Identity by descent (IBD) matrix estimation is a central component in mapping of Quantitative Trait Loci (QTL) using variance component models. A large number of algorithms have been developed for estimation of IBD between individuals in populations at discrete locations in the genome for use in genome scans to detect QTL affecting various traits of interest in experimental animal, human and agricultural pedigrees. Here, we propose a new approach to estimate IBD as continuous functions rather than as discrete values.Estimation of IBD functions improved the computational efficiency and memory usage in genome scanning for QTL. We have explored two approaches to obtain continuous marker-bracket IBD-functions. By re-implementing an existing and fast deterministic IBD-estimation method, we show that this approach results in IBD functions that produces the exact same IBD as the original algorithm, but with a greater than 2-fold improvement of the computational efficiency and a considerably lower memory requirement for storing the resulting genome-wide IBD. By developing a general IBD function approximation algorithm, we show that it is possible to estimate marker-bracket IBD functions from IBD matrices estimated at marker locations by any existing IBD estimation algorithm. The general algorithm provides approximations that lead to QTL variance component estimates that even in worst-case scenarios are very similar to the true values. The approach of storing IBD as polynomial IBD-function was also shown to reduce the amount of memory required in genome scans for QTL.In addition to direct improvements in computational and memory efficiency, estimation of IBD-functions is a fundamental step needed to develop and implement new efficient optimization algorithms for high precision localization of QTL. Here, we discuss and test two approaches for estimating IBD functions based on existing IBD estimation algorithms. Our approaches provide immediately useful techniques for use in single QTL analyses in the variance component QTL mapping framework. They will, however, be particularly useful in genome scans for multiple interacting QTL, where the improvements in both computational and memory efficiency are the key for successful development of efficient optimization algorithms to allow widespread use of this methodology. Variance component analysis  is a fleSeveral methods have been proposed to estimate IBD matrices in a wide range of population structures. -5 Some mWe have recently reported an efficient method for variance component estimation in QTL mapping . Here, wTo illustrate the generality of the new approach to estimate IBD functions, we have worked out two algorithms for IBD-function estimation. First, we show one example of how to re-implement an existing IBD estimation algorithm  to compuWe re-implemented an algorithm for deterministic IBD matrix estimation  to compuWe developed a general algorithm for approximating IBD-functions in marker brackets (CF-IBD – Continuous Function IBD) from a limited set of IBD matrices from any existing IBD estimation algorithm and evaluated how well it approximates the IBD matrices between markers that were obtained using the software LOKI .This particular software was chosen because it is a stochastic algorithm based on Monte Carlo Markov Chain (MCMC) iteration procedure, which means that the computation strategy of this algorithm is completely different from the one of Ricardo Pong-Wong's method. By using this method to illustrate the efficiency of the curve-fitting approach, we also highlight the important aspect of the generality of this approach.The input to our algorithm was IBD values at marker positions calculated by LOKI and the IBD matrices obtained from the IBD functions were compared to LOKI's IBD-matrices as they are the ones approximated by CF-IBD.In the optimal scenario, i.e. all markers are fully informative , the IBD values are all either 0; 0.5; 1; (or 2 in the case of inbreeding). Consequently, a more informative IBD matrix has a higher density of those values than of intermediary values. None of the methods presented here requires fully informative markers, but we use the property mentioned above to postulate that the algorithm attribute the values 0, 0.5 or 1 when the IBD is estimated without uncertainty and that intermediate values then exist only when there is uncertainty. Our comparisons show that the IBD to be used for QTL mapping are likely to be much more influenced by the assumptions made when estimating IBD from data  rather than by the estimation of an IBD function from discrete IBD at marker locations. Figure et al. produce less informative IBD matrices. To quantify the similarity between the methods, we have computed the average difference (i.e. the absolute pairwise difference between the non-null elements of two matrices) of the matrices computed by LOKI (L) to those of CF-IBD (CF) and the Pong-Wong (PW) based algorithm for several positions within the marker-bracket. Since the average expected value of the IBD between two individual is determined by their relationship , the pairwise differences have been weighted by this relationship coefficient. The differences between the methods are nearly constant within the marker brackets and is always bigger between PW and L, than between CF and L (weighted average difference of non-null elements in the matrices is 0.001 for CF vs L and 0.03 for PW vs L). Our approximations of LOKI are always closer to the original LOKI estimates than the estimates obtained using Pong-Wong's method, indicating that the effect of the assumptions used when calculating IBD at marker locations has a greater impact on the obtained IBD-matrices than the use of CF-IBD to approximate IBD in marker brackets.The distribution of the estimated IBDs were compared in a region close to marker KITL1  located As the ultimate aim of estimating the IBD-matrices in QTL mapping applications is to use them for estimating genetic variance components of QTL, it is important to evaluate the potential decrease in statistical power from using an approximate IBD-estimation method rather than the exact method. Thus, we compared the variance component estimates obtained by REML  based onm) can be calculated as:Since our method considers the IBD values as a function of the recombination probability between two markers on the chromosome, it is possible to store the output of the algorithm as a function instead of a traditional IBD matrix. The IBD-function for each pair of individuals is calculated from a set of recombination functions Table , and cann is the number of individuals in the pedigree, nz is the proportion of non-zero elements in the matrix and p is a constant relating to the memory required to store two individual ID's as integers and a real valued IBD value with single (p ≈ 2 × 10-5) or double (p ≈ 2.9 × 10-5) precision.where m = 8722 * 0.5 * 2 * 10-5 ≈ 7 Mb, as there are 872 individuals in the pedigree and around 50 % non-zero elements. If on the other hand the IBD functions for a marker interval are stored in the form of a sparse matrix that contains the three parameters  of a second degree polynomial (ax2 + bx + c) for each bracket and pair of individuals with single precision, then the constant p in [-5 and the memory requirement to store one IBD-function matrix to about m = 8722 * 0.5 * 4 * 10-5 ≈ 14 Mb. The advantage of this storage approach for varying number of IBD matrices in a marker bracket is illustrated in Figure Storing a typical IBD-matrix in the chicken pedigree used in this study at single precision requires approximately Storing a large number of individual IBD-matrices representing an interval or and entire genome can thus require large amounts of memory if one wants to identify the QTL location with high precision. A single QTL genome scan with 1 cM resolution in the chicken genome requires approximately 2300 matrices corresponding to a memory requirement of 16 Gb, whereas the IBD-function based storage in a genome with on average 15 tested locations in a marker bracket (here corresponding to 15 cM marker spacing) requires about 2 Gb. In an epistatic analysis including two loci with a single interaction effect, the variance-covariance matrix for the interaction effect is usually constructed as the Hadamard product of the respective marginal effects IBD matrices . An exhaTo facilitate development of methods for mapping multiple interacting QTL using variance component models, it is necessary to significantly improve the computational efficiency. To do so, three key components need to be addressed: estimation of relationship between relatives (i.e. IBD matrix estimation), variance component estimation of QTL at a given genomic location (or combination of locations) and the genome scan for genomic locations (or combinations of locations) to evaluate . Recently, more efficient algorithms for variance component estimation  and optiWe have shown that it is possible to estimate the IBD function for a marker bracket exactly by re-implementing existing IBD algorithms using Pong-Wong algorithm  as an exIn situations where re-implementation of the original algorithm is not achievable, but where IBD functions could be useful, we provide a general curve-fitting based algorithm to estimate these functions from IBD matrices provided by any existing algorithm. To illustrate the applicability of this approach, we have made an in-depth study of its properties. By using input from one IBD estimation method , we showIBD probabilities can theoretically be calculated exactly, but in real datasets this will be computationally prohibitive. The methods used in practice thus aim to approximate the true IBD and depending on the assumptions made, methods will generally obtain different IBD matrices for a specific dataset. Here, we clearly show that differences in underlying assumptions in the algorithms may have a much greater impact on both the direct estimates if IBD's, as well as on variance component estimation, than from using IBD functions as approximations of a specific IBD estimation method.By re-implementing an already fast deterministic IBD estimation algorithm by Pong-Wong et al  to conduOur new approach facilitates the development of efficient algorithms for multi-dimensional genome scans for interacting QTL  by signiThe computational efficiency in QTL mapping in general can be improved considerably by replacing the commonly used grid search with a more efficient optimization algorithm ,7. UsingAn existing optimization algorithm that would be interesting to explore for single- and multi-dimensional QTL searches in the variance component framework would be DIRECT . This alHere, we describe how and why to estimate IBD relationships in a population as a set of IBD functions between individuals. This approach immediately improves the efficiency in IBD matrix estimation in genome-scans for QTL, but also to facilitate further improvements by resolving methodological bottlenecks for implementing existing and developing new and more efficient algorithms to screen the genome for single or multiple QTL. We have provided two examples of how to estimate continuous IBD functions between a pair of markers, instead of a set of discrete IBD values, based on existing discrete IBD estimation algorithms. The first, and most efficient, method is a re-implementation of an existing IBD estimation algorithm to provide exact IBD functions that give the same values as the original algorithms. The second, more general method, approximates IBD-functions using IBD values from any algorithm of choice without the need for re-implementation of algorithms or software. The loss of information in this estimation is minor and it only affects QTL variance component estimates marginally. The estimation of continuous IBD-functions, rather than discrete IBD estimates, was shown to improve the computational efficiency (when re-implementing) and memory usage (for both strategies) in genome-scans for QTL. These improvements are expected to be of practical importance in particular in genome scans for multiple interacting QTL, where much higher computational and memory efficiency is required. The presented approaches will also has a positive impact on the use of more efficient search algorithms for QTL and the advent of continuous IBD functions opens new opportunities for developing new strategies for high-precision estimation of QTL location in single- and multidimensional genome scans.et al [Pong-Wong et al  describei, in a pedigree, paternally (Pi) and maternally (Mi) inherited chromosomes are assigned based on marker information and pedigree. IBD probabilities are calculated recursively for all chromosomes in the pedigree and summarized in an IBD-matrix. As individuals are diploids, the resulting matrix is of size 2n × 2n, (rather than n × n), where n is the number of individuals in the pedigree.The original algorithm proceeds as follows. For each individual Note that the deterministic algorithm estimates gametic IBD matrices (G) (of size 2n*2n), but the object that is used in the variance component method is an individual matrix (of size n*n), where the four element of G, corresponding to IBD between the four chromosomes of a pair individual, are summed up to obtain a single IBD value between the two individual.In the result part, when comparing values given by the different IBD estimation methods, we always refer to the individual IBD matrix.i, his father j and a given ancestor k, The IBD probability between the paternal chromosome, (Pi), of