Phenotypic variation, including that which underlies health and disease in humans, results in part from multiple interactions among both genetic variation and environmental factors. While diseases or phenotypes caused by single gene variants can be identified by established association methods and family-based approaches, complex phenotypic traits resulting from multi-gene interactions remain very difficult to characterize. Here we describe a new method based on information theory, and demonstrate how it improves on previous approaches to identifying genetic interactions, including both synthetic and modifier kinds of interactions. We apply our measure, called interaction distance, to previously analyzed data sets of yeast sporulation efficiency, lipid related mouse data and several human disease models to characterize the method. We show how the interaction distance can reveal novel gene interaction candidates in experimental and simulated data sets, and outperforms other measures in several circumstances. The method also allows us to optimize case/control sample composition for clinical studies. The primary focus of GWAS is the reliable extraction of relevant genetic markers such as SNPs and indels that are associated with a complex phenotype. Numerous studies of gene regulatory networks, protein interaction networks, and other biological networks have made it clear, however, that genetic interactions are widespread and therefore important for full genetic analysis The rapid progress of sequencing technology, in both accuracy and cost, has enabled comprehensive Genome-Wide Association Studies (GWAS) which have identified many genetic contributions to complex phenotypes in humans  with increasing biological and computational complexity and evaluate the results using statistical permutation tests.interactions between QTLs with marginal effects, modifier interactions, and synthetic interactions. The first type spans only loci with individual effects on the phenotype and are the easiest to detect. Modifier interactions, on the other hand, link genetic markers, some of which exhibit marginal effects and some are markers with no effect by themselves, called modifiers of the significant loci (QTLs). The most difficult kind of interaction to detect is synthetic, which links genetic markers that have no marginal effect on the phenotype when present alone, but have an effect when present together. Formally, this corresponds to having no pairwise dependence between either gene variable, genetic interaction as a general short term referring to the interaction among genetic variants. Note that while we fully recognize the differences among the terms locus, QTL, modifier locus and gene, we use the term “genetic interaction” in the interest of brevity where the meaning is sufficiently clear.In human genetics a typical dataset consists of a large heterogeneous population characterized genetically by a series of polymorphisms or genetic markers, and phenotypically by a set of specific trait variations. To decipher the relationship between genetic markers and to construct a gene regulatory network we have to detect the dependency of the phenotypes on multiple variants in the population. Formally, detecting a genetic interaction corresponds to detecting a statistical dependence of Synthetic and modifier interactions are the focus of this work. In order to identify these interactions we have devised the Interaction Distance (ID) measure that extends the concept of interaction information (II) first proposed by McGill We define interaction information for three variables, The conditional entropy, In contrast to the mutual information, which is a non-negative quantity, the interaction information can take both positive and negative values. With our sign convention positive values of the interaction information indicate presence of an informational “synergy”, or collective dependence among The interaction information can then, in principle, indicate synergy among variables. However, this measure is affected significantly when the entropy of the single variables is extreme. This situation arises in genetics, for example, when the minor allele frequency is particularly low, making the entropy also low. To account for this situation we adjust the interaction information by normalizing its components and define interaction distance as a new measure.Recall that two random variables, Following the notion of recursion as used in Unlike A striking difference between ID and the interaction information is that ID is a normalized measure. Normalization allows ID to adjust its information components so as to tackle the problem of small minor allele frequencies as we will illustrate.The current section has presented the basic idea of interaction distance. More detailed discussion of the theoretical properties of ID and a further exposition of the theory are provided in the Methods section.Saccharomyces cerevisae, then we looked at a somewhat more complex system, the mouse. I.e., the genetics of the crosses are equivalent between yeast and mouse in general, but the number of states in the genetic loci of the progeny is two in yeast  and three in the mouse (diploid progeny). Finally we use models to simulate various characteristics of human genetics that are accessible using ID. The exposition and comparison of these applications is the central theme of this paper.To illustrate our approach we applied ID to diverse genetic data sets. First we considered data from a relatively simple genetic system, the budding yeast We analyzed a data set of the genetics of yeast sporulation efficiency Saccharomyces cerevisae strains with very different sporulation efficiencies (3.5% and 99%). The sporulation efficiency of each strain was characterized by a real value between 0 and 1, which we binned into four integer values. We considered two binning strategies: optimal and uniform (see Methods and the discussion below). Each strain was genotyped at 225 markers distributed along the genome. Each marker is a binary variable since the strains from the cross are haploid and the alleles correspond to either parent A or parent B.The dataset consists of 374 yeast progeny of a cross between two th marker on the 7th chromosome Initially we focused on single gene effects by calculating the conditional entropy rker see . The thrFor genetic interaction analysis we calculate both the conditional phenotype entropy arkers,  . Recall  entropy . Note alions see . QTLs wiheat map  clearly We calculated the interaction distance after binning the phenotype using two different binning strategies, uniform and optimal , and evaluated the significance of interactions using four permutation tests with increasing stringency. Note that although the number of strains is not large enough to allow for consistent detection of subtle interactions  the interaction distance can be used as an effective filter for identifying the most interesting pairs that can be subjected for further analysis. We can compare our findings from −5 as shown in −7 . The top three pairs of n −7 see .Although at first one might suspect some redundancy between markers 7.9 and 13.6, due to a negative ID value (−0.014), a careful statistical analysis indicates that this ID value is not significant  and hence there is no interaction between 7.9 and 13.6.www.yeastgenome.org). There are other markers that modify QTL 10.14, such as markers 16.2 and 16.7 : i) phenotype data is more subtle and therefore noisier and harder to bin, ii) data sets are typically smaller (due to a limited number of progeny), and iii) the genotype data is three-valued  – QTLs can be homozygous (AA and BB) or heterozygous (AB). All these differences can cause a lack of statistical power when searching for interactions between small effect QTLs.The data set, kindly provided by Jake Lusis As for the yeast analysis, we start with the analysis of the conditional phenotype entropy  of mice . To anal map of   shows a We computed ID values for all pairs of markers and selected the top pairs based on p-values and the missing data threshold (see Methods for details). Although these criteria are strict, resulting in selection of a very small subset of candidate pairs, they lead to interesting observations and frame useful hypotheses. Researchers from the Lusis lab characterized this mouse cross for 27 different phenotypes. In particular, they included fat mass ID detects locus 388 interacting with loci 454 in LDL and 281 in MW see . AlthougPrior results show that locus 361 is recognized as a QTL strongly affecting the arterial plaque lesion size Markers detected in the previous work appear prominently in the set of interactions for weight and LDL detected using ID. In other words, our ID-based analysis provides evidence that strong effect QTLs in one phenotype can have significant effects on other phenotypes by participating in additional interactions. Moreover, these examples demonstrate that results provided by the analysis based on the information theory measure ID can lead to biological hypotheses for further investigation.We increase further the complexity of the system considered now and apply ID analysis to human data. Our aim here is to carefully consider different challenging properties of human data and explore the modifications required for ID to handle them. The primary challenges are the variations in allele frequency, and the high levels of diversity encountered. To have close control of the data parameters, we use simulated data obtained from several well-established models representing pairwise interactions. For comparison, we simultaneously apply interaction information (II), a non-normalized measure previously used for genetic data filtering Most human genetic studies are medically motivated. It is increasingly evident that most disease phenotypes involve complex genetic interactions To illustrate and benchmark the use of ID and II, we use several well-established models for pairwise genetic interactions For all examples shown, the interaction distance has higher values than interaction information (the standard deviation bars indicate significance information). This difference, however, can be attributed to the re-scaling due to the normalization . More interesting is the fact that although the interaction information is either negative or close to zero when For example, consider model M15 with MAFs of A and B equal to 0.5% and 2.5% respectively. In this case the average values of Recall from When the values of To examine the effect of different levels of penetrance on interaction detection we add an extra tunable parameter to a disease model: all the values of the “risk” genotypes  in the disease penetrance table are replaced with this parameter, thus controlling how often the risk genotypes result in a disease. We studied the behavior of ID for various values of the penetrance parameter across