Genome-wide association studies have helped us identify thousands of common variants associated with several widespread complex diseases. However, for most traits, these variants account for only a small fraction of phenotypic variance or heritability. Next-generation sequencing technologies are being used to identify additional rare variants hypothesized to have higher effect sizes than the already identified common variants, and to contribute significantly to the fraction of heritability that is still unexplained. Several pooling strategies have been proposed to test the joint association of multiple rare variants, because testing them individually may not be optimal. Within a gene or genomic region, if there are both rare and common variants, testing their joint association may be desirable to determine their synergistic effects. We propose new methods to test the joint association of several rare and common variants with binary and quantitative traits. Our association test for quantitative traits is based on genotypic and phenotypic measures of similarity between pairs of individuals. For the binary trait or case-control samples, we recently proposed an association test based on the genotypic similarity between individuals. Here, we develop a modified version of this test for rare variants. Our tests can be used for samples taken from multiple subpopulations. The power of our test statistics for case-control samples and quantitative traits was evaluated using the GAW17 simulated data sets. Type I error rates for the proposed tests are well controlled. Our tests are able to identify some of the important causal genes in the GAW17 simulated data sets. However, these variants explain only a small proportion of the phenotypic variance or heritability of a trait [Genome-wide association studies have helped us to understand the genetic basis of several complex diseases and have identified thousands of variants associated with such diseases [ a trait ,2. The f a trait ,4. CurreOne active area of research in statistical epidemiology is the development of efficient statistical tests to detect associations involving rare variants. In resequencing studies, the number of variants available for testing within a genomic region is generally large. Selecting rare variants for association tests presents a challenge. Using statistical tests of association for each rare variant is not an optimal strategy because of multiple test issues and the typically low statistical power of the test for rare variants. Therefore many of the procedures proposed in the literature involve some kind of pooling or the use of a weighted combination of the rare variants to establish the joint association . King etRecently, we developed a test statistic, the kernel-based association test (KBAT), to test the joint effect of multiple variants based on the genetic similarity between individuals . In the Here, we present methods for testing the joint association of several rare and common variants with binary and quantitative phenotypes. To construct the test with all variants, rare variants are pooled and common variants are considered individually. We introduce first a pooling strategy used for rare variants, followed by the measures of genotypic and phenotypic similarity between individuals. Next we describe the KBAT statistic, which is based on pooled rare variants and common variants for multiple subpopulations. Finally, we present the statistic for quantitative traits (the QT-KBAT).n individuals on K single-nucleotide polymorphisms (SNPs) within a genomic region or gene. To test the joint association of all SNPs, the rare SNPs are pooled, as described in what follows.Suppose that genotyping is done for kth SNP for the ith individual. Based on a minor allele frequency (MAF) threshold, without loss of generality we assume that 1 to rK are rare SNPs and that the rest, cK = K − rK, are common SNPs. The combined genotype of the rare variant is given by:Let The pooled genotype is the total number of minor alleles among the rare variants truncated at 2. For the gene-level test, the truncation does not involve much loss of information because the proportion of individuals with more than two rare variants within a gene is usually small. This coding is only a convenience and not a constraint for the current analysis. Other types of pooling strategies can be devised based on the nature and function of the SNPs. However, in tests with several rare variants, as in a pathway-level or whole-genome-level test, different thresholds for pooling can be used. But, this issue is not discussed here because it is beyond the scope of this paper.We use two types of similarity measures: the allele match kernel for genotype similarity and the Euclidean distance for phenotype similarity.ig and jg of two individuals i and j. The score is 4 if ig and jg are the same, 2 if one is a heterozygote and the other a homozygote, and 0 if they do not share any common alleles (Table For any SNP (pooled rare or common), the allele match kernel score determines the number of alleles common to genotypes es Table . The powiT =  and jT =  denote a vector of Q quantitative trait values for two individuals i and j. The square of the Euclidean distance between two vectors is used to define the phenotypic similarity between two individuals, which is given by:Let Two individuals who have similar trait values will therefore have a lower similarity score compared to individuals with different trait values. Although there are several measures of phenotypic similarity, only the Euclidean distance metric is considered here.n1 case subjects and n2 control subjects for K SNPs. Details of the KBAT statistics for testing the joint association of all K SNPs with the disease are given by Mukhopadhyay et al. [Assume that we have genotypes for y et al. . Here, wkth SNP, P denotes the subscripts for the pooled rare variants i and j within group l (l = 1 denotes a case subject and l = 2 denotes a control subject) with the respective genotypes U statistic, denoted as:For the The within-group sum of squares (SSW) is represented by:The between-group sum of squares (SSB) is represented by:where:and:The statistic:based on the analysis of variance (ANOVA) model, is used to test the joint effect of all SNPs with the disease status.k, consider the genotypic similarity scores of the allele match kernel, given in Table n samples can be assigned to one of these three similarity groups. Individuals from group 1 are those pairs whose genotype similarity value is 4, that is, those pairs with genotypes , , and . Individuals from groups 2 and 3 can be similarly identified.Based on the measures of genotypic and phenotypic similarity, we introduce a KBAT-type method to test the association of genotypes with quantitative phenotypes. For a SNP Let:and:denote the pairs of individuals in groups 1, 2, and 3, respectively. Then, the number of individuals in these groups is given by:and:f0, f1, and f2 denote the frequency of the genotype counts 0, 1, and 2, respectively, for SNP k. Note that the number of groups m (here m = 3) and the genotype pairs within the groups can vary depending on the definition of the kernel function used to score the genotype similarity.respectively, where If genotypes are associated with the trait values, then a strong correlation would be expected between the genotypic and the phenotypic similarity. Therefore the average similarity for the pairs across these three groups should be different. In general, a higher average phenotypic similarity is expected in groups with higher genotypic similarity. Thus genotypic similarity is compared across three groups using the one-way ANOVA model, as in KBAT. Consider the model:i, j) in group k; μ is the overall mean or the general effect for pairs of individuals; lα denotes the group-specific treatment effect for similarity scores over the general effect; and le(ij) are the error components.Here, K SNPs with the traits, the QT-KBAT statistic is defined as:To test the joint association of the where:and:k,denote the within-group sum of squares and the between-group sum of squares, respectively, for SNP andp-value of the ℑ statistic in Eq. (16). The test statistics for the quantitative trait are slightly different from the KBAT statistics for case-control samples (sum of F ratios compared to the ratio of the sum of the SSBs to the sum of the SSWs). Although it is straightforward to define QT-KBAT similarly to KBAT, the form in Eq. (16) is used so that the F statistics of the one-way ANOVA from any standard computer program can be directly used.The usual assumptions of the ANOVA model are not valid because the observations within each of the groups are correlated. In addition, the distribution of phenotypic similarity values, as defined earlier (Eq. 2), may not follow a normal distribution. Therefore permutations are used to compute the iN  are obtained from L different populations. When performing tests of association using regression models, it is possible to use subpopulation indicator variables to account for population stratifications. Here, we adopt a similar approach by taking the weighted combination of the population-specific F statistics. The combined statistic for the whole data set is given by:Suppose that samples of sizes where:i = 1, 2, …, L, and the ℑi are the population-specific case-control (KBAT) or quantitative trait (QT-KBAT) association statistics defined by Eq. (8) or Eq. (16), respectively.The GAW17 mini-exome genotype data consist of subsamples from several ethnic cohorts. The samples from Europeans, Asians, and Africans are well separated in the plot of principal components derived with and without rare variants (data not shown). From the simulation model, we know that the phenotypes were simulated without reference to the ethnicity of the samples. However, we performed the analysis twice, once assuming a single population and once assuming three subpopulations . Furthermore, we considered two categories of SNPs: (1) only nonsynonymous SNPs, because the phenotype simulations are based on only nonsynonymous SNPs; and (2) all SNPs, to examine changes in the power of the test statistic for additional noncausal variants. We used the KBAT statistic to test the association of genes with case-control status, and we used the QT-KBAT statistic to perform three separate tests for the quantitative traits, namely, Q1, Q2, and a multivariate trait . There are 3,205 genes for the all-SNPs analysis and 2,194 genes for the nonsynonymous-SNPs-only analysis. For the case-control analysis (KBAT) we tested all the genes, and for the quantitative traits analysis (QT-KBAT) we tested only the causal genes  and no pooling was done for common SNPs. For example, if a gene contained seven rare SNPs and three common SNPs, then our gene-level test would have four SNPs (one pooled using the seven rare SNPs and the three common SNPs). We used a pooling MAF threshold of 0.05 because we wanted to have a reasonable frequency of the three counts of the kernel score for pooled SNPs for our QT-KBAT statistics. A reduced threshold increases the number of SNPs for our test statistics and hence may reduce the power.p-values for the permuted data set were obtained with an additional 2,000 permutations of the phenotypes.We also checked the type I error rates of the KBAT and QT-KBAT statistics. For type I error computation, we first selected one of the 200 replicates of phenotypes at random and then permuted the original phenotype; p-values of association tests using 2,000 permutations of the trait values. Assuming that the phenotypes in the 200 iterations were generated under the alternative hypothesis, the power of the test at a 5% level of significance was computed here as the proportion of times the unadjusted p-values were less than 0.05 out of the 200 iterations.For each of the 200 iterations, we computed the Type I error rates for the KBAT statistic for rare variants were well controlled Figure . To undeFLT1, 0.751 for KDR, 0.338 for VEGFC, and 0.29 for PIK3C3.The quantile-quantile (Q-Q) plot based on the random sample of 1,000 genes shows that the type I error rate for QT-KBAT is well under control  with MAF = 0.167, which has an r2 value of 0.0142  with the