individual i, and the chromosome Pk of the ancestor k, is equal to:Considering an individual Figure Pi) from the parent j is inherited either from j's paternal chromosome (Pj) or j's maternal chromosome (Mj). Consequently, the sum of the PDQ for the two parental chromosomes is equal to one  is the probability that a particular locus of a chromosome  Search for the closest informative markers flanking gii) Determine which allele  that have been inherited for each of the two flanking markersiii) For each pair of individuals in the pedigree that have a parent-offspring relationship, calculate the PDQs at g, using a pre-specified recombination probability, r, between the markers that is estimated from the genetic map.iv) For each pair of chromosomes, from ancestor to descendant, calculate the IBD values recursively.Since IBD within the same bracket are considered independently, steps i)-iv) are performed successively each time an IBD matrix is calculated. Figure et al. [Existing methods for estimating IBD matrices consider IBD values in the same bracket to be independent. However, IBDs within a marker bracket are not independent and this fact can be used to simultaneously infer all IBD values in a bracket. An updated version of the algorithm of Pong Wong et al. , that doi) and ii) of the original algorithm are completed once for a marker bracket. When the parental inheritance has been inferred in step ii), one can select the recombination functions for calculation PDQs between pairs of individuals. This function is the same for all positions between the markers. We give these functions in Table a) Steps g, within the marker bracket (represented by ra and rb in the formulas in Table b) For all parent-offspring pairs, PDQs are computed "horizontally" for each pair of chromosomes Figure , using tc) For all pairs of individuals in the pedigree, IBD probabilities are computed horizontally, for all positions between the markers, using the Pong-Wong original recursive strategy illustrated in Figure The updated Pong-Wong algorithm thus computes several IBD matrices simultaneously, which improves the computational efficiency by minimizing the computational cost of reading data and searching for informative markers.LOKI  is a freWe propose a general algorithm for estimation of marker bracket IBD functions that as input use IBD values at marker locations calculated using an external IBD estimation algorithm suitable for the experimental data that is analysed. Using information from the IBD-matrices at the markers, a continuous function of IBDs is estimated in the brackets between markers. This approach is possible as the IBD values within a marker bracket are a continuous function of their location in the bracket and strongly correlated with the IBD values at the flanking marker positions. From the estimated set of IBD functions in the pedigree, IBD matrices can then be directly computed at any location between the markers at a very low computational cost.IBD-functions for each pair of individuals in a marker bracket can be estimated using curve fitting based on the functions of Table LM); IBD(RM)) plus one position between the parkers (IBD(M) are calculated using any algorithm of choice (e.g. LOKI). Expressed in terms of the distance between the two makers (d), the position between the two markers where an IBD will be calculated (x) and setting M = d/2 we have:The algorithm can be summarized as follows: For each pair of individuals, the IBD values at the three positions: two flanking markers (IBD(x) = ax2 + bx + c) can then be fitted by solving the equation system:A polynomial function (IBD(which gives:Polynomial functions representing the IBD relationship for each pair of individuals in the pedigree are then estimated as illustrated in Figure An AI-REML algorithm  programmÖC initiated the study. ÖC and FB designed and developed the project and wrote the paper together. All authors have read and approved the manuscript.


We used data reduction and clustering methods to identify five phenotypically homogeneous groups of study participants with similar profiles for cardiovascular disease risk factors. We constructed both qualitative (binary subgroup membership) and quantitative traits (probability of subgroup membership) for each individual. The Cluster 1 comprised individuals who were generally healthy and had some history of smoking. Cluster 2 was dropped from the analyses due to the preponderance of missing data. Cluster 3 was used as the control group, healthy non-smokers. Members of Cluster 4 had features of the metabolic syndrome and were generally not as obese as Cluster 5. Obesity was the hallmark of Cluster 5, the members of which also had some features of the metabolic syndrome.We then examined the genetic associations with both qualitative and quantitative representations of these empirically derived traits. Genetic analyses of the qualitative traits were conducted, comparing each of the affected groups with the unaffected cluster alone and, to increase statistical power, the unaffected group and healthy smokers combined. One single-nucleotide polymorphism on chromosome 4 met a conservative genome-wide significance level, but the effect was muted when we accounted for population stratification. The results for the quantitative traits were similar, with a small number of genome-wide significant findings muted by control for admixture. The directional findings will provide the basis for hypothesis generation for syndromes such as the metabolic syndrome and obesity. The identification of subtypes of disease using data reduction and clustering methods has been helpful for identifying genetic associations in schizophrenia, rheumatoid arthritis, and other disorders -3.The Framingham Heart Study (FHS) is an ongoing longitudinal study focusing on the development of coronary heart disease and associated risk factors in Framingham, MA. The long history of research conducted in Framingham has contributed to the contemporary understanding of cardiovascular and related diseases. One advantage of this study is that the ascertainment criteria do not require disease at the time of study entry. In addition, this is a random sample of households and not restricted to those seeking healthcare for any specific complaint.Our analyses of the data made available to participants in the Genetic Analysis Workshop (GAW) 16 were restricted to members of the Offspring Cohort because some measures of metabolic function were not available in the Original Cohort and were not available for more than one measurement period in the Generation 3 Cohort.The objectives of this study were 1) to identify subgroups of study participants with similar phenotypic characteristics (or syndromes) including patterns across measurements at multiple visits, and 2) to construct qualitative and quantitative representations of the new phenotypic subgroups, and 3) to conduct genome-wide association analyses of the quantitative traits; and for qualitative traits, to compare the newly identified affected phenotypic groups with unaffected controls.We created categorical variables using the Centers for Disease Control (CDC) definitions for each of the following for each exam: body mass index , high cholesterol (measured or on lipid lowering medication), low high-density lipoprotein (HDL), high triglycerides, and hypertension (measured or on medication) . These vThe strategy for the development of qualitative traits included nonparametric data reduction, iterative two-staged clustering on the observed dimensions, and the assignment of binary membership in each cluster for each individual. Quantitative traits (probability of cluster membership) were estimated using logistic regression with cluster membership as the outcome and all variables used for clustering as predictor variables in the models.We used multiple-correspondence analysis (MCA) for data reduction instead of the more traditional principal-components analysis (PCA). PCA is a method commonly used for data reduction. These data did not meet the distributional assumptions for a Pearson correlation, the basis for PCA. A similar method designed for use with categorical data was employed. MCA is a nonparametric data reduction method free of the assumptions underlying PCA and was developed for qualitative data. The objective of MCA is to identify a low-dimensional subspace that comes closest to all of the data points. It is analogous to graphing the results of a factor analysis in a multidimensional euclidean space. However, the space identified in MCA is not euclidian. The coordinates of each individual in the identified multi-dimensional space served as the basis for the identification of subgroups or clusters .k-means clustering with different random cluster seeds and used a larger k (number of clusters) than we expected in the data. Groups that consistently clustered together across all of the initial analyses were identified as intact clusters. An agglomerative hierarchical clustering algorithm was then implemented using the intact clusters and the remaining individuals in the sample. An examination of the change in Ward's aggregation criterion and the nature of the groups was used to choose the final cluster structure ). Cluster 2  comprised individuals who were missing information on two or more measurements. This group was omitted from the genetic analyses. Cluster 3 comprised healthy non-smokers . This group served as the control group in our genetic association studies. Cluster 4  comprised individuals with features of the metabolic syndrome (MS) who were generally not obese , the members of which also had some features of the MS. MS, as identified by the National Cholesterol Education Program's Adult Treatment Panel III, is a clustering of risk factors that can lead to cardiovascular disease (CVD). The risk factors include: abdominal obesity, atherogenic dyslipidemia, raised blood pressure, insulin resistance with or without glucose intolerance, proinflammatory state, and prothrombotic state [Among the 2,760 study participants in the Offspring Cohort, five clusters were identified. Cluster 1 comprised individuals who were generally healthy and had some history (current or past) of smoking ( greater ). Obesitic state .The distributions of sex, age, diabetes ever and myocardial infarction ever are as follows : healthy smokers ; healthy controls ; MS ; obese . By Visit 7, nearly 90% of the MS group was on treatment for hypertension. The same was true for 45% of the Obese group, as well as 12% of the healthy smokers and 18% of the controls. Figure p-values at or beyond the genome-wide significance level (data not shown). There was one genome-wide significant SNP on chromosome 4 when we used the larger comparison group. However, this finding was muted when we accounted for population stratification.We compared the MS and Obese groups with the healthy control group, and, to increase power , with the healthy control and healthy smokers combined. We used a fairly conservative approach to correct population stratification . Neither of the analyses using the controls alone revealed SNPs with Figure p < 1 × 10-5; similarly, there were eight for the Obese group. In both cases the findings were somewhat muted when we accounted for admixture.Our analyses of the 50 k chip showed similar results (data not shown). For the quantitative trait for the MS-like group there were five results across the genome with We identified five phenotypic clusters using a limited set of measures pertaining to medical history, metabolic function, and environmental exposures. One group was omitted due to missing data. Two groups appeared to be relatively healthy, one of which was more inclined toward tobacco use than the other. The remaining two groups were characterized by elevated measures related to MS and obesity. Interestingly, those characterized by features of MS were not as heavy, nor did they gain weight as quickly over time as did the Obese group. Features associated with MS were not as prevalent in the group characterized by obesity as they were in the other affected cluster.p-value instead.There are several limiting factors in our analyses. We used a conservative approach for the correction of population stratification. We also used a somewhat conservative approach for genome-wide significance levels. It would be interesting to see these results using an empirical Next steps in these analyses will be to use the results for hypothesis generation and to also examine regions with suggestive findings for genes that have been implicated in metabolic disorders and obesity.We identified distinct clusters of individuals with different manifestations of metabolic disorders. Genetic association analyses revealed several regions for further investigation.BMI: Body mass index; CVD: Cardiovascular disease; FHS: Framingham Heart Study; GAW: Genetic Analysis Workshop; GWA: Genome-wide association; HDL: High density lipoproteins; MCA: Multiple correspondence analysis; MS: Metabolic syndrome; PCA: Principal-components analysis; SNPs: Single-nucleotide polymorphismsThe authors declare that they have no competing interests.MW designed the study, conducted the phenotype analyses, and drafted the manuscript; QL carried out all of the genetic analyses and GWAS figure preparation; YS participated in discussions about the analytic methods; PS and JB participated in discussions about the phenotype definitions and facilitated acquisition of the data; DW participated in discussions about the analytic methods for the GWAS.


