CYP1A1) and 1A2 (CYP1A2) gene region (rs2472297) and one near the aryl-hydrocarbon receptor (AHR) gene (rs6968865), have been associated with habitual caffeine consumption. We sought to establish whether a more refined and comprehensive assessment of caffeine consumption would provide stronger evidence of association, and whether a combined allelic score comprising these two variants would further strengthen the association. We used data from between 4,460 and 7,520 women in the Avon Longitudinal Study of Parents and Children, a longitudinal birth cohort based in the United Kingdom. Self-report data on coffee, tea and cola consumption (including consumption of decaffeinated drinks) were available at multiple time points. Both genotypes were individually associated with total caffeine consumption, and with coffee and tea consumption. There was no association with cola consumption, possibly due to low levels of consumption in this sample. There was also no association with measures of decaffeinated drink consumption, indicating that the observed association is most likely mediated via caffeine. The association was strengthened when a combined allelic score was used, accounting for up to 1.28% of phenotypic variance. This was not associated with potential confounders of observational association. A combined allelic score accounts for sufficient phenotypic variance in caffeine consumption that this may be useful in Mendelian randomization studies. Future studies may therefore be able to use this combined allelic score to explore causal effects of habitual caffeine consumption on health outcomes.Two genetic loci, one in the cytochrome P450 1A1 ( CYP1A1) and 1A2 (CYP1A2) gene region on chromosome 15 and one near the aryl-hydrocarbon receptor (AHR) gene on chromosome 7, have been found to be associated with habitual caffeine consumption across a number of studies CYP1A1 and CYP1A2, and rs6968865 51 kb upstream of AHR, provide the strongest signals, each with an effect equivalent to an increased consumption of ∼0.2 cups per day per risk (T) allele. The genes are biologically plausible candidates for caffeine consumption phenotypes as they both encode members of the same biochemical pathway. AHR is known to induce CYP1A1 and CYP1A2 by binding to the DNA in the region between these two genes Caffeine is one of the most widely-consumed psychoactive substances world-wide, and while coffee and tea consumption dominate, it is also present in some soft drinks A limitation of studies to date is that they have typically used a single measure of caffeine consumption . One study http://www.bristol.ac.uk/alspac/researchers/data-access/data-dictionary/. Ethics approval for the study was obtained from the ALSPAC Ethics and Law Committee and the Local Research Ethics Committees .The Avon Longitudinal Study of Parents and Children (ALSPAC) sample is a longitudinal birth cohort that comprises 20,248 pregnancies. The mothers of 14,541 (71.8%) pregnancies were recruited antenatally during 1990–92 (Phase I). Post-natal recruitment to the ‘Focus@7’ clinical assessment at the age of ∼7 years recruited a further 456 children from 452 (2.2% of eligible) pregnancies (Phase II). Recruitment during ages 8–18 years (Phase III) added a further 257 children from 254 (1.2% of eligible) pregnancies, giving an overall total of 15,247 (75.3% of eligible) enrolled pregnancies; from these pregnancies there were 14,775 live-born children of which 14,701 were alive at one year of age. The phases of enrolment are described in more detail in the cohort profile paper Data on coffee and tea consumption were collected via self-report during pregnancy at 8, 18 and 32 weeks gestation and 2, 47, 85, 97 and 145 months after delivery. Participants were asked to report “current daily coffee and tea drinking”, as number of drinks, separately for weekdays and weekends. Similar questions were asked for cola consumption in drinks per week. For cola consumption, questions were open format at 8, 18, and 32 weeks gestation, and 2 months after delivery, and closed format at later time points . Closed format responses were recoded to 0, 0.5, 2, 5.5 and 7 drinks per week, and cola consumption values further recoded to reflect daily consumption. Outlying daily consumption values  were coded as missing data. Similar questions were also asked for decaffeinated coffee, tea and cola consumption at the same time points, and coded in the same way. In order to obtain a measure of total daily caffeine consumption, number of cups of tea and coffee were summed with drinks per day of cola, weighted with respect to approximate caffeine content  CYP1A1 (rs2472297) and AHR (rs6968865) loci were available from GWAS genotyping data. A total of 10,015 ALSPAC mothers were genotyped on the Illumina 660K quad chip at the Centre National de Genotypage, Paris, resulting in 557,124 directly genotyped SNPs before quality control. Genotypes were called with Illumina GenomeStudio and PLINK (v1.07) was used to carry out quality control steps.Genotypes at the P>5×10−6) were considered for further use. Population stratification was assessed by means of multidimensional scaling of genome-wide identity by state (IBS) pairwise distances using the four  HapMap populations as a reference. Cryptic relatedness was assessed using estimates of the proportion of SNPs expected to be identical by descent given estimates of IBS. Subject with a relatedness of 0.1 or higher were excluded. Genotypes were imputed with Markov Chain Haplotyping software (MaCH 1.0.16) (45) using CEPH individuals from phase 2 of the HapMap project as a reference set (release 22). SNP rs2472297 was directly genotyped, had a MAF of 0.27, HWE P-value of 0.1 and 0.02% missingness before imputation. SNP rs6968865 was imputed with an imputation quality of 0.96, and MAF of 0.39. After imputation genotypes were available for 8,340 subjects. The frequencies of the T allele were 0.27 in rs2472297 and 0.61 in rs6968865.Individuals were excluded from further analysis on the basis of having incorrect sex assignments; minimal or excessive heterozygosity, disproportionate levels of individual missingness (>5%); evidence of cryptic relatedness  and being of non-European ancestry . SNPs with a minor allele frequency of <1% and call rate of <95% were removed. Furthermore, only SNPs which passed an exact test of Hardy–Weinberg equilibrium . Best-guess genotypes were used for analysis.To obtain joint effects to take into account genotypes at both SNPs simultaneously, following Sulem and colleagues We examined within-locus non-additivity by testing the significance of a second heterozygote term, and between-locus non-additivity by testing for a joint effect beyond the sum of the effects of both SNPs individually. Our results indicated that these SNPs act additively, and their effects are independent .alspac-exec@bristol.ac.uk). The ALSPAC data management plan  describes in detail the policy regarding data sharing, which is through a system of managed open access.Data used for this submission will be made available on request to the ALSPAC executive committee . Consumption of tea (means 2.73 to 3.18 drinks per day) and cola (means 0.60 to 2.31 drinks per week) varied across time points, but with no clear pattern of change. As a result, total daily caffeine consumption tended to increase across time points (means 206.8 mg to 306.1 mg). These data are shown in CYP1A1  and AHR  genotypes (Ps  = 1.15×10−3 to 3.74×10−14).Across all time points, total caffeine consumption was associated with both enotypes . SimilarCYP1A1, the proportion of phenotypic variance explained ranged from 0.15% to 0.88%, while for AHR it ranged from 0.04% to 0.48%. However, the combined SNP score accounted for a somewhat higher proportion of phenotypic variance on average, ranging from 0.16% to 1.28%.In general, the proportion of phenotypic variance explained across all time points was small, as would be expected for the association of common variants with complex behavioural phenotypes. For Estimates of the proportion of phenotypic variance obtained using GCTA Ps  = 2.34×10−2 to 5.46×10−5) and tea , but not cola   –4. InterAHR or CYP1A1 genotypes, or the combined SNP score, was associated with consumption of decaffeinated coffee, tea or cola . These indicated no evidence of association , suggestAHR and CYP1A1 are associated with caffeine consumption, and extend previous findings in two important ways. First, our results are the first to show association in a sample where caffeine consumption via caffeinated beverages other than coffee is common. Moreover, we show that a combined caffeine consumption phenotype derived from measures of consumption of three caffeinated beverages  provides a stronger signal than any one of these measures separately. Second, our results also confirm that these results are due to caffeine consumption, rather than some other common characteristic of caffeinated beverages. By using measures of consumption of decaffeinated drinks as negative controls we show no evidence of association with either AHR or CYP1A1. While our results hold for both SNPs individually, our strongest results are obtained when both SNPs are combined to create a 2-SNP genetic risk score.Our results confirm that two SNPs in Observationally, caffeine  consumption has been shown to be associated with a number of health outcomes Mendelian randomization (MR) offers one approach to better understanding the causal nature of the observed associations between caffeine consumption and health outcomes. Genetics variants are randomly assorted during gamete formation and conception, and therefore should be unrelated to other lifestyle factors associated with coffee consumption which may confound observational associations AHR and CYP1A1, and our 2-SNP genetic risk score, are not associated with a range of potential confounders that may give rise to spurious associations in studies of health-outcomes putatively related to caffeine consumption. This, together with the clear evidence of association with caffeine consumption, indicates that the 2-SNP genetic risk score could be used as an instrumental variable in MR analyses. The greater variance explained by the combined score would increase statistical power and reduce the sample size required to detect associations with health outcomes, compared to using either SNP individually. The risk score explains up to 1.3% of the variance in caffeine consumption, which although small in absolute terms is relatively large by the standards of common genetic variants. This is comparable to the variance explained in body mass index (BMI) by variants in the FTO gene, and in cigarette consumption by variants in the CHRNA5-A3-B4 gene cluster Critically, we have shown that the two SNPs in all soda consumption. To better understand whether this lack of association is genuine will require the study of populations where levels of cola consumption are higher. Third, our sample was restricted to women only. Rates of caffeine consumption may differ between men and women, although there are no clear reasons to expect that the pattern of results we observed would differ in males. While patterns of consumption during pregnancy may not be typical, our data extend to ∼12 years post-pregnancy. It is likely that the women in our sample reverted to pre-pregnancy patterns of caffeine consumption over time. Fourth, we only included 2 SNPs in our analysis. These were chosen on the basis of being those for which there is the clearest evidence from recent GWAS of caffeine consumption. Future studies may extend our 2-SNP score by including further variants. Fifth, although we are optimistic that these genotypes, and the 2-SNP score, can be used as instrumental variables in MR analyses, potential pleiotropic effects will need to be considered. Metabolic enzyme genotypes typically relate to several metabolic differences with may give rise to associations with health outcomes. In principle, this can be tested by examining the association of genotype with health outcome separately in those who do and do not consume caffeinated drinks There are some limitations to this study that should be considered when interpreting our results. First, caffeine consumption was measured using a food frequency questionnaire, and these may have modest reliability and validity AHR and CYP1A1 genotypes with caffeine consumption, and extend previous work by showing that this association holds for tea consumption as well as coffee consumption. Moreover, no association is observed for decaffeinated tea or coffee consumption. This strengthens the argument that the association is mediated via caffeine consumption, although it remains possible that other compounds present in both tea and coffee mediate this association. Future work, perhaps selecting participants on the basis of AHR and CYP1A1 genotype, could explore this possibility through the administration of caffeine in a laboratory setting. Finally, the relatively large proportion of variance in caffeine consumption accounted for by the combined SNP score, and the lack of association of this with potential confounders, means that it could be used in Mendelian randomization studies to explore the causal effects of habitual caffeine consumption on health-related outcomes.In conclusion, our data confirm the association of Figure S1Distribution of total caffeine consumption (mg).(TIF)Click here for additional data file.Figure S2Distribution of total coffee consumption (cups per day).(TIF)Click here for additional data file.Figure S3Distribution of total tea consumption (cups per day).(TIF)Click here for additional data file.Table S1Variance in total caffeine consumption explained using linear regression and GCTA.(DOCX)Click here for additional data file.Table S2Association of CYP1A1 rs2472297, AHR rs6968865 and combined genetic score with decaffeinated coffee consumption.(DOCX)Click here for additional data file.Table S3Association of CYP1A1 rs2472297, AHR rs6968865 and combined genetic score with decaffeinated tea consumption.(DOCX)Click here for additional data file.Table S4Association of CYP1A1 rs2472297, AHR rs6968865 and combined genetic score with decaffeinated cola consumption.(DOCX)Click here for additional data file.


