensionality reduction (MDR) is one of most commonly used methods for GGI detection. The generalized multifactor dimensionality reduction (GMDR) method is an extension of MDR method that is applicable to various types of traits, and allows covariate adjustments. Our previous Fuzzy MDR (FMDR) is another extension for overcoming simple binary classification. FMDR uses continuous member-ship values instead of binary membership values 0 and 1, improving power for detecting causal SNPs and more intuitive interpretations in real data analysis. Here, we propose the fuzzy generalized multifactor dimensionality reduction (FGMDR) method, as a combined analysis of fuzzy set-based analysis and GMDR method, to detect GGIs associated with diseases using fuzzy set theory.Through simulation studies for different types of traits, the proposed FGMDR showed a higher detection ratio of causal SNPs, compared to GMDR. We then applied FGMDR to two real data: Crohn’s disease (CD) data from the Wellcome Trust Case Control Consortium (WTCCC) with a binary phenotype and the Homeostasis Model Assessment of Insulin Resistance (HOMA-IR) data from Korean population with a continuous phenotype. The interactions derived by our method include the pre-reported interactions associated with phenotypes.http://statgen.snu.ac.kr/software/FGMDR.The proposed FGMDR performs well for GGI detection with covariate adjustments. The program written in R for FGMDR is available at  In many genetic association studies, despite successful identification of genetic factors that govern various phenotypes, parts of heritability remained unexplained as the ‘missing heritability’ . For exaAmong efforts to explain the missing heritability, the analysis of gene-gene interactions (GGIs) has been studied to understand the etiology of common complex traits using statistics and machine learning . Among tAmong the many extensions of MDR, GMDR tests GGIs using residuals of a generalized linear model as score statistics. This idea permits adjustment of covariates, addressing both binary and continuous phenotypes . In manyIncluding GMDR, MDR-based methods reduce a dimensionality of genetic information of multiple SNPs to one dimension with binary values (high-risk: H or low-risk: L). This basic idea of MDR, makes all binary interaction models detectable and potentially extending to numerous types of data and methods. However, these extensions, like MDR frameworks, are based on traditional classifications that allow each genotype combination to belong to only one of high/low risk groups. In traditional classification, the class membership value is binary, and an object is a member of a class or not. Such traditional classification may not reflect real phenomena in biological and medical studies, because traditional classification approach can imply the following shortcomings. On one hand, genetic variants with similar characteristics can be classified into different risk groups. On the other hand, genetic variants with different characteristics can be classified into the same risk groups. For example, in GMDR , each ceThe fuzzy set, introduced by Zadeh , handlesIn our previous study , we wereIn this paper, we propose fuzzy set-based generalized multifactor dimensionality reduction (FGMDR) to detect GGIs while allowing for covariate adjustment. Since FGMDR is based on the generalized linear models, it can be applied to both quantitative and binary traits. FGMDR serves as a generalized MDR framework, including Fuzzy MDR, MDR, and GMDR. Through simulation studies with different epitasis models, as listed by Velez et al. , we compThe remainder of this paper is organized as follows: the GMDR framework is briefly reviewed, and the algorithm of FGMDR is proposed. The power of the proposed FGMDR, using several simulations under different epitasis models, is presented. We then present the results of FGMDR applied to Crohn’s disease (CD) dataset and a homeostatic model assessment of insulin resistance (HOMA-IR) dataset. Finally, the results are discussed and put into logical context.yidenote the phenotype of individual i with expectation E(yi) = μi. In general, this can be represented by the following generalized linear model (GLM): l(μi) is the link function, α is the intercept, and xi is a vector that expresses possible genotype combinations of interest. The variable zi is a vector representing environmental factors, and β and γ are coefficient vectors. In the first step, the residual based on GLM, is calculated from the null model β = 0. At the second step, the average value of residuals is calculated within each multifactor cell of contingency table to classify each SNP combination. Cells are then classified either as “high risk group H”, if the average value is nonnegative (or meets or exceeds a preassigned threshold T) or as “low risk group L”, if the average value is negative (or does not exceed threshold T). At the third step, the balanced accuracy (BA) is calculated using the sum of residuals. Through a 10-fold cross-validation, the best k-way model having the minimum prediction error and maximum cross-validation consistency is selected.Lou et al.  proposedGMDR framework is based on traditional classification, allowing each genotype combination to belong to only one of H/L groups. However, this classification may not reflect characteristics of genotype