all the models and found no significant effect of the penetrance on ID. Indeed, since all the risk genotypes have the same penetrance, the genetic composition of all the cases in the case-control data set stays the same when we change the penetrance parameter from 1 to a lower value. Similarly, the genetic composition of all the controls also stays the same  when we change the penetrance. Note that with the penetrance parameter less than 1, a number of individuals with the risk genotype will not have a disease and will be added to the set of controls. However the number of such individuals is very small due to low MAFs of the risk genotypes.One should be aware that the majority of pairs of SNPs has no effect on the phenotype, and thus, they can contribute to noise that will have an impact on the results provided by ID (or by any other measure). Let us now briefly consider the effect of this background noise. We investigate this by generating background distributions (see Methods section for more details) for various allele frequencies where both SNPs are independent and have no marginal or pairwise effect on the phenotype. We generated 20 million pairs of SNPs with various MAFs ranging from one to fifty percent, one million for fixed frequencies. The maximal observed value of the ID score was 0.019, the average was about 0.002. This seems to be significantly different from the ID scores observed in many cases illustrated in In the current paper we do not provide any analysis of a false positive rate or ratio of false to true positives. This follows from our philosophy of how ID (or II) should be applied in practice, which is, as stated above, a filtration approach. We have to remember that in order to analyze false positive rate, one has to make a final decision whether a pair of markers interacts or not. While a filtration approach is based on ID threshold, which is not too stringent, a final decision should be based on a permutation test. But this is strongly dependent on the context of available data. For instance, if we have only 200 markers which results in 20 000 pairs an ID score with p-value as high as So far we have been assuming independence of alleles from different loci. Let us now briefly discuss a case when two SNPs are not independent and are in so called Linkage Disequilibrium (LD). In other words this means that alleles of two different loci are correlated. LD between markers makes proper permutation testing of interaction between these markers challenging. Although both markers in LD have marginal effects on the phenotype, Test III is not applicable, since both markers are correlated and the randomization performed during the testing procedure affects that correlation. Hence, formally speaking, we cannot make any statement about potential interaction between such markers based on this test.We analyzed the same models as depicted in We conclude that although in many cases LD makes detection of potential interactions more difficult, it does not lead to false positives since non-interacting markers that are linked result in low ID scores and are not selected for further investigation during the filtration step. The main scope of this paper is to examine interactions where at least one marker has no marginal effect, which does not happen in case of markers in LD. A detailed analysis of the influence of LD on detection of interactions between markers with marginal effects will be a topic of our future work. The problem of constructing a permutation test seems to be especially interesting.The model analysis shown in Information theory approaches to data analysis can be used to good effect in addressing genetics problem of several kinds. The information measure called interaction information has been recently used for finding associations between two SNPs in human data We have applied our measure to both yeast and mouse cross data sets, with equal allele frequencies across the genome, which works reasonably well. It is particularly useful, however, in application to human data. We show that for several models the normalized measure, ID, can be particularly useful for low MAF cases by providing a compensatory effect for large differences in the entropies of the marker variables. The yeast and mouse examples demonstrate a powerful use of the interaction distance as a tool for choosing pairs and markers for further investigation. Even though we developed strict methodology for statistical testing of the ID and II scores, the number of strains/samples available may often be insufficient to provide p-values in a useful range. We argue, therefore, that the results provided by the ID analysis should be compared with results related to similar or related phenotypes. In both, yeast and mouse, cases we were able to identify QTLs that are very likely to be involved in several of the phenotypes analyzed. We observed that very often a QTL having strong effect on one phenotype may have a small effect on another, related phenotype, often due to interactions. This kind of analysis can result in biological hypotheses and potentially better understanding of the architecture of genotype–phenotype relationships, and is an important and promising direction.We will be applying ID analysis to more sophisticated models and to real data sets. Since, the genetic data typically consists of hundreds of thousands SNPs, the problem is computationally challenging but our test cases demonstrate that it is tractable. The analysis is completely parallelizable and with proper parallelization we can perform all computations on a million SNPs data set within a relatively short time on multi-core desktop machines. Of course, this refers to calculations of the ID only. The permutation tests are computationally expensive and can usually be performed only for selected sets of pairs.Finally, it is important to point out that the extension of genetic analysis here to the three-variable case, while at the edge of current capability is just the first step. We have developed a general, multi-variable formulation based on information theory, to be published elsewhere, that will allow us to perform analyses of the interaction of a larger number of markers, and of more complex interactions.Let us briefly discuss the normalization issues. The information distance  is normaSincelet us assume that When the mutual information is normalized by ance see  to guaraIt is unclear at this stage which type of normalization would be preferable in our biological applications. We observed that normalization plays a marginal role in case of mouse and yeast data: II and ID produce almost equal results because of the homogeneity of the variables (50% MAFs). The variables representing genetic markers in these types of experiments have similar entropies, thus, the effect of normalization is negligible. The statistical significance of the observed results is also the same. On the other hand, there is a significant difference in performance between II and ID in case of the human models where the MAFs are not uniform.In Theorem 1. If The first property shows the range of II scores. The next two properties suggest the interpretation of the II values: positive II scores suggest synergistic effects while negative scores indicate redundancy. The functional relation in the second property means that the knowledge about the states of two variables results in the knowledge about the state of the third one, on one hand. On the other hand, any two variables are independent. The relation presented in the third point represents a situation when We now present the counterpart of these results that establish the basic properties of the interaction distance. Large parts of the proof remain the same as for II. Remember that we consider only finite random variables, i.e., maps defined on a probability space with finite state space.Theorem 2. The interaction distance has the following three properties:If a sequence Proof. 1. By definition, the ID is a difference between two measures normalized to the interval 2. Suppose The proof of the converse statement is trivial. Note that 3. The convergence of interaction distances implies that (a) the unconditional distance converges to zero and (b) the conditional distance converges to one.Since the unconditional distance converges to zero, This implies that Since the conditional distance converges to 1, the ratio According to property 1 of Theorem 2, ID is normalized to the range from −1 to 1, as opposed to II that ranges between This can be further illustrated by an example where both According to property 3 of Theorem 2, ID values close to −1 indicate that the value of Test 0. Note that Test 0 gives only a rough estimate of p-values since it does not account for missing data. Moreover, the Test 0 background distribution is generated under the null hypothesis that i) both markers have no effect on the phenotype and ii) there is no interaction between the markers. Therefore, formally speaking, a rejection of such a hypothesis does not imply the presence of an interaction between markers. For example, the null hypothesis may be rejected if one of the markers has an effect on the phenotype. To increase the accuracy of the significance testing we propose three permutation tests (Test I-III) with different null hypotheses.The current section presents detailed description of the permutation tests used for calculating the p-values of the interaction distance between two markers. We can make a rough estimate of p-values by generating 1 million pairs of random markers with no missing alleles and the same allele frequency as in the original data. Computing ID for these 1 million pairs and the phenotype, which remains untouched, generates a background distribution used to test the significance of the original ID values. We refer to this statistical testing as The tests are performed for each pair of markers separately. Let us suppose that The background distribution is obtained by generating vectors Test I is performed by randomizing The most difficult is the case of two markers having marginal effects. Test III is designed to preserve the effects of the markers but randomize a possible interaction between them, so the null hypothesis is that two markers with effects During Test III, we randomize each sub-vector Since the approximation of information theory measures, such as mutual information, for continuous variables may be difficult and requires large number of samples, we binned the phenotype into four bins. In the current work we use two binning strategies: i) a uniform binning with 0.25, 0.5 and 0.75 quantiles as thresholds and ii) a so-called optimal binning proposed in We are aware that binning, discretization of continuous variables, leads to loss of information. Nevertheless, the information content of the binned variables is sufficient to detect the signal of interacting pairs of markers. In practice, we