In genome-wide association studies (GWAS) for complex diseases, the association between a SNP and each phenotype is usually weak. Combining multiple related phenotypic traits can increase the power of gene search and thus is a practically important area that requires methodology work. This study provides a comprehensive review of existing methods for conducting GWAS on complex diseases with multiple phenotypes including the multivariate analysis of variance (MANOVA), the principal component analysis (PCA), the generalizing estimating equations (GEE), the trait-based association test involving the extended Simes procedure (TATES), and the classical Fisher combination test. We propose a new method that relaxes the unrealistic independence assumption of the classical Fisher combination test and is computationally efficient. To demonstrate applications of the proposed method, we also present the results of statistical analysis on the Study of Addiction: Genetics and Environment (SAGE) data.Our simulation study shows that the proposed method has higher power than existing methods while controlling for the type I error rate. The GEE and the classical Fisher combination test, on the other hand, do not control the type I error rate and thus are not recommended. In general, the power of the competing methods decreases as the correlation between phenotypes increases. All the methods tend to have lower power when the multivariate phenotypes come from long tailed distributions. The real data analysis also demonstrates that the proposed method allows us to compare the marginal results with the multivariate results and specify which SNPs are specific to a particular phenotype or contribute to the common construct.The proposed method outperforms existing methods in most settings and also has great applications in GWAS on complex diseases with multiple phenotypes such as the substance abuse disorders. In the past decade, genome-wide association studies (GWAS) have produced rich single-nucleotide polymorphism (SNP) data available to researchers. Among them, the large scale studies including the HapMap project  and the Existing statistical methods for complex diseases with multivariate phenotypes can be categorized into three types of approaches. The first approach is to conduct a GWAS for each marginal phenotype and then aggregate the results. The major issue with this approach is that it does not make use of the correlation structure among phenotypes. The second approach is to summarize multiple phenotypic traits into a composite score and then conduct a GWAS on the score. This approach, however, may have difficulty in identifying proper summary scores. The third approach involves multiple phenotypic traits simultaneously. Thus, it may gain power as well as avoid the issue of multiple testing. However, it is based on stronger assumptions that may not be satisfied in some practical settings.In this study, we provide a comprehensive review of existing statistical methods for conducting GWAS on complex diseases with multiple phenotypic traits. We also propose a new statistical method based on the Fisher combination function. The performance of competing methods is evaluated by a simulation study. In order to demonstrate applications of the proposed method, we conduct statistical analysis on the database of the Study of Addiction: Genetics and Environment (SAGE).Xi be the number of reference alleles corresponding to a candidate SNP and Yi=′ be the measures of multiple phenotypes for the individual i. In this study, we conduct a comprehensive review of existing statistical methods that can be used to test the association between Xi and Yi.Let When the phenotype is univariate, we can use the one-way analysis of variance (ANOVA) with three levels of the genotype for GWAS. When we have correlated multivariate phenotypic traits, the natural extension of the one-way ANOVA is the one-way multivariate analysis of variance (MANOVA) . SimilarThe principal component analysis (PCA) , 10 is aThe method of generalized estimating equations (GEE)  was devep-value based on individual p-values of association tests for marginal phenotypes. Specifically, for m-variate phenotypic traits, one can conduct m tests of the association between a candidate SNP and each marginal phenotypic trait and derive mp-values: p1,…pm. Let p(1),…pm)<jα/m . Hence, n matrix , 21. SinCombining independent tests of significance to form a join statistic has been used as an alternative approach to tackling complex multivariate location problems . This apm-multivariate phenotypes, we have m marginal p-values: p1,…,pm. The Fisher combination statistic is defined as Based on the Fisher combination test , to testT is used to infer the association between the SNP and multivariate phenotypic traits. When the marginal p-values are independent, the statistic T follows a chi-squared distribution with 2m degrees of freedom so the p-value of T can be obtained straightforwardly. In reality, however, the phenotypic traits are always correlated so the chi-squared distribution with 2m degrees of freedom tends to underestimate the variance of the T statistic. The resulting chi-squared test is, therefore, too liberal.p-value of the T statistic in Eq. . The following are technical details of the derivation of the mean and variance of T statistic when the marginal p-values are based on two-sided tests , where Φ is the standard Gaussian distribution function. Under the null hypothesis of no association between a SNP and multivariate phenotypic traits, the distribution of T statistic is approximated by a Gaussian distribution with the mean of T as T as Without loss of generality, we assume that the association test statistic for the T, we need to calculate the covariance for each pair  which can be expressed as F is the standard bivariate Gaussian distribution. Let Therefore, in order to calculate the variance of δjk is a function of the correlation between zj and zk: ρj,k. We explore the relationship between δjk and ρj,k by calculating δjk numerically for the values of ρj,k from -0.99 to 0.99 with the step of 0.01. The results are shown in Fig. y-axis, we can approximate the relationship between δjk and ρj,k using a tenth-order polynomial: Thus, c1=3.9081,c2=0.0313,c3=0.1022,c4=−0.1378 and c5=0.0941; the maximum residual was less than 0.0001.Using the adapt function in the R package fCopulae , we obtaδjk in Eq. ,,4), the μ and σ2, the global p-value of T statistic can be computed efficiently using the gamma distribution function as follows: Γ is the gamma distribution function with the shape parameter v/2 and the scale parameter 2γ.Given the proposed estimators of τ. Kendall and Gibbons [ρ and τ as In this study, we compare two alternative methods to calculate  Gibbons  have shoτ to derive τ in comparison to Pearson’s sample correlation coefficient.Thus, we can use Kendall’s p-value of the Fisher combination test: m degrees of freedom under the independence assumption.FC-FC-Permutation: the permutation method based on 1000 permutes.FC-Pearson: the proposed method with the correlation τ.FC-Kendall: the proposed method with We conducted a simulation study to evaluate the performance of the four methods reviewed  as well as the methods based on the Fisher combination function. In the simulation, we adopted four different approaches to calculating the x, and the multivariate phenotypes, Y, was defined as β is the effect of the SNP on the phenotypes and ε is the error term. In this simulation study, we simulated x based on a minor allele frequency of 0.25. We evaluated the performance of competing methods under three different settings of effect sizes: Null hypothesis (no effects)β=′. Moderate equal effect sizesβ=′. Varied effect sizesβ=′.For each subject, the relationship between the SNP, 0 and the variance-covariance Σ, which is a compound symmetry matrix with the value of 1 on the diagonal and the value of ϱ=0,0.25,0.5, or 0.75 on the off-diagonal. In the second case, the error term was simulated from a mixture of two multivariate normal distributions: 90 % from the same multivariate normal distribution in the first case and 10 % from the multivariate normal distribution with the mean 0 and the variance-covariance matrix 5Σ. The purpose of the second case was to simulate long tailed distributions of phenotypic traits which are common in real data. We generated simulated data of 100 subjects under each configuration. In addition, each configuration was repeated 10,000 times. The nominal type I error rate was set at 0.05 and the power was calculated as the proportion of p-values less than 0.05.For the error term, we considered two cases. In the first case, the error term was simulated from a multivariate normal distribution with the mean ϱ varied from 0 to 0.75. The numbers in each cell are the mean (standard deviation) of the indicator variable for p-value <0.05 among the 10,000 replications. The top panel corresponds to the case of β=′ (i.e. when the null hypothesis is true) and thus can be used to evaluate if each of the competing methods was able to control the type I error. The results indicate that GEE and the Fisher combination test with ϱ>0 while all the other methods did quite well under all values of ϱ. We, thus, did not find it meaningful to further compare these two methods with the other methods in terms of the statistical power.Table β=′. The power of MANOVA decreased rapidly as the correlation ϱ increased. When ϱ=0.75, for instance, all the other methods had the power of at least 0.4 but the power of MANOVA was only 0.2. Further, PCA and TATES had higher power than MANOVA when ϱ>0. Yet, none of these three methods can beat the three Fisher combination tests  that performed equally well.The middle panel of Table ϱ increased. Furthermore, the Fisher combination tests had higher power than the other three methods  in most conditions. MANOVA only beat the Fisher when ϱ=0.75; TATES had higher power than the Fisher when ϱ≥0.5.The bottom panel of Table Table http://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/study.cgi?study_id=phs000092.v1.p1. The institutional review board of the University of Michigan has approved this secondary data analysis project (HUM00084927). The SAGE is a case-control study that aggregated together the data from three large scale studies in the substance abuse field: the Collaborative Study on the Genetics of Alcoholism (COGA), the Family Study of Cocaine Dependence (FSCD), and the Collaborative Genetic Study of Nicotine Dependence (COGEND). The total number of individuals with individual level data available is 4121. Each individual was genotyped using the Illumina Human 1M-Duo beadchip which contains over 1 million SNP markers.The National Center for Biotechnology Information (NCBI) has been managing and distributing the large database of Genotypes and Phenotypes (dbGaP) for scientific investigation of various human diseases . In ordeWe selected unrelated individuals that passed the quality control measures according to the Gene Environment Association Studies Initiative (GENEVA) quality control report. The final number of unrelated individuals is 3,741  and the total number of SNP markers is 917,694. Because the purpose of our analysis is to identify the genes that are associated with addiction, we used the symptomatology variables of four highly comorbid substance use disorders as the phenotype outcomes: the number of alcohol dependence symptoms endorsed , the number