causal variant C10S3110 with MAF = 0.002. Similarly, the higher power values observed with the inclusion of all SNPs in genes ARNT, SHC1, and INSIG1 may be due to these genes having synonymous common variants with MAFs of 0.43, 0.09, and 0.04, respectively. Therefore in some cases the synergic effect of common and rare variants may have improved the power, and we thus suggest examining the powers of the two analyses: one that considers only the rare variants and the other that includes both rare and common variants.We proposed joint association tests for both rare and common variants in a genomic region. A comparison of the analysis of all SNPs with the analysis involving only nonsynonymous SNPs showed that the inclusion of additional nonassociated SNPs in general reduced the power of the test. However, if moderately common variants are close to the rare causal variants, it is possible that the latent correlation between these common and rare variants may improve the power of the test when these common variants are included. For example, the gene Our quantitative association test can be used for testing the pleiotropic effect of genes on multiple phenotypes. Our multivariate analysis did not identify any additional causal genes because the GAW17 simulation model treats each trait independently with separate sets of causal genes. In future studies we will examine the power of this approach for testing pleiotropy.Because of time constraints, our analysis was restricted to only the allele match kernel for scoring the genotypic similarity and the squared Euclidean distance for scoring the phenotypic similarity between individuals. However, several choices of kernel functions for determining genotypic and phenotypic similarity could be examined. It is important to identify optimal similarity metrics, because the interpretation of the analysis and its power may depend on the similarity measure used.The association tests proposed here are not adjusted for other covariates. However, we could perform a stratified analysis on the subcategories of certain covariates if sufficient samples were available. We are also exploring the possibility of testing the joint association of both quantitative and binary traits using a single test statistic. In addition, we have not included any weights, such as the frequency or functional weights, of rare variants. Furthermore, we used a MAF threshold of 0.05 for pooling rare variants. In our method, implementing various subsets for pooling SNPs, such as using MAF thresholds and the nature of SNPs, is straightforward.p-value using permutation takes a long time if there are many SNPs. Therefore we restricted the quantitative trait analysis to only 36 causal genes. We are currently exploring the possibility of efficiently implementing the permutation procedure for QT-KBAT in a genome-wide analysis.The permutation procedure for the case-control test statistic KBAT can be efficiently implemented. Using the KBAT method, we have performed a gene-based analysis even for data sets from genome-wide association studies. The current implementation of the QT-KBAT to compute the One of the main drawbacks of our QT-KBAT statistics is the unequal distribution of observations within groups. The number of observations in each group depends on the allele frequency of the SNPs (either pooled or common), and if the MAF is small, then one of the group sizes may become small or zero.Future genetic association studies for common complex genetic disorders will involve the analysis of rare and common variants. Therefore efficient statistical techniques are needed for their integrated analysis. In the current study we proposed methods for testing the joint association of rare and common variants that underlie quantitative and qualitative phenotypes. We developed a test statistic using measures of genotypic and phenotypic similarity between two individuals under the ANOVA framework. This approach can also be used for combining multiple phenotypes and multiple subpopulations. We applied our statistics to the GAW17 simulated data sets and identified important simulated causal genes.The authors declare that there are no competing interests.AT conceived of the study, performed the statistical analysis and wrote the manuscript. JZ, GTHK participated in the design and statistical analysis. VK, IM participated design of the study, and in the draft of the manuscript. All the authors read and approved the final manuscript.


This study was carried out to estimate (co)variance components and genetic parameters for live weight of Nellore cattle from Performance Test of Young Bulls using random regression models. Data of weights and ages of 925 weaned males was used. The animal model included the fixed effect of contemporary group, age of the animal at weighing as a covariate and as random effects it was considered the effect of additive genetic and permanent environment of the animal. The residue was modeled considering four classes of variances. The models were compared based on the Bayesian information criteria of Akaike and Schwartz. The model polynomial of fourth and sixth order for the direct additive genetic effects and permanent environment of the animal, respectively was the most appropriate to describe the changes in the variances of the weights during the period in which the animals participating in the performance test young bulls. Heritability estimates showed moderate magnitudes and indicated that direct selection will promote improvement of selection criteria adopted. Furthermore, due to high positive correlation between the estimated weights, it was suggested selecting the best animals before at 365 days of age, because it is the period in which the animals have a higher growth rate and thus you can select animals heavier and less delayed. The beef cattle production has an important role for the Brazilian agribusiness. This production goes to increase around of 4.4% until 2015. This is possible due to great potential of Brazilian herd production, which is first in world exports and second largest producer of beef cattle and third in world food consumption Brasil, .In Brazil, the Zebu breeds are more than 80% of national bovine herd. But, these animals have presented difference in growth and production in each Brazilian region due to diversity of climate and management system variance components and genetic parameters for live weight of Nellore cattle from Performance Test of Young Bulls using random regression models.Data set used in this study was composed by 5,550 records of weight-age of 925 Nellore cattle from Performance Test of Young Bulls, whit general mean and standard-deviation of 294.14±60.80. A summary of the data is given in Table This farm is located in Santo Antônio de Goiás City, Goiás State, latitude of 16°28´00”S, longitude of 49°17´00”W and altitude of 823 m. In accordance with Köppen this city is classified as Aw climate, savanna tropical and megathermic. The mean air temperature is 23.0°C and the minimum and maximum temperature occurs in July and September with average temperature of 14.2°C and 31.7°C, respectively. It present two distinct season, rainy season, between October and April; and dry season, between May and September. The annual average precipitation is 1,485 mm and the annual average of relative humidity is 71%. The soil is dystrophic red latosol and clayey and your environment is savanna biome and relief plan.The annual distribution of average, minimum and maximum temperature (°C), precipitation (mm), relative humidity (%) and temperature humidity index were collected in Climatological Station of Embrapa Rice and Bean, during the years 2001 to 2012 are presented in Figure Brachiaria brizantha, cv Marandu and cv Xaraés, in rotational grazing (9 days of occupation by 36 days of rest) defined in accordance with forage offer. It was supplied 7 kg of forage to each 100 kg of life weight (LW). Thus, the animals can to select the good forage during the test and were fed with mineral and proteic supplement of minimum consumption during de dry season. Average weight and age at the beginning of the test was 200 kg and 8 to 10 months, respectively. Animals were weighed regularly during the period of 294 days of performance test.In the performance test the animals were fed with The animals’ weights of performance test were analyzed using random regression models. It was considered as fixed effect the contemporary groups  and age of animals as covariate adjusted using Legendre polynomials of third order. As random effects it was considered the direct additive genetic effect and animal environment permanent. This model was:yij is the ith measure of jth animal; FE is the set fixed effect; bm is the regression coeficient; ϕm(ti) = Legendre polynomial regression function; ϕm(tij) = Legendre polynomial regression function of each animal j for age ,(ti) in accordance with additive direct genetic effect and animal environment permanent; αjm and δjm= random regression coefficients for additive genetic and permanent environment effects, respectively, for each animal; kb, kA and kC = Legendre polynomial order; eij= random error associated with each age i of animal j. The random regression model in its matrix form is:Where, Where E[y] = Xβ, E[α] = 0, E[β] = 0 and E[e] = 0 and (co)variance matrix was:y is the measure vector; β is the fixed effects vector (FE and bm); α is the random coefficients vector of direct addictive genetic effects; δ = coefficient vector of animal permanent environment; X, Z1 and W1is the matrix of fixed effect, direct addictive genetic and animal permanent environment effects, respectively; KA and KC is the (co)variance matrix of direct addictive genetic and animal permanent environment effects, respectively; A is the relationship matrix; I is the identity matrix; Nα = number of animals with valid records; R is the residual variance matrix; and e is the error vector.Where, Wombat device variance matrix explained 100% of total additive variance. The first eigenvalue accounted for 89.82%. The first eigenfunction was negative down of 400 days and positive up to 400 days of age; but the linear trend indicates a positive correlation between weights. Thus, selection for greater weights at any age will increase other weights. There was decrease of second eigenfunction up to 400 days of age and increased thereafter. This change indicates that selection for this component will have opposite effects at the beginning and at the end of the trajectory. However, the genetic change due to selection for this component would be small because it was responsible for only 9.55% of additive variance. Similar results were reported by Albuquerque & Meyer  for growEstimates of (co)variance between the random regression coefficients for M46 model are showed in Table The variance components and heritability coeficients estimated for animals weight are show in the Figure There was quadratic trend for heritability estimates with major values at beginning and end of performance test. The estimatives of heritability ranged from 0.35 to 0.46. Similar results were reported by Souza et al.  which shEstimative of variance components and heritability coefficients for weight at 240, 365, 450 and 550 days of age are showed in Table The genetic correlations were greater than 0.70 at most ages. Similar estimates, varying from 0.43 to 0.88, were described by Boligon et al. (2009). These results suggest that selection for greater weight at any age should increase mature weights (Boligon et al., The model that polynomials of fourth and sixth order for the direct additive genetic and animal permanent environment effects, respectively, were the most appropriate to describe the changes in the variances of the weights during the period in which the animals participating in the performance test young bulls.Heritability estimates showed moderate magnitudes and indicated that direct selection will promote improvements in the selection criteria adopted. Furthermore, due to the high positive genetic correlation, we suggest selecting the best animals in the previous 365 days of age, because this is the period in which the animals have a higher growth rate and thus it can select precocious and heavier animals.