combinations corresponding to “tied” cells. To overcome this drawback, fuzzy set allows for partial membership for H/L groups, in the GMDR framework.x, μA(x)) | x ∈ X }, where μA(x) on  represents the “degree of membership” of x in the fuzzy set A. When A is a classical set, its membership value is to be 1 or 0, as to whether or not an element is a member of a set. Thus, a classical set is considered a special case of a fuzzy set. GMDR, therefore, uses the traditional classification based on classical set, to reduce the dimensionality of genotype combinations, by grouping cells into H/L groups. By adopting the fuzzy set theory, we propose an FGMDR representing H/L groups by two fuzzy sets, which are identified by the membership functions μH and μL, respectively. By introducing these membership functions, FGMDR allows each genotype combination to partially belong to both H/L groups, while GMDR restricts each genotype combination to belong to only one of H and L groups.The fuzzy set proposed by Zadeh has been employed to handle the concept of partial membership of elements in a set . The onlyi for individual i, GMDR uses a studentized (standardized by a sample-based estimate of a population standard deviation) residual based on GLM as a score for GMDR, as follows: Si=For the phenotype p SNPs, let j = {1, ⋯, 3k},  k is a number of SNPs in an interaction model (interaction order). Since GMDR uses balanced accuracy based on classical sets of H/L groups having membership values 0 and 1, the magnitude of the value tl, th (tl ≤ th) need to be determined a priori.For a dataset having j-th multifactor cell, and j-th multifactor cell. Then, the balanced accuracy (BA) using the membership function, BAFUZZY, is defined as the arithmetic mean of SENFUZZY and SPEFUZZY introduced in . We tested various coefficients of disease models. In the results from tests, consistent patterns were seen. Therefore, in each scenario, coefficients of disease models are adjusted to about a 50~ 60% average success rate for all the methods.Common to all three scenarios, genetic effects are based on 70 penetrance models , without marginal effect , and powYi = α + XiTβ + ZiTγ + εi, where Yi represents a phenotype value, Xi represents a genetic effect, Zi represents covariates of the ith individual, and εi represents the error. Xi is randomly selected based on normal distribution of the mean: a penetrance value corresponding to the genotype value of the ith individual and standard deviation 0.1. Zi is randomly selected on a normal distribution of mean 0, and standard deviation 0.7. εi is randomly selected on a normal distribution of mean 0, and standard deviation 1. All values of coefficients  are the same as one. Simulation results of scenario 1 are summarized in Fig. Since scenario 1 simulates a continuous phenotype, we used the following model with an identity link function p-value: 2.82e-10), FMDR .In Fig. Since scenario 2 simulates a binary phenotype determined by a continuous value, we calculated a continuous value, and discretized as 1 for higher than a specific threshold , or 0 for the others. For a continuous value calculation, we used the same identity link function in GLM as scenario 1. All other parameter values, except the standard deviation of error (0.5), are the same as in scenario 1. The simulation results of scenario 2 are summarized in Fig. p-value: 1.35e-07), FMDR  and also higher than that of GMDR, but not significantly .In Fig. Since scenario 3 simulates a binary phenotype with a probability using a logit model given below:β is reduced to 0.5, and the standard deviation of the error increased to 2. The simulation results of scenario 3 are summarized in Fig. In this scenario, the value of p-value: 1.31e-07), FMDR (1.36e-07) and GMDR .The simulation results of scenario 3 in Fig. p-value of likelihood ratio test: 6.93E-11). Among SNPs, we selected 30 SNPs reported to associate with the CD phenotype [P-values and their rank of Table The CD data in Wellcome Trust Case Control Consortium  dataset,henotype , 30, 31 We next performed FGMDRs with/without covariate adjustment, with 10-fold cross validation, from two to five-locus SNP combinations, as summarized in Table BAFUZZY in testing data, and its relatively high cross-validation consistency (CVC) value. Interaction of this SNP combination is represented in Fig. Among these results, we selected a four-locus order: 4) SNP combination as the best SNP combination, based on the best  SNP combIn Fig. Figure Additionally, the interaction between IR23R (SNP1) and ATG16L1 (SNP5) for CD, was reported and in a case-control study within a cohort study , and revP-values and their rank in Table We next analyzed HOMA-IR data from the Korea Association REsource project (KARE) to illustrate FGMDR in the context of quantitative traits. A total of 8577 samples are available, after removing subjects with at least one missing phenotype value. The genomic DNAs were genotyped using Affymetrix Genome-Wide Human SNP Array 5.0. For GGI analysis using our FGMDR, we used only 10 candidate SNPs identified in earlier studies –36 from Using the residuals calculated from (2), FGMDR was then performed.p-value in Table We performed FGMDRs with/without covariate adjustment, with 10-fold cross validation from two to five-locus SNP combinations and summarized the results of HOMA-IR in Table BAFUZZY in testing data and CVC. Three SNPs in the selected SNP combinations except SNP10 are located in ROR1, JAK1, and nearby SOCS5 (about 19.8 k BP). For these three genes, several biological evidences of interactions are pre-reported: 1) ‘Jak1 has previously been implicated in adipocyte insulin resistance.’ [Among the results of FGMDR, we selected the four-locus SNP combination as the best SNP combination based on stance.’ , 2) ‘Mosstance.’ , 3) ‘Whestance.’ , 4) ‘RORstance.’ , 5) ‘Resstance.’ .BAFUZZY in testing. These BAFUZZY values are not directly comparable to ordinary BA because the Fuzzy set theory has been implemented. BAFUZZY is more concentrated near 0.5, compared to ordinary BA. Nevertheless, BAFUZZY values in testing HOMA-IR data were lower than those of CD. This seems to be caused by heritability differences. The heritability of CD is 53% [CVCs decreased by increase of order, in both Tables D is 53%  but the D is 53% , and 22%D is 53% .In this study, we proposed a FGMDR, a fuzzy extension of GMDR to detect GGIs. FGMDR can handle both binary and quantitative traits, and allows adjustment for covariates. Thus, FGMDR is a method to overcome shortcomings due to the traditional classification commonly used in MDR-based frameworks by allowing partial membership degrees of high and low risk groups for each cell, and provides more flexible interpretations for results. Our proposed FGMDR is based in the generalized linear models (GLMs), it can handle any distributions of phenotypes from the exponential family including normal, binomial, Poisson and gamma distributions. Further, our FGMDR does not require any balancing of H/L risk groups. Three simulation scenarios for power comparison were made under one continuous phenotype, and two binary phenotypes with adjustment for covariates. Simulation studies showed that FGMDR outperformed MDR and GMDR in most scenarios. Through two real applications to CD and HOMA-IR data, we identified the best SNP combinations associated with two diseases. In our applications, we found several biological evidences of two-order interactions included high-order interactions identified by FGMDR (four-way interaction for CD and four-way interaction for HOMA-IR).The existing MDR extensions using classical sets as groups for classification, can be extended to any fuzzy set-based MDR methods. These fuzzy set-based MDR methods may contribute to identify important interactions in the biological systems, through reflecting the vagueness of classification due to objects that can seldom be classified uniquely.


To quantify polygenic effects, i.e. undetected genetic effects, in large-scale association studies, we propose a generalized estimating equation (GEE) based estimation framework. We develop a marginal model for single-variant association test statistics of complex diseases that generalizes existing approaches such as LD Score regression and that is applicable to population-based designs, to family-based designs or to arbitrary combinations of both. We extend the standard GEE approach so that the parameters of the proposed marginal model can be estimated based on working-correlation/linkage-disequilibrium (LD) matrices from external reference panels. Our method achieves substantial efficiency gains over standard approaches, while it is robust against misspecification of the LD structure, i.e. the LD structure of the reference panel can differ substantially from the true LD structure in the study population. In simulation studies and in applications to population-based and family-based studies, we illustrate the features of the proposed GEE framework. Our results suggest that our approach can be up to 100% more efficient than existing methodology. While genome-wide association studies (GWAS) led to the discovery of many genetic risk loci for complex diseases and traits, the overall magnitude of the combined genetic effects on the disease phenotype of interest that is explained by GWAS findings is much smaller than classical heritability studies suggested , we can stateThe derivation of this approximation can be found in Appendix A of the Biostatistics online). Equation , , 2.5), aIf we utilize the assumption of normally distributed genetic effects for the moment, we can conclude that the equation  is valid of GCTA  by assumBiostatistics online, we show how the original GEE assumption of independent clusters can be relaxed in such a way, i.e. the sandwich-variance estimator can be extended to the scenario of sparsely correlated clusters and how the classical asymptotic results can be reestablished. Now, we describe the GEE objects to estimate the parameters in the mean model for the association test statistics. The following description and the derivations in the Biostatistics online are strongly related to Biostatistics online). We denote the true, unknown parameter by Now, we propose our PolyGEE approach to estimate the parameters ond 1 cM , we can atistics . GEE appatistics . The obsur model  and 2.2 e design  is fixedfunction . We use Biostatistics online), we can reestablish the following results for the asymptotic existence of the estimator Biostatistics online. The two key points in the proofs of Propositions 3 and 5 are the utilization of a central limit theorem for weakly dependent data and the modification of the sandwich-variance estimator to