of nicotine dependence symptoms endorsed (nic_sx_tot), the number of marijuana dependence symptoms endorsed (mj_sx_tot), and the number of cocaine dependence symptoms endorsed (coc_sx_tot).The values of phenotype variables range from 0 to 7. Figure p-values from the positive component of the hurdle model were used for further analysis. For each phenotype of addiction, the estimated p-values are summarized using QQ-plots in Fig. p-values fall on the diagonal line and this indicates that there is no SNP associated with nicotine symptoms; (2) For the symptoms of the other three substances, all the p-values deviate from the diagonal lines (except for 0 and 1), with the p-values of alcohol symptoms furthest away from the diagonal line and the p-values of the remaining two symptoms closer to the diagonal line. Since the four symptomatology variables are moderately correlated and they all measure the common construct of addiction, we used them as the multivariate phenotype and applied the proposed Fisher combination approach  to identify the SNPs associated with it. The QQ-plot of the p-values for this multivariate analysis is shown in Fig. For each trait, we conducted a genome-wide association test using the hurdle model  because −6 to account for multiplicity. Based on the results of the marginal tests, the numbers of SNPs identified to be associated with the individual phenotypes are 917 for alcohol dependence symptoms, 0 for nicotine symptoms, 9 for marijuana symptoms, and 0 for cocaine symptoms. Using the proposed Fisher combination method , on the other hand, we identified 6 SNPs associated with the multivariate phenotype of addiction. Among them, 5 SNPs were also identified by the marginal test for alcohol symptoms. This implies that if we ignore the correlations among the 4 phenotypic traits and conduct the marginal tests, we would identify many SNPs that may be specific to alcohol dependence. Thus, if our goal is to identify the genes associated with the construct of addiction that contributes to the 4 types of substance dependence symptomatology, the proposed method is a better approach.To identify the SNPs associated with the phenotypes, we adopted the commonly used significance level for GWAS of 10p-values such as the PCA, MANOVA, and GEE, on the other hand, would require a complete re-analysis.In GWAS for complex diseases, the association between a SNP and each phenotype is usually weak. Combining multiple related phenotypic traits can increase the power of gene search and thus is a practically important area that requires methodology work. This study provides a comprehensive review of existing methods for conducting GWAS on complex diseases with multiple phenotypes including MANOVA, PCA, GEE, TATES, and the classical Fisher combination test. Built upon the Fisher combination test, we proposed a new method that relaxes the unrealistic independence assumption and is also computationally efficient. Particularly, in an exploratory study where multiple sets of phenotypes may be of interest, when the set is changed, our proposed methods only require re-calculation of the correlation between phenotypes and then the available marginal p-values for each SNP can be re-used. The competing methods which do not involve marginal τ) performed as well as the permutation method and yet only required 10−2 computational time. In most settings of the simulation, these three Fisher combination tests outperformed the other methods. The real data analysis also demonstrated that the Fisher combination tests allow us to compare the marginal results with the multivariate results and specify which SNPs are specific to a particular phenotype or contribute to the common construct.We conducted a simulation study to compare the performance of the competing methods. The GEE and the Fisher combination test with the independence assumption did not control the type I error rate and thus are not recommended. In general, the power of the methods decreased as the correlation between phenotypes increased. Furthermore, all the competing methods tended to have lower power when the multivariate phenotypes come from long tailed distributions. The proposed method (with the correlation being estimated by the Pearson’s sample correlation coefficient or the Kendall’s In our simulation study, we only considered continuous multivariate phenotypes. Future studies may extend the methodology work to the case of correlated discrete phenotypes. For example, in the substance abuse field, many outcomes are zero-inflated count data  or ordin


We also review nonparametric methods including Nadaraya-Watson estimator, reproducing kernel Hilbert space, support vector machine regression, and neural networks. We assess the relative merits of these 14 methods in terms of accuracy and mean squared error (MSE) using simulated genetic architectures consisting of completely additive or two-way epistatic interactions in an F2 population derived from crosses of inbred lines. Each simulated genetic architecture explained either 30% or 70% of the phenotypic variability. The greatest impact on estimates of accuracy and MSE was due to genetic architecture. Parametric methods were unable to predict phenotypic values when the underlying genetic architecture was based entirely on epistasis. Parametric methods were slightly better than nonparametric methods for additive genetic architectures. Distinctions among parametric methods for additive genetic architectures were incremental. Heritability, i.e., proportion of phenotypic variability, had the second greatest impact on estimates of accuracy and MSE.Parametric and nonparametric methods have been developed for purposes of predicting phenotypes. These methods are based on retrospective analyses of empirical data consisting of genotypic and phenotypic scores. Recent reports have indicated that parametric methods are unable to predict phenotypes of traits with known epistatic genetic architectures. Herein, we review parametric methods including least squares regression, ridge regression, Bayesian ridge regression, least absolute shrinkage and selection operator (LASSO), Bayesian LASSO, best linear unbiased prediction (BLUP), Bayes A, Bayes B, Bayes C, and Bayes C Understanding the complexity of these traits and accounting for the effects that are contributed by these genes and their interactions is not trivial.Complex quantitative traits are measured on a continuous scale and are controlled by a network of many genes, by the environment, and by genetic by environment interactions. Most traits of economical interest in agriculture (The gene-by-gene interaction or epistasis is an important research topic in quantitative genetics. Epistasis can be modeled in different ways . PhysiolMost simulation studies of genomic selection (GS) methods , adaptati.e., QTL mapping and for marker-assisted selection (MAS). In MAS, markers and phenotypic information are used to guide indirect selection of a trait of interest  greatly exceeds the number of individuals, n, that have been evaluated in the study], difficulty of interpretation of effects due to collinearity among the explanatory/predictor variables, model assumptions that cannot be satisfied, and nonadditivity among genetic effects.With advanced molecular techniques that provide dense marker maps, it is possible to overcome some shortcomings of MAS. In GS, individuals with both phenotypic and marker information  are used to model the association between the phenotype and the genotype. The model is used to predict the phenotypic value of individuals for which only the marker information is available . In GS, all available markers are included in the model, not just those above a significant threshold, thus eliminating the problem of multiple testing.Effort is underway to find ways to model epistasis, the gene-by-gene interaction. In the presence of epistasis, the effect of one locus changes the effect of another locus on the phenotype. Usually several loci are involved, which means that multiway interactions may need to be modeled. Because the volume of marker data points available is huge, the number of epistatic interactions can be overwhelming and computationally intractable to estimate with parametric methods .e.g., normality, linearity, independent explanatory variables), which suggests the use of nonparametric methods. Also, the convenient partitioning of genetic variance into additive, dominance, additive × additive, additive × dominance, etc., only holds under conditions of linkage equilibrium, random mating of male and female parents, no inbreeding, no assortative mating, no  selection, and no genotyping errors. In breeding programs, these conditions are all violated. More recently, π). We also explain the nonparametric kernel regression using the Nadaraya-Watson estimator (NWE) and the semi-parametric reproducing kernel Hilbert space (RKHS) regression. Finally, we describe support vector machine (SVM) regression and neural networks (NN) applications to GS. Herein, we review some existing statistical methods used in GS. First, we discuss the parametric methods in more detail. Then, we focus on the nonparametric and semi-parametric methods. Among parametric methods, we review linear least squares regression, penalized ridge regression, Bayes ridge regression, least absolute shrinkage and selection operator (LASSO), and Bayes LASSO methods, best linear unbiased prediction (BLUP), and some Bayesian alternatives used in GS  populations. We simulate the F2 and the BC populations with low and high heritabilities and compare the two extreme genetic architectures. One architecture had only additive genetic effects from alleles at 30 loci, and the other had only two-way epistatic genetic effects among 30 loci. The performance of the methods is illustrated by comparing the accuracy of prediction, which we defined by the correlation between the true phenotypic value and the predicted phenotypic value and the mean squared error (MSE). We demonstrate the advantage of some nonparametric methods for the epistatic genetic architecture. Because the results for the F2 and BC populations were similar, we only illustrate the F2 population in this article. In the Supporting Information, we provide accuracy and MSE values for a simulated BC population with low and high heritabilities and with two extreme genetic architectures.Here, we use simulated data to compare the performance of the parametric models with the nonparametric procedures for predicting the genetic value for individuals in a i = 1…n individual, j = 1…p marker position/segment, iy is the phenotypic value for individual i, μ is the overall mean, ijX is an element of the incidence matrix corresponding to marker j, individual i, jm is a random effect associated with marker j, and ie is a random residual. Typically, the residual term, e, is chosen to have a normal distribution with mean of 0 and variance of y can be written as:μ, m), we can use least squares to minimize the sum of squared vertical distance between the observed response and the estimated response, which can be represented as |y − Xm|2 (where | denotes the norm of a vector). The estimate of m obtained by solving the linear equations X′Xm = X′y. Then, it is estimated as Linear Models in Statistics   exists among the markers.One obvious problem with linear regression is that usually the number of markers (explanatory variables) available is much greater than the number of individuals with phenotypic