ANGPTL4 from the Dallas Heart Study. The performance of the proposed procedures is further assessed via simulation studies. The methods are implemented in a freely available R package BhGLM (http://www.ssg.uab.edu/bhglm/).Complex diseases and traits are likely influenced by many common and rare genetic variants and environmental factors. Detecting disease susceptibility variants is a challenging task, especially when their frequencies are low and/or their effects are small or moderate. We propose here a comprehensive hierarchical generalized linear model framework for simultaneously analyzing multiple groups of rare and common variants and relevant covariates. The proposed hierarchical generalized linear models introduce a group effect and a genetic score  for each group of variants, and jointly they estimate the group effects and the weights of the genetic scores. This framework includes various previous methods as special cases, and it can effectively deal with both risk and protective variants in a group and can simultaneously estimate the cumulative contribution of multiple variants and their relative importance. Our computational strategy is based on extending the standard procedure for fitting generalized linear models in the statistical software R to the proposed hierarchical models, leading to the development of stable and flexible tools. The methods are illustrated with sequence data in gene  ANGPTL4 from the Dallas Heart Study and are further assessed via simulation studies. The methods have been implemented in a freely available R package BhGLM (http://www.ssg.uab.edu/bhglm/).Complex diseases and traits are likely influenced by many common and rare genetic variants and environmental factors. Next-generation sequencing technologies have provided unparalleled tools to sequence a large number of individuals, allowing for comprehensive studies of both common and rare variants. However, detecting disease-associated rare variants and common variants of small effects poses unique statistical challenges. We propose here a comprehensive hierarchical generalized linear model framework for simultaneously analyzing multiple groups of rare and common variants and relevant covariates. The proposed hierarchical generalized linear models introduce a group effect and a genetic score for each group of variants, and jointly they estimate the group effects and the weights of the genetic scores. This framework includes various previous methods as special cases, and it can effectively deal with both risk and protective variants in a group and can simultaneously estimate the cumulative contribution of multiple variants and their relative importance. The methods are illustrated with sequence data in gene  Many common human diseases and complex traits are highly heritable and are believed to be influenced by multiple genetic and environmental factors. Genome-wide association studies (GWAS) represent a powerful way for discovering disease-associated factors and investigating the genetic architecture of complex diseases Next-generation sequencing technologies have provided unparalleled tools to sequence a large number of individuals in candidate genes, exomes, or even the entire genome, allowing for comprehensive studies of both common and rare variants. In addition to the common problems of handling large numbers of variants, however, detecting disease-associated rare variants and common variants of small effects poses unique statistical challenges Several approaches along this line have been proposed All the existing methods have been developed to assess only one group of variants at a time. Since common diseases are likely caused by a complex interplay among many genes and environmental factors, however, it is more appropriate to simultaneously model multiple groups of variants and covariates ANGPTL4 from the Dallas Heart Study, and are further assessed via simulation studies. Finally, we conclude this article by highlighting some areas of future research.We propose here a comprehensive hierarchical generalized linear model (GLM) framework for simultaneously analyzing multiple groups of rare and common variants and relevant covariates. The proposed hierarchical GLMs introduce a group effect and a genetic score  for each group of variants, and jointly estimate the group effects and the weights of the genetic scores. This framework includes various previous methods as special cases, and can effectively deal with both risk and protective variants in a group and can simultaneously estimate the cumulative contribution of multiple variants and their relative importance. The methods are illustrated with sequence data in gene n unrelated individuals, phenotyped for a continuous or discrete disease trait and genotyped for a number of rare and/or common genetic variants in one or multiple candidate genes or genomic regions. The observed values of the response variable are denoted by y = . We assume that the genetic variants can be divided into K groups, kG, k = 1, ···, K, and the k-th group kG contains kJ variants, where K≥1 and kJ>1. The groups can be constructed based on candidate genes in which the variants are located and the types of the variants . We assume that some non-genetic variables  are also measured for each individual and will be included as covariates in the model to control for possible confounding effects.Suppose that a population-based association study consists of η, the link function h, and the data distribution pi is expressed as the multiplicative form:β0 is the intercept, ijx and jβ represent covariate j and its coefficient, respectively, ijz is the main-effect predictor for individual i at genetic variant j in group kG, equaling to the number of minor alleles for an additive coding , the common coefficient group effect for kJ variants in the k-th group, and the individual coefficients weights or relative effects of individual variants.We extend the hierarchical generalized linear model (GLM) of Yi and Zhi kJ individual main-effect predictors for variants in group kG. The linear combination provides a way to combine the genetic variation across the kJ individual variants, referred to as genetic score. Therefore, the common coefficient kJ individual variants in the k-th group, hence referred to as the group effect, and the weights kG.The common coefficient h:The mean of the response variable is related to the linear predictor via a link function kg = 0, k = 1, ···, K. We treat the weights k[j] indexes the group k that variant j belongs to.Our main goal is to estimate the group effects The above hierarchical prior assumes that If the number of groups is not large, the group effects For the covariate effects kJ individual variants in group k with the disease. In our hierarchical model, the multiplicative term ijz ultimately gets the coefficient Our hierarchical GLMs include multiplicative parameters, a common coefficient kJ variants as a predictor . Such an approximate calculation has been routinely applied in statistical practice Our Bayesian hierarchical GLMs can be fitted using Markov chain Monte Carlo (MCMC) algorithms that fully explore the joint posterior distribution by alternately sampling each parameter from its conditional posterior distribution Our algorithm updates the coefficients tth iteration, and −5).We initialize our iterative algorithm by setting the parameters . We have implemented the EM-IWLS algorithm by inserting the E-step for updating the missing values  that is an extensible and interactive environment for genetic association analysis of common and rare variants and gene-gene and gene-environment interactions.Our model fitting strategy is based on extending the well-developed IWLS algorithm for fitting classical GLMs to our Bayesian hierarchical GLMs. The IWLS algorithm is executed in the glm function in R  with the prior mean Our hierarchical multiplicative GLMs include various models as special cases. Although less comprehensive, these reduced models can be useful in some situations, and thus can be used as alternative approaches to analysis of multiple groups of rare and common variants. We here consider two types of reduced models. The first ignores the group effects and directly models the main effects of individual variants. Thus, the linear predictor (1) is reduced toThe second alternative approach is to preset the weights of individual variants using the previous methods ANGPTL4 in lipid metabolism. The study included the 3,551 participants of the Dallas Heart Study (DHS) from whom fasting venous blood samples were obtained. The DHS is a population-based random sample of Dallas County residents, consisting of 601 Hispanic (H), 1,830 African American (AA), 1,045 European American (EA) and 75 other ethnicities. The 75 participants from other ethnicities will be excluded from our analysis. The phenotype analyzed in our study is the log-transformed plasma levels of triglyceride. The top panel of continuous and binary traits. Our analyses included race (a three-level factor), age, sex, and BMI as covariates in the model. We excluded individuals with any missing values of the covariates from the analysis, resulting in 3008 and 1499 individuals in the analyses of the continuous and binary traits, respectively.Romeo et al. ANGPTL4, and identified a total of 93 sequence variations. After removing variants that were not segregating in the sample, the numbers of variants reduced to 82 and 63 for the analyses of the continuous and binary traits, respectively. Most of these variants were rare: only 12 and 13 variants had a minor allele frequency above 1%, and 33 and 26 variants were found only in one object in the two analyses, respectively  with the proposed hierarchical prior distributions (Equations 4 and 5). For comparisons, we then used three alternative methods: 1) Setting all the scale parameters All the analyses simultaneously fitted all the non-genetic variables , the two common non-synonymous variants  and the three groups of variants. We used a normal regression and a logistic regression for the continuous and the binary traits, respectively. We set the prior means jμ set to 1 for all the grouped variants. All the non-genetic covariates and the common non-synonymous variant 8191_R278Q were found to significantly affect both the continuous and the binary traits. These effects were also significant under the other models considered . The right panel of jμ set to the functional probabilities for the rare non-synonymous variants, which were estimated to range from 0.16 to 1. These models produced qualitatively identical results as the above analyses, but slightly lower p-values for the significant effects. Price et al. We used simulations to validate the proposed models and algorithm and to study the properties of the method. Although most published simulation studies of rare variants generated genotypes assuming a population genetics model for the propagation of rare variants, the best way will be to take real sequence data obtained from many individuals and simulate phenotypes based on variants in those sequences, making assumptions only about genetic effects of variants We evaluated some factors that may affect the performance of the methods:Sample size: We considered two sample sizes, including all observations in the real data (n = 3008) or individuals in the bottom and top quartiles of the real continuous phenotype (n = 1499), respectively.Number of groups and number of variants in each group: We first considered the three groups of variants  as in our real analyses. We then considered the second scenario with six groups by randomly partitioning each group into two with equal number of variants.Genetic effect sizes and directions of variants: For each group of variants, we first assumed the total heritability (h) explained by the variants and the proportion of negative additive effects (p.neg) for the variants. We then randomly sampled an additive effect jβ for each variant from the region  and