suggest exploring various binning strategies with different number of bins. In the presented yeast example, the number of bins (four) was “optimal” both in terms of statistical power and number of discovered candidate pairs. Since we are mainly interested in applications to human data where phenotypes are often discreet (case-control studies), we do not analyze approaches based on approximating mutual information and entropy from continuous data such as kernel based approximations.We applied ID to two mouse phenotypes: the LDL cholesterol level and weight. We observed that male weight is considerably more different than female weight  and for weight  and then estimated their Test 0 p-values. Note that whenever we observed blocks of consecutive markers interacting with each other, we selected the pair with the highest interaction distance as a representative of similar interactions. For example, the pair  in LDL represents a group of interactions between markers 691–693  and 267–273 , since  has the highest ID score among all of these pairs. Note that although the distance between markers within a block can be large (up to 11 Mbp), these markers are strongly correlated. We might mix two or more biological interactions together, however we are not able to distinguish these interactions in a block of such highly correlated markers.In order to simulate a clinical cohort data with the same number of cases and controls, we generate independently two versions of each SNP according to the probabilities In order to estimate the expected values of II and ID we calculate the expected frequencies of each genotype within a healthy and affected population . For example, the probability of observing genotype AABB given an affected individual can be written as:If the two SNPs are in LD, then the above formula needs to be written as:In case of allele configuration Obviously, with the growing number of samples, the observed frequencies are getting closer to the conditional probabilities calculated as presented above. Therefore, given the allele frequencies, we use these theoretical values to approximate the expected values of ID and II for the case-control data models. Table S1Comparison of p-values of example pairs for different tests and both binnings in the yeast example.(DOC)Click here for additional data file.Table S2Comparison of p-values of example pairs for different tests in mouse LDL phenotype.(DOC)Click here for additional data file.Table S3Comparison of p-values of example pairs for different tests in female weight phenotype.(DOC)Click here for additional data file.Table S4Comparison of p-values of example pairs for different tests in male weight phenotype.(DOC)Click here for additional data file.Table S5Coordinates of selected markers from the mouse example.(DOC)Click here for additional data file.Code S1This file contains Matlab scripts that will enable readers to reproduce results presented in the paper and apply presented tools to their data.(ZIP)Click here for additional data file.


The corresponding repeatability estimates were 0.39, 0.38, 0.39, 0.49, 0.51, and 0.16, respectively. The genetic and phenotypic correlations between different production traits ranged from −0.86 to 0.95 and from −0.88 to 0.98, respectively. The genetic and phenotypic correlations between production traits and Log10SCC were weak almost nonsignificantly differentl from zero. The results imply that milk traits for Jersey dairy cattle in Zimbabwe are more heritable. Therefore, these traits may be preferred by breeders as selection criteria for development of effective genetic improvement programme.Genetic and phenotypic parameters for production traits and somatic cell count (SCC) for Jersey dairy cattle in Zimbabwe were estimated. A total of 10986 lactation records were obtained from Zimbabwe Livestock Identification Trust, with cows calving in the period from 1996 to 2008. An ASReml program fitting an animal model was used for the analyses. Heritability estimates for milk yield, fat yield, protein yield, fat percentage, protein percentage, and Log Knowledge of genetic parameters is the basis of sound livestock improvement programmes. Estimates of heritabilities and genetic correlations are essential population parameters required in animal breeding research and in design and application of practical animal breeding programmes. Genetic parameters are a characteristic of the population in which they were estimated and may change overtime due to selection and management decisions .The Jersey is one of the dairy breeds found in Zimbabwe. It is the second most important breed of dairy after Holstein-Friesian. It is a small breed of dairy cattle, with mature live weight ranging from 360 to 540 kg. Originally bred on the British Channel Island of Jersey, the breed is popular for the high butterfat content of its milk and the low maintenance costs incurred due to its lower bodyweight, as well as its genial disposition. In addition, Jerseys can thrive on locally produced food. They can tolerate high temperatures; heifers mature more quickly than those of other breeds and can be mated at 13 to 15 months; they produce an average herd milk production of 3500 to 5000 kg .Bos indicus to produce a hardy, disease-tolerant, dairy-type cow which does not need a high plane of nutrition to produce reasonable milk yield and is suited to dairying in the communal areas [Jersey is an ideal breed for crossbreeding with al areas . Howeveral areas . Furtheral areas . In addiIt thus becomes imperative to genetically evaluate the Jersey production traits as a way of increasing productivity through selection. The objective of this study was to analyse lactation records in order to fit an appropriate model for evaluation of the Zimbabwean Jersey dairy cattle regarding production traits and somatic cell count (SCC). Genetic and phenotypic parameters were also estimated. This led to the development of a selection index .2, and it is divided into five agroecological regions. Rainfall patterns and crop production progressively deteriorate from region I to V. However, livestock production including dairying is practised in all the regions. In the regions with low rainfall, dairying is assisted by production of drought-resistant fodder crops. Most dairy farms are located within 40 km of the major cities and towns [Zimbabwe is located in Southern Africa in the tropical savannah region. The total land area is 390,759 kmnd towns . The standard 305-day milk production records of pure bred Jersey were obtained from Zimbabwe Livestock Identification Trust (LIT). The file contained production records of 25 herds, with cows calving during the period 1996 to 2008. The following production traits were recorded: 305-day milk, fat and protein yields; milk yield over the complete lactation period; length of lactation; calving interval; days dry; days open and somatic cell count. Identity of herd, cow, dam, sire, date of birth, and date of calving were also recorded. The records were edited using Statistical Analysis of Systems version 9.1.3  and PediMultivariate analyses to estimate variance components, heritability, predicted breeding values (EBVs), and genetic and phenotypic corrections were done using animal models and the ASReml program developed by . ijkl is the observed value for all traits ; μ is the overall mean common to all observations; HYSi is the fixed effect of herd year of calving-season of calving i  ; CIj is the fixed effect of classes of calving interval j  ; DDk is the fixed effect of classes of days dry k  ; b1AC and b2AC are the linear and quadratic regression coefficients, respectively, on age at calving in months; al is the random animal effect of cow l; cl is the random permanent environmental effect of cow l; and eijkl is the random residual effects, eijkl ~ N.Two models were fitted to the data. Model 1  was for data sets 1 and 4. The model is described as follows:Model 2 was used for data sets 2 and 3. This model was similar to model 1. The only difference is that for data set 2 fixed effects of the previous calving interval and days dry were excluded because these values were just the same (zero) since cows had calved for the first time. The random effects of the permanent environmental effects were also excluded, since there were no repeat records within parity. For data set 3, the permanent environmental effect was excluded but the fixed effects of the previous calving interval and days dry were included in the mixed animal model. Variance components, heritability, and repeatability estimates are given in Estimates of the additive genetic variances for yield traits in multiparous cows were dramatically smaller than those estimates from first lactation cows. As a result, heritability estimates for yield traits with repeated records were lower. The reason behind this is that in multiparous cows a repeatability animal model was used. One of the assumptions behind the use of repeatability models in the present study is that all lactations are genetically the same trait. This implies that the same set of genes exerts a common influence on the phenotypic expression of the first and later lactations . This asThe heritability and repeatability estimates for yield traits were somewhat lower than the average estimates of such estimates given by , 12, buth2 = 0.08) of SCC implies that substantial improvement has been made of Jersey herds in Zimbabwe from mastitis. As a result, this has led to good quality milk. The heritability estimates for all records for SCC obtained in this study were not significantly different from zero. However, heritability estimates for SCC were lower in the first lactation and increases the later lactations, but they were also not significantly different from zero. This suggests that little genetic progress can be made by selection of this trait. High values of SCC are highly associated with mastitis ; therefoThe heritability estimates for all production traits are highest during the first lactation and decreases towards the later lactations. This is in agreement with the results reported by various researchers , 15, 16.The phenotypic and genetic correlations of 305-day production and SCC variables are given in This study also shows that the genetic and phenotypic correlations between production traits and SCC were weak almost nonsignificantly, different from zero. Juozaitiene et al.  and NěmcVariance due to permanent environmental effects was an important source of variation. The milk traits for Jersey dairy cattle in Zimbabwe were found to be more heritable than the SCC, and the correlation among the milk traits was significantly high. The correlation between the production traits and SCC was weak almost nonsignificantly different from zero. Therefore, these production traits may be preferred by breeders as selection criteria for development of effective genetic improvement programme.