information (response variables), which means that ocedure) , but it y = μ + jXjm + e, where y is the vector of the phenotypic information, μ is the overall mean vector, jX is the thj column of the design matrix corresponding to the thj segment, jm is the genetic effect associated with the thj segment, and e is the vector of the error terms. By plotting the log likelihood of this model, segments with significant effects were found. The segments with significant effect (QTL) were used for simultaneous estimation by the model: q is the number of QTL. With this approach, they eliminated the problem of having more predictor (explanatory/independent) variables than regressands (response/dependent variables), but it does not fully take advantage of the whole marker information because only markers with a significant effect are included in the final model. To overcome some of the drawbacks of the linear regression approach, other methods for GS have been introduced.X′X is close to zero due to column dependencies. Using a penalized regression model  were random, and they were drawn from a normal distribution with kn is the number of marker loci , where ∑β is a known covariance matrix  has a normal distribution with mean σ2, the Bayesian estimator of m is the mean of the posterior distribution, and it is given by Ridge regression can also be viewed from a Bayesian perspective. In this case, we assume that the parameter vector The BLUP theory and the mixed model formulation were first discussed by y is the (n × 1) phenotypic data vector, μ is the (n × 1) overall mean vector, jZ is the thj column of the design matrix, jm is the genetic effect associated with the thj marker, and p is the number of markers. The intercept, μ, is fixed, and jm is the random effects with E(jm) = 0, Var(e) = σ2I, and Cov = 0. In the statistical literature, the vector of random effects is usually denoted by u instead of m. If other covariates are available, then we replace the intercept μ by Xβ to include all the fixed effects. Then, we can write:β is a p1 × 1 vector of unknown fixed effects where usually the first element is the population mean, and X is the incidence matrix that relates y to β. The above equation is generally called a mixed model (or mixed effects model). The vector β is estimated by the best linear unbiased estimator (BLUE). In the biological literature, the term BLUP is occasionally used loosely and refers to both BLUE and BLUP. BLUP is the predictor of the random effects. It is a linear function of the data vector y. Within the linear functions of the data it is unbiased, which means that the expected value of the prediction is the same as the population parameter, and it can be formulated as y, it is unbiased among the linear estimators, and it is best in the sense that it minimizes the MSE.BLUP was proposed as a tool in GS by β, m) be obtained by maximizing the joint likelihood of  given by:L with respect to β, m and equating it to zero, we obtain a set of linear equations [known as Henderson’s mixed model equations (MME)]:R = Var(e) and G = Var(m). The solution to the MME is the BLUE of β and the BLUP of m. Henderson’s derivation assumes that m and e are normally distributed and maximizes the joint likelihood of  over the unknowns β and m. Maximizing the likelihood implies an optimization criterion of (y − Xβ − Zm)′R−1(y − Xβ − Zm) + m′G−1m, and it can be viewed as the “ridge regression formulation” of the BLUP  can be used with degrees of freedom ν and scale parameter S2 as the prior distribution. This is a convenient choice because it is a conjugate prior so the posterior distribution is in the same family of distributions as the prior distribution. The posterior distribution is also a scaled inverse chi-square distribution jn is the number of haplotype effects at marker position j.The difference between Bayes A and Bayes B lies in the way in which we model the variances of parameters. In both methods each marker position has its own variance. The Bayes A approach applies the same prior distribution for all of the variances of the marker positions. The scaled inverted (or inverse) chi-squared probability distribution χmjσ = 0 and an inverted chi-square distribution when mjσ > 0. It can be summarized as mjσ = 0 with prob= π and mjσ ∼ χ−2 with prob=(1 − π).The Bayes B approach seems more realistic for GS than Bayes A. The only difference between the two methods is the prior for the variance components. Bayes B assumes that not all markers contribute to the genetic variation. It has a prior density on the variance that is a mixture. It has a high probability mass at m′m > 0, then one cannot sample jm and jm.For the Bayes B method if To sample from the distribution1.Sample 2.π and Bayes Dπ  denotes the normal density with mean μ + Xm and variance σ2I where I is an n × n identity matrix. The prior distribution on the marker effects j = 1…p can be written as jτ is p(jτ|λ) ∼ Exp(λ) where Exp(λ) denotes the exponential distribution with rate parameter λ. λ2. The prior has a Gamma distribution with shape parameter α1 and scale parameter α2, and it can be written as p(λ2) ∼ Γ ; λ has two interpretations. In the Bayesian formulation, it is the rate parameter that controls the shape of the prior distribution of the λ controls the penalty for minimizing the MSE.In this section, we review some of the nonparametric estimation methods that have been proposed for the case where the form of the relationship between a response variable and a set of predictors is unknown. A popular approach, at least in terms of usage, is based on the kernel method proposed by n is the number of observations, K is the kernel function that satisfies the condition h is positive real-valued smoothing parameter , x is the focal point, and iX is the p × 1 dimensional vector of dummy covariates for observation i. We can calculate x, and the observations that are closer to the focal point will get a higher weight in the calculation, so the kernel function K is usually chosen to be a symmetric unimodal density, so the kernel density estimator xi coordinates closer to the focal point x are weighted more strongly in the computation of the fitted value The kernel density estimator included . Figure h = 0.1, the data have strong influence on the density estimate, resulting in little bias and large variability among estimates. It is called an undersmoothed estimate. As we increase the bandwidth value, the estimates become smoother. When h = 10, the spread is too big and even the bimodal feature of the data disappears, which implies that the estimate is oversmoothed. Setting the bandwidth too large results in a large bias with little variance.When i = 1, 2, …, n where iy phenotypic measurement on individual i, xi is a p × 1 vector of dummy SNP covariates observed on individual i, g(.) is some unknown function relating genotypes to phenotypes, g(xi) = E(iy|xi), and ie is a random residual effect for individual i where ie ∼  and is independent of xi.In the context of GS, p(x). The estimator has the form:xi is the observed p-dimensional SNP genotype of individual i, i = 1, 2, …, n. Similarly,iy, i = 1…n and is called the NWE.The conditional expectation function can be written in the form:stimator  can be uh, including the leave-one-out cross-validation (CV), penalizing functions, and plug-in methods. thi observation  and fit the model to the other n − 1 observations. Using the marker information, predict n observations. The CV criterion is .The selection of the bandwidth, h, value is challenging. erion is :CV(h)=∑ii = 1, 2, …, n, β is a vector of fixed unknown effects , u is a q × 1 vector of additive genetic effects, g(xi) is an unknown function of the SNP data and the vector of residuals, and e is assumed to have a u is distributed as A is the additive relationship matrix.The authors suggested two different methods for estimation in this model. The first strategy, denoted “Mixed Model Analysis,” consists of a two-step approach with a “corrected” data vector g(.) function” approach, where it is assumed that β, u are known. In this case:β(j), u(j) can then be obtained from the distribution The other method they suggested is the “Random i = 1, 2, …, n. The penalized sum of squares is given by:g(x)‖ is a function of the second derivatives of g(x). The goal is to find g(x) that minimizes the penalized SS. K is the reproducing kernel.Finally, ix) ∈ pR, then the separator is a hyper-plane in pR−1.SVM was proposed by Vapnik and discussed by Because fitting a regression model essentially consists of finding an optimal projection of the observations on a lower-dimensional hyper-plane, the idea can be used to estimate the unknown regression function subject to restrictions. The reader can refer to S = {, xi ε Rn,i ε Ry, i = 1…n}, where xi is a p dimensional vector containing the genotypic values for the p markers for individual i, and iy is the phenotypic value for individual i. A model that describes the relationship between the phenotype and the genotype of an individual can be written as:b is a constant and w is a vector of unknown weights. The constant b reflects the maximum error we are willing to commit when estimating the weights w. We learn about the function f(x) by minimizing the expression L(.) denotes the loss function that measures the quality of the estimation. The regularization parameter λ quantifies the trade-off between the sparsity and the complexity of the model. Increasing λ implies a higher penalty on the error. The norm ‖w‖ of vector w is inversely associated with model complexity; by choosing w to minimize ‖w‖, we reduce model complexity.Suppose that we have a training sample ε-insensitive loss. Here, we present these loss function formulations.1.L(y − f(x)) = (y − f(x))2. It scales the loss quadratically by the size of the error. Using this loss function indicates that outliers are also weighted quadratically, which requires the user to deal with the outliers before the regression analysis.The squared loss function has the form 2.L(y − f(x)) = |y − f(x)|. The absolute loss function scales the loss linearly by the size of the error eliminating the difficulty of using data sets with outliers.The absolute loss function has the form 3.ε-insensitive loss function has a form:The where ε determines the number of support vectors used in the regression function. By definition (xi that satisfies the equation iy(wxi + b) = 1. Increasing ε implies that fewer support vectors are used in the fitting. The ε-insensitive loss function ignores the errors in the regression that have size less than ε. When the error is greater than ε, the loss is |y − f(x)| − ε.There are many loss functions used for SVM regression. Some of the popular loss function choices include the squared loss, absolute loss, and the finition , a suppoε-insensitive loss functions as a function of the error y − f(x).