changed the sign of jβ with the probability p.neg. The upper bound hβ was calculated using the method of Yi and Zhi h. We considered several combinations of different h and p.neg (see .neg see  and 2. Tiy = 1) and the other individuals as ‘unaffected’ (iy = 0) to create a binary trait We simulated a continuous and a binary phenotype. As in our real data analyses, we simultaneously fitted all the non-genetic variables , the two common non-synonymous variants  and the grouped variants. We assumed the non-genetic coefficients and the additive effects of 8155_T266M and 8191_R278Q to be their estimated values in the continuous trait analysis . The conclusions described above generally held (data not shown). Yi and Zhi p-values for the simulated group effects than the previous methods. This was also true for the proposed method in our simulation studies. This indicates that our method usually provides stronger evidence of association if the variants really influence the disease.We also evaluated power and type I error at several different levels  but also assess the relative importance of single variants. Second, our hierarchical model includes various existing methods for rare variants as special cases. This shows that the proposed method is theoretically more advantageous than the existing methods, and allows us to conveniently analyze data using different ways. Third, any external information about variants, for example, the functional prediction, can be easily incorporated into our hierarchical model by specifying the prior means of the weights for variants. By doing so, our approach has the additional advantage of accounting for uncertainties about the prior assumptions. Fourth, our approach is based on the generalized linear model framework and thus can deal with various types of continuous and discrete phenotypes and covariates, and can fit any generalized linear models. Finally, the proposed algorithm extends the standard procedure for fitting classical generalized linear models in the general statistical package R to our Bayesian model, leading to the development of stable and flexible software.Our approach is highly extensible; we have planned several extensions to the proposed method, some of which have been initially implemented in our software BhGLM. The key to our approach is the use of hierarchical prior distributions for the weights and the group effects, so that these multiplicative parameters are identifiable and can be simultaneously estimated from the data. We have proposed to use the hierarchical expression of the half-Cauchy distribution with the innovation of introducing both group- and variable-specific parameters. The half-Cauchy prior is an excellent default choice for many problems offset in the model. This method updates coefficients in a conditional manner, significantly reducing the number of parameters in each M-step of the EM-IWLS algorithm, and thus can deal with large number of variables.Although demonstrated with only several groups of variants, our method can be adapted to deal with large-scale sequencing data involving thousands of exomes or candidate genes. For these high-dimensional settings, we need to modify the prior distributions of the group effects and the computational algorithm. We can place a shrinkage prior on the inverse scale in the gamma prior of in silico predictions of function or evolutionary conservation Our third extension could incorporate external gene or pathway level information into the hierarchical model. Candidate genes or pathways studies usually consist of data at different levels, i.e., genetic variants within multiple candidate genes or pathways which may be functionally related Our fourth extension could incorporate genetic interactions (gene-gene and gene-environment interactions) into the model. Just as interactions must be considered in standard GWA studies The proposed hierarchical generalized linear models may provide efficient tools for disease risk prediction and personalized medicine. GWA studies have raised expectations for predicting individual susceptibility to common diseases using genetic variants Text S1The EM-IWLS Algorithm for Fitting Hierarchical Generalized Linear Models.(DOCX)Click here for additional data file.


Epistatic genetic interactions are key for understanding the genetic contribution to complex traits. Epistasis is always defined with respect to some trait such as growth rate or fitness. Whereas most existing epistasis screens explicitly test for a trait, it is also possible to implicitly test for fitness traits by searching for the over- or under-representation of allele pairs in a given population. Such analysis of imbalanced allele pair frequencies of distant loci has not been exploited yet on a genome-wide scale, mostly due to statistical difficulties such as the multiple testing problem. We propose a new approach called Imbalanced Allele Pair frequencies (ImAP) for inferring epistatic interactions that is exclusively based on DNA sequence information. Our approach is based on genome-wide SNP data sampled from a population with known family structure. We make use of genotype information of parent-child trios and inspect 3×3 contingency tables for detecting pairs of alleles from different genomic positions that are over- or under-represented in the population. We also developed a simulation setup which mimics the pedigree structure by simultaneously assuming independence of the markers. When applied to mouse SNP data, our method detected 168 imbalanced allele pairs, which is substantially more than in simulations assuming no interactions. We could validate a significant number of the interactions with external data, and we found that interacting loci are enriched for genes involved in developmental processes. Elucidating non-additive (epistatic) interactions between genes is crucial for understanding the molecular mechanisms of complex diseases. Even though high-throughput, systematic testing of genetic interactions is possible in simple model organisms, such screens have so far not been successful in mammals. Here, we propose a computational screening method that only requires genotype information of family trios for predicting genetic interactions. We tested our framework on a set of more than 2,000 heterozygous mice and found 168 imbalanced allele pairs, which is substantially more than expected by chance. We confirmed many of these interactions using data from recombinant inbred lines. The number of significant allele pair imbalances that we detected is surprisingly large and was not expected based on the published evidence. Our framework sets the stage for similar work in human trios. The simultaneous perturbation of two epistatically interacting genes leads to a phenotype that is not expected based on the phenotypes of the individual genes. Understanding these phenomena is indispensable for explaining multi-factorial traits and diseases Epistasis is always defined with respect to a specific phenotype and describes a non-additive interaction effect of two genes on that phenotype. Most gene interaction studies explicitly measure a phenotype such as growth rate or viability Whereas several such incompatibilities are known in plants ”, relies on sequence data only, making it applicable to the many already available SNP studies without the need for additional phenotype measurements. ImAP is based on inspecting In a population of We have made the top The core step of ImAP consists of a We applied ImAP to search for potential epistatic interactions using an outbred heterogeneous stock (HS) of mice that was established by crossing eight inbred lines After removing individuals with more than We did not have to discard any SNPs due to lack of Hardy-Weinberg equilibrium as is generally done in genome-wide association studies. Instead, ImAP corrects for the disequilibrium see . In the When applying ImAP to the HS mouse data, we limited our analysis to markers residing on different chromosomes in order to exclude local LD Most of the loci only interact with one other locus, only ractions . Not surractions . HoweverIn order to rule out the possibility of false positive findings due to increased numbers of missing values or small MAF on some markers, we compared the distributions of missing values and MAF between block representatives from significant block pairs to those of non-significant pairs  and S4. The histograms in This difference between the real and simulated data can now be quantified to make suggestions about the number of true allelic incompatibilities in the HS mouse population. For example, at As can be seen in the inset of For the second step of the analysis we chose all LD blocks that were involved in at least one significant interaction. There might be one or more interacting markers within each LD block and the above analysis does not reveal which markers within a region are involved in the interactions. We repeated the calculation of the test statistics, null distribution and p-values with all markers in those blocks to find the SNP pairs with the highest signal in each significant block pair. This resulted in  with a   and S4, Only few allele incompatibilites in mouse have been reported so far www.genenetwork.org) and recalculated the We therefore investigated the distant LD of markers that were genotyped in the RIL as well as in the HS mice. We downloaded the genotype data for a priory there is no other rigorous way of performing such GO enrichment. This strategy also prevents a bias in GO enrichment due to local gene clusters with similar annotation.We investigated if the genes mapping to loci that participate in high ranking interactions are enriched for relevant Gene Ontology (GO) categories We further restricted the enrichment analysis to interacting pairs whose The enrichment test was conducted using the topGO algorithm The top ranking GO biological process terms for the original data as well as for an exemplary simulation are shown in the Supporting Material  and S2. Epistatic interactions affecting the viability of an organism often bridge parallel pathways Here, we are interested in markers having a significant number of common interactors. In order to find such groups of markers with similar interaction profiles, we compared the marker interaction profiles from the ImAP analysis using the congruence score Since here we are analyzing interaction profiles  we chose a less stringent cutoff value for interacting block pairs  signaling, which is involved in the development of the lung and the telencephalon Fgf10 and Shh signaling are involved in development of anatomical structures and are known to influence each other Among the Fgf10 leads to a reduced expression of this gene while individuals carrying the minor allele of Btrc show a higher expression than individuals with the major allele. Since Btrc is an inhibitor of Shh signaling, this implies that both minor alleles reduce Hedgehog signaling.According to our gene expression analysis, the minor allele of Btrc and Fgf10 loci share Nkx2.1, which is indispensable for lung and telencephalon development. Depending on the cell type and developmental stage Nkx2.1 either interacts with the Fgf10 and Shh pathway Btrc or Fgf10 alleles may be rescued by a fully functional Nkx2.1. The ImAP analysis suggests that the combination of the minor allele at the Nkx2.1 locus together with minor alleles at either the Btrc or Fgf10 locus leads to an embryonic lethal phenotype, presumably due the loss of the buffering effect of Nkx2.1.The We present a new approach to infer epistatic interactions on a genome-wide scale in family data using sequence information only. The method scans all marker pairs in the genome for deviation from the expected allele pair frequencies resulting in a list of putative pairs featuring an allele incompatibility. Relying on sequence data only is an advantage compared to existing methods for the inference of gene-gene interactions, since the approach can readily