account for the specified dependence structure of the clusters.Under the assumption of sparsely correlated clusters and reasonable technical conditions , schizophrenia (SCZ), major depressive disorder (MDD), attention deficit disorder (ADHD), and autism spectrum disorder (ASD). We analyzed the data from the PGC-Cross-Disorder Group for SCZ  Consortium association study for autism spectrum disorder (ASD)  implies equation ), assumiAt a genome-wide level, we derived a general marginal model for single-variant association test statistics of complex diseases. We proposed the PolyGEE methodology to estimate the amount of polygenic effects and confounding biases from the association test statistics of large-scale association studies using this model. We showed via simulation studies and the application to real data that our approach is substantially more efficient than the existing LD Score regression framework, i.e. the estimates are more precise. The increased efficiency of our approach is achieved by incorporating detailed local LD information from the external reference panels, e.g. 1000 Genomes project, into the estimation step. However, our GEE approach does not require that the reference panel and the study data are exact matches in terms of the LD structure. The approach is robust against deviations of the sample LD structure from the reference panel and can compute asymptotic valid standard errors. Our theoretical derivations and assumptions lead to general valid results for the estimation framework. For further research, this makes it possible to extend the mean model to incorporate more components or estimate the genetic correlation between two traits.Supplementary DataClick here for additional data file.


Milk composition was evaluated as average test-day measurements adjusted for environmental effects. Next, we genotyped a subset of 8080 cows  with the BovineSNP50 Beadchip. For each breed, genotypes were first imputed to high-density (HD) using HD single nucleotide polymorphisms (SNPs) genotypes of 522 MON, 546 NOR, and 776 HOL bulls. The resulting HD SNP genotypes were subsequently imputed to the sequence level using 27 million high-quality sequence variants selected from Run4 of the 1000 Bull Genomes consortium (1147 bulls). Within-breed, multi-breed, and conditional GWAS were performed.Genome-wide association studies (GWAS) were performed at the sequence level to identify candidate mutations that affect the expression of six major milk proteins in Montbéliarde (MON), Normande (NOR), and Holstein (HOL) dairy cattle. Whey protein  and casein  contents were estimated by mid-infrared (MIR) spectrometry, with medium to high accuracy  and those with effects reported for the first time here .Thirty-four distinct genomic regions were identified. Three regions on chromosomes 6, 11, and 20 had very significant effects on milk composition and were shared across the three breeds. Other significant effects, which partially overlapped across breeds, were found on almost all the autosomes. Multi-breed analyses provided a larger number of significant genomic regions with smaller confidence intervals than within-breed analyses. Combinations of within-breed, multi-breed, and conditional analyses led to the identification of putative causative variants in several candidate genes that presented significant protein–protein interactions enrichment, including those with previously described effects on milk composition  contains supplementary material, which is available to authorized users. In cattle, milk protein composition is mostly influenced by genetic factors –4 and isIn Run4 of the 1000 bull genome reference population, a database containing more than 56 million SNPs and small insertions/deletions (InDel) was constructed by analyzing whole-genome sequences (WGS) from 1147 bulls representing 27 different breeds, including 288 HOL, 28 MON and 24 NOR bulls. These data can then be used to impute WGS from experimentally or routinely obtained 50 K SNP genotypes . In thisSince WGS contain almost all the genomic variants, they should contain the causal mutations for a given trait and, thus they provide a much higher GWAS resolution. However, due to the long-range linkage disequilibrium that exists within dairy cattle breeds, the resolution of within-breed GWAS is often limited. For causal mutations that are shared among breeds, a multi-breed model can be used to refine regions that harbour quantitative trait loci (QTL). This approach takes advantage of the historical recombination events that have occurred in each breed, resulting in linkage disequilibrium over shorter distances and better resolution .Here, we report the results of a GWAS at the sequence level for six major milk protein contents, namely α-lactalbumin and β-lactoglobulin and αs1, αs2, β, and κ caseins from HOL, MON, and NOR cows. The results of within-breed, multi-breed, and conditional analyses, that fit the most significant variant in addition to other tested variants, are examined together in order to pinpoint potential candidate variants in each genomic region.s1-CN, αs2-CN, β-CN, κ-CN, α-LA, and β-LG) were predicted in g/100 g protein. Total casein content and total whey protein content were also analyzed . We applied the following quality control filters: the individual call rate had to be higher than 95%, the SNP call rate