ε-insensitive loss function, which needs a more robust representation to account for the noise in the data.In the remainder, we focus on the ξ constrained as follows  by introducing non-negative “slack variables”  follows :ξ1i ≥ iy − f(xi) − ε, where i = 1, …, n n is the number of training observations,ξ2i ≥ f(xi) − iy − ε, where i = 1, …, n.xi, xj〉, which is a linear function of the observations.We can now re-write the objective function to be minimized as:1nαixix+b. The solTo take advantage of higher dimensional feature spaces, we can introduce the data via nonlinear functions. For example, we can replace the inner product of the data by a kernel function:1.k = 〈x, z〉The linear kernel 2.k = exp(−σ‖x − z‖2), where σ is the bandwidth parameterThe Gaussian radial basis function 3.k = exp(−σ‖x − z‖).The Laplace radial basis function λ, ε, and σ are not straightforward. Because optimizing SVMs is not the focus of this article, we refer the reader to The solution to the minimization problem can also be written as a function of the kernel function. The resulting expression is NNs represent a nonparametric prediction procedure that captures additivity and epistasis by being able to model linear and complex nonlinear functions. The original idea of NN came from the theory of how neurons in the human brain work and interact, and how the brain conducts computations. In the NN, every unit is analogous to a brain neuron and the connections between them are analogous to synapses , M is the number of output layer units, L is the number of hidden layer units, lβ  are the output layer weights connecting the thl hidden layer unit to the output layer units, σ is the activation function modeling the connection between the hidden layer and the output layer, and l ε RKw and l ε Rb are the unknown learning parameters of the hidden layer unit l  connecting the thk neuron in the input layer to them ( to them .kI represents a vector of predictors (marker genotypes or other information) collected on individual k , where K is the number of individuals in the analysis. The activation function σ is typically chosen to be the sigmoid (logistic) or the Gaussian radial basis function.In GS, typically http://www.r-project.org, the qtlbim package can be accessed by library(qtlbim) in R, and the description of the package can be found at http://cran.r-project.org/web/packages/qtlbim/qtlbim.pdf. The reader can refer to F2 population with specifications listed in For the purpose of illustrating the parametric and nonparametric prediction approaches, simulated data were created by the R  package F2 and a BC population. The results for the BC population can be found in the supporting information section. For each set we created 20 replicates, which yielded to a total of 80 phenotypic and 80 genotypic data sets. Within each replicate we created 25 different training–testing data sets. Half of the data sets assume only additive effects and half assume only epistatic effects without any additive effects. We only evaluated the two extreme genetic architectures. Finally, for each genetic architecture, we generated data with two different narrow sense heritabilities. The low heritability was determined to be 0.30, and the high heritability was 0.70. For each of the simulated combinations of population, genetic architecture, and heritability, the data contain phenotypic information for 1000 individuals and genotypic information for 2000 biallelic markers  for each individual. Out of the 1000 individuals, 800 were chosen randomly to be in the training set to fit the model, and 200 individuals were in the testing set. We predicted the phenotype for the individuals in the testing set. The qtlbim package uses Cockerham’s model as the underlying genetic model. The simulated genome has 10 chromosomes, each having a specified length. The 2000 markers were distributed throughout the genome in such a way that each chromosome had 200 markers and the markers were equally spaced over the chromosomes. We assumed no missing genotypic values and no missing phenotypic values. The phenotypic values are normally distributed.We simulated four sets of phenotypic and genotypic information for a For the additive model, we placed two QTL on each of the 10 chromosomes with either positive or negative additive effect. For the additive model we assumed no epistatic interaction.P = G + E model. For the epistatic model, we only considered two-way interactions between the QTL. The interacting QTL were at the same genomic location as the QTL for the additive model, and only neighboring QTL were associated with each other, resulting in 10 two-way epistatic interactions, with each having either positive or negative epistatic effect on the phenotype. For the epistatic model, we assumed that the QTL contributed no additive effect. The phenotypic values were drawn from a normal distribution and are based on the π. We also evaluated nonparametric methods, including NWE, RKHS method, SVM, and NN. To implement the parametric and nonparametric methods, the statistical software R and software written in C++ provided by the Animal Science Department at Iowa State University were used. Specifications of the parameters and inputs for each method are described below.To compare the performance of the methods, we used cross-validation, where we divided the data into training sets and testing sets. The training sets were used to fit the models, and the testing sets were used to determine the performance of the particular method. The performance of the methods was calculated by the accuracy of prediction and the MSE. We define accuracy of prediction as the correlation between the true phenotypic values and the predicted phenotypic values. We evaluated parametric methods including parametric least squares regression, ridge regression, Bayesian ridge regression, BLUP, LASSO, Bayesian LASSO, Bayes A, Bayes B, Bayes C, and Bayes Clm function was used that can be found in the stats package  is the sum of the 2000 marker variances. For all of the scenarios, we used h2 = 0.4.For this method, we used Gustavo de los Campos’ software written in R. The code implements the calculation of the ridge regression estimation of the marker effects discussed in the ridge regression section and uses these estimates to perform the prediction for the individuals in the testing set. For the procedure, all of the available phenotypic and marker information is used. In the estimation of the marker effect, the penalty parameter χ2 distribution with degrees of freedom BRdf = 5 and scale parameter BRS = 0.01. The residual variance Eσ has a scaled inverse χ2 distribution with degrees of freedom Edf = 4 and scale parameter ES = 1. The BLR function implements the Gibbs sampler, and the number of iterations is specified to be 20,000. We used 2000 iterations for the burn-in period without any thinning. To fit the Bayesian ridge regression, we used all available phenotypic and genotypic data.To fit the Bayesian ridge regression model, the function Bayesian linear regression (BLR) was used, which can be found in the BLR package  in R. Fomixed.solve function in R that can be found in the rrBLUP package .To predict phenotypic values in the testing set using the LASSO method, we used the λ, is specified to be random and has a Gamma prior distribution with shape parameter α1 = 0.53 and rate parameter α2 = 0.00005. The residual variance Eσ has a scaled inverse χ2 distribution with degrees of freedom Edf = 4 and scale parameter ES = (fd − 2)(1 − h2)Var(y), where we specify that fd = 4, h2 = 0.5, and Var(y) is the phenotypic variance. The Gibbs sampler was applied with 20,000 iterations, and 2000 iterations were in the burn-in period. The chain was not thinned.To fit the Bayesian LASSO method, the function BLR of the BLR package  in R wasπ models, software called Gensel (version 2.12) was used. GenSel was written by To implement the Bayes A, Bayes B, Bayes C, and the Bayes Cπ have been implemented. We used the settings for the four methods that are listed in In GenSel Bayes A, Bayes B, Bayes C, and Bayes Ch, that minimized the criteria. To use the NWE for predicting the phenotypic values in the testing set, first we formed the cross-validation criteria and we evaluated it on a grid of values. We examined the cross-validation criteria between 1 and 1000 and chose the value to be the bandwidth, The RKHS regression was based on methods and algorithms described by ksvm function of the kernlab package  kernel as the kernel function. After fitting the model, the predict function was used to perform the prediction of the phenotypic values for the testing set. For the other input parameters, the default values were used.To implement the SVM regression, we used the brnn function of the brnn package . Two-dimensional scans of the whole genome had previously revealed that the genetic architecture of this trait is composed primarily of interactions involving many loci. Thus, the simulated architectures used in our study are reasonable for many quantitative traits.The clear distinctions of estimated accuracies and MSE values between parametric and nonparametric methods when underlying genetic architecture is epistatic suggest that data analyses consisting of a combination of parametric and nonparametric GS methods could be used as a diagnostic to reveal the prevalent genetic architecture of the trait. It is likely that the true underlying genetic architecture consists of mixtures of additive and epistatic genetic effects, so the inferential limits of applying pairs of GS methods to data from samples of breeding populations as a diagnostic needs further investigation. However, the first step is to look at the extremes in terms of genetic architecture.i.e., inferences about specific allelic contributions to the trait are limited. If the goal is genetic improvement and the underlying genetic architecture is known to be additive, then parametric GS methods will provide better predictions for selection. It has previously been hypothesized (Our results also suggest that if the goal of the research is to accurately predict the genotypic value of an individual, particularly for purposes of selection, and if the underlying genetic architecture of the traits are not known, then it is best to use the nonparametric NWE, the RKHS, or the SVM. Unfortunately, these methods do not provide interpretable inferences about relative weighting that is being applied to various regions of the genome, Although heritability did not affect the ability to distinguish among GS methods, it did affect estimated accuracies. When heritability is high and genetic architecture is additive, predictions are more accurate than for low heritability. When genetic architecture is based on epistasis and the trait exhibits low heritability, predictions are not very accurate for almost all GS methods. Even when the heritability is 0.70, the highest mean for prediction accuracy is 0.35, which indicates that further improvement of the models is necessary. Also, further research is needed to determine the affects of more complex plant breeding population structures. Typically, plant breeding population structures consist of inbred progeny derived from multiple crosses involving related and unrelated inbreds (In practice, plant breeders do not know the genetic architecture responsible for quantitative traits and the dynamics of selection for genetic improvement will tend to favor alleles that contribute to additive components. Genetic improvement is affected not only by the underlying genetic architecture but also by additional types of unpredictable genetic contributions including intra-locus dominance and genotype by environment interactions. Herein, we have demonstrated the superior ability of the nonparametric NWE, the RKHS, and the SVM methods to accurately predict phenotypes for additive by additive inter-locus interactions. We hypothesize that nonparametric GS methods also will enable more accurate predictions of individual genotypic value for traits that are affected by dominance and genotype by environment interactions.