Family-based association tests are important tools for investigating genetic risk factors of complex diseases. These tests are especially valuable for being robust to population structure. We introduce a tool, EFBAT, which performs exact family-based tests of association for X-chromosome and autosomal biallelic markers.The program EFBAT extends a network algorithm previously applied to autosomal markers to include the X-chromosome and to perform tests of association under the null hypotheses "no association, no linkage" and "no association in the presence of linkage" under additive, dominant and recessive genetic models. These tests are valid regardless of patterns of missing familial data.The general framework for performing exact family-based association tests has been usefully extended to the X-chromosome, particularly for the hypothesis of "no association in the presence of linkage" and for different genetic models. Family-based association tests (FBATs) are widely used in studies of the genetic risk factors of complex human diseases. These tests avoid identifying spurious associations that may result from population structure. The transmission/disequilibrium test (TDT)  compareset al.[Schneiter et al. describeWe describe the software package EFBAT which incorporates the tests for autosomal markers, but which also extends this exact testing procedure to markers located on the X-chromosome. EFBAT implements exact FBATs for biallelic markers located on either the X or autosomal chromosomes under the additive, dominant, or recessive models of inheritance and remains valid for arbitrary patterns of missing data. An exact test of "no linkage and no association" for X-chromosome markers has been implemented by  for the The EFBAT software implements exact tests of the null hypotheses of "no linkage and no association" and of "linkage but no association" between the marker and disease. EFBAT can be run interactively via a menu , in which the distribution of offspring genotypes is identified either from parental genotypes or from sufficient statistics for the parental genotypes when one or both are unavailable. Their algorithm extends to X-chromosome markers. The hypothesis of "no linkage and no association", the conditional distribution of children's genotypes is given in Table AB, A), or the father is known and the mother can be inferred as AB.1. Genotypes switch if both parents are known, (AB, sons switch; daughters also switch if there are two genotypically distinct daughters.2. If the mother is known to be 3. If neither parent is known, daughters switch if there are two genotypically distinct daughters; sons switch if there are two genotypically distinct sons.S = ∑XT, where X is a function of an individual's genotype and T is a function of the individual's trait. The product of X and T is summed over all offspring in all families. For the exact test, we assume T is 1 for affecteds and otherwise 0 since allowing T to be continuous is straightforward in theory but computationally difficult.The statistic used to implement the exact test for both X-linked and autosomal markers is derived from the conditional distribution of offspring genotypes. It is given by S is a count of the allele of interest for that individual. Analyses can also be performed assuming dominant or recessive models, with sons treated as in the additive case and daughters coded as in autosomal markers. EFBAT assumes sons are coded as homozygous for each marker, although only the first allele is used.By default, EFBAT assumes additive inheritance, i.e. for each child, S is a count the allele of interest among all affected offspring in all families. Under dominant inheritance, X is a count of all genotypes that include at least one copy of the allele of interest among affected children. Assuming recessive inheritance, S is a count of the genotypes homozygous for the allele of interest among all affected children. Parental data are pertinent solely to the identification of the distributions of offspring genotypes and do not contribute to the value of the test statistic.Assuming additive inheritance, S is obtained by identifying the probability of each possible value of S. A p-value is calculated by summing the probabilities of S more extreme than the observed value. To identify all possible values of S explicitly is very time consuming for any but very small datasets as the number of possible values increases multiplicatively across families. The modified network algorithm implicitly identifies these values, resulting in rapid production of exact p-values. For a dataset of 300 families, EFBAT computes the exact p-value in less than one second.The exact distribution of The EFBAT software implements exact FBATs of the hypotheses "no linkage and no association" and "linkage but no association" for biallelic markers from either autosomal or X chromosomes. These procedures are valid under the additive, dominant, and recessive models of inheritance and for data consisting of families with or without available parental genotypes.• Project name: EFBAT• Project homepage: • Programming language: C++• License: Freely available• Any restrictions on use by non-academics: NoKS developed the software algorithm, helped revise the conditioning algorithm, implemented the software, and drafted the manuscript. JD helped develop and tested the software, developed the conditioning algorithm, and helped draft the manuscript. NL conceived of the software and participated in its design and implementation. CC helped develop the software algorithm. XX helped test and revise the software. All authors have read and approved the final manuscript.efbat.cpp. The file contains C++ source code for efbat program and can be compiled under Linux, Unix, or Windows XP.