It is an assumption of large, population-based datasets that samples are annotated accurately whether they correspond to known relationships or unrelated individuals. These annotations are key for a broad range of genetics applications. While many methods are available to assess relatedness that involve estimates of identity-by-descent (IBD) and/or identity-by-state (IBS) allele-sharing proportions, we developed a novel approach that estimates IBD0, 1, and 2 based on observed IBS within windows. When combined with genome-wide IBS information, it provides an intuitive and practical graphical approach with the capacity to analyze datasets with thousands of samples without prior information about relatedness between individuals or haplotypes. We applied the method to a commonly used Human Variation Panel consisting of 400 nominally unrelated individuals. Surprisingly, we identified identical, parent-child, and full-sibling relationships and reconstructed pedigrees. In two instances non-sibling pairs of individuals in these pedigrees had unexpected IBD2 levels, as well as multiple regions of homozygosity, implying inbreeding. This combined method allowed us to distinguish related individuals from those having atypical heterozygosity rates and determine which individuals were outliers with respect to their designated population. Additionally, it becomes increasingly difficult to identify distant relatedness using genome-wide IBS methods alone. However, our IBD method further identified distant relatedness between individuals within populations, supported by the presence of megabase-scale regions lacking IBS0 across individual chromosomes. We benchmarked our approach against the hidden Markov model of a leading software package (PLINK), showing improved calling of distantly related individuals, and we validated it using a known pedigree from a clinical study. The application of this approach could improve genome-wide association, linkage, heterozygosity, and other population genomics studies that rely on SNP genotype data. High-density microarrays measuring single nucleotide polymorphisms (SNPs) provide information about the genotypes across many loci. SNP genotypes observed for any two individuals can be compared in terms of identity-by-state (IBS), in which two individuals are observed to have 0, 1, or 2 alleles in common at a given locus, across a chromosomal region, or throughout the genome. These alleles may be shared identical-by-descent (IBD) in which 0, 1, or 2 alleles are inherited from a recent common ancestor, or they may be identical by chance because the allele is frequent in the population. The expected proportion of genome sharing between two individuals varies as a function of their genetic relatedness. We introduce a method to estimate IBD that can be used to analyze relatedness in pedigrees or in large-scale population studies with thousands of individuals. This can be combined with observed IBS to distinguish a variety of types of relatedness, providing theoretically justified results that are graphed in a manner that is straightforward to interpret. The methods we introduce are relevant to a variety of SNP applications including linkage and association studies and population genomics studies. Single nucleotide polymorphism (SNP) genotyping is used to delineate the extent and nature of chromosomal variation, examine population genetic structure, and find loci that contribute to disease. SNPs are used as proxies for the unobserved sequence variants in the surrounding DNA, allowing measurement of the flow of genetic material through populations There are important limitations for using SNPs to identify causal disease variants. Genome-wide association studies (GWAS) rely on representative sampling of a subset of individuals from a population. Therefore, calculations testing the association between alleles, the frequency of alleles in the population, and the contribution of alleles to a phenotype must use estimates of the population allele frequency based on the representative sampling. These estimates of allele frequencies are sensitive to inflation or deflation when the genotyping data are derived from individuals with unreported familial relationships or with admixed ancestry .For any given pair of individuals with genotype information, identity-by-state (IBS) can be observed at a given locus with three possible outcomes: the individuals have two different alleles (IBS0) or they share one (IBS1) or two (IBS2) alleles in common. For example, a pair of individuals with genotypes AA and BB are IBS0 at this locus whereas a pair with AA and AB are IBS1. Two individuals who share 1 or 2 alleles IBS at a given locus may have inherited the shared allele(s) from a recent common ancestor, in which these allele(s) are identical-by-descent (IBD). IBD approaches have been applied to linkage mapping In this study we demonstrate an approach that combines our IBD method with IBS information to estimate the relatedness between individuals in pedigrees and/or in large population-based studies. There are two main aspects of this work. First, we introduce plots based in part on methods suggested by Lee Combining these IBS and IBD approaches, our analyses simultaneously reveal previously unknown familial relationships and population substructure in large-scale SNP data. Notably, our method applies to pedigrees but does not rely on prior knowledge of relationships or ethnicity. While other exploratory techniques such as principal components analysis (PCA) of genotype data can indicate outliers, the nature of such relationships is not explicitly described. In contrast, our method is useful to define relationships. We observed differences within and between populations (and pedigrees) due to multiple factors including familial relationships, autosomal heterozygosity rate, chromosomal anomalies, and population admixture. We analyzed data from the Coriell Institute's National Institute of General Medical Sciences (NIGMS) Human Genetic Cell Repository Human Variation Panel (referred to as the Human Variation Panel), an extensively used data source, and found undocumented familial relationships.Our IBD method uses an overlapping window approach see  and is cThe Human Variation Panel consists of four populations . All samples from these individuals were submitted to the NIGMS repository with annotation indicating they were unrelated. Using autosomal SNP genotype data  from these samples, we analyzed all pairwise IBS relationships in each population group .For the four within-population comparisons we generated a plot (referred to as an IBS2* plot) having x-axis values referred to as IBS2*_ratio and based on the ratio of IBS2*/(IBS0 + IBS2*), suggested by Lee We used a Z-test . Surprisingly, we observed 25 data points with values >0.70 that potentially corresponded to familial relationships . TypicalPutative parent-child relationships  also hadThe y-axis (percent informative SNPs) of the IBS2* plot  providedInferred sibling comparisons (n = 5) were evident with IBS2*_ratio values ranging from 0.92 to 0.95. We defined this group boxed in  as sibliIBS2* plot x-axis (IBS2*_ratio) values less than 2/3 reflected differences in heterozygosity values in pairwise comparisons either within or between geographic groups. For example, CAU individual NA17251 in comparison with other CAU individuals had a sum of heterozygosity of 56.7 +/- 0.01% in contrast to other CAU comparisons having values of 59.7 +/- 0.1% , arrow. For 9 of the 25 comparisons with IBS2*_ratio values greater than 0.70 (but less than 0.714), chromosomal IBS analysis using SNPduo failed to reveal any regions lacking IBS0 calls that would imply the presence of IBD1. Additionally, these samples represented all CAU within-group comparisons and had among the highest pairwise sums of heterozygosity rates , arrow 3Comparisons between individuals from different Human Variation Panel geographic groups were all expected to represent pairs of unrelated individuals having IBS2*_ratio measurements <2/3. Consistent with this expectation, there were no IBS2*_ratio values >0.69  to infer IBD from IBS data For the 13 relationships that were previously inferred to be second-degree related or more based on the IBS2*_ratio, our method revealed expected estimates of Cotterman coefficients of relationship. For identical samples that are expected to have a K2 of 1.0, our method estimated zero IBD1  so that IBD1 estimates K1,  were comFor the great majority