The generalized linear mixed model (GLMM) is a useful tool for modeling genetic correlation among family data in genetic association studies. However, when dealing with families of varied sizes and diverse genetic relatedness, the GLMM has a special correlation structure which often makes it difficult to be specified using standard statistical software. In this study, we propose a Cholesky decomposition based re-formulation of the GLMM so that the re-formulated GLMM can be specified conveniently via “proc nlmixed” and “proc glimmix” in SAS, or OpenBUGS via R package BRugs. Performances of these procedures in fitting the re-formulated GLMM are examined through simulation studies. We also apply this re-formulated GLMM to analyze a real data set from Type 1 Diabetes Genetics Consortium (T1DGC). However, despite the popularity of these software packages, they often lack many options such as choosing different algorithms for maximizing the likelihood or specifying various particular type of the covariance structures that are available in standard statistical software packages such as procedures “proc mixed,” “proc glimmix,” and “proc nlmixed” in SAS  or OpenBUGS in R . One major obstacle in using these standard software packages to fit the GLMM is the requirement of the same correlation structures across all the families (or clusters). A few recent studies also suggested fitting the GLMM by using SAS  provides a rich class of statistical models to model correlated data with responses from the exponential family of distributions including Gaussian, Binomial, Poisson, etc.  response variable for a disease phenotype; ijz be some fixed environmental covariates that need to be adjusted for; ijg denote the observed genotypes at certain targeted genetic marker loci, for family members j = 1, 2, ···, in and i = 1, 2, ···, N. To model the family data and test for the association of ijg with the phenotypic response ijy, we need to account for the genetic correlation among family members induced by identity-by-descent (IBD) alleles shared by the family members at some putative disease susceptible loci (DSL). In addition, family members may share certain common environmental factors which could also contribute to the disease phenotypes. Let ijv denote the random genetic effect from those putative (unobserved) DSL on the phenotypic response ijy, and ie be the shared environmental effect such as diets for members in a family i. Define μij = E. Then a generalized linear mixed model (GLMM) to model the family data can be written asSuppose that we have a randomly collected sample of g(·) is a known link function, m is an intercept for the baseline, α is a p-dimensional vector of parameters for the fixed effects of environmental covariates ijz, x(ijg) is a q-dimensional vector with its components being defined by certain coding functions for marker genotypes , ···, g(μiin), which is often induced by identity-by-descent (IBD) alleles shared by the i-th family members at the putative DSL.where g is often chosen as an identity function  = x). Assuming that there is no inbreeding between parents, it has been known that the genetic covariance between a pair of relatives j, k within a family i can be expressed as Kempthorne . Let μi = . From model Equations (1), (2), we havewhere σin is a in × in matrix of 1's, Φi = (ϕjk) and Δi = (δjk) are in × in kinship and double coancestry matrices, respectively, for the i-th family members. In the absence of inbreeding, the expected kinship and double coancestry coefficients for various common relatedness have been well established , it is possible to estimate the actual genome-wide kinship and double coancestry coefficients among family members using external programs such as PLINK  being the identity link function, the GLMM becomes a linear mixed model (LMM).The above GLMM provides a rich class of statistical models, which is applicable to both quantitative and qualitative traits. Depending on the family structures, however, not all the variance components in a GLMM are always estimable. For example, for a quantitative trait with i = 1, …, N, j = 1, … in, ϵij ~ N are the model residuals, with σ2 being the residual variance of random effects from other risk factors not captured by ijz, ijg, ijv, and ie. When each family comprises two parents and one offspring, the expected Δi = I3, which is a 3 × 3 identity matrix for any family i. We havewhere 2D and the residual variance σ2 are completely confounded. Similarly, if each family consists of only siblings , thenThus, the dominant variance component σ2D = σ2c = 0). For a sample of unrelated individuals, it is also easy to see that all the four variance components σ2A, σ2D, σ2c, and σ2 are inseparable.In this case, we cannot distinguish the four variance components from each other unless some of them are negligible , the full likelihood isThe main goal in fitting GLMM Equation (1) is to make statistical inference on the fixed effects α, β as well as assessing the variance components σg(·), based on which the pseudo-likelihood or restricted pseudo-likelihood for optimization can then be derived  using “proc nlmixed” and “proc glimmix” procedures in SAS, or OpenBUGS as an alternative choice.In practice, several common statistical software packages are available in fitting a standard GLMM. These include but not limited to “proc nlmixed” and “proc glimmix” procedures in SAS, and OpenBUGS for Bayesian approach. The “proc nlmixed” mainly conducts integral approximations using an adaptive Gauss-Hermite quadrature as default, and then directly maximizes the approximately integrated likelihood. In contrast, the “proc glimmix” primarily performs several model linearization based pseudo-likelihood methods, although it can also fit the GLMM using Laplace or adaptive Gauss-Hermite quadratures for integral approximations . In addition, the model fitting algorithm in R package “lme4” consists of an iteration between two sub-optimization procedures. One is to determine the conditional modes of the fixed and random effects, given the current deviance and variance components using penalized iteratively re-weighted least squares (PIRLS). The other is to obtain MLE of the deviance and variance components based on a profile likelihood from Laplace approximation, given the current conditional modes of the fixed and random effects  into a GLMM with random regression coefficients. First, we apply separate Cholesky decompositions on the kinship and double coancestry matrices: 2Φi = LiΦLTiΦ, Δi = LiΔLTiΔ, for i = 1, ···, N. We then re-parameterize the random genetic effects within family i as iv = LiΦia + LiΔid, where ia = T ~ N with ir = rank(Φi), and id = T ~ N with is = rank(Δi). Besides, we assume that ia's are independent of id's. By replacing iv by LiΦia + LiΔid in model Equation (1), we haveTo deal with different families sizes or varied kinship or double coancestry matrices, we apply a Cholesky decomposition based re-parameterization of the random genetic effects ia and id are orthogonal within families as well as across families, even though the dimension of these random effects may still vary from family to family because the families may have different sizes.Note that the random effects r = imax{ir} and s = imax{is} be the maximum number of columns in {LiΦ, i = 1, ···, N} and {LiΔ, i = 1, ···, N}, respectively. For those families with ir < r (or is < s), we further expand their design matrices LiΦ (or LiΔ) to r (or s) columns by adding r − ir (or s − is) columns of 0's at the right end (or any places). Then, we obtain a re-formulated GLMM with all the families having the same dimension r (or s) for their random effects ia (or id) as the following.Next, to deal with possible different family sizes, let iZ = T, X(ig) = (x(gi1), ···, x(iing))T, ia ~ N, id ~ N and ie ~ N. Finally, we construct the design matrix LΦ of ia from LiΦ, i = 1, ···, N, by stacking them one above the other; and similarly build the design matrix LΔ of id by stacking LiΔ, i = 1, ···, N, one above the other. Now, the columns of the two matrices LΦ and LΔ can be treated as (r + s) ordinary fixed continuous covariates with ia's and id's being their random regression coefficients. Within each family, all the family members share the same set of slopes. Across different families, the ia, i = 1, ···, N , are independent but share the same variance component σ2A (or σ2D).where e/subject = famid,” “random La1 … Lar/subject = famid type = TOEP(1);” and “random Ld1 … sLd/subject = famid type = TOEP(1);” to specify the correlation structures for the random effects ie, ia, and id, respectively. Here, La1,…, rLa represent the columns of the stacked matrix LΦ; Ld1 … sLd denote the columns of the stacked matrix LΔ. The option “TOEP(1)” can force all the elements in ia (or id) to share the same variance component σ2A (or σ2D). For “proc nlmixed,” currently it only allows to have one “random” command. But we can specify a joint multivariate normal distribution for ie, ia, and id via “random e a1 … r da1 … sd ~ normal subject = famid,” where v is a diagonal matrix with one σ2c, r elements of σ2A's and s elements of σ2D's on its diagonal. As an example, the SAS codes for specification of a GLMM using both “proc glimmix” and “proc nlmixed” for families with two parents and two full sibs  are provided in Appendices lme4 to fit the re-formulated GLMM. But it appears that the functions “lmer” and “glmer” provided in lme4 do not have an option that can force all the elements in ia (or id) to share the same variance component σ2A (or σ2D).The re-formulated GLMM above can be easily specified by “proc nlmixed” or “proc glimmix” procedures in SAS. With “proc glimmix,” we can use three separate “random” commands “random iΣ = iσ2A2Φ + i σ2DΔ involves two unknown variance components σ2A and σ2D. As a result, we cannot directly specify the covariance matrix iΣ in OpenBUGS. With the re-formulated GLMM, we can pass LΦ and LΔ as fixed covariates to OpenBUGS with ia and id being their random regression coefficients. Since the Bayesian approach often treats all the model parameters as random, it appears especially suitable for fitting the re-formulated GLMM. We can also extract from the MCMC the posterior distributions of random effects for each individual, which allow us to assess the variation contribution from the putative random genetic effects {ijv} to the total variance of {g(μij)}. The R codes for specification of a re-formulated GLMM using BRugs + OpenBUGS are also provided in Appendix This re-formulation also makes it more convenient to fit GLMM Equation (1) using the Markov chain Monte Carlo (MCMC) based Bayesian approach. We use R package BRugs to get access to OpenBUGS software . The three genetic markers are assumed to be unlinked  and have allele frequencies p1 = 0.5, p2 = 0.2, p3 = 0.1 for alleles “1” at locus 1, 2, and 3, respectively. We first generate a pairs of haplotypes independently for each parent, where each haplotype is comprised of three alleles randomly generated from Bernoulli(jp) for j = 1, 2, and 3, respectively. Each child inherits one haplotype from father, and the other from mother. The haplotype from father (or mother) consists of three alleles with each allele being selected from the two paternal  alleles at the same locus with 50% chance. In case where a family has more than one child, the above random process is repeated for each child independently. For simplicity, we use the expected values to construct the kinship and double coancestry matrices for each family.In this section, we examine the performances of procedures “proc nlmixed” and “proc glimmix” in SAS (version 9.3) as well as R packages BRugs + OpenBUGS in fitting the re-formulated GLMM through simulation. We consider three biallelic genetic markers with alleles “0” or “1,” and one fixed explanatory covariate gij =  be the genotypes of the j-th subject in family i, where ijkg ϵ {0, 1, 2} counts the number of alleles “1” at locus k = 1, 2, 3. The quantitative trait values are simulated from the following LMM.We first consider simulating quantitative traits from a LMM. Let x(ijkg) = (ax(ijkg), dx(ijkg))βk with ax(ijkg) = ijkg, dx(ijkg) = 1 (or 0) if ijkg = 2 (or otherwise) based on the allele-coding T being the effects of marker k for k = 1, 2, 3. We further set σ2c = σ2 = σ2A = 1 and σ2D = 0.5. The other true values of parameters are listed in Table n1 families with two parents and one child and n2 families with two parents and two children. We