higher than 90%, the minor allele frequency (MAF) higher than 5%, and genotype frequencies had to be in Hardy–Weinberg equilibrium with The 50 K SNP genotypes of the 7907 cows were imputed to whole-genome sequence (WGS) using FImpute software, which accurately and quickly analyzes large datasets . A two-sPrecision of imputation from HD to sequence was assessed by comparing imputed genotypes with those obtained by re-genotyping a subset of the same cows with a custom chip. This additional information was not used in the imputation process. Two datasets were available: (1) a group of 168 Holstein cows that were genotyped with the first version (V1) of the EuroG10k Illumina chip, with 721 additional markers; and (2) a group of 2142 Montbéliarde cows that were genotyped with the fourth version (V4) of the same EuroG10k chip containing 3082 additional SNPs. Only SNPs with good technical quality  were used. Imputation accuracy was measured by the squared correlation between true and imputed genotypes and by the genotypic and allelic concordance rate.In order to remove SNPs with the lowest accuracies of imputation, only variants with a MAF higher than 0.02 were retained for further association analyses. Thus, about 11 million variants were included in each within-breed analysis and around 13 million were included in multi-breed analyses ; I the identity matrix and All association analyses were performed using the  variant :1\documeenotypes , and \dov to Model (1), with W as the incidence matrix relating phenotypes to breed effect (three levels), and x, b, u, and e as defined previously:P value of 3.7 × 10−9 (−log10(P) = 8.4). QTL regions were identified by grouping significant results that were located within the same 2 million base-pair (Mbp) interval in a single genomic region, regardless of the breeds or traits under study. QTL regions were determined by considering positions of variants included in the upper third of the peak. For a given trait in a given breed, when two consecutive QTL regions had overlapping confidence intervals, or when the distance between the limits of the confidence intervals was less than 1 Mbp, only the confidence interval that presented the most significant results was retained.For multi-breed association analyses, Model (2) was applied by adding a fixed breed effect cojo option of GCTA [2 < 0.9).In the most significant QTL regions, conditional analyses were carried out using the  of GCTA  in order10(P) higher than 8.4 and located within confidence intervals were annotated. To avoid missing important genes, confidence intervals were extended by 100 kb on each side.Sequence-derived polymorphisms were extracted for candidate mutation regions from the corresponding VCF files . All varIn addition, functional protein–protein interactions (PPI) encoded by candidate genes were investigated, as well as gene ontology (GO) enrichment, using the STRING Genomics 10.0 database of protein–protein interaction (PPI) networks . This daThe results of imputation accuracy at the sequence level for SNPs used in the GWAS analyses (MAF ≥ 2%) are in Table 10(P) ≥ 8.4) in at least one within-or multi-breed analysis and for at least one milk protein composition trait.Among the 13 million tested variants, 71,755 had genome-wide significant effects  for β-LG, κ-WP, and For each trait, the number of significantly associated variants was relatively consistent between breeds. It was lower (from 193 to 2394) for αQTL regions were defined by merging the overlapping QTL regions obtained for the different traits and breeds and by grouping the corresponding significant results. Confidence intervals of these regions were defined as described in the Methods section. Thus, 34 QTL regions with significant effects on one or several milk protein composition traits were identified in within-breed and/or multi-breed analyses , BTA11 (103.3 Mbp), and BTA20 (58.3–58.4 Mbp), were detected in three breeds. One additional region on BTA29 at about 9.6 Mbp was common to MON and HOL, and another region on BTA14 at 1.7–1.8 Mbp was common to HOL and NOR  ≥ 20; P value < 10−13 after Bonferroni correction).The 34 QTL were distributed on 17 of the 29 bovine autosomes, with one to seven QTL per chromosome. Almost all of them (31) were detected in multi-breed analyses while 11, 8, and 11 QTL regions were found in MON, NOR, and HOL within-breed analyses, respectively. Four QTL regions, located on 10(P) value of the most significant (top) variant was almost always higher in multi-than in within-breed analyses; this was true even for most of the regions that had significant effects in only one within-breed analysis. For these QTL, the mean −log10(P) value of the most significant (top) variant was 64 in multi-breed analyses versus 49, 46, and 42 in MON, NOR and HOL within-breed analyses, respectively. In addition, the QTL confidence intervals generated by multi-breed analyses contained a smaller number of variants than those produced by within-breed analyses. For the 17 QTL regions, an average of 134 variants (2–374) were found in multi-breed analyses versus 189 (39–335), 287 (61–872), and 308 (9–1236) in MON, NOR, and HOL within-breed analyses, respectively. However, in some QTL regions, specifically those located on BTA2 (131.8 Mbp), 6 (38 Mbp), and 19 (61 Mbp), the number of significant variants was smaller in within-breed