be applied to existing SNP data. There is no need for resource- and cost-intensive phenotype measurements.Regression and We identified substantially more interacting loci than expected by chance, which is first evidence that we detect true ‘signal’. Further, we could show that interacting marker pairs are enriched for genes involved in developmental processes and a significant number of interactions could be validated using independent external data. Due to the very large number of pairs tested, finding a large number of interactions with low p-values even in the simulations is expected. However, at low p-values we observed significantly more interactions in the original data than in any of the simulations; e.g. at Future work should also include haplotype information. Local haplotypes have been inferred for the HS population in terms of probability of inheritance from any of the eight founder strains An epistatic interaction is always defined with respect to a specific phenotype. In this study the phenotype is implicit, hidden. Indeed, looking for allele pairs that are underrepresented in the HS population reveals the genotype of the non-existing individuals. Therefore, the hidden phenotypes should relate to any biological processes affecting the fertilisation, the development or the viability of an individual and thus prevent its appearance in the population. Interestingly, top scoring marker pairs are enriched for genes involved in these expected phenotypes.It is not immediately obvious how our findings translate to human populations The calculation of the test statistic can be divided into several steps which are depicted in Panel A of 1. Let 2. For each individual in Let The expected genotype of individual 3. Correct the expected genotypes for possible confounding factors such as segregation distortion. There might be a preference in the inheritance of a certain genotype on one marker in the population which is independent of interaction effects, e.g. if this genotype leads to increased fitness. In order to correct the expected frequencies for allele selection that is independent of other loci we multiply each individual's expected genotype by the ratio of the sample-wide observed and expected frequencies for the corresponding marker :Normalize the corrected expectation so that the probabilities for each marker sum up to This guarantees an adjustment of expected allele frequencies in cases where the observed frequency of a marker in the population deviates from the theoretically expected values.4. Next, the observed and expected number of times each combination of genotypes of two markers appears, can be inferred.Let Using the product of the marginal probabilities of each single marker genotype for calculating the probability of the genotype combination mimics the assumption of no epistatic effects under the null hypothesis. This step results in the 5. Finally, a The significance of the imbalances observed for each marker pair is assessed with a permutation approach based on pseudo-controls. This approach has already been adopted in related problems The general outline of the procedure is shown in We use these pseudo genotypes to assess the significance of the test statistics of each marker pair by calculating an empirical marker-specific null distribution based on In an earlier version of our analysis pipeline we calculated the p-values based on the However, we found that the distribution of these parametric p-values differed conditional on the minor allele frequencies (MAF) of the markers, as shown in  the MAF . This phIn order to speed up the calculations but still retain an acceptable resolution of loci with potentially interacting genes, we pursued the following strategy.In a first run of ImAP we split the data into blocks of high linkage disequilibrium (LD). This is again done with the package trio, which provides an algorithm to estimate LD block borders in parent-offspring data. Afterwards, one representative marker is chosen randomly among all markers with a minimum number of missing values in each LD block and the test is applied to all possible combinations of these representatives on different chromosomes. The restriction to markers on different chromosomes is applied to rule out false positive results due to local linkage disequilibrium. Subsequently, we identify all block pairs which were assigned an FDR below The pseudo-control data was used to compute p-values. In order to also correct for multiple hypothesis testing and for testing for any other possible biases in our data we simulated the mating process in the mouse population assuming independence of the markers but adhering to the original pedigree structure.The simulation starts with the first generation of mice for which we have genotype information (F0 generation). Using fastPHASE Obtaining the haplotypes of the F0 generation allows us to initialize the mating process. For each mother and father of an F1 individual we start with randomly choosing whether they pass on the maternal or the paternal allele of the first marker on a chromosome to the offspring. Then, using either general or sex-specific recombination rates Click here for additional data file.Figure S2Exemplary distributions of the test statistics depending on the MAF of the markers. The scores follow a (PDF)Click here for additional data file.Figure S3Cumulative distribution functions of the proportion of missing values of representative markers of significant and non-significant LD block pairs.(PDF)Click here for additional data file.Figure S4Cumulative distribution functions of the MAF of representative markers of significant and non-significant LD block pairs.(PDF)Click here for additional data file.Figure S5Number of interactions for each of the (PNG)Click here for additional data file.Figure S6Cumulative distribution function of the overall distant linkage disequilibrium in the RIL (grey) and RIL marker pairs with ImAP p-value (PDF)Click here for additional data file.Figure S7Fraction of congruence scores (PDF)Click here for additional data file.Table S1GO enrichment of top ranking marker pairs in the original data. All genes between the flanking markers are considered.(PDF)Click here for additional data file.Table S2GO enrichment of top ranking marker pairs in the simulated data. All genes between the flanking markers are considered.(PDF)Click here for additional data file.Table S3Top ranking ImAP interactions. The first two columns contain the IDs of all (XLSX)Click here for additional data file.Table S4Mapping from marker IDs to Ensembl gene IDs. For each marker (XLSX)Click here for additional data file.


Genome-wide analysis of gene-gene interactions has been recognized as a powerful avenue to identify the missing genetic components that can not be detected by using current single-point association analysis. Recently, several model-free methods  were developed for detecting non-linear dependence between genetic loci, but they are potentially at the risk of inflated false positive error, in particular when the main effects at one or both loci are salient. In this study, we proposed two conditional entropy-based metrics to challenge this limitation. Extensive simulations demonstrated that the two proposed metrics, provided the disease is rare, could maintain consistently correct false positive rate. In the scenarios for a common disease, our proposed metrics achieved better or comparable control of false positive error, compared to four previously proposed model-free metrics. In terms of power, our methods outperformed several competing metrics in a range of common disease models. Furthermore, in real data analyses, both metrics succeeded in detecting interactions and were competitive with the originally reported results or the logistic regression approaches. In conclusion, the proposed conditional entropy-based metrics are promising as alternatives to current model-based approaches for detecting genuine epistatic effects. Over the past years, genome-wide association analysis has greatly facilitated the identification of common genetic factors that were relevant to complex diseases. Based on the common disease common variants hypothesis i.e. robust to model misspecifications) being used in a genome-wide interaction analysis. Random forest Numerous methods or algorithms for detecting gene-gene interactions have been developed. In general, they can be categorized into two classes: model-based and model-free, based on whether a explicit interaction model is assumed. The former is to explicitly model and estimate, e.g. the odds ratio or relative risk, to measure the interaction effects in a dichotomous phenotype (e.g. affected or unaffected). This class of methods is usually based on the generalized linear model framework, and is superior when the true interaction pattern is prior known. However, substantial bias and power loss might occur if the model is incorrectly assumed, which is often the case in practice where the true model is unknown. In this sense, a model-free method has its own merits (GenoMI in this study), which measures the association between a SNP pair (e.g. loci G and H) and the disease status (D). By defining S as the joint genotypes of G and H, which have 9 possible values, GenoMI can be formulated as: S and D, marginal probability mass functions of S and D, respectively. If cases and controls have the same joint genotype distribution, GenoMI equals to zero. With large sample size, GenoMI approximately follows N is the total sample size and the degree of freedom, v, equals to 8. GenoMI is plausible to detect interaction because interaction can generate discrepancies in joint genotype distribution between cases and controls. Indeed, GenoMI has shown its power for constructing genetic association network for rheumatoid arthritis The entropy-based methods have received increased attention in recent years. Several entropy-based statistics or metrics have been developed for detecting non-linear dependence G1 and G2 can be expressed as: However, many entropy-based metrics have potentially suffered from a risk of inflated false positive error (type 1 error) in testing the interaction hypothesis, induced by the main effects of loci. In an interaction analysis, the false positive error is an undesirable error and requires great efforts to control. Nevertheless, this issue has received limited attention. Very recently, it was in detail discussed by Ueki and Cordell β12 equals to 0, which, however, is very complicated in model-free methods. Some widely used model-free methods are not able to distinguish the interaction effect from the joint effect. For example, the rationale for GenoMI to detect interaction is the presence of the interaction-induced variation in the joint genotypic distribution between cases and controls. However, such genotypic fluctuation can also be generated by the main effect, even if the interaction does not exist. Thus, there is a potential risk of inflated false positive error in GenoMI, as well as other entropy-based statistics, in the presence of main effect. This problem and its solution, unfortunately, have not been well addressed.A reasonable way to detect interactions is to partition the joint effect of the two loci and directly test whether GenoMI metric to detect interaction via extensive computer simulations. Then, to remedy this problem, we proposed and derived two conditional mutual information (CMI) based metrics  to enhance the capability of the information-based metrics to control type 1 error induced by the main effect. Finally, we systematically evaluated the two proposed metrics for detecting genuine interactions (epistasis), in terms of type 1 error control and statistical power, by using large-scale simulations and applications to two real data examples.Inspired by the work of Ueki and Cordell X due to the knowledge of variable Y when variable Z is given. In statistical perspective, the null hypothesis under CMI is conditional