consider three cases: (a) n1 = 500, n2 = 500; (b) n1 = 1000, n2 = 1000; (c) n1 = 0, n2 = 2000.where Table 3. We also explored using “proc glimmix” to fit the simulation data but abandoned it due to some severe unconvergence problems. In running BRugs + OpenBUGS, we choose the following priors for initialization of the parameters: m ~ N, α ~ N, βka ~ N and βkd ~ N for k = 1, 2, 3, τa ~ Gamma, σD ~ Uniform, τc ~ Gamma, and τ ~ Gamma, where τa = 1/σ2A, τd = σ2D, τc = 1/σ2c, and τ = 1/σ2. We use the first 10,000 updates as burn-in, and another 10,000 updates to estimate parameters as posterior means. The parameter estimates and their standard deviations (SD) from the 200 simulations are summarized in Table For each simulation data set, we fit the GLMM by using two methods: (1) adaptive Gaussian quadrature (AGQ) via “proc nlmixed” in SAS; 2) Bayesian approach via BRugs + OpenBUGS. Based on 200 simulation data sets, the parameter estimates are summarized in  Bayesianka, and βkd  as well as the variance components with improved accuracy as sample size increases. As expected, the estimate of allelic interaction βkd has a larger SD than that of the additive allelic effect βka at each locus k, for k = 1, 2, 3. The estimates of σ2D and σ2 are slightly biased, which could be caused by the potential confounding between these two variance components. Besides, it appears that “proc nlmixed” tends to over-estimate the dominant variance σ2D and under-estimate the residual variance σ2. Meanwhile, the Bayesian method heads to the opposite way. From Table 2D and σ2. The average lengths of the 95% confidence (or probability) intervals for σ2D and σ2 are also greater than the ones for other two variance components. In addition, the average length of the 95% confidence (or probability) intervals for βd3 is substantially larger than that for βd1 and βd2, which is likely caused by the fact that the allele “1” at marker locus 3 is rare and the homozygous genotypes “11” at marker locus 3 are much less present in a simulation data set than those at the other two loci.From Table 2 and σ2D are confounded in single-child families, and only two-child families are informative for distinguishing them. In case (b), where there is an increased number of 2-child families, the estimates of these two variance components are improved with both methods. In case (c), the accuracy in estimates of these two variance components is further improved. In terms of the computational speed, it appears that the “proc nlmixed” runs much faster than BRugs + OpenBUGS. We ran BRugs + OpenBUGS on one laptop installed with Intel(R) Core(TM)i7-3520M CPU @ 2.90GHz. It took about 19 h, 50 h and 38 min, and 59 h and 15 min of CPU time to complete the 200 simulations (including data generation and MCMC iterations) in cases (a), (b), and (c), respectively. The “proc nlmixed” procedure in SAS was performed on a UNIX workstation which has a compatible speed with the laptop, and it took about 2 and a half hours, 6 and a half hours, and 7 h 18 min of CPU time to complete the 200 simulations in cases (a), (b), and (c), respectively. By contrast, the Bayesian approach can provide the posterior distributions for all the parameters rather than just the modes (MLE) and their variance or covariance estimates. When σ2 and σ2D are almost completely confounded, we found that the “proc nlmixed” may give unreliable estimates of the model parameters or encounter unconvergence problem caused by the nearly singular Hessian matrix, while the Bayesian approach can still provide reasonable estimates at least for other model parameters except σ2 and σ2D.Overall, the results from “proc nlmixed” and BRugs + OpenBUGS are quite comparable in all three cases. As we have mentioned before, the variance components σWe also consider binary traits and simulate phenotypic values from the following mixed logistic regression model.i = 1, …, n and j = 1, …, in. In order to have enough number of events, we choose m = −3. Meanwhile, we keep the same true values as before for other model parameters. We explored various options on using “proc nlmixed” and “proc glimmix” but failed to fit the above model appropriately due to severe unconvergence problems. By contrast, BRugs + OpenBUGS can still provide reasonable estimates for most of the model parameters. Using the similar priors in the previous setting, 10,000 burn-in and 10,000 updates for parameter estimation, we compute the means and standard deviations (SD) of the parameter estimates and the average lengths (AL) and coverage rates (CR) of the 95% probability intervals from 200 simulations as summarized in Table for Under our simulation setting, it appears that the SAS procedures “proc nlmixed” and “proc glimmix” perform unexpectedly poorly especially for binary outcomes. But we have to admit that, although we have explored many options that are available in running “proc nlmixed” and “proc glimmix,” our exploration is surely not exclusive based on our limited knowledge. Besides, we use SAS version 9.3 on our Linux computer system. Recently, a newer version 9.4 of SAS has become available for PC users, which may provide improved performance in fitting the GLMM. Unfortunately, we do not have access to this new version of SAS yet based on our current Linux computer system. The performance of “proc nlmixed” and “proc glimmix” in SAS 9.4 on fitting the re-formulated GLMM needs further exploration.2D and σ2) are noticed. Typically, the auto-correlation could be reduced by thinning the Markov chains. Otherwise, appropriate adjustment is needed in computing the SD of parameter estimates.While the frequencist approach implemented in “proc nlmixed” and “proc glimmix” often require numerical approximations to the full likelihood, the Bayesian approach directly maximizes the full likelihood via MCMC. From our simulation study, it seems that the Bayesian approach can better handle the re-formulated GLMM especially for binary traits. However, it should be pointed out that the choice of priors in using the Bayesian approach could have a significant impact on convergence of an MCMC procedure. Besides the gamma priors for all the τ's, we also tested using uniform priors on σ's and obtained similar results. Throughout our simulation, the Monte-Carlo errors for all the parameter estimates appear to be acceptable. But some auto-correlations in certain Markov chains . The data set includes five cohorts: Asia-Pacific (AP), Danish Steno Diabetes Center (DAN), European (EUR), Sardinian (SAR) and United Kingdom Genetic Resource Investigating Diabete (UK). For simplification, we only adopt nuclear families and exclude some grand-parents or grand children . Most of the families (about 80.4%) consist of 2 parents and 2–3 children. There are 13 families that have more than 9 family members, and they all belong to the DAN cohort. The actual numbers of subjects and families we used in the five cohorts are listed in Table http://www.who.int/genomics/about/Diabetis-fin.pdf” also provides a nice review of the T1D. From our previous study  at HLA-DQB1, male and “age≤18” as a baseline, we fit each cohort separately using the following mixed logistic regression model.Our research interest is to test for association of HLA-DQB1 locus with Type 1 Diabetes (T1D) incidence, while appropriately controlling for other potential genetic risk factors on the incidence of T1D. The adjustment for random additive and dominance effects is important because it has been known that T1D is a polygenic disease. Some studies have suggested that other genes such as INS and CTLA4 could be implicated with T1D , eiα  are odds ratios of having T1D in different age groups comparing with the youngest age group of age ≤18, ejβ  are odds ratios of having T1D in different HLA-DQB1 risk groups comparing with the high risk group, e6β is the odds ratio of T1D in females versus males, and {ijv} have the covariance structures as specified in GLMM Equation (1).where LΦ and LΔ) cannot exceed 12 even though the total number of family members can still exceed 12. We also find that some of the parameter estimates become unstable when we actually use 10–12 columns. So we adopt using 8 columns of the stacked matrices LΦ and LΔ in fitting the Dan cohort. The estimates of odds ratios and variance components in the fitted models from running BRugs + OpenBUGS are summarized in Table To simplify the calculation, we use the expected values to construct the kinship and double coancestry matrices for each family. As the actual kinship and double coancestry matrices should be close to their expected matrices, we would expect only minor deviations from the fitted GLMM. In running “proc glimmix” and “proc nlmixed” to fit the GLMMs, we encountered some severe unconvergence problems (data not shown). In running BRugs + OpenBUGS, we use 10,000 burn-in and another 10,000 updates to estimate the parameters. For the DAN cohort, the current BRugs + OpenBUGS cannot accommodate more than 12 additive or dominant random coefficients in the model specification—see part (2) of Appendix The results have confirmed that age is significantly associated with T1D incidence. In most cohorts except AP, the odds of T1D occurrence reaches the highest in the youngest group of age < 18 and then decreases quickly as age increases. The females appear to have a less chance of having T1D than males in DAN and EUR cohorts. For HLA-DQB1, it appears that in each cohort there is a significant increase of T1D risk in the HLA-DQB1 very high risk group comparing with the high risk group, and the high risk group is also significantly different from the low risk and moderate risk groups after adjusting for the age and gender effects. The odds of T1D for the high or very high risk groups appear significantly higher than that for the low or moderate risk groups, which are consistent with the relative risk of 3–45 of the HLA-DQB1 susceptibility variant reported in “Genetics and Diabetes” from WHO.2D in the models, we also fit the GLMM models without σ2D and compare them with our previous models. Based on the deviance information criterion (DIC), which is an estimate of the expected predictive error (lower deviance is better) and it can account for both model fitness and model complexity, the models with both additive and dominance variances included have their DIC values of 658.6, 647.3, 1525, 342.2, and 316.6 which are lower than the DIC values of 669.5, 658.9. 1543. 354.1, and 322.3 in the reduced models without σ2D in AP, DAN, EUR, SAR, and UK cohorts, respectively. So the models with both additive and dominance variances included are preferable. We also estimate the variations contributed by the random putative genetic effects {ijv}, which account for 22%, 26%, 9%, 27%, and 12% of the total variation in logitP for the AP, DAN, EUR, SAR, and UK cohorts, respectively.Regarding the variance components, the estimates of the additive and family shared variances appear reasonably well, although the dominance variances in SAR and AP cohorts have relatively large variation due to perhaps the lack of information in their family data. To see whether we should not include σThe T1DGC family data was collected from the observational retrospective sampling, which likely had over-sampled families with T1D children. It is well known that fitting a mixed logistic regression model with random effects to a retrospective data set may no longer provide the equivalent maximum likelihood estimates of the model parameters as the ones defined in the same model for a prospective cohort. Therefore, the results above are only applicable to the families we obtained from T1DGC. For general populations, an adjustment for the retrospective sampling strategy is needed in order to avoid the bias in the parameter estimates.In this study, we propose a Cholesky decomposition based re-formulation of the GLMM to fit family data with varied sizes and diverse genetic relatedness. Assuming that there is no genetic correlation between different families, we first apply separate Cholesky decompositions on the genetic kinship (or double co-ancestry) matrices within each family. Next, we stack these Chelosky decomposition matrices from different families into columns to form two stacked matrices. The columns of the stacked matrices can then be treated as fixed covariates with random regression coefficients of equal variances. It should be pointed out that applying separate Cholesky decompositions on each family does not provide