of pairwise comparisons, for which IBS2*_ratio values were centered on 2/3, we expected to observe IBD0 but little (if any) IBD1 or IBD2. Our method revealed extensive IBD0  and very limited amounts of IBD1 and IBD2, as expected for unrelated individuals . Based oIn our initial analyses using IBS measurements , 25 compPLINK's HMM reported dramatically more IBD1 for comparisons having IBS2*_ratio values approaching 2/3 and above , arrow 6Our method revealed little IBD2  as expecTo further characterize IBD relationships, we calculated IBS2* and IBD metrics for the Human Variation Panel between-group comparisons, which serve as a gold standard for authentically unrelated individuals. These plots again indicated very high levels of K0 . The K1 and K2 estimates were very low (data not shown). PLINK is not designed for the analysis of IBD involving members of different populations The IBD method we introduce lacked apparent false positive results that occurred using PLINK's HMM for the determination of distant genetic relatedness. The 21 comparisons  that we identified as related were among those having the 22 highest K1 values. The most distantly related pair that we present as related (NA17632/NA17695) had a K1 value of 0.0323, while an unrelated pair had a comparable K1 value (0.0326). This is slighter higher than a theoretical 1/64th relationship (with a K1 of 0.03125) that appears to be the limit of detecting relatedness inferred by our IBD method. Some comparisons lacked an IBS2*_ratio greater than 0.70.Based on our IBS and IBD analyses we identified previously unannotated familial relationships  and recoWe validated the combined IBS and IBD approach by analyzing data from individuals with known familial relationships and compared it to PLINK's HMM. We used genotype data from a pedigree in a study of metachondromatosis As the number of genome-wide association studies, SNP datasets, and meta-analyses increase, proper characterization of familial relationships and underlying population structure increases in importance. As an example, Pemberton et al. recently identified a series of unexpected relationships in phase 3 HapMap data In this study we applied visualization and analysis of autosomal IBS and IBD measurements to geographic populations. We introduced a metric based on informative IBS combinations of IBS0 and a subset of IBS2, termed IBS2*, in which AB genotypes are aligned in two individuals at a given chromosomal position. These IBS analyses were complemented by an IBD method that revealed the extent of genome sharing in unannotated related (or known related) pairs of individuals. The main significance of these analyses is that (1) we identified first, second and third degree relationships that were unexpected (for the Human Variation Panel) or consistent with prior annotation ; (2) we identified population substructure for the geographic groups; (3) we identified individuals who accounted for dramatic variations in the population substructure; (4) knowledge of these individuals may inform future studies that use these datasets; and (5) these combined methods do not require prior information about allele frequencies, ethnic background, or haplotype structure. Our approach is scalable to the study of datasets of any size.In terms of our IBS approach, Rosenberg Allele-sharing methods based on IBS metrics have been widely used Plots of mean versus standard deviation of IBS values, such as those by Abecasis and colleagues with Graphical Representation of Relatedness (GRR) Apparent genetic relatedness between two individuals could have two independent explanations: shared ancestry (e.g. the two are third cousins) or membership in geographic or ethnic groups that have varying population allele frequencies. The combined IBS and IBD method allowed us to visualize and determine relatedness in the context of either (or both) explanations. The dimension of IBS2*/(IBS0 + IBS2*) reveals relatedness in a manner that is largely independent of population allele frequencies. Our analyses indicated that atypical heterozygosity levels can lead to high IBS2*_ratio values (e.g. 0.70–0.75). Such cases do not necessarily imply familial relatedness and are characterized by two features: (1) relatedness between a given individual and large numbers of others in the population, beyond what is observed in typical pedigrees, and (2) a lack of IBD1 regions on a chromosome-by-chromosome basis, confirming that the individual with atypical heterozygosity is not related to others despite the deviation from a 2/3 IBS2*_ratio value.While IBS variability can be attributed to familial relatedness and/or to population allele frequencies, another source of variability is the SNP selection process which Clark et al. have shown is subject to ascertainment bias The genotypes of 400 individuals from the NIGMS Human Genetic Cell Repository obtained on the Affymetrix Genome-Wide Human SNP Array 6.0 using the Birdseed algorithm were obtained from the Coriell Cell Repositories . These collections (n = 100 each) were from AA (HD100AA), CAU (HD100CAU), CHI  and MEX . In all cases, these individuals were reported to be unrelated and apparently healthy. The data set is available from dbGaP (study accession phs000211.v1.p1). Only autosomal data were used for analysis.For a validation dataset, we obtained SNP genotype data from a published study that included 12 individuals of ‘known’ relationship We measured autosomal IBS values using the freely available, cross-platform SNPduo++ software (v1.02) which measures IBS2* from pairwise relationships as well as measurements of IBS0, IBS1, and IBS2 As suggested by Lee p and q, where ip+iq = 1 denotes the allele frequency for the ith informative locus, with informative markers i = 1, 2, … m . Conditional probabilities for concordance under the null hypothesis H0  are given by Lee as follows We consider alleles A and B having frequencies T1), variance, and statistic Z1 are given in Z1 statistic. Results based on an exact binomial test for p = 2/3 were quantitatively and qualitatively similar (data not shown).A notable feature of this approach is that the probabilities are expected to be independent of the population allele frequencies for each SNP and should reduce to 2/3 based on the genome-wide sums of IBS2* and IBS0. The test statistic  are positively correlated with heterozygosity rates (shown in In contrast to the x-axis (IBS2*_ratio), the allele frequencies which are represented by shown in . The morWe implemented a method for IBD estimation. We present a simplified graphical overview in For each window, we calculated a likelihood of each IBD state given the observed IBS values. For example, we expected to observe an IBS0 ratio (i.e. IBS0/(IBS2* + IBS0) of 1/3 for IBD0 states, and an IBS0 ratio of 0 for p and q.An observation error rate is also incorporated into the likelihood model. This is fixed for all of the analyses. The probabilities used to compute likelihoods given an IBS call at a SNP for each IBD state assignment are in To estimate P(S | D0), we limit our likelihood calculations to only the IBS0 and IBS2* calls. The observed frequencies of these two IBS states are independent of allele frequency at each SNP when conditioned on an observed IBS0 or IBS2 call, with P  = 1/3.We use the following distribution assumptions to create the likelihood of observing the IBS0 frequency. The error (set by default at 0.01) reflects the genotyping error.From this, we have P(S | D0) and P(S | c is discussed below. The error rate is set at 0.01.We use this estimate of P(D0 | S) as our estimated IBD0 probability. The next step is to divide P and P(S | D2). Since all the three terms of P(S) above are estimated after specifying a prior probability, we can compute the three IBD probabilities, P(D0 | S), P(D1 | S), and P(D2 | S) using Bayes' rule.w. Pw denotes a probability value across a given window.We estimate the K coefficients using the estimates of P(D0 | S), P(D1 | S), P(D2 | S) for each window, w, spanning genome length lc parameter (equation 9) which is used for distinguishing IBD1 and IBD2 for regions that are f, taking the allele frequency p, as defined in The windowed approach used observed IBS to identify regions that were IBD0 or p), to calculate an expected f(p).We can integrate over the allele probability function, P(p) to represent any prior belief about the allele frequency of those SNPs that have observed variation. We used a one-step empirical Bayes' estimate of P(p) to