independence, assuming that X and Y are mutually independent across each stratum of Z.CMI is a fundamental concept in information theory, defined as the reduction of the uncertainty of variable A and B as the risk alleles of the unlinked SNP loci G and H, respectively. Let i and j  represent the genotypes of loci G and H, respectively, based on the count of risk alleles. Let D denote the disease status, where D = 1 (D = 0) indicates affected (unaffected). CMI based on genotypes (GenoCMI) between two SNP loci can be defined as Denote allele AP cases and 1-AP controls. Let ijP, iP., and Pj. be the joint genotype frequency P, the marginal frequencies P(G = i) and P(H = j), respectively, in the general population. Let K be the disease prevalence. Then, after some algebra . Note thatSuppose that the case-control study has GenoCMI metric measures the quantities of interactions between genotypes of loci G and H in terms of relative risk. If the interaction between genotypes i and j is absent, the relative risk of the joint genotype ij is expected to equal the multiplication of their marginal relative risks such that GenoCMI equals to zero. On the other hand, any departure of GenoCMI from zero indicates the presence of interaction between two loci. Under the null hypothesis, the GenoCMI metric approximately follows v) is 8 and N is number of individuals  to the abovementioned GenoCMI to measure conditional allelic dependence between two unlinked loci. Consider two unlinked loci G and H with alleles A and B being the risk alleles. Let klh be a gamete of loci G and H, where k and l  indicates the carrier states of the risk alleles A and B in the gamete, respectively. The four possible gamete for the diallelic loci G and H, h00, h01, h10 and h11, represent for gamete ab, aB, Ab and AB, respectively. Analogous to GameteCMI metric can be defined as: P(G = k|d) and P(H = l|d) are the frequencies of alleles k and l under disease status d, respectively. Under the rare disease assumption, the Several previous studies have successfully utilized gametic disequilibrium as a measure to detect interaction G and H) located on different chromosomes to be the disease loci for a dichotomous disease phenotype. The risk allele frequencies at loci G and H in the general population were 0.30 and 0.42, respectively. To evaluate the two proposed metrics, we performed extensive simulations in varieties of parameter settings. The publicly available software, genomeSIMLA K = 0.02) and 1∶1 case/control ratio (1000 cases and 1000 controls). To assess the influence of disease prevalence, a rare disease assumption with prevalence K = 0.0001 (in Schemas 4 and 5) was also considered. Moreover, to assess the influence of case/control ratio, a 1∶2 design with 1000 cases and 2000 controls was carried out (in Schemas 6 and 7). Finally, for power evaluation, two schemas were designed. In Schema 8, neither of the loci had main effect, and in Schema 9, one locus had a main effect.Schemas 1–7 were used to assess the capability of the two proposed metrics to control type 1 error under varieties of parameter settings. Schema 1 assumed lack of both main effects and interaction effects at both loci . Schema 2 assumed that only one locus had main effect, and Schema 3 assumed that both loci had main effects. Both schemas were assumed under a common disease assumption  as the truly simulated, then used a 1 df Wald test (named logit_1df in this study) to test the only interaction term. The second model coded genotypes at each locus as two independent factor levels, and as a result, this model had four independent interaction terms (i.e. a saturated model). Therefore, the Wald test (named logit_4df) had 4 degrees of freedom. In essence, the second model can be considered “model-free”. Several methods  require estimation of gamete frequencies, which was carried out by using the E-M algorithm implemented in R library “haplo.stats” (version 1.5.6). The nominal significance of 0.01 was used as the criterion as no multiple tests in each round of simulation were involved. The empirical type 1 error rate or the statistical power was calculated as the percentage of significant results (P<0.01) among the 1000 replicates.We compared our proposed metrics with several previously developed statistics. Original Wu et al. statistic GenoCMI and GameteCMI), we first simulated a general population of 1,000,000 individuals with two independent diallelic loci. Then, we randomly sampled 1,000 individuals from the general population and randomly assigned their disease status (independent of genotypes) according to the specified population prevalence. We repeated 10,000 times to obtain the empirical distributions of GenoCMI and GameteCMI and compared with their corresponding theoretical distributions, the central χ2(8) and χ2(2), respectively (GenoCMI (P = 0.961) and GameteCMI (P = 0.848) with their theoretical ones, respectively.To empirically determine the asymptotic distributions of the CMI-based metrics (ectively . The KolG) had a main effect. The GenoCMI and GameteCMI showed good agreement between the observed and expected values under two different main effect settings . However, an obvious departure was identified for the GenoMI metric when one locus (G) had a main effect, supporting our perspective that GenoMI is unable to distinguish the genuine interaction effect from the joint effect well. All three logistic-based statistics also showed satisfactory performance in these parameter settings except for the original Wu et al. statistic that showed a marked deviation. Except for GenoMI, genetic models appeared having only subtle influence on various metrics to control type 1 error, with GameteCMI and adjusted Wu statistic showing slight inflation in the dominant model when GOR = 3.0.  for testing interaction in a common disease model (K = 0.02) are given in GenoMI could not maintain correct type 1 error when one locus had a main effect, indicating that it is very sensitive to any departure from the global null hypothesis of no main effect at both loci and no interaction effect. It is also interesting to note that the original Wu et al. statistic had a fairly consistently inflated type 1 error over a range of main effects and genetic models. All the remaining six metrics or statistics, including our proposed two conditional information based metrics, adjusted Wu statistic, the joint effect statistic, the logistic regression model with 1 df Wald test and the logistic regression model with 4 df Wald test, had well maintained correct type 1 error rates over three common genetic models and different sizes of main effect. In short, these simulations demonstrated that the two proposed CMI-based metrics could tolerate the influence on type 1 error rate from the large main effects (e.g. OR = 3) at one locus.The estimated type 1 error rates (at K = 0.02). GenoMI and the original Wu et al. statistic. All the remaining metrics or statistics, including the proposed GenoCMI and GameteCMI, had maintained the correct type 1 error rate. However, in the dominant-dominant interaction model, the gamete-based statistics (GameteCMI and the adjusted Wu statistic) appeared not being so effective to maintain the correct type 1 error as well. This inflation was more salient when the sizes of main effect became larger  on the type 1 error rate, we also considered the scenarios of a rare disease (K = 0.0001) in Schemas 4 (with one locus having a main effect) and 5 (with both loci having main effects). K = 0.02, 0.0001 when one locus had a recessive main effect (GOR = 2.0). The size of K appeared to have little effect on the Q-Q plots. A similar pattern also figured in both the dominant model , disease prevalence also had little effect on the Q-Q plots for the recessive-recessive genetic model  on type 1 error under a common disease model . Little effect of the case/control ratio on Q-Q plots for all seven metrics or statistics was observed in the presence of main effect at one locus (data not shown). When both loci had main effects (GOR = HOR = 2.0), only subtle differences in Q-Q plots between different sizes of AP were observed for different genetic models  for power evaluation. Since the original Wu et al. statistic suffered from the risk of inflated type 1 error even in the scenario of no main effect, we excluded it from power evaluation. Given the fact that GenoMI appeared to have highest power, followed by GameteCMI, the logistic regression with 1 df test and GenoCMI. The proposed GenoCMI and GameteCMI achieved comparable performance to the logistic regression with 1 df test in this scenario, with >80% power at GHOR = 2.0 or above. The adjusted Wu statistic, the joint effect statistic and the logistic regression with 4 df test appeared to be more conservative in this case. In the presence of main effect at one locus, GameteCMI and the logistic regression model with 1 df test achieved the best performance. GenoCMI had comparable or relatively higher statistical power compared to the adjusted Wu statistic, the joint effect statistic and the logistic regression model with 4 df test.GenoMI outperformed other approaches as well, but was closely followed by GameteCMI. GenoCMI appeared to have comparable performance to the adjusted Wu statistic, the joint effect statistic and the logistic regression model with 1 df test. The logistic regression model with 4 df test, which did not assume any genetic models, was the most conservative in this case. All metrics or statistics, except for the logistic regression model with a 4 df test for interaction, achieved >90% power at GHOR = 1.5 or above. The same pattern was also identified in the presence of main effect at one locus.GenoCMI appeared to have the best performance among all metrics or statistics, with >80% power at GHOR = 3.0 or above, followed by the logistic regression with 1 df test and GameteCMI in order. Interestingly, GenoMI performed relatively poor in this scenario. In the presence of main effect at one locus, GenoCMI again outperformed other approaches. GameteCMI seemed to have comparable performance to the logistic regression with 1 df test under the correct genetic model and better performance than the adjusted Wu statistic and the joint effect statistic as well.Hb) and α+-thalassemia genes on protecting against malaria. The S variant of Hb gene, located on chromosome 11, is known to be responsible for sickle-cell disease. The α+-thalassemia gene, located on chromosome 16, can cause the α+-thalassemia if defective alleles are produced. Although the hemoglobin (Hb) and α+-thalassemia genes can result in undesirable blood disorders, numerous evidence indicate a protective role for the mutant genes against the more severe manifestations of malaria. However, the mechanisms underlying this protection are poorly understood, in particular much less is known about the α+-thalassemia gene and how it exerts anti-malaria effects when inherited in combination with the Hb gene. Here we analyzed a datum from a birth cohort study of the incidence of hospital admission with malaria and severe malaria from Kilifi District Hospital on the coast of Kenya in Africa Hb gene was genotyped as HbAA and HbAS, representing the wild type and mutant type, respectively The wild-type homozygote, heterozygote and mutant homozygote genotypes of the α+-thalassemia gene were encoded as αα/αα, −α/αα, and −α/−α, respectively. For comparison, the logistic regression models based on different genotypic coding , and based on coding the Hb and α+-thalassemia gene as dummy variables, were also provided. The results of three entropy-based metrics, including our proposed two CMI-based metrics, are summarized in P-values obtained by using the Poisson regression, originally reported by Williams et al. P-values obtained by using logistic regression with 1 df test and 4 df test, respectively. Our results showed that P-values obtained by using GenoMI, GenoCMI and GameteCMI were smaller than those obtained by using Poisson regression