computational or storage benefits comparing with the regular Cholesky decomposition on the whole covariance matrix of the random genetic effects because the non-diagonal blocks in the sparse matrix decomposition are actually not saved. However, with the reduced number of columns in our stacked matrices, this re-formulated GLMM can be specified more conveniently using “proc nlmixed” and “proc glimmix” in SAS, or OpenBUGS via R package BRugs. Theoretically, the re-formulated GLMM is equivalent to the original GLMM. Therefore, it should retain the same validity and power in testing the fixed and random genetic effects as the original GLMM. From our simulation, it appears that this re-formulated GLMM can be fitted reasonably well by “proc nlmixed” and BRugs + OpenBUGS for quantitative traits with moderate family sizes. For binary traits, our simulation and real data example shows that at least BRugs + OpenBUGS can appropriately fit the re-formulated GLMM for families of sizes not exceeding 12.This Cholesky decomposition based re-formulation of GLMM in fitting family data is somewhat analogous to Haseman and Elston's regression method for sib-pairs  usually do not asymptotically follow the standard Chi-square distributions under the null because the zeros under the null hypothesis are located on the boundary of the parameter space for the variance components, where the standard regularity conditions no longer hold. As pointed out in Wang et al. , in the It should be pointed out that this Cholesky decomposition based re-formulation in fitting the GLMM has some limitations. For example, in order to synchronize the varied family sizes, the number of random effects in smaller families needs to be expanded to match that in the largest family. The Cholesky decomposition also requires that the kinship and double coancestry matrices be positive. When the kinship matrix is calculated from genome-wide genotypes, the kinship or double coancestry matrices could become singular for some of the families. One possible solution to this problem is to apply a different type of decomposition to the kinship and double coancestry matrices for these families. Note that the decomposition matrices can have a reduced number of columns as long as a good approximation to the kinship and double coancestry matrices is maintained. Finally, it appears that the computational speed in fitting the re-formulated GLMM via SAS or OpenBUGs is slow. The proposed GLMM re-formulation is probably more suitable for a refined analysis on certain targeted loci rather than a genome-wide scan for a large number of genetic markers.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


We apply a family-based extension of the sequence kernel association test (SKAT) to 93 trios extracted from the 20 pedigrees in the Genetic Analysis Workshop 18 simulated data. Each extracted trio includes a unique set of parents to ensure conditionally independent trios are sampled. We compare the empirical type I error and power between the family-based SKAT and the burden test under varying percentages of causal single-nucleotide polymorphisms included in the analysis. Our investigation using simulated data suggests that, under the setting used for Genetic Analysis Workshop 18 data, both the family-based SKAT and the burden test have limited power, and that there is no substantial impact of percentage of signal on the power of either test. The low power is partially a result of the small sample size. However, we find that both the family-based SKAT and the burden test are more powerful when we use only rare variants, rather than common variants, to test the association. Genome-wide association studies (GWAS) have proven to be a powerful approach to identify novel common single-nucleotide polymorphisms (SNPs) contributing to the etiology of complex traits . HoweverA major limitation of population-based association analyses is the potential for unrecognized population heterogeneity as a result of population stratification. This problem, however, can be well addressed through the use of family-based studies, which use related individuals in association studies. Family-based controls eliminate the need to adjust for population structure ,5. AnothThe data set for the Genetic Analysis Workshop 18 (GAW18) consists of whole genome sequence data from a pedigree-based sample. These pedigrees are drawn from the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Sample Project 2 (T2D-GENES Project 2). The T2D-GENES Project 2 is designed to identify low-frequency or rare variants influencing susceptibility to type 2 diabetes using information from whole genome sequencing of 1043 individuals from 20 Mexican American pedigrees enriched for type 2 diabetes from San Antonio, Texas. The pedigree data are drawn from 2 San Antonio-based family studies: the San Antonio Family Heart Study (SAFHS) and the San Antonio Family Diabetes/Gallbladder study (SAFDGS).A variance component test, known as a sequence kernel association test (SKAT), is proposed for testing associations of rare variants in population-based designs ,6. SKAT The data set for GAW18 includes 959 individuals out of 1043 individuals from 20 Mexican American pedigrees of the T2D-GENES Project 2. We conducted our analysis using the simulated phenotypes, baseline systolic blood pressure (SBP) and diastolic blood pressure (DBP), and the whole genome sequenced and imputed genotypes of the 959 correlated individuals. To keep the notation simple and make our discussion transparent, we considered trio designs, acknowledging the fact that the method is applicable to more general family structures.Conditionally independent trios were extracted from the 20 extended pedigrees. Each extracted trio included a unique set of parents. For nuclear families with more than 1 offspring, we randomly selected 1 offspring and formed a trio with the parents. Specifically, the individuals were grouped into families by the parents' identifications, and 1 offspring was selected with equal probability to form a trio. We only selected trios that had complete genotype data for all 3 family members. Finally, 93 conditionally independent trios were extracted from the GAW18 data.ith trio, denote the specific region of the genome by G, the offspring trait by jth variant in G by Xij (1 ≤ j ≤m), where m is the number of variants in the region G. We assume a generalized linear mixed effects model (GLMM) as follows: α is the regression coefficients for the potential confounders β is the vectors of regression coefficients for the m variants βs equal to 0, is equivalent to testing X represents the offspring genotype matrix, i at variant j. It can be shown that the test statistic β coefficients are independently distributed. To allow for possible correlation of effects among different variants, a family kernel was proposed [β coefficients are assumed independent. When ρ = 1, the test statistic becomes p value was calculated using the moment matching approach [The family-based SKAT proposed recently  can be dt (FBAT) . The p vapproach  or inverapproach , as consThe goal of our analysis was to assess the power of detecting association between the simulated quantitative phenotypes (baseline SBP and DBP) and the causal genes (from the simulation answer sheet) on chromosome 3 by the family-based SKAT and the burden test, whether or not adjusting for different proportions of causal variants. To ensure a fair comparison of power, the empirical type I error rates of all tests were evaluated by using the variable Q1 . To evaluate the power of tests, we conducted the family-based SKAT and the burden test for each causal gene using, respectively, baseline SBP and DBP as the response trait. Each of the 2 tests was conducted separately by including rare variants only, common variants only, and all variants of each gene. Therefore, 12 tests were conducted at each gene. Table ij denote the number of successful rejection of the null hypothesis for the jth test of the ith gene, and i = 1,2,...31, j = 1,2,...12. We treated the Yijs as correlated measures for the ith gene, and then we constructed a binomial regression model using GEE method to compare the power for each test:For the analysis without adjusting for proportion of causal variants in the gene, we used the generalized estimating equation (GEE)  method tj = 1,2,...,36.where Using the approach stated in the methods section, we extracted a total of 93 trios from the GAW18 data. Our analysis focuses on chromosome 3 only. With knowledge of the simulating model, the 31 causal genes were available for the family-based SKAT and the burden test of all SNPs. When examining different power to detect the association under different proportions of causal SNPs, only the 16 causal genes that contain at least 1 causal rare variant and at least 1 causal common variant were included.We applied the family-based versions of the burden and SKAT tests on the 93 trios for the gene-based association test of the 31 causal genes, using the 200 simulated data sets. Variable Q1 (a quantitative trait in the simulation data set that is not associated with any of the SNPs) was used to test for type I error and the empirical type I error rates are close to the nominal level of 0.05 with a range of (0.043 to 0.059), which is within the 95% confidence interval of the nominal level, that is, . An earlier study  also repMAP4 has a strong signal. Our results show both the family-based SKAT and the burden test are able to detect MAP4.Figure PROK2 and SERP1. For both genes, the peaks were observed only when common variants were included in the test. This finding is consistent with the underlying simulating model, in which almost all the causal SNPs in these 2 genes are common variants. However, the family-based SKAT did not show any power beyond type I error to identify these two genes.In Figure We conducted both the family-based SKAT and the burden test in scenarios containing different proportions ie, 10%, 25%, 50%) of causal SNPs in the analysis. Figure 5%, 50% oMAP4 and MLH1. In Figure MAP4. The possible explanation is that the causal rare variants in MAP4 affect SBP in different directions , thus the burden tests lose considerable power because causal variants β coefficients are in mixed directions. When we included common variants in the analysis, the power of the family-based SKAT decreased. This is a result of very few common SNPs being causal in MAP4; therefore, adding common variants increases the number of noncausal SNPs and dilutes the causal signals. The burden test, however, had a slightly higher power than the family-based SKAT in gene MLH1. Examination of the simulating model suggests that almost all variants in MLH1 affect DBP and SBP in the same direction, so it is not surprising that the burden tests had comparable or better performance than the family-based SKAT for testing the variants in MLH1.We did not observe a substantial impact of percentage of causal SNPs on the power of both tests from the plot, whether or not the analysis included rare variants. The power of both the family-based SKAT and the burden test is comparable across most genes. Two prominent exceptions are the genes PTPLB1  in power between the family-based SKAT and the burden test across all scenarios, whether we adjusted for the proportion of causal variants or not. However, after adjusting for proportions of causal variants, we found that on average, the tests using common variants only had less power compared to those using rare variants only, followed by the tests using both common and rare variants. The test for overall difference in power yields a p value of 0.04.In addition to visually comparing power as presented in Figures Our analysis using the GAW18 simulated baseline phenotypes and sequence genotypes with sample size of 93 conditionally independent trios shows limited power of both the family-based SKAT and the burden test. The low power is most likely the result of using a small number of trios and the weak signals in the simulating model. However, we found that both models adequately controlled the type I error rates with only 93 trios. This agrees with the results of simulation studies in , where aRecently, 2 methods using SKAT for family data have been proposed ,16. BothThe authors declare that they have no competing interests.Jing Huang and Yong Chen drafted the manuscript. Jing Huang conducted statistical analyses. All authors revised, read, and approved the final manuscript.