suggest a practical value.We can specify P(Assume that an event O occurs when a SNP has observed variation in the two comparison samples.p) in a region of IBD1 as P(O | p) = 5+p (1-p). This is the complement of the chance to observe identical homozygous genotypes in a region of IBD1.We define P(O | p) is specified as a uniform prior over p. This initial uniform prior is used to compute a posterior P(p | O), which is then used as the empirical prior to compute Pe(p | O).An initial estimate of P(e (p | O) in place of P(p) in the calculation of c above results in the expectation c = .518. The value of c should be chosen greater than the error and less than or equal to than the maximum allowable value of 2/3. The assumed binomial model considers all SNPs to be drawn with the same allele frequency which is not accurate, as the binomial parameter p varies across SNPs as a function of allele frequency. The binomial model described above was chosen for its simplicity and the experimental insensitivity to choice of c across a wide range of reasonable values. Future work may use the beta binomial distribution to better account for the distribution of allele frequencies. We tested a range of c values and observed consistent results for K1 and K2 estimates given c≥0.25 (data not shown).Using PThe IBD algorithm  is available as an executable at the authors' website For the Human Variation Panel, we applied PLINK's HMM, given by the "--genome" option, to each group independently  because a homogeneous population is recommended PLINK's HMM analysis for the validation dataset was run with incomplete pedigree information since data for both parents, which must be specified in a tped file, were not available. Only one trio was specified. We analyzed the 12 family members' data in PLINK using the same quality control measures as above, and no samples were excluded based on low genotyping rate. 7 SNPs were removed due to ≤99% genotype call rate, while 87,234 were removed due to MAF ≤0.01 leaving 450,924 SNPs for analysis.Figure S1Reconstruction of pedigrees. Pedigrees inferred from IBS and IBD sharing are shown for individuals from CAU, CHI, and MEX populations. Abbreviations: K1, estimate of IBD1 using our method; x, IBS2*_ratio value. For panel B, the relationship of NA17626 to two males (NA17655 and NA17656) is via a father for whom SNP data were unavailable; these relationships are indicated with a dashed line.(PDF)Click here for additional data file.Figure S2Pedigree for validation dataset. Numbers correspond to individuals listed in (PDF)Click here for additional data file.Figure S3c). Region 3, having observed IBS1 calls, is inferred to be IBD1. Region 4, lacking observed IBS1 calls, is inferred to be IBD2. K0, K1, and K2 estimates are defined in the figure and correspond to Cotterman coefficients of relatedness k0, k1, and k2, respectively.Overview of the IBD method. Upper panels: representative IBS observations between two related individuals across a typical chromosomal segment (x-axis). Each data point corresponds to a single locus for which IBS0, IBS1, or IBS2 is observed and plotted (y-axis). We delineate four regions using an iterative approach . In Region 1, having IBS0, we detect IBS0 and infer an IBD0 region. In Region 2, lacking IBS0 calls, we define the locus as “not IBD0.” We infer the IBD1 versus IBD2 state depending on the IBS1 calls Click here for additional data file.Table S1Samples with low genotype rate. Genotype rates for individuals that were removed from PLINK HMM analysis due to low genotype call rate (≤0.980).(TXT)Click here for additional data file.Table S2Quality control information for each ethnic group. Quality control was performed using PLINK to remove individuals with ≤98% genotype call rate, SNPs that failed a HWE (p <0.0001), SNPs that had ≤99% genotype call rate, and SNPS with a MAF ≤0.01. The first MEX, CHI, CAU, and AA columns represent the quality control measures in which individuals with low genotyping rates were removed. The second MEX, CHI, CAU, and AA columns represent the quality control measures in which individuals with low genotyping rates were not removed.(TXT)Click here for additional data file.


Genome-wide association studies (GWAS) have identified many common polymorphisms associated with complex traits. However, these associated common variants explain only a small fraction of the phenotypic variances, leaving a substantial portion of genetic heritability unexplained. As a result, searches for "missing" heritability are drawing increasing attention, particularly for rare variant studies that often require a large sample size and, thus, extensive sequencing effort. Although the development of next generation sequencing (NGS) technologies has made it possible to sequence a large number of reads economically and efficiently, it is still often cost prohibitive to sequence thousands of individuals that are generally required for association studies. A more efficient and cost-effective design would involve pooling the genetic materials of multiple individuals together and then sequencing the pools, instead of the individuals. This pooled sequencing approach has improved the plausibility of association studies for rare variants, while, at the same time, posed a great challenge to the pooled sequencing data analysis, essentially because individual sample identity is lost, and NGS sequencing errors could be hard to distinguish from low frequency alleles.A unified approach for estimating minor allele frequency, SNP calling and association studies based on pooled sequencing data using an expectation maximization (EM) algorithm is developed in this paper. This approach makes it possible to study the effects of minor allele frequency, sequencing error rate, number of pools, number of individuals in each pool, and the sequencing depth on the estimation accuracy of minor allele frequencies. We show that the naive method of estimating minor allele frequencies by taking the fraction of observed minor alleles can be significantly biased, especially for rare variants. In contrast, our EM approach can give an unbiased estimate of the minor allele frequency under all scenarios studied in this paper. A SNP calling approach, EM-SNP, for pooled sequencing data based on the EM algorithm is then developed and compared with another recent SNP calling method, SNVer. We show that EM-SNP outperforms SNVer in terms of the fraction of db-SNPs among the called SNPs, as well as transition/transversion (Ti/Tv) ratio. Finally, the EM approach is used to study the association between variants and type I diabetes.The EM-based approach for the analysis of pooled sequencing data can accurately estimate minor allele frequencies, call SNPs, and find associations between variants and complex traits. This approach is especially useful for studies involving rare variants. Finding genomic variants associated with complex traits is one of the most important problems in modern genomics. Genome-wide association studies (GWAS) based on common variants have been the dominant approach to achieve this objective . HoweverStudies of rare variants are complicated by the low minor allele frequencies of rare variants. The development of next generation sequencing (NGS) technologies such as Illumina and Roche 454 has made it possible to sequence a large number of reads economically. Despite such important progress, sequencing a large number of individuals separately is still costly for most biological laboratories. One frequently adopted approach to reduce sequencing cost in the search of rare variants is pooled sequencing, where mixtures of genetic materials from several individuals are grouped together to form a pool for a single sequencing. While this design greatly lowers the sequencing cost, it also makes it hard to distinguish true genetic polymorphisms from sequencing errors, estimate minor allele frequencies at the polymorphic loci, and perform association studies on the rare variants.Several research groups have used pooled sequencing to detect rare variants that are associated with complex traits such as retinitis pigmentosa, diabetes, cancer, and inflammatory bowel disease -8. Thereet al.[et al.[et al.[et al.[et al.[Several groups have developed SNP calling methods based on pooled sequencing data ,9-12. Ouet al. modeled l.[et al. developel.[et al. improvedl.[et al. developeet al. [et al.