analysis. GameteCMI achieved lower P-values compared to logistic regression with either 1 df test or 4 df test, whereas P-values obtained by using GenoCMI were comparably higher than those obtained by using the two logistic regression based tests.To evaluate the utility of the entropy-based metrics in detecting gene-gene interactions in the real settings, we analyzed several published datasets. The first example was about the epistatic interaction between the hemoglobin  and codon72 polymorphism in p53 gene  predisposing to different cancers. The p53 gene encodes a tumor suppressor protein (p53) that is involved in cell cycle regulation as a trans-activator and acts as a tumor suppressor in many cancers. The mutant p53 72Arg>Pro is credited with a more efficient transcriptional activation and suppression of cell growth MDM2 gene is a target gene of the TP53 protein, meanwhile, an important negative regulator of the p53 tumor suppressor. The mutant of MDM2 309T>G is proposed to elevate the expression of the product of MDM2, which binds the p53 protein and targets its proteasomal degradation, thereby attenuating its response and diminishing its tumor suppressor function p53-MDM2 interaction could provide valuable insight into the mechanisms underlying tumor oncogenesis afforded by these genes. In this example, we evaluated four phenotypes in three different populations, including gastric cardia cancer in Han populationp53 and MDM2 are Pro and G, respectively. P-values obtained by using three entropy-based metrics including two proposed CMI-based metrics. The P-values from the two logistic regression methods with either 1 df test or 4 df test were used for comparison. The P-values of GameteCMI in all the four cancer phenotypes were smaller than that of logistic regression analyses, while the P-values of GenoCMI were smaller than or similar to those of logistic regression with 1 df test and were smaller than those obtained by using logistic regression with 4 df tests. In conclusion, these two real data applications demonstrated that GameteCMI and GenoCMI achieved optimal or nearly optimal performance compared to the benchmark logistic regression based test.The second example was composed of several studies concerning the interaction between SNP309 in GenoMI) was not able to distinguish the genuine interaction effect from the joint effect. In particular, when the main effects at two loci were large, the inflation of type 1 error in testing interaction effect was much salient. To remedy this issue, we proposed to use two conditional mutual information (CMI) based metrics . By mathematical derivations and extensive simulations, we showed that the two CMI-based metrics worked well to control false positive error in various two-locus interaction models.In this study, we attempted to solve the potential false positive problem associated with current model-free or data mining approaches for genome-wide interaction analysis, by using the entropy-based metrics as examples. Inspired by the work of Ueki and Cordell GenoCMI and the previous joint effect statistic could control false positive error well under the recessive-recessive or dominant-dominant genetic models, whereas GameteCMI as well as the adjusted Wu statistic exhibited noticeable inflated type 1 error under the dominant-dominant genetic model. Unfortunately, we are not very clear about why the GameteCMI appeared having an inconsistent type 1 error rate under dominant-dominant interaction models. Although this phenomenon was also observed (but without explanations) in the previous studies . Fortunately, both our proposed CMI-based metrics and several previous statistics (adjusted Wu statistics and the joint effect statistic) were able to control type 1 error with moderate main effects at both loci, e.g. OR = 2.0, which is often the case for most complex diseases. Furthermore, based on our simulation studies, the disease prevalence (K) and case/control ratio (AP), two important factors underlying a case-control study, appeared to have subtle influence on the control of type 1 error.To evaluate the capability of the proposed CMI-based metrics to control the type 1 error, we considered varieties of parameter settings in a systematic way. In absence of both main effect and interaction effect or presence of main effect at one locus, the proposed two CMI-based metrics achieved adequate control of type 1 error in both common and rare disease models, with only feebly influenced by genetic models and the sizes of main effect. In the scenario of a common disease and both loci having main effects, GenoMI performed best in detecting the additive-additive or dominant-dominant interaction effect. However, this advantage was somewhat tempered by its possibility of inflated type 1 error. The trade-off between the gain of power and the risk of false positive error becomes a difficult balance to strike in real data analysis. One feasible consideration is to apply multistage tests or an ensemble approach, in which we first use GenoMI to filter out non-informative gene pairs, followed by refining gene pairs that are with genuine interaction effects by using GenoCMI and GameteCMI or other alternatives mentioned in this study.In terms of statistical power for detecting gene-gene interaction, the proposed CMI-based metrics in general had better performance than the adjusted Wu statistic and the joint effect statistic, and had comparable performance to the benchmark logistic regression. In the absence of main effect at both loci, the Finally, we should recognized that several issues still remained unattended in this study, although we have very carefully explored the statistical properties of the two proposed CMI-based metrics for detecting gene-gene interaction. First, we did not concern varieties of definitions of interaction, which can be very diverse from statistical standpoints to biological ones Figure S1Chi-squared Q-Q plots for the dominant model with main effect at one locus (Schema 2). Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Figure S2Chi-squared Q-Q plots for the additive model with main effect at one locus (Schema 2). Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Figure S3Chi-squared Q-Q plots for the dominant-dominant model with main effect at one locus, when disease prevalence varied (Schema 4). Assuming a main effect at single locus (ORG = 2.0) and 1∶1 case/control ratio. Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Figure S4Chi-squared Q-Q plots for the additive-additive model with main effect at one locus, when disease prevalence varied (Schema 4). Assuming a main effect at single locus (ORG = 2.0) and 1∶1 case/control ratio. Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Figure S5Chi-squared Q-Q plots for the dominant-dominant model with main effects at both loci, when disease prevalence varied (Schema 5). Assuming main effects at both locus (GOR = HOR = 2.0) and 1∶1 case/control ratio. Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test(TIFF)Click here for additional data file.Figure S6Chi-squared Q-Q plots for the additive-additive model with main effects at both loci, when disease prevalence varied (Schema 5). Assuming main effects at both locus (GOR = HOR = 2.0) and 1∶1 case/control ratio. Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Figure S7Chi-squared Q-Q plots for the dominant-dominant model with main effects at both loci, when case/control ratios varied (Schema 7). Assuming main effects at both locus (GOR = HOR = 2.0) and disease prevalence 0.02. Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Figure S8Chi-squared Q-Q plots for the additive-additive model with main effects at both loci, when case/control ratios varied (Schema 7). Assuming main effects at both locus (GOR = HOR = 2.0) and disease prevalence 0.02. Top panels: A. GenoMI; B. GenoCMI; C. GameteCMI. Middle panels: D. original Wu et al statistic; E. adjusted Wu statistic; F. joint effect statistic. Bottom panel: G. logistic regression model with 1 df test; H. logistic regression model with 4 df test.(TIFF)Click here for additional data file.Text S1Mathematical derivation of GenoCMI and GameteCMI metrics.(DOC)Click here for additional data file.Text S2Mathematical derivation of asymptotic distributions of GenoCMI and GameteCMI metrics.(DOC)Click here for additional data file.


Genome-wide association studies (GWAS) are routinely conducted for both quantitative and binary (disease) traits. We present two analytical tools for use in the experimental design of GWAS. Firstly, we present power calculations quantifying power in a unified framework for a range of scenarios. In this context we consider the utility of quantitative scores (e.g. endophenotypes) that may be available on cases only or both cases and controls. Secondly, we consider, the accuracy of prediction of genetic risk from genome-wide SNPs and derive an expression for genomic prediction accuracy using a liability threshold model for disease traits in a case-control design. The expected values based on our derived equations for both power and prediction accuracy agree well with observed estimates from simulations. In the last five years, GWAS have been published for both quantitative traits  of a 1) NCP for quantitative responses in population samples, 2) NCP for binary traits in population samplesz is the height of the normal curve truncating the proportion K, where K is the proportion of the population that are cases.3) NCP for binary responses in ascertained case-control samples, P the proportion of cases in the case control sample and 4) NCP for quantitative responses in ascertained case-control samples, i is the mean liability in cases and t is the threshold on the normal distribution which truncates the proportion of disease prevalence K, and from Daetwyler et al. The covariance between disease liability and genetic values in an ascertained case-control sample isTherefore, from (5), (6) and (7),ccl on ccg isThe regression coefficient of Finally, the proportion of variance attributable to the SNPs or set of SNPs of a quantitative response in an ascertained case-control sample can be obtained as the squared regression coefficient multiplied by the genetic variance in the case-control sample and scaled by the variance of disease liability in the case control sample, i.e.5) NCP for quantitative responses for cases only in ascertained case-control samples,When underlying continuous quantitative responses are available only for cases in the ascertained case-control sample, i.e. the recorded values follow a mixture distribution of zero for controls and truncated normal distribution for cases. An example may be a GWAS of major depressive disorder in which cases are recorded for a quantitative severity score, whereas controls have not been scored. In this situations,i and t are the same as defined above. There is an assumption here is that the quantitative trait is the phenotypic liability.The variance of the mixed zero and truncated normal values in an ascertained case-control sample is,ccg in an ascertained case-control sample is,The covariance between cc can be derived analytically as,From the equations above, regression coefficient of Therefore, the proportion variance attributable to the variance in the SNPs from mixed zero and quantitative response in an ascertained case-control sample . Following Daetwyler et al. jth SNP effect isjβ of the phenotype on the jth SNP genotype, ijx  = 0, 1 or 2 for the ith individual, N is the number of individual records and 2σ is the residual variance.For a quantitative trait, g)) explained by the set of M SNPs is 2h. Following Daetwyler et al. M SNPs in the predictor  and the residual variance (2σ) asAssuming a phenotypic variance of one, the genetic variance (var(2σ = 1 (phenotypic variance) as in Daetwyler et al. τ defined as the ratio of the number of samples (N) over the number of SNPs (M), the accuracy can also be written as The squared correlation coefficient between the true