P = 0.003), but not CRAE . Using admixture mapping, we did not detect significant admixture association with either CRAE (genome-wide score = −0.73) or CRVE (genome-wide score = −0.69). An a priori subgroup analysis among hypertensive individuals detected a genome-wide significant association of CRVE with greater African ancestry at chromosome 6p21.1 . Each additional copy of an African ancestral allele at the 6p21.1 peak was associated with an average increase in CRVE of 6.14 µm in the hypertensives, but had no significant effects in the non-hypertensives (P for heterogeneity <0.001). Further mapping in the 6p21.1 region may uncover novel genetic variants affecting retinal vascular caliber and further insights into the interaction between genetic effects of the microvascular system and hypertension.Retinal vascular caliber provides information about the structure and health of the microvascular system and is associated with cardiovascular and cerebrovascular diseases. Compared to European Americans, African Americans tend to have wider retinal arteriolar and venular caliber, even after controlling for cardiovascular risk factors. This has suggested the hypothesis that differences in genetic background may contribute to racial/ethnic differences in retinal vascular caliber. Using 1,365 ancestry-informative SNPs, we estimated the percentage of African ancestry (PAA) and conducted genome-wide admixture mapping scans in 1,737 African Americans from the Atherosclerosis Risk in Communities (ARIC) study. Central retinal artery equivalent (CRAE) and central retinal vein equivalent (CRVE) representing summary measures of retinal arteriolar and venular caliber, respectively, were measured from retinal photographs. PAA was significantly correlated with CRVE (ρ = 0.071,  Retinal vessels provide a window to microvascular systems elsewhere in the body. The diameter of retinal vessels varies between racial/ethnic groups, being generally wider in African Americans compared to European Americans. To determine whether genetic background may contribute to this observed difference, we scanned the entire genomes of 1,737 African Americans, searching for genomic regions where individuals with either wider retinal venular or narrower retinal arteriolar caliber have a difference from the average percentage of African ancestry. We find that the percentage of African ancestry is positively correlated with retinal venular caliber, particularly in the hypertensive individuals. We detect substantive evidence of association between excess African ancestry and wider retinal venular caliber on chromosome 6p21.1 in the hypertensives, but not in the non-hypertensives. The 6p21.1 region contains genes that are known to be involved in development and modulation and of retinal vessels. Our results suggest that genetic factors may contribute to the observed difference in retinal vascular caliber between African Americans and European Americans. Further fine-mapping studies of the genomic region may identify variants affecting retinal vascular caliber. Changes in retinal vascular caliber provide unique information regarding the structure and state of the microvasculature in the eye, possibly reflecting pathophysiological processes in the microvascular systems elsewhere in the body. Narrowed retinal arteriolar caliber has been known to be predictive of hypertension Retinal vascular caliber has been observed to vary between racial/ethnic groups. In the Multi-Ethnic Study of Atherosclerosis, African Americans and Hispanics had wider retinal arteriolar and venular caliber compared to Whites and Asian Americans, even after controlling for cardiovascular risk factors To identify chromosomal regions which may harbor genes that modulate retinal vascular caliber, we utilized admixture mapping, a technique that scans the genomes in recently admixed populations, such as African Americans, for regions which may contain variants that not only differ in frequencies between the two genetically diverse ancestral populations (Europeans and West Africans in this case), but can also partially explain differences in phenotypes between populations The main hypothesis of the present study was that some alleles affecting retinal vascular caliber are present at higher frequency in Africans than Europeans. We thus conducted a genome-wide admixture mapping scan for retinal arteriolar and venular caliber using approximately 1,365 ancestry informative markers in self-identified blacks from the ARIC study. In addition, hypertension is known as one of the most important risk factors for cardiovascular and cerebrovascular diseases. Previous studies indicated the importance of interactive effects between genes and vascular risk factors, in particular hypertension, on the occurrence of cardiovascular diseases Central retinal artery equivalent (CRAE) and central retinal vein equivalent (CRVE) were measured from retinal photographs of study subjects in the ARIC study to represent the average retinal arteriolar and venular caliber, respectively see . Table 1P = 0.003), while the correlation between PAA and CRAE was weaker and not statistically significant . To assess the potential interaction between genetic effects and hypertension, we examined the correlations limited to hypertensive individuals as an a priori subgroup analysis. We found that among hypertensive individuals, the correlations with PAA became stronger for both CRAE  and CRVE .The correlations between retinal vascular caliber and PAA are shown on We carried out genome-wide admixture mapping scans for both CRAE and CRVE using case-only and case-control approaches, using up to 1,365 ancestry-informative SNP markers. Cases (N = 261) and controls (N = 260) for CRAE were defined as the extreme 15% of samples with narrowest caliber and the 15% with widest caliber, respectively, after adjustment for age, sex, study site, 6-year mean blood pressure, and fasting glucose level. For CRVE, the extreme 15% of samples with widest caliber was defined as cases (N = 260), while the extreme 15% with narrowest caliber as controls (N = 261). The mean CRAE was 46.2 µm (±15.3) narrower in cases compared to controls. On the other hand, the mean CRVE was 51.0 µm (±16.1) wider in cases compared to controls. Using 18 pre-specified European ancestry risk models, we did not detect significant admixture association with either CRAE or CRVE . The gen5.47 = 295120.9. The likelihood ratio after corrected for two hypothesis testing was 295120.9/2 = 147560.5, corresponding to a LOD score of 5.17, which still exceeds the threshold of >5 for significance. No other locus exceeded a LOD score of 5.To examine whether hypertension modified the effects of genetic ancestry on retinal vascular caliber, we performed admixture mapping scans by hypertension status. Cases and controls were defined as the extreme 15% (after adjustment for the covariates as described above) in the hypertensive subset and included about 150 subjects in each group. On average, CRAE was 46.5 µm (±15.5) narrower and CRVE was 51.6 µm (±16.7) wider in the cases, compared to the controls. We found genome-wide significant evidence of associations with CRVE in the hypertensive subset . The genTo evaluate the stability of our results, we carried out a longer analysis with 10-times more iterations in our Markov Chain Monte Carlo run. We obtained a similar strength of signal with a genome-wide score of 2.28 and a peak locus-specific LOD of 5.43 at the same location. The risk model with the strongest score corresponded to a risk of 0.5 due to one copy of an European ancestral allele with the inverse risk for carrying zero copies  after conservatively correcting for multiple hypothesis testing .The admixture-generated signal for CRVE in the hypertensive subset was further supported by the case-control analysis. At the 6p21.1 peak, the cases had a highly statistically significant increase in African ancestry compared to controls . The highest locus-specific LOD was 2.37, arising from chromosome 5, followed by the second highest LOD of 2.04 on chromosome 6 . AdmixtuP = 2.9×10−8; Model 1). To assess whether continuous CRVE was associated with local ancestry at 6p21.1 due to their associations with global ancestry , we modeled the transformed CRVE as a function of local, global and regional ancestry. We found that the residual association of local ancestry with the transformed CRVE after adjustment for both global and regional ancestry remained significant (P = 3.9×10−6), indicating that there may be a gene in the region of 6p21.1 that is associated with CRVE above and beyond the fact that variants in this locus are highly differentiated between ancestral populations and thus correlated with global ancestry. The association was nominally genome-wide significant (P = 7.8×10−3) after correcting for multiple hypotheses tested . The results were similar when CRVE was additionally adjusted for other covariates (Model 2), including high-density lipoprotein (HDL) cholesterol, low-density lipoprotein (LDL) cholesterol and plasma triglyceride levels, body mass index (BMI), smoking, and alcohol consumption. We estimated that each additional copy of an African ancestral allele at the 6p21.1 peak was associated with a CRVE increase of 0.37 Z-score units on average .To further assess the robustness of the admixture-generated signal, we extracted the local estimate of African ancestry at the 6p21.1peak, and tested for association with continuous CRVE by hypertension status. This enabled us to increase power by including all samples in a quantitative analysis. We carried out a series of linear regression analysis, with the normal-quantile transformed CRVE (after adjusted for covariates as described above) as a dependent variable and the local ancestry as an independent variable. As shown in P<0.001) between the hypertensive and non-hypertensive groups, which was in line with the results of the above dichotomous admixture scans.In contrast, in the non-hypertensive subset, the local ancestry effect at the 6p21.1 peak on CRVE was weak and did not reach significance . The effGiven the significant statistical evidence for peak at 6p21.1, we constructed 95% credible interval (CI) for the loci identified. The 95% CI spanned from 40.8 to 43.9 Mb on build 35 of the human genome reference sequence . The locWe used admixture mapping to search for genomic regions that may account for inter-individual variations in retinal vascular caliber. We found evidence for association with retinal venular caliber at 6p21.1 in hypertensive African Americans and observed concordant results when venular caliber was examined as a continuous variable, with higher levels of African ancestry being significantly associated with wider retinal venular caliber. The significant evidence of association with the local ancestry at the 6p21.1 peak was above and beyond the contribution of both global and regional ancestry. Methodologically, these results are interesting in that subset analysis was required in order to detect the association. We note that subsets analysis has previously been very successful in admixture mapping. For prostate cancer, the chromosome 8q24 locus was not detected until the analysis was limited to a subset of individuals with a younger age at diagnosis Our findings in persons with hypertension in the subgroup analysis imply that genes associated with hypertension may have exerted their effects on retinal vascular caliber. While we cannot exclude the possibility of a chance finding, we were able to show consistent results in the local ancestry analysis, which included the total population. Furthermore, from a physiological perspective, our finding that there is a genetic association with retinal venular caliber specifically in