analyses than in the multi-breed analysis.Multi-breed analyses led to the detection of a larger number of QTL regions than within-breed analyses: 14 of the 31 QTL detected in multi-breed analyses were not found in within-breed analyses. For the 17 QTL regions found in both within-and multi-breed analyses, the −logManhattan plots of three of the most significant QTL regions are in Fig. PAEP  and DGAT1  genes.All variants included within confidence intervals (+100 kb on each side) were functionally annotated Table  and (seeSLC37A1), BTA2 (ALPL), BTA5 (MGST1), BTA6 , BTA11 (PAEP), BTA14 , BTA20 (ANKH), BTA27 (AGPAT6), and BTA29 (PICALM).In 29 QTL regions, annotation led to the identification of candidate genes for milk protein composition. In total, 47, 72, and 82 candidate genes were identified in MON, NOR, and HOL within-breed analyses . Some of these were shared across breeds: 12 were found in both MON and NOR, 15 in MON and HOL, 39 in HOL and NOR, among which 10 were common to the three breeds , as well as GO enrichment, were investigated for the 22 most plausible candidate genes of our study. Network proteins encoded by these genes had significantly more interactions than expected  and also possibly because of the more stringent significance thresholds applied here. For GWAS on WGS data, the very stringent threshold that we used  probably reduced the detection power but minimized the number of false positive QTL.Instead, the better resolution of the WGS data, combined with the power of the multi-breed GWAS approach, led to the detection of 12 QTL that were not previously found in the 50 K SNP study. To evaluate the impact of marker density on GWAS results, we extracted 50 K and HD GWAS results from the WGS results. In several genomic regions, for example the regions on BTA2, 6, and 20 Fig. , the incMGST1 and AGPAT6 genes described below). For QTL that were detected in both within-and multi-breed analyses, the multi-breed approach provided smaller confidence intervals of the QTL than within-breed analyses. The three French breeds used in our study are not strongly related. Based on 50 K SNP data, Gautier et al. [Furthermore, the WGS resolution enables the use of a multi-breed approach, which is expected to better estimate the effects of rare variants and to reduce LD between neighboring variants. With the multi-breed analysis, we detected 14 QTL that were not detected in any of our within-breed analyses . In three other genes, we suggest the presence of a breed-specific candidate variant (MGST1 on BTA5 and PICALM on BTA29) or several candidate causative variants (ANKH on BTA20). Finally, four regions, including the DGAT1 region on BTA14 and three regions on BTA6 , were more complex, because they contained several candidate genes, each with several candidate variants. Eight of these candidate genes  are known to be overexpressed in the mammary gland compared to other 17 tissues [On average, depending on the breed, between 60 and 73% of the QTL variants that we detected in the GWAS were located in genes; this is about twice as high as the percentage of genic variants at the whole-genome scale . The mo tissues  and betwSLC37A1  gene, which encodes a glucose-6-phosphate transporter that is involved in the homeostasis of blood glucose, is highly expressed in the mammary gland [SLC37A1. One downstream variant was detected for αs1-CN in the MON analysis, which ranked 104th among the significant variants, while multi-breed analyses revealed three downstream variants that ranked 81st, 87th, and 103rd. All intronic variants that are located at the top of the peaks are in strong LD but only one variant (indel), located at 144,397,274 bp, was common to all three TOP10 lists; it was 1st in the αs1-CN/MON ranking, 9th in the αs1-CN/multi-breed ranking, and 4th in the α-LA/multi-breed ranking. The top1 intronic variant detected in the αs1-CN/multi-breeds analysis, at 144,398,814 bp, ranked 75th in the αs1-CN/MON peak and 76th in the α-LA/multi-breed peak.The ry gland . It coulSLC37A1 gene variants on milk production traits. In an analysis of HD SNP genotypes, Kemper et al. [10(P) = 10.2), but it ranked 101st. Two other intronic variants in strong LD in the SLC37A1 gene, at 144,367,474 and 144,377,960 bp, were previously proposed as the best candidate mutations for changes in phosphorus concentration and milk production traits [10(P) value lower than 6 for all analyzed traits. In another study of targeted QTL regions after imputation to WGS level, the variant with the most significant effects was located at 144,381,564 bp [Two previous studies described the effects of r et al.  describen traits . However1,564 bp . This vaThe conditional analyses that we performed included the two best candidate variants as well as the candidate variant described by Kemper et al. . These r10(P) values did not reach the stringent threshold of 8.4 that we applied in this study  gene, which encodes a member of the alkaline phosphatase family of proteins. The most significant variant differed among the three within-breed analyses: it was located at 131,806,882 bp in NOR, 131,850,456 bp in MON, and 131,808,301 bp in HOL sequences. Instead, the top-ranked variant in the peak detected in the multi-breed GWAS was located at 131,806,882 bp. All three single-breed conditional analyses that included each of these variants