and estimated genetic value is the ratio of the true genetic variance over the estimated genetic variance 2σ approximated as 2σ = K(1–K) , the prediction error variance for the jth SNP effect can be written asβ is allele substitution effect on the 0, 1 observed scale and β from regression of the 0,1 discrete phenotypes on SNP coefficients. The estimated genetic variance on the observed scale of the SNP predictor (var(u) or In binary disease traits, with The squared correlation coefficient between the true and estimated genetic values isBecause genetic variance as a proportion of phenotypic variance on the observed scale can be transformed from that on the liability scale as jth SNP can be written as jth SNP effect can be expressed asβ is allele substitution effect on the 0, 1 observed scale and β from regression of the 0,1 discrete phenotypes on SNP coefficients in the case-control sample. The estimated genetic variance on the observed scale in a case-control design can be derived asAscertainment in case-control samples often results in over-representation of cases compared to the case prevalence in the population. The variance of the explanatory variable is inflated by a factor of ijβjx) is the genetic variance on the observed scale due to the jth SNP effect transformed to the liability scale 2σ = P(1–P) , and the value for f is close to 1 . Therefore, the genetic variance in a case-control sample isThe squared correlation coefficient between the true and estimated genetic values isFor binary traits, area under the receiver-operator characteristic curve (AUC) is a useful statistic for the genomic prediction accuracy g) were simulated from an additive multilocus model of M = 100 independent SNPs with equal allele effects and allele frequency of 0.5. Residual values (e) were independently generated from a random normal distribution with a mean of zero and variance of 2h was obtained. We simulated 2h of 0.01, 0.05 and 0.1 so that each SNP explained 0.0001, 0.0005 and 0.001 of the phenotypic variance /K. In population or case-control sample, we used both binary (BT_POP or BT_CC) and quantitative responses (QT_POP or QT_CC). We conducted 100 replicates for each simulation scenario, therefore 10000 association tests were carried out. Power was calculated as the proportion of the 10000 association tests in which the association p-value less than 0.05 and was compared to power calculated from the NCP using a function in R package, i.e. power  = 1– pchisq  where T is the normal distribution threshold corresponding to the significance level 0.05.In order to check the analytically derived equations of NCP for BT_POP, BT_CC, QT_POP and QT_CC, we carried out a simulation study. Individual genetic values (variance . Liabilig) were simulated from an additive multilocus model with equal allele effects  and residual values (e) independently generated from a random normal distribution with a mean of zero and variance of 2h was obtained. Liability phenotypes for each individual were simulated as y  =  g + e. Affected individuals were those with liability phenotype that exceeded a threshold determined by population prevalence. Population prevalences of K = 0.001, 0.01, 0.1, 0.2 and 0.5 were used with N = 2000 and M = 2000. To vary τ =  N/M, N = 2000 and M = 400 were used for τ = 5, and N = 100 and M = 5000 were used for τ = 0.02. Following Daetwyler et al. g) and estimated genetic values Simulations were carried out to verify the validity of K, there was an excellent agreement between the observed and expected power with a small empirical standard error. When using a higher variance explained by each locus (2h), although the empirical standard error increased slightly, the observed value also agreed well with the expected value (Exp and Obs in The power of association tests observed in simulation and expected from theory agreed well under a range of scenarios . Whether2h). Generally, the power increases when the variance explained by SNPs increases, and when the ascertained case-control design is used. For BT_POP, the power decreases as K decreases, reflecting the smaller number of cases in a given population sample. For QT_POP, the power is, of course, constant across a–c in K decreases, which reflects the greater over-sampling of cases with lower K for the same sample size and hence the difference in mean liability between cases and controls increases. There is a moderate difference between BT_CC and QT_CC when using population prevalence K = 0.1 Click here for additional data file.Appendix S2R code for the prediction accuracy derivations described in the paper.(DOC)Click here for additional data file.


Most connections between phenotype and genetic variants detected in genome wide association studies (GWAS) of human longevity-related traits did not reach the genome-wide level of statistical significance. These estimates also suffer from the lack of replication of research findings in studies of independent populations  done at baseline. Studies included in the CHARGE Consortium [except for the Original cohort of the Framingham Heart Study (FHS) where the bio-specimen collection was done some time after baseline] are typical examples. Since data on lifespan or other durations are often incomplete , the Cox's type regression models is usually implemented in GWAS of these data . Note that using follow up data alone and conditioning on age at baseline may be required by the goal of the study . However, conditioning on the ages at bio-specimen collection when some of these ages are high enough may diminish or even totally exclude a substantial part of the genetic variation in longevity, i.e., eliminate the effects we are looking for. This is because the oldest old individuals participating in bio-specimen collection are precisely those who passed the process of mortality selection in the genetically heterogeneous population, and, therefore, are likely to carry genetic variants linked with “longevity” alleles . Thus conditioning on the age at genotyping, especially when the oldest old study participants are in the sample, may leave little hope that associations of remaining genetic variants with human longevity will reach genome-wide significance.Such conditioning, however, seems to be a common practice in GWAS of human longevity-related traits dealing with prospective data. For example, a large group of researchers performed comprehensive genetic analyses of human lifespan and free of major diseases lifespan using data from nine studies of the CHARGE Consortium . Such a situation characterizes the FHS data, which is the part of the CHARGE Consortium.It turns out, however, that in cases when the ages at bio-specimen collection include young adults and the oldest old individuals, the additional information about the role of genetic variants in lifespan can be obtained from the age patterns of genetic frequencies evaluated for any genetic variant even without using the follow up data. The approach based on comparison of genetic frequencies among individuals of different age categories is typically used in genetic studies of centenarians Weir, . A monotOne such method for joint analyses of the two types of data is based on maximizing the joint likelihood function of the combined data comprised of the cross-sectional age patterns and follow-up data. This method is a special case of a more general approach , and non-carriers, μ(x|G = 0), and the initial proportion of the index variant, P0 = P(G = 1)  data  = μ0(x)eGγ , where G = 0 for non-carriers and G = 1 for carriers, and the baseline mortality μ0(x) is the Gompertz function, i.e., ln μ0(x) = ln a + bx. The Gompertz parameters a and b correspond to the initial mortality at birth and the rate of exponential growth in mortality with age in non-carriers. The regression parameter γ modifies the initial mortality for carriers so that it becomes aeγ. We used the Gompertz parameters ln a = −9.0 and b = 0.08 to produce reasonable survival patterns corresponding to human populations, and the proportion of carriers at birth P0 = 0.25. The parameter γ varied from −0.5 to 0.5 with the interval 0.05 to simulate scenarios with different effect sizes.We assumed that carriers and non-carriers of some hypothetical allele in a population have mortality rates μeGγ  for carriers and μ0(x) for non-carriers, with the parameters defined above]. Then we assigned the hypothetical “age at entry” into the study to each individual in the population generated as a discrete random variable uniformly distributed over the interval 40–100 years. Such “ages at entry” were assumed to be the same as age at biospecimen collection, in line with the LLFS design. We collected a sample of 4500 individuals  whose life spans exceeded their hypothetical “age at entry.” Individuals with simulated life spans exceeding “age at entry” plus 6 years were considered censored at that age.First, we generated a large  “general population” assigning the genetic status  to individuals in the general population in accordance with the initial proportion The above procedure was repeated 1000 times (in each scenario with respective γ) to generate 1000 datasets which were subsequently estimated using the likelihood that uses only follow-up information and the likelihood that takes into account information on ages at biospecimen collection in addition to follow-up data.Z-test of the mean and found the values of the standard deviations that produced the best fit to the empirical power curves for each method (0.056 for “FU + A” and 0.084 for “FU”), see solid and dashed lines in Figure 10(α) for better visibility] that yields power w = 0.8, as a function of the effect size in both methods . Figure We calculated the average values and standard deviations of parameter estimates in 1000 simulated datasets in both methods  for each γ. The results showed that the standard deviations of γ were smaller in the method that takes into account ages at biospecimen collection . Although such a difference may look small in absolute value and not worth mentioning at first sight, it corresponds to a substantial gain in power, if we translate these observations into power curves and curves for the level of the test that yields specific power as a function of the effect size (regression parameter). Figure p-value reduces from 10−2 to 10−5 and for the effect size equal to 0.4 p-value drops from 10−4 to 10−9. This means that many genetic variants which would be not genome-wide significant in GWAS of follow-up data using the traditional Cox-type approach would become highly significant if the proposed approach is used. Our simulations with different follow-up periods (data not shown) reveal that this effect diminishes with an increase of a follow-up period. It is clear intuitively that, in the case of a growing follow-up period, information from this long follow-up makes an increasing contribution compared to information hidden in the distributions of ages at biospecimen collection. Conversely, in the case of a shorter follow-up period, distributions of ages at biospecimen collection play a more important role in differentiating the allele- or genotype-specific survival patterns from the data. Thus our results show that the additional use of information on ages at biospecimen collection may have important implications for GWA studies of longevity especially in cases with relatively short follow-up periods. The LLFS data provide a good example of a study which could potentially benefit from the addition of information on ages at biospecimen collection in its GWAS of longevity . As longevity is known to run in families, family-based genome-wide association studies can provide additional advantages compared to analyses of independent samples. However, performing GWAS in cohort studies that contain both unrelated individuals and family members requires a special consideration of the analytical approach to take the full advantage of such data (Manichaikul et al., The results shown in Figure