people with hypertension is sensible. Hypertension is known to have profound effects on the retinal microcirculation In hypertensive persons, we detected significant genetic association for retinal venular caliber, but not arteriolar caliber. Because retinal arterioles and venules likely possess distinct genetic determinants P = 1.2×10−4) and 8q21 , respectively. A recent linkage analysis in the Australian Twins Eye Study identified 8p23.1 (LOD = 2.24) and 2p14 (LOD = 2.69) as suggestive loci for CRAE and CRVE, respectively To our knowledge, there have been only two prior studies examining the genetic basis of retinal vascular caliber with a genome-wide approach, and both of them used linkage analysis One major concern of the present study is a potential measurement error on retinal vascular caliber itself, which has been suggested to account for some of the observed racial/ethnic difference in retinal caliber VEGF) gene. VEGF is an endothelial-cell selective mitogen intimately associated with vasculogenesis, angiogenesis and vascular permeability. In the retina, VEGF plays a crucial role in the induction of retinal vasculature and its expression is regulated by hypoxia during embryonic development VEGF gene indeed contribute to variations in retinal vascular caliber. If proven, this may help explain why we detected the association at 6p21.1 only in hypertensive persons.The 95% CI for the 6p21.1 locus, a ∼3.1 Mb region, contains genes that may have biological relevance to the development and modulation and of retinal vessels . One sucVEGF, there are two other genes in the 95% CI that may potentially be associated with retinal phenotype: peripherin 2 (PRPH2/RDS) and guanylate cyclase activator 1A (GUCA1A), both of which are also expressed in retina VEGF appears to be the strongest candidate gene based on its known function. However, it does not exclude the possibility that the admixture-generated signal is due to other genes.In addition to The human leukocyte antigen (HLA) loci, located on chromosome 6p21.3 and about 10 Mb away from the 95% CI, are gene-rich and highly polymorphic In summary, using a genome-wide admixture mapping scan in 1,737 African Americans, we detected a risk locus influencing retinal vascular caliber in hypertensive individuals at 6p21.1, where the association between local ancestry and retinal venular caliber was strong, suggesting the presence of a genetic effect beyond the effects of global ancestry. Follow-up fine mapping or haplotype tagging across the peak will be necessary to determine whether this region harbors genetic variations that may interact with hypertension to modulate retinal venular caliber. Understanding the genetic basis of retinal vascular caliber may provide novel insight into the development and remodeling of the microvasculature in the brain and elsewhere in the body.This study was conducted according to the principles expressed in the Declaration of Helsinki. All sample collections were carried out according to institutionally approved protocols for study of human subjects and written informed consent was obtained from all subjects.Subjects of the present study were from the 2,997 African-American participants of the ARIC study at the third examination. The ARIC study is a prospective epidemiologic study that examines clinical and subclinical atherosclerotic disease in a cohort of 15,792 persons , aged 45 to 64 years at their baseline examination. Participants were selected by probability sampling from four U.S. communities: Forsyth County, NC (12% African American); Jackson, MS (100% African American); the northwest suburbs of Minneapolis, MN (<1% African American); and Washington County, MD (<1% African American). The sampling procedure and methods used in ARIC have been described in detail elsewhere 1) African-American subjects who lived in Minneapolis, MN, or Washington County, MD, or 2) did not consent to genetic studies or did not have DNA samples available, 3) samples that were not genotyped successfully or that failed to pass quality control (see “Elimination of problematic samples”), 4) subjects who had no retinal photographs, ungradable photographs or retinal vascular occlusions, and 5) subjects who had missing data on the covariates used in the main admixture mapping scans (see “Admixture mapping”).The final sample for the present analysis included 1,737 African Americans after excluding the following samples : The retinal photography procedure and its assessment have been described in detail elsewhere Trained graders who were masked to participant measured the calibers of all arterioles and venules coursing through a specified area surrounding the optic disc according to a standardized protocol Current blood pressure was defined as measurements at the time of retinal photography at visit 3, and 3- and 6- year past blood pressure was defined as measurement taken at visit 2 and visit 1, respectively. Mean arterial pressure was defined as two thirds of diastolic plus one third of systolic blood pressure. Mean arterial pressure across the three visits was averaged to obtain the 6-year mean arterial pressure. BMI was calculated as weight (in kg)/height (in meters) squared. Blood collection and processing followed a standard protocol We genotyped a total of 1,536 SNPs included in the phase 3 admixture panel . This panel was constructed by using the panel of ancestry informative markers previously published by Smith et al. (phase 1 panel) We used previously published genotyping data to estimate the frequency of each SNP in West Africans and Europeans 1) samples with low (<94%) call rate, 2) samples showing gender mismatch between self-reported data and genetically estimated gender based on 50 markers on the X chromosome, and 3) duplicate samples  We eliminated SNPs (N = 15) if they did not meet the requirement for Hardy-Weinberg equilibrium (P>0.01) in both ancestral West African and European populations. 2) We applied a “freqcheck” filter that examined whether the observed frequency of a SNP in African Americans was statistically consistent with being a mixture of the frequencies observed in the West Africans and European American samples that we used to represent the ancestral populations 3) Lastly, we applied a “ldcheck” filter that for each sample, iteratively eliminated SNPs that were less informative (in terms of the information content about ancestry) until none were within 200 Kbs of each other or in detectable LD with each other in the ancestral West African or European populations To decrease the likelihood of false-positives in our admixture scans, we applied a series of filters that had the goal of detecting and removing any SNPs with problematic genotyping, as described previously . First, SNPs were dropped if there were atypical clustering patterns, ill-defined clusters, or relatively low genotyping success rate (95%). This left us with 1,416 SNPs . We then applied three previously described filters to further eliminate SNPs from the analysis Using the ANCESTRYMAP software P<0.01) and known to be associated with retinal vascular calibers We used the ANCESTRYMAP software For the purpose of this dichotomous admixture analysis, study participants were ranked by the adjusted values for each trait. For CRAE, the 15% of individuals with the lowest values were defined as cases, and the 15% with the highest values as controls. For CRVE, conversely, the 15% with the highest values for were defined as cases, and the 15% with the lowest values as controls. Because ANCESTRYMAP uses Bayesian statistics, a prior distribution of risk models is required locus-specific score and a case-control scorelocus-specific score was obtained in cases  by calculating the likelihood of the genotyping data at the SNPs at the locus under the risk model and comparing it to the likelihood of the genotyping data at the SNPs at the locus assuming that the locus is unassociated with the phenotype. The ratio of these two likelihoods is the “likelihood ratio”, and the log-base-10 of this quantity is the “LOD” score. A locus-specific LOD score of >5 has been recommended as criterion for genome-wide significance ANCESTRYMAP provided two scores to assess statistical significance: a case-control score was calculated by comparing locus-specific deviations in European ancestry in cases versus controls at each locus across the genome. This score tests whether any deviation in ancestry from the genome-wide average is significantly different comparing cases with controls P<5×10−5, or a corrected nominal P<0.05 after conservatively correcting for 1,000 hypotheses tested .A The main admixture scans were based on a dichotomous phenotype  in a subset of our samples. To check whether the results were consistent in our entire sample, we used the ANCESTRYMAP software to obtain local estimates of African ancestry at the admixture peak To determine a 95% CI for the position of a trait locus, we obtained the likelihood ratio for association at each marker on the chromosome where we identified an association. This provided a Bayesian posterior probability for the position of the underlying causal variant assuming a flat prior distribution across the region for the position of the trait locus. We defined the CI as the central region of this peak containing 95% of the area.Figure S1Genome-wide admixture mapping scans for loci affecting retinal arteriolar caliber in hypertensive individuals.(0.19 MB PDF)Click here for additional data file.Figure S2Posterior probability distribution for the position of the locus of retinal venular caliber.(0.37 MB PDF)Click here for additional data file.Table S1Summary of the initial admixture scan results by chromosome.(0.06 MB DOC)Click here for additional data file.Table S2Admixture scan results and association of the CRVE to local African ancestry at regions near 6p21.1 in the hypertensive African Americans.(0.05 MB DOC)Click here for additional data file.


HLA-DRB1 alleles (NIMA) have been implicated as a rheumatoid arthritis (RA) risk factor. Using genotype data from North American Rheumatoid Arthritis Consortium study participants and the maternal-fetal genotype incompatibility (MFG) test, we find evidence for offspring allelic effects but no evidence for NIMA as a RA risk factor. We discuss possible reasons why our result conflicts with several previous studies (including one of our own) that used RA patients from northern Europe.Non-inherited maternal antigens encoded by specific  HLA-DRB1 *0101, *0102, *0401, *0404, *0405, *0408, *0423, *10, *1001, *1402, and *1406 alleles that encode a shared epitope (SE)  is the indicator function, M is the event that the offspring does not inherit an SE coding allele carried by the mother , and Z denotes the number of SE coding alleles present in the offspring genotype. The parameter p denotes the population baseline disease incidence rate, which ultimately cancels from Eq. (1). The parameter μ is the relative risk due to NIMA; ρ1 and ρ2 are the relative risks when one or two copies of the SE coding allele are present in the offspring genotype, respectively, relative to zero copies. Note that the relative risks  range from 0 to infinity, and each have a null value of 1. The numerical maximization of the log-likelihood is better conditioned when we use the natural logarithm of the relative risks as parameters so we actually estimate α = log(μ), β1 = log(ρ1), and β2 = log(ρ2) [α, β1, and β2, are therefore defined on the real line from negative infinity to infinity and each have a null value of zero.where  log(ρ2) . Each ofS for the SE coding risk allele and N for the non SE coding allele. Thus, there are three possible genotypes . The penetrance function as expressed by Eq. (2) is short-hand for three mutually exclusive maternal-offspring genotype combinations in which the indicator functions determine which relative risk corresponds to the genotype combination. These mutually exclusive combinations are: the offspring carries two copies of the S allele; the offspring carries one copy of the S allele; and the offspring does not carry S allele but the mother does. Each of these combinations has an associated risk that we define in a standard manner, that is, as relative to a reference category in which both the offspring and mother are N/N.Because we assume that all SE coding alleles confer the same risk to RA susceptibility (as was assumed in earlier studies -8), the Assuming mating symmetry under the null hypothesis, there are six possible mating types . NeitherHLA-DRB1 genotypes must be available to test for NIMA effects. The ethnic background of the NARAC families is representative of United States and Canadian families affected with RA .Of the four GAW15 RA study samples, only the NARAC data set was suitable for this analysis because at least some parental S and N, we used the pedigree trimming option of Mendel (version 6.5.0) to remove individuals without HLA-DRB1 genotypes who are not necessary to define the relationships among genotyped individuals [HLA-DRB1 error rate was less than 1%. Individuals' genotypes that were in error were omitted and when several family members could be in error, the entire family's HLA-DRB1 genotypes were omitted. Most of the families are nuclear families; however there are a few extended families. For each extended pedigree, a single nuclear family was selected using the following protocol: whenever a nuclear family has complete parental genotypes available, they are selected over those families with zero or one parent genotyped. Whenever multiple nuclear families within the extended pedigree have the same parental genotype availability, one of the nuclear families is randomly selected. At this point, there were 708 nuclear families.Before recoding the alleles into ividuals ,13. We tividuals  and founHLA-DRB1 genotypes is too low to allow certain classification of their alleles as S or N. For example the allele *01 could be *0101 (S) or *0103 (N). These ambiguous genotypes were treated as missing in the analysis.About half the families have neither parent genotyped. These families provide very little power for detecting maternal-fetal genotype interactions  and werep = 0.946).The final data set had 263 nuclear families with at least one parent and one affected offspring genotyped. The distribution of affected offspring and genotyped parents is given in Table α = β1 = β2 = 0) and 2 (α = 0) in Table p = 1.5 × 10-28). The relative risk for individuals with S/N is ρ1 = 4.3 and the relative risk for individuals with S/S is ρ2 = 12.7, similar in magnitude to the results of earlier association studies (see for example [α = 0) and 3 (no parameter restrictions) shows there is no evidence for the NIMA effect on RA risk in the presence of offspring allelic effects (p = 0.8956).We fit three models by placing different constraints on the parameters. These models are compared by constructing likelihood-ratio test statistics. Comparison of Models 1 . ComOne possible explanation for our failure to find evidence for the NIMA effect is that our sample size is too small. The earlier studies found the MFG effect to be ~4, approximately equal to the S/N risk to offspring -5. We siMFG incompatibility results when specific maternal-fetal genotype combinations produce an adverse effect on the developing fetus that ultimately increases offspring disease risk. The exposure of fetuses who do not carry SE coding alleles to a maternal SE antigen is an example of a putative MFG incompatibility event and so can be examined using the MFG test.The highly significant offspring allelic effects are consistent with numerous studies ,15,16. HHLA-DRB1 genotypes, only 263 nuclear families were used in our analysis. However when the genotypes are missing at random, it has been shown that MFG test produces unbiased parameter estimates and accurate hypothesis tests when only approximately 25% of the families in the sample have both parental genotypes [There were originally 708 nuclear families available in the NARAC data set. After excluding families without parental genotypes and individuals with low-resolution enotypes . The misHLA loci could have been misattributed to a HLA-DRB1 NIMA effect (an issue of model misspecification); b) HLA-DRB1 is difficult to accurately genotype and mistyping could lead to both false-positive and negative NIMA results; c) there could be population differences in the susceptibility to microchimera; d) differences in ascertainment or diagnostic criteria used in the different studies might also contribute to the variability in the results. The NARAC families are multi-case families and these families may have a different genetic background than single-case families [The following reasons might account for the failure to find a significant MFG incompatibility effect: a) the previous studies with positive findings -5 could HLA-DRB1 NIMA effect is a risk factor for RA. Although we can not rule out NIMA as a RA risk factor of small effect in the NARAC families, we find no evidence for NIMA as a RA risk factor of moderate effect.The MFG test ,9-11 canThe author(s) declare that they have no competing interests.


The Genetic Analysis Workshop (GAW) 16 Problem 3 comprises simulated phenotypes emulating the lipid domain and its contribution to cardiovascular disease risk. For each replication there were 6,476 subjects in families from the Framingham Heart Study (FHS), with their actual genotypes for Affymetrix 550 k single-nucleotide polymorphisms (SNPs) and simulated phenotypes. Phenotypes are simulated at three visits, 10 years apart. There are up to 6 "major" genes influencing variation in high- and low-density lipoprotein cholesterol , and triglycerides (TG), and 1,000 "polygenes" simulated for each trait. Some polygenes have pleiotropic effects. The locus-specific heritabilities of the major genes range from 0.1 to 1.0%, under additive, dominant, or overdominant modes of inheritance. The locus-specific effects of the polygenes ranged from 0.002 to 0.15%, with effect sizes selected from negative exponential distributions. All polygenes act independently and have additive effects. Individuals in the LDL upper tail were designated medicated. Subjects medicated increased across visits at 2%, 5%, and 15%. Coronary artery calcification (CAC) was simulated using age, lipid levels, and CAC-specific polymorphisms. The risk of myocardial infarction before each visit was determined by CAC and its interactions with smoking and two genetic loci. Smoking was simulated to be commensurate with rates reported by the Centers for Disease Control. Two hundred replications were simulated. The Framingham Heart Study (FHS) is a rich platform for the study of cardiovascular disease and the application of novel, imaginative analytic strategies. For Genetic Analysis Workshop (GAW) 16, we use a semi-simulated approach using actual genotypes from the 500 k Affymetrix platform and the 50 k candidate gene chip and building phenotypes on the observed genetic variation. Because blood lipid levels are a major risk factor in the development of cardiovascular disease . Several® Human Mapping 500 k Array Set (Nsp and Sty), and the 50 k Human Gene Focused Panel) comprised the genotypes for GAW16 Problem 3. Novel fictitious phenotypes were simulated for subjects.The FHS pedigrees, distributed as GAW16 Problem 2, formed the basis of our simulation . In totaAlthough family members of the FHS attended various exams at different times, depending on the generation, we modeled our study as if all subjects were recruited at one time, calculated the family member's relative ages at one particular exam, and then assigned a simulated age for everyone at three time points, with 10-year intervals. The mean age in years (range) for the simulation, by generation and visit, is shown in Table β4), for which the minor allele frequency was 1%. We simulated an overdominant effect (γ1) because there appears to be evidence supporting this possibility and this mode of inheritance is rarely, if ever, modeled. The gene α4 is pleiotropic for HDL and TG and interacts with β5 in determining LDL  represents the mean of HDL in FHS, given a 5-year age interval and sex;  is standard deviation of HDL in FHS, given a 5-year age interval and sex; hα1 is the square root of simulated heritability for the α1 SNP  or (+1) with the purpose of randomly changing the contribution direction of polygenes; apolyκ represents an instance of each of the 1,000 SNPs effects , selected as polygenes for HDL; hapolyκ is an instance of the of square roots of heritabilities for 1,000 SNPs selected as polygenes for HDL; aε represents the environmental effect that contributes to HDL; and hε is the square root of HDL variance explained by environmental causes.where, for example, per se. Instead, we simulated the complicating effects of medication. The simulated value for LDL at each visit for each subject was checked, and individuals in the upper tail of the distribution were simulated as medicated. The proportion of subjects that are medicated increased across visits to comprise 2%, 5%, and 15% of the subjects in Visits 1, 2, and 3, respectively. These proportions were estimated from the FHS data, and reflected the secular increase in the proportion of individuals being treated for elevated cholesterol levels. The response to treatment is governed by two loci (δ1 and δ2) as pharmacogenetic processes. The δ1 variant has a marginal effect on both HDL and TG levels via additive effects but also, individuals that are homozygous for the minor allele are non-responders to the treatment. Responders  exhibit a 10% increase in their HDL levels and a 15% decrease in TG levels. Similarly, δ2 is a variant with an additive marginal effect on LDL, and homozygotes for the minor allele are non-responders to treatment. Responders exhibit a 30% decrease in LDL levels. Total cholesterol (CHOL) level is calculated as 0.8*(HDL + LDL + TG/5), and has no independent genetic effects except those influencing the component phenotypes.As individuals progressed to the next visit 10 years later, their phenotypes were scaled by the appropriate age-sex means and variance, but there are no genes governing longitudinal trends AI) was modeled as a function of total CHOL, HDL, and five other genes (τ1-τ5) having direct effects on its development. CACAI was simulated under the modelCoronary artery calcification (CAC) was simulated as a quantitative phenotype that takes many years to develop. For this reason, CAC was modeled in two stages. First, age-independent CAC  while τ2 displays an additional measurable additive main effect; PE is the joint effect from τ3 and τ4, a pair of purely epistatic SNPs, each with no main effect; Het is an effect from τ5, a SNP that displays heterosis (over-dominance); and ε is the residual variation not explained by the factors mentioned above. The term ε, 300 times a random draw from a normal distribution with mean 0 and variance 1 ), represents the sum of normal deviations from the mean of each of the modeled genetic effects and "noise" from unmeasured environmental and genetic effects. Because CAC cannot be negative, CACAI = 0 if the generated value was negative. The models for the effects on CACAI due to the ME and PE genotypes are illustrated in Tables τ1-τ4 is ~0.5. SNP τ5, which determines the Het effect, has a MAF of 0.2. SNP τ5 genotype 1/1 (common homozygote) increases CACAI on average by 25, genotype 1/3 decreases CACAI by 100, and genotype 3/3 increases CACAI by 400. CAC is derived from CACAI by using a piecewise linear age adjustment: subjects under 20 years have not developed measurable levels of CAC, CAC buildup is linear from the ages of 20 to 60, and for subjects older than 60, CAC = CACAI. Table where φ1 and φ2. No MIs were fatal in our data. Smoking and φ1 have an interactive effect on risk of MI. The effect of smoking is to constrict blood vessels, thus increasing the risk that CAC will lead to an MI. The risk of MI for a smoker with the most common φ1 genotype (3/3) is the same as that of an equivalent non-smoker whose CAC is 10% higher. The risk of MI for a smoker with either of the other φ1 genotypes is the same as that of a non-smoker whose CAC is 40% higher. The φ1 genotype has no effect on risk of MI in non-smokers. Carrying the most common φ2 genotype (3/3) has the same effect on risk of MI as reducing CAC by 5%. The effect of any other genotype is the same as increasing CAC by 5%. The final model for MI risk isWhether a subject smoked during the period before a visit influenced the risk of a myocardial infarction (MI). At first visit, men had a 27% chance to be smokers and women had a 23% chance. Each smoker had an 8% chance of permanently quitting smoking before each subsequent visit. The resulting smoking rates are commensurate with rates reported by the Centers for Disease Control for 1998. The risk of an MI before each visit is determined by CAC and its interactions with smoking and two genetic loci, smoke is the joint effect of smoking and φ1 ; and the value of the event variable is -0.05 if the φ2 genotype is 3/3 and 0.05 otherwise. The MAFs for φ1 and φ2 are ~0.3. MI_risk was calculated for each visit and a draw from a uniform distribution determined whether the risk resulted in an MI. The SNPs for CAC and MI event were chosen from the 50 k SNPs in the Gene Focused Panel based on desired MAF, completeness of genotyping, and lack of linkage disequilibrium between the SNPs. The specific identities of the SNPs τ1-τ5, φ1 and φ2, and their chromosomes are listed in Table where ∂The phenotypic simulated files are named simphen#.txt, where # stands for a number from 1-200, representing the replication number. The simulated data are archived in the dbGAP of the National Center for Biotechnology Information under the name "GAW16 Framingham and Simulated Data" . The 200We tested all the simulated traits and causative SNP heritabilities as well as the respective association models. Analyzing and interpreting data obtained as part of a genome-wide association study presents numerous challenges, as well as the promise of improved understanding of the genetic factors influencing complex traits. For validation and a detailed analysis of the simulated model see the Online Supplemental Materials for GAW16 . Many geCAC: Carotid arterial calcification; CHOL: Cholesterol; FHS: Framingham Heart Study; GAW: Genetic Analysis Workshop; HDL: High-density lipoprotein; LDL: Low-density lipoprotein; MAF: Minor allele frequency; MI: Myocardial infarction; SNP: Single-nucleotide polymorphism; TG: Triglyceride.The authors declare that they have no competing interests.All the authors contributed equally.Heritability targets and other characteristics for each polygene affecting HDL.Click here for fileHeritability targets and other characteristics for each polygene affecting LDL.Click here for fileHeritability targets and other characteristics for each polygene affecting TG.Click here for file


Purely epistatic multi-locus interactions cannot generally be detected via single-locus analysis in case-control studies of complex diseases. Recently, many two-locus and multi-locus analysis techniques have been shown to be promising for the epistasis detection. However, exhaustive multi-locus analysis requires prohibitively large computational efforts when problems involve large-scale or genome-wide data. Furthermore, there is no explicit proof that a combination of multiple two-locus analyses can lead to the correct identification of multi-locus interactions.p-value determination and a progressive search for the best ensemble. 2LOmb is benchmarked against an exhaustive two-locus analysis technique, a set association approach, a correlation-based feature selection (CFS) technique and a tuned ReliefF (TuRF) technique. The simulation results indicate that 2LOmb produces a low false-positive error. Moreover, 2LOmb has the best performance in terms of an ability to identify all causative single nucleotide polymorphisms (SNPs) and a low number of output SNPs in purely epistatic two-, three- and four-locus interaction problems. The interaction models constructed from the 2LOmb outputs via a multifactor dimensionality reduction (MDR) method are also included for the confirmation of epistasis detection. 2LOmb is subsequently applied to a type 2 diabetes mellitus (T2D) data set, which is obtained as a part of the UK genome-wide genetic epidemiology study by the Wellcome Trust Case Control Consortium (WTCCC). After primarily screening for SNPs that locate within or near 372 candidate genes and exhibit no marginal single-locus effects, the T2D data set is reduced to 7,065 SNPs from 370 genes. The 2LOmb search in the reduced T2D data reveals that four intronic SNPs in PGM1 (phosphoglucomutase 1), two intronic SNPs in LMX1A , two intronic SNPs in PARK2  2, parkin) and three intronic SNPs in GYS2 (glycogen synthase 2 (liver)) are associated with the disease. The 2LOmb result suggests that there is no interaction between each pair of the identified genes that can be described by purely epistatic two-locus interaction models. Moreover, there are no interactions between these four genes that can be described by purely epistatic multi-locus interaction models with marginal two-locus effects. The findings provide an alternative explanation for the aetiology of T2D in a UK population.The proposed 2LOmb algorithm performs an omnibus permutation test on ensembles of two-locus analyses. The algorithm consists of four main steps: two-locus analysis, a permutation test, global An omnibus permutation test on ensembles of two-locus analyses can detect purely epistatic multi-locus interactions with marginal two-locus effects. The study also reveals that SNPs from large-scale or genome-wide case-control data which are discarded after single-locus analysis detects no association can still be useful for genetic epidemiology studies. Complex diseases cannot generally be explained by Mendelian inheritance  because χ2 and odds ratio tests  is defined as the disease penetrance of the two-locus genotype ij that consists of genotype i at locus A and genotype j at locus B. The marginal penetrances MAi for genotype i at locus A and MBj for genotype j at locus B are given byThe pure epistasis model of interest is proposed by Culverhouse et al. . The modandpA and pB are the major allele frequencies. Equations 4 and 5 are usually represented by a penetrance table as illustrated in Table where K is the disease prevalence. Obviously, many combinations of penetrance fij satisfy the condition given in equation 6. Culverhouse et al. [h2) of the two-locus interaction model is defined bywhere e et al.  suggest VT = K(1 - K) is the total variance of the dichotomous phenotypes in the population and VI is the epistatic variation attributable to the genotypes. VI is defined bywhere fij that also maximises the heritability or other variance-based objectives can be treated as a constraint optimisation problem. Many algorithms including a double description method [The search for feasible penetrance n method  and a gen method  have beepA = pB = 0.5 with the penetrances in Table Culverhouse et al.  have ideVI must also be expanded to cover additional genotypes while the expression for VT remains unchanged. With the necessary model expansion, the maximum heritability of a three-locus interaction model for pA = pB = pC = 0.5 with the penetrances in Table When a two-locus interaction model is expanded into a multi-locus interaction model, the marginal penetrance equality constraint must be extended to cover all loci. Furthermore, the expression for pA = pB = pC = pD = 0.5 with the penetrances in Table Similarly, the maximum heritability of a four-locus interaction model for Additional details about the maximum heritability and the corresponding two-locus and multi-locus penetrance tables for other values of disease prevalence can be found in Culverhouse et al. . In thisdudek@chgr.mc.vanderbilt.edu.genomeSIM is a simulation package for generating case-control samples in large-scale and genome-wide association studies . The pacnd) is set to the number of SNPs in the control samples that deviate from HWE. Two test statistics are subsequently calculated for each SNP: an allelic association statistic and a statistic for the deviation from HWE of each SNP in the case samples. The allelic association statistic is a χ2 statistic which is calculated from the contingency table of alleles or genotypes with disease status. On the other hand, a χ2 statistic for the deviation from HWE of each SNP in the case samples indicates the level of association. A large deviation from the equilibrium usually signifies strong association between a SNP and the disease. However, an excessively large deviation may be the result of genotyping errors. nd SNPs with largest test statistics for the deviation from HWE are hence excluded from the consideration.A set association approach (SAA) is an association detection technique based on an omnibus permutation test on sets of candidate SNPs . The tesS statistic for each remaining SNP. SNPs are then ranked according to their S statistics. A preset number of SNPs with highest ranks are considered for association. The first candidate SNP set contains only the SNP with the highest rank (the highest S statistic). The p-value for this first set is determined from a permutation simulation where the case and control labels are randomly permuted while the numbers of case and control samples remain unchanged. In each permutation replicate, a new genotype contingency table is constructed and a new S statistic is subsequently obtained. The p-value is given by the fraction of permutation replicates with an S statistic greater than or equal to the S statistic from the original data. The second candidate SNP set consists of the first two SNPs in the rank list. The test statistic for this SNP set is the sum of S statistics from both SNPs. The p-value for the second candidate SNP set is also obtained through the permutation simulation. By progressively adding the remaining SNP with the highest rank to the previously considered candidate set and performing the permutation simulation, p-values for all candidate SNP sets are estimated. The sizes of candidate SNP sets have the range of one to the preset number. Among all candidate sets, the SNP set that best describes genetic association has the lowest p-value.The test statistics for the allelic association and deviation from HWE are multiplied together to form a single p-value for the selected candidate set must be evaluated. This is achieved through a permutation simulation in which the current raw p-value for the chosen candidate set is now used as the test statistic. The existing permutation replicates, created for the early estimation of the raw p-value, can be reused and a nested permutation simulation is hence avoided. In this study, the maximum allowable size of the candidate SNP set is the total number of available SNPs while the number of permutation replicates for p-value evaluation is set to 10,000. The allelic association statistic employed in the study is the χ2 statistic that is obtained through the contingency table of genotypes with disease status. A PASCAL program for the set association approach can be obtained from the website for S statistic in gene mapping [Since multiple hypotheses are postulated during the construction of candidate SNP sets, the global  mapping .A correlation-based feature selection (CFS) technique  is an atMeritF is the heuristic merit of an nc-attribute subset F, cf is the average feature-class correlation and ff is the average feature-feature inter-correlation. An attribute subset receives a high merit score if it contains features that are highly correlated with the class and at the same time have low inter-correlation among one another. An application of a best-first search for the best subset identification is carried out to avoid searching through all possible attribute subsets. CFS has been integrated into a Weka package [where  package ,86.nk nearest neighbours from the same class and another nk nearest neighbours from the opposite class. The attribute values--the genotypes in this application--of the neighbour samples are compared to that of the randomly picked sample and are subsequently used to update the relevance score for each attribute (genetic marker). This process is repeated for a specified number of samples, which is limited by the total sample size. The rationale of ReliefF is that an attribute which is important for the classification should have different values for samples from different classes and have the same value for samples from the same class. The relevance score of an attribute have a range from -1 (not relevant) to +1 (highly relevant). TuRF exploits the capability of ReliefF by repeatedly executing ReliefF and removing a portion of worst attributes at the end of each execution. This leads to the reevaluation of remaining attributes and, hence, reduces the effects of attribute noise on the attribute screening. In this study, the number of repetitions for random sample picking in the ReliefF part is equal to the total number of case-control samples while the neighbourhood size (nk) for the relevance score calculation is set to ten. Furthermore, the worst 1% of SNPs is removed at the end of each ReliefF iteration (TuRF 1%). TuRF has been integrated into the current distribution of multifactor dimensionality reduction (MDR) software.A tuned ReliefF (TuRF) algorithm is a ranking algorithm for identifying genetic markers which are important in case-control classification . TuRF isnf -fold cross-validation technique provides a means to determine the prediction accuracy of the candidate marker model. Basically, the combined case and control samples are randomly divided into nf folds where nf - 1 folds of samples are used to construct a decision table while the remaining fold of samples is used to identify the prediction capability of the constructed decision table. The decision table construction and testing procedure is repeated nf times. Hence, the samples in each fold are always used both to construct and to test the decision table. The number of cells in a decision table is given by nc is the number of candidate markers selected from possible markers and G is the number of possible genotypes according to the marker. For a SNP, which is a bi-allelic marker, G is equal to three. During the decision table construction, each cell in the table is filled with case and control samples that have their genotype corresponds to the cell label. The ratio between numbers of case and control samples provides the decision for each cell whether the corresponding genotype is a protective or disease-predisposing genotype. An example of decision table construction is illustrated in Figure A multifactor dimensionality reduction (MDR) method is a wrapper-based technique that is capable of identifying the best genetic marker combination among possible markers for the separation between case and control samples . Similarnm is the total number of available markers in the study. The best genetic marker combination is determined from two criteria: prediction accuracy and cross-validation consistency. Each time that a testing fold is used for the prediction accuracy determination, the accuracy of the interesting marker combination model is compared with that from other models that also contain the same number of markers. The model that consistently ranks the first in comparison to other choices with the same number of markers has high cross-validation consistency. Prediction accuracy is the main criterion for decision making while cross-validation consistency is only used as an auxiliary measure. Cross-validation consistency generally confirms that the high rank model can consistently be identified regardless of how the samples are divided for cross-validation. In a situation where two or more models with different number of markers are equally good in terms of prediction accuracy and cross-validation consistency, the most parsimonious model--the combination with the least number of markers--is chosen as the best model.The prediction accuracy of the decision table is subsequently evaluated by counting the numbers of case and control samples in the testing fold that their disease status can correctly be identified using the constructed decision rules. The process of decision table construction and evaluation must be cycled through all or some of possible p-value is denoted by the fraction of permutation replicates with the prediction accuracy greater than or equal to the prediction accuracy obtained from the original data. MDR software, which incorporates many additional features including interaction visualisation via dendrograms and genetic marker screening via a χ2 test, an odds ratio test, ReliefF and TuRF, is available from its homepage [After the best model has been selected, a permutation test is used to assess the probability of obtaining prediction accuracy that is at least as large as or larger than that observed in the original data from randomised data. This represents the probability that the null hypothesis of no association is true. Each permutation replicate is constructed by randomly assigning the case/control status to each sample with the numbers of case and control samples remaining fixed. MDR analysis is subsequently carried out to obtain the prediction accuracy of each permutation replicate. The empirical homepage .D' [r2 [JLIN or a Java LINkage disequilibrium plotter is a computer program for visualisation of linkage disequilibrium analysis . The proD'  and r2 [D' [r2 . The proD' [r2 .An interaction dendrogram is a graphical tool for the visualisation of relationships among attributes (SNPs) ,69. The .The 2LOmb program for Windows platforms and examples of simulated data are available at DIO2: deiodinase, iodothyronine, type II; E2LA: exhaustive two-locus analysis; EGFR: epidermal growth factor receptor  oncogene homolog, avian); FAMHAP: software for single-marker analysis and joint analysis of unphased genotype data from tightly linked markers ; FUSION: Finland-United States Investigation of NIDDM Genetics; genomeSIM: simulation package for generating case-control samples in large-scale and genome-wide association studies; GYS2: glycogen synthase 2 (liver); HNF4A: hepatocyte nuclear factor 4, alpha; HuGENet: Human Genome Epidemiology Network; HWE: Hardy-Weinberg equilibrium; JLIN: Java LINkage disequilibrium plotter; KCNJ11: potassium inwardly-rectifying channel, subfamily J, member 11; LD: linkage disequilibrium; LIM domains: protein structural domains that are named after their initial discovery in the proteins Lin11, Isl-1 and Mec-3; LMX1A: LIM homeobox transcription factor 1, alpha; MDR: multifactor dimensionality reduction; NIDDM: noninsulin-dependent diabetes mellitus; PARK2: Parkinson disease  2, parkin; PGM1: phosphoglucomutase 1; PPARG: peroxisome proliferator-activated receptor gamma; RXRG: retinoid X receptor, gamma; SAA: set association approach; SNP: single nucleotide polymorphism; T2D: type 2 diabetes mellitus; TuRF: tuned ReliefF; UCP2: uncoupling protein 2 ; Weka: Waikato environment for knowledge analysis; WTCCC: Wellcome Trust Case Control Consortium.2LOmb: omnibus permutation test on ensembles of two-locus analyses; ALT: alanine transaminase; ANOVA: analysis of variance; AST: aspartate transaminase; CFS: correlation-based feature selection; CI: confidence interval; CVC: cross-validation consistency; WW conducted the literature survey, formulated the research question, implemented the proposed algorithm, designed the experiment, and collected and interpreted the computational results. AA conducted the literature survey, formulated the research question, designed the experiment and secured the access to the genomeSIM package. TP performed the statistical analysis and interpreted the statistical results. SS monitored and oversaw the execution of computer programs on the Beowulf cluster. CL provided additional comments about the genetic association study of T2D. NC conducted the literature survey, formulated the research question, designed the proposed algorithm, designed the experiment, secured the access to the T2D data from WTCCC, selected the candidate genes for the T2D association study, discussed all results, drew the conclusions and wrote the manuscript. All authors read and approved the final manuscript.WW is a Ph.D. student at the Department of Electrical Engineering, Faculty of Engineering, King Mongkut's University of Technology North Bangkok. He also received his B.Eng. and M.Eng. degrees in electrical engineering from King Mongkut's University of Technology North Bangkok. His current research interests include machine learning, evolutionary computation and bioinformatics.AA is a Ph.D. student at the Department of Immunology, Faculty of Medicine Siriraj Hospital, Mahidol University. He also received his B.Sc. degree in pharmacy from Mahidol University. His current research interests include human genetics, genetic epidemiology, population genetics and bioinformatics.TP is a Ph.D. student at the Department of Electrical Engineering, Faculty of Engineering, King Mongkut's University of Technology North Bangkok. He also received his B.Eng. and M.Eng. degrees in production engineering from King Mongkut's University of Technology North Bangkok. His current research interests include evolutionary multi-objective optimisation and machine learning.SS is a part-time research assistant at the Department of Electrical Engineering, Faculty of Engineering, King Mongkut's University of Technology North Bangkok. He received his B.Eng. and M.Eng. degrees in electrical engineering from Thammasat University and King Mongkut's University of Technology North Bangkok, respectively. His current research interests include machine learning and genetic epidemiology.CL is the Head of Division of Molecular Genetics at the Department of Research and Development, Faculty of Medicine Siriraj Hospital, Mahidol University. He also received his M.D. degree from Mahidol University. His current research interests include human genetics and genetic diseases.NC is an associate professor of electrical engineering at King Mongkut's University of Technology North Bangkok and an adjunct professor of genetic epidemiology at Mahidol University. He received his B.Eng. and Ph.D. degrees from the Department of Automatic Control and Systems Engineering, University of Sheffield. His current research interests include evolutionary computation, machine learning and genetic epidemiology.List of SNPs for the association study of T2D. This Excel spreadsheet file contains the information about 7,065 SNPs which are explored during the genetic association study of T2D. Bonferroni-corrected and uncorrected χ2's p-values from single-locus analyses are also provided.Click here for filegenomeSIM parameters. This text file contains an example of parameter setting in the genomeSIM simulation package.Click here for file