as fixed effects lacked peaks . In particular, although the αs2-CN-associated peaks were detected in all within-and multi-breed analyses, even if in the MON and HOL analyses, the maximal −log10(P) < 6), whereas the HOL analysis detected a single peak with a −log10(P) = 8, which was close to the significance threshold of 8.4. Only one gene, MGST1 , was present within the confidence interval obtained in the multi-breed analysis. Fifty-one variants were located in intronic (29), exonic (1 synonymous), 5′-UTR (2), or regulatory (19 in the upstream region) regions of the gene. The variant with the most significant effects was located at 93,950,211 bp in the upstream region and its −log10(P) value was 9.3, versus a value of 8.0 for the variants that ranked 2nd , which were located, respectively, in the 5′-UTR and upstream regions of the gene. The MAF value for these variants was low in the MON population  and ranged from 0.08 to 0.12 in NOR, from 0.37 to 0.42 in HOL, and from 0.19 to 0.22 in the multi-breed population. Thus, the fact that peaks were detected only in HOL (close to significance) and multi-breed (significant) analyses could be due to the relatively low MAF for these variants in MON and NOR. The most significant variants in our study are located near a variant that was reported by Raven et al. [MGST1 has causative pleiotropic effects on milk composition . These authors failed to identify causative variants in the gene but they pointed to a cluster of 17 variants that were grouped in a 10-kbp segment of the MGST1 gene . Only one of these 17 variants is located in the confidence interval of the QTL that we detected and this is an intronic variant  that ranked 7th in the peak in spite of having a higher MAF (0.32) than the most significant variants (MAF = 0.19–0.22). Thus, our study highlights three new candidate mutations in the MGST1 gene, which are located very close to each other, in the 5′-UTR region  or in the upstream region  of the gene.One region on BTA5 that contains 63 variants affected PC in the multi-breed analysis. The MON and NOR within-breed analyses revealed no peaks  that was described by Cohen-Zinder et al. [10(P) values of 31 and 21, respectively; these values were higher than those of the top variant in the peaks after filtering for MAF . However, among the sires of the HOL cows, six bulls were previously found to be heterozygous for the QTL detected in this region, but homozygous for this mutation [Several QTL were found on BTA6. The first one, detected in HOL and multi-breed analyses, was located in the 37.6–38.4 Mbp region, which contains the Y581S polymorphism of the r et al.  as a cauABCG2 gene, at 38,015,146 and 38,020,110 bp. Other variants, which are located in three other genes, i.e. MEPE (one downstream), PKD2 (one intronic), and HERC3 (two intronic), also had highly significant effects on αs1-CN. Due to the relatively low MAF of the candidate variants located in this region, these results require further analyses, including a larger number of animals and more accurate imputation or direct genotyping.In the HOL analysis, nine variants with MAF ranging from 0.022 to 0.041 were located within the confidence interval of the QTL. The most significant variants were located in intronic regions of the SEL1L3 gene . In NOR, the top 116 variants in the peak were intergenic, while the genic variant with the most significant effects was located in an intron of the SEPSECS gene . In HOL, the most significant genic variants (DHX15) ranked 16th in the peak . Finally, among the top 80 variants detected by the multi-breed analysis, only one was genic, which was located in an intron of the SEL1L3 gene . There is insufficient concordance among these results to propose a single set of candidate variants.In all within-breed and multi-breed analyses, the αs1-CN phenotype was affected by another region of BTA6 at 45.8–46.9 Mbp. However, the most likely candidate genes differed among breeds. In MON, the nine variants with the most significant effects were located in intronic regions of the CSN3 gene, which encodes the κ casein: at 87,376,747 bp (upstream) in NOR, 87,392,592 bp (5′-UTR) in MON and multi-breed, and 87,394,293 bp (downstream) in HOL. Each of these variants, as well as the κ casein A/B variant , was therefore included as a fixed effect in the conditional analyses. The results were breed-specific: in MON, the κ-CN-associated peak disappeared after fixing the upstream, missense, or 5′-UTR variant; in HOL, the peak disappeared after fixing the upstream, 3′-UTR, or downstream variant; but in NOR, the peak remained with the inclusion in the model of any of the four variants. Thus, none of the four candidate variants succeeded in explaining all the effects observed on κ-CN in the three breeds.On BTA6, we found a QTL that affected both the overall protein content of milk and the content of all four individual caseins in all three breeds. Variants with the most significant effects were located in an 840-kb interval that contains the 250-kb casein gene cluster ; other variants with effects on αs1-CN and β-CN in MON were located at 85.2 Mbp. In all within-and multi-breed analyses, the most significant effects were detected for the κ-CN phenotype, followed by αs1, αs2, or β-CN depending on the breed. In each analysis, the variant with the most significant effects on κ-CN