[In order to estimate minor allele frequencies in pooling studies, several groups developed statistical models for the sampling of individuals and the sampling of reads from the individuals in the pools ,15. Theset al.  studied . [et al. considerhttp://www.ncbi.nlm.nih.gov/projects/SNP/) among the SNPs called by EM-SNP is higher than that of SNVer. Similarly, the transition/transversion ratio of rare variants called by EM-SNP is higher than that of SNVer. These observations show that EM-SNP outperforms SNVer at calling rare variants with minor allele frequencies less than 1%.In this paper, we develop new methods for estimating minor allele frequencies, SNP detection, and association studies using pooled sequencing data based on the models in -16. In cf be the frequency of the minor allele (denoted as "1"), and 1 - f be the frequency of the major allele (denoted as "0") in a population. We also consider the following potential sequencing error rates:Consider a locus along the genome. Let G pools of individuals are sequenced and each pool contains K/2 individuals (K chromosomes). For each pool g, a total of g nreads are mapped to this locus, with Assume that a total of C be the number of chromosomes carrying the minor allele among the K chromosomes in a pool. Then C follows a binomial distribution, i.eLet C = k, the probability that a sequence read covering a variant carries the minor allele isConditional on g-th pool is,Thus, the probability of observing the data for the Since the pools can be considered independent, the likelihood of observing the data for all the pools isGiven the above likelihood expression and the data f, α, β).• Find the maximum likelihood estimate of  or minor (1)) of read i in the g-th pool;• gir: the observed allele state (major (0) and minor (1)) of the i-th read in the g-th pool.• We also use the following notation:Based on the above notation, the complete log-likelihood is:f, α, β) at the t-th iteration is Θt) (= (ft). The maximization (M)-step gives:Suppose that the value of Θ = (Et) (is taken when the parameters are at Θt)(.Note the expectation The expectation (E)-step is formulated as follows:andt-th step.where all the parameters in the equations are of the values taken at the f.From Equations 3 and 4, we are able to obtain the recursive formula for Et)((T11|Data). Note thatNext we calculate i, and we denote it as which does not depend on Et)((T10|Data), Et)((T01|Data) and Et)((T00|Data), and the recursive formulas for α and β can be derived from them.Similarly, we can derive the formulas for i.e. loci that are not truly polymorphic. Thus, before testing for associations with phenotypes, we need to determine the true polymorphic sites. This step is especially important in the case of rare variants since the sequencing error rates for NGS could be close to or even higher than the minor allele frequencies.Due to sequencing errors, the observed variants may contain a significant amount of false positives, f1 and f0 be the minor allele frequencies at a locus among the cases and controls, respectively. Denote f =  and 0 = . We can test if an observed variant is a true SNP using the likelihood ratio test for H0 : f0 = f1 = 0 vs. H1 : f0 ≠ 0 or f1 ≠ 0:Consider a case-control study with a group of case individuals and another group of control individuals. Let lf is the maximum log-likelihood of the observed data for both the cases and the controls. Note that the null hypothesis f = 0 is on the boundary of the region of the parameters of interest. Therefore, the asymptotic distribution of Λ is I0 is the point mass at 0 and i = 1, 2 are the chi-square distributions with i degrees of freedom. When the number of pools is relatively small, simulation approaches for the null distribution of Λ are needed to obtain the asymptotic distribution.where rding to , where IH0 : f0 = 0 vs. H1 : f0 > 0. Similarly, we replace f0 by f1 for the case pools. We then use the statisticWe can also test if an observed variant is a true SNP using cases or controls separately. For the control pools, we conduct a likelihood ratio test for i f= 0 is on the boundary of parameter region i f> 0, the statistic Λi has an asymptotic distribution to test each hypothesis, where rding to . We refeH1 : f1 ≠ f0 versus the null hypothesis H0 : f1 = f0. This association test is conducted by the likelihood ratio test statistic:We test if a SNP is associated with a phenotype of interest using the likelihood ratio test again. Here we test the alternative hypothesis This statistic has an asymptotic chi-square distribution with 1 degree of freedom.f), sequencing error rate (α), the number of chromosomes in each pool (K), the number of pools (G) and the relative risk for a disease (λ).We use simulations to evaluate our approaches for allele frequency estimation, SNP detection and test for association. A large range of parameter space is considered to see how different parameters affect the performance of our methods. These parameters include minor allele frequency  values from a normal distribution with mean equal to the starting values of α, and variance equal to 0.1 times the starting values. Finally, we generate 1000 pooled data sets with each combination of the five parameters  based on iα(= iβ), i = 1, ⋯, 1000.Since the sequencing error rate can vary from locus to locus and from one pool to another, we generate 1000 For each of the 4 × 4 × 2 × 3 × 3 = 288 combinations, we do the following:i-th simulation, we use the EM algorithm derived above to estimate the parameters , denoted as 1. In the 2. Repeat Step 1) for R = 1000 times.f,3. Compute the mean squared error (MSE) of ffrac = C/(KG) in the pools,4. Compute the MSE of f by the fraction of reads carrying the minor allele.We use both MSE and Cg to compare the accuracy of the EM algorithm with the naive approach of estimating f and that the locus is under Hardy-Weinberg equilibrium. When generating the genotypes of the case individuals, we assume that the penetrance  of the genotypes 00, 01 and 11 are g0 = 0.01, λg0, and λ2g0, respectively. In our simulations, we choose λ = 1.2, 2, and 4.In order to evaluate the power of SNP identification using EM-SNP and test for association, we simulate case-control data as follows. When generating the control data, we assume that the minor allele frequency is i if we use case or control samples separately) defined in equation 6 is used to test if an observed variant is a true polymorphic locus or not. For a given type I error γ (= 0.05 in our study), we claim that the variant is truly polymorphic if Λ >γt, where γ tis the threshold corresponding to type I error γ. If the threshold can not be found theoretically, we can do parametric bootstrap to find the threshold. Firstly, assume that the variant site is not polymorphic, estimate the allele frequency and error rate using the maximum likelihood approach. Secondly, generate the reads data as in our model a large number of times (R = 1000), and obtain the empirical distribution of Λ. For a given type I error γ, we find the upper γ percentile γt. Finally, the null hypothesis is rejected if the value of Λ is at least γt. For a given relative risk λ, we repeat these steps 1000 times and the power is the fraction of times that the locus is called as polymorphic.We can use the case or control samples separately or combine them for SNP detection as in the "SNP identification using EM" subsection. For example, we consider both the cases and controls jointly. The log-likelihood ratio statistic Λ  to map the reads to the human reference genome (hg19) with parameters -hs 15 -p 12 -mmp 0.05 -act 26 - mhp 100 -bw 51 as recommended in its documentation. MOSAIK is a widely used reference-guided assembler that hashes the whole reference genome and locate information in the hash table using a 'jump database' [http://samtools.sourceforge.net/) [We use our method to study the pooled sequencing data related to type 1 Diabetes dataset (T1D) in  and compgenome hg with paratabase' . Then wege.net/)  to pileuge.net/)  to annotWe first present our results on the effects of various parameters on the estimation accuracy of the minor allele frequency using the EM algorithm. We then present the results on the power of SNP detection and association studies. Finally, we present our results on the analysis of the data in  using thf. Table f ≤ 0.1% or , which covers the typical situations of rare variant studies under current NGS technologies. Moreover, the advantage of the EM method increases as allele frequency f decreases and as sequencing error rate α increases, which is reasonable