was located within or in the immediate vicinity of the CSN2 gene (three downstream and three intronic) and in the upstream region of the ODAM (odontogenic ameloblast-associated) gene (between the CSN1S2 and CSN3 genes).Instead, the peaks associated with the αs2-CN and β-CN phenotypes in NOR and the PC and αs2-CN phenotypes in MON could be explained by two distinct groups of six SNPs in complete LD, which were respectively located in the CSN2, which ranked 147th and 86th, respectively, for their effects on PC and αs2-CN in NOR, were responsible for the αs2-CN phenotype in NOR but not for any other effect on the other traits or in the other breeds.Finally, the A1/B and A2 variants of CSN1S1-CSN2-CSN1S2-CSN3 . The polymorphisms of the amino-acid sequences of caseins are well known, and the effects on milk composition and cheese-making abilities have been well described . NevertPAEP  gene, also named LGB gene, which encodes the β-LG protein. The β-LG protein variants A and B, which are common in most cattle breeds, are associated with different β-LG levels in milk [PAEP gene, was shared by the four analyses. Moreover, this variant is one of the most significant in each analysis, ranking 6th, 4th, 1st, and 3rd, respectively, in the MON, NOR, HOL, and multi-breed analyses. The inclusion in conditional analyses of one of the causal missense variants or the most probable upstream variant identified in our study led to similar results in MON and HOL but not in NOR  and four in HOL (three in RECQL4 and one in MROH1). In this region, no variant remained significant in the conditional analyses for NOR when the DGAT1 (K232A) or BOP1  variants were included, and for HOL when the variants in RECQL4  or MROH1  were included. In contrast, a less significant peak persisted when the DGAT1 or BOP1 variant was included in the HOL analyses and when the RECQL4 or MROH1 variant was included in the NOR analyses  in bones and α-LA exhibits a high affinity to metal ions, calcium in particular. In addition, ANKH is highly expressed in mammary tissue in Holstein and Jersey cows [ANKH and ALPL (candidate gene on BTA2 for effects on αs2-CN), which suggests a functional link between these two genes  or in an intergenic region. In MON and HOL, for which the most significant variants were intergenic, ANKH intronic variants ranked 2nd  and 13th , respectively. After fixing the most significant variant from each within-breed analysis, a peak remained in all conditional analyses  < 8.4), and no peak was observed in HOL . When the κ-CN phenotype was conditioned on the effect of any of these mutations, the association signals completely disappeared in the MON, NOR, and multi-breed analyses  = 7.9)  and one variant that ranked 10th in the HOL analysis, is located in the upstream region of the gene . When conditional GWAS analyses were performed, the inclusion of the intronic variants removed the peak in MON but not in HOL analyses, and conversely, inclusion of the upstream variant removed the peak in HOL but not in MON analyses. In NOR, the peak in question persisted when either intronic or upstream variants were fixed  and in genes for which no such relationship was known . In the future, functional analyses will enable the establishment of causative links between these candidate variants and milk protein phenotypes. However, even before such studies are completed, our results offer the opportunity to improve cheese-making properties through the identification of genetic variants associated with changes in milk composition. Direct consequences of these results on practical selection are not obvious and depend on potential premiums on protein composition and on incentives proposed by the milk processing industry. Nevertheless, it would be desirable to favour caseins against whey proteins at least for milk collected for cheese production. Such an option could be implemented by including variants that affect individual proteins in genomic evaluation models.Our study provides evidence that a GWAS-based approach applied to fine-scale phenotypes, whole-genome sequences, and multiple breeds provides enough resolution to identify candidate genes and directly pinpoint a limited number of candidate variants in most of these genes. Several variants, some shared among breeds, were identified as plausible candidate mutations for changes in milk protein composition in the three main French dairy cattle breeds. They were located both in genes that had previously been found to affect milk composition . .Additional file 2: Table S2. Number of variants included within confidence intervals for each QTL region and trait, regardless of breed.Additional file 3: Table S3. Description of the pleiotropic QTL regions detected in within-breed  or multi-breed (Multi) analyses.Additional file 4: Table S4. Description of other significant QTL regions detected in within-breed  or multi-breed (Multi) analyses.Additional file 5: Table S5. Functional annotations of variants included within confidence intervals (± 100 kb) of the 34 QTL for each trait in the three within-breed Montbéliarde (MON), Normande (NOR), and Holstein (HOL) or in multi-breed analyses.Additional file 6: Figure S1. −log10(P) plotted against the position of variants detected by GWAS (in grey) and conditional GWAS  a On BTA1, b BTA2, c BTA5, d BTA11, e BTA14, f BTA20, g BTA27 and h BTA29