since it becomes more difficult for a naive estimate such as f ≥ 1%, the simple naive estimation method works reasonably well.We compare our EM estimate αstart = 1%, n = 3000, K = 100, G = 10, and a minor allele frequency of f = 1%. The upper left panel shows that f and ffrac, while f. The upper right panel shows the histogram of ffrac might be responsible for the majority of the variance of f and ffrac in the lower right panel quantitatively demonstrates the superiority of Figure f across all replications. The log values of the RE of all 288 simulations for both n in each pool, the number of chromosomes K and the number of pools G have little effect on the RE of f and the sequencing error rate α play a dominant role in affecting RE. To further explore their effects, we demonstrate the average effects of f and α on the RE by computing the average RE based on the values of f and α across all different  in Table α, the RE of f ; while fixing f, the RE of α. Thus, f is significantly lower compared to that of f ≤ 1%.We measure the bias of an estimator by the relative error (RE) defined as K, G, n, f, α) on the estimation accuracy of minor allele frequency f using the EM algorithm. We note that the estimation of β = f(0|1) is highly unreliable (data not shown). This phenomenon can be explained as follows. When minor allele frequency f is low, the expected number of chromosomes having the minor allele in each pool is also low. When the number of pools G is small, the estimation of β can be difficult with a small number of chromosomes carrying the minor allele. Thus, we do not show detailed results on the estimation of β. Despite the fact that β cannot be reliably estimated, the other two parameters f and α can be reliably estimated using the EM approach.Next we present our results for the effects of  = . The histograms of f and α are shown in Figure S2 of the additional file f, but the variance of f is relatively large. The source of this variance, however, is largely due to the variance of ffrac, rather than the algorithm itself, as shown in Figure S3 of the additional file ffrac. Thus, ffrac than of f. We also observe as a general trend that f and ffrac, and its variance appears to be affected less by α but significantly by f. This observation is also confirmed in Table f and α are shown.To study the effects of minor allele frequency 100, 10, 000. The κ% of its values and recalculate the values of MSE and Cg. The results on the modified measures are presented in additional file To reduce the effect of a few outliers of K, G, n) on the estimation accuracy of G and n as expected. However, the accuracy decreases with the number of individuals K in each pool.We also studied the effects of  on the power of SNP detection using the likelihood ratio approach for the case and the control samples, respectively. The number of reads in each pool (n) is set at either 1000 or 3000 as in the above simulations. We start from default values of the parameters  = . Then we change one of these parameters and keep all the other parameters at default. Figure We next study the effects of  on the power of detecting associations between SNPs and phenotypes using the likelihood ratio approach for the case and control samples together. The parameter setting is similar to that in the above subsection except that here we also let the relative risk λ to be 1.2, 2, and 4, respectively. Figure We also study the effects of  ratio. It is well known that transitions are much more frequent than transversions in evolution, and the number of transitions over the number of transversions, referred to as Ti/Tv ratio, in known SNPs is expected to be between 2 and 4 . Figure We also consider the top 150 ranked SNPs and the corresponding figures and tables are shown as Figure S11-S12 and Tables S7-S8 in the additional file -5. The p-value obtained through the likelihood ratio test reflects the true p-value better because it takes the variation in estimating the allele frequency into account.We then study the association of the identified variants with type 1 diabetes (T1D). We first look at the common SNPs with estimated minor allele frequencies above 1% in the controls as in  and wante - 7. This SNP was also found to be associated with a preliminary p-value of 5e - 7 in the original study [http://www.genome.gov/gwastudies/) to date. Nevertheless, other SNPs within the IL2RA gene were found to be associated with T1D by Cooper et al. [The SNP rs3184504 residing within gene SH2B3 has an EM p-value of 8.4r et al.  before tr et al. ,28 afterr et al.  used genr et al.  that wasFor rare polymorphisms In this paper, we developed an EM algorithm based unified approach for minor allele frequency estimation, SNP calling and association studies, applicable to pooled sequencing data where genetic materials of multiple individuals are pooled together. This study differs from previous studies in that we estimate sequencing error rate for each position while previous studies generally assume a pre-specified sequencing error rate across all sequenced regions. Since sequencing error rate depends on the genomic context, it is essential that the sequencing error rate be estimated specifically for different loci. In a pooling design without tagging, the origin of the reads is not known, and it is impossible to obtain the individual genotypes from the pooled data. Therefore, we modelled the pooled sequencing data as a "missing value" problem and designed an EM algorithm to estimate the minor allele frequency and sequencing error rate.We first studied the effects of minor allele frequency, sequencing error rate, number of pools, number of individuals in each pool, and the sequencing depth in each pool, on the estimation accuracy of the minor allele frequency. It was shown that the naive approach, which estimates the minor allele frequency by the fraction of observed minor alleles in the reads, can significantly over-estimate the true minor allele frequency, and that the effect is most severe for rare variants. The EM based algorithm, on the other hand, can estimate the minor allele frequency in a relatively unbiased manner. Although the variation of this estimation seems to be relatively large, a major part of the variation comes from the sampling of individuals from the population rather than the algorithm itself. We also show that the estimation accuracy of the EM algorithm increases with the number of pools and sequence depth as expected. However, the estimation accuracy decreases with the number of individuals in each pool, most likely because a more extensive pooling induces greater loss of information. Secondly, we used a likelihood ratio statistic based on the estimated parameters from EM to call SNPs. With the real data from , in termWe made several simplifying assumptions in our study. First and foremost, we did not consider errors introduced by mapping the reads to the reference genome. The mapping of Roche 454 data still has many challenges, in particular, in regions around homopolymers, and further development of algorithms for mapping is needed. Secondly, although we assumed that the amount of genetic materials from each individual is the same for each pool, this assumption can be violated. To overcome this problem, one approach would assume that the fractions of genetic materials from individuals follow a Dirichlet distribution . Thirdlyhttp://www-rcf.usc.edu/~fsun/Programs/EM-SNP/EM-SNP.html.Software can be downloaded from The authors declare that they have no competing interests.Both authors participated in the development of methodology, simulations, real data application, revisions, and manuscript preparation. Both authors read and approved the final manuscript.The publication costs for this article were funded by US NIH 1 U01 HL108634.BMC Genomics Volume 14 Supplement 1, 2013: Selected articles from the Eleventh Asia Pacific Bioinformatics Conference (APBC 2013): Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/14/S1.This article has been published as part of Supplementary materials. Supplementary methods and results.Click here for file


In classical pedigree-based analysis, additive genetic variance is estimated from between-family variation, which requires the existence of larger phenotyped and pedigreed populations involving numerous families (parents). However, estimation is often complicated by confounding of genetic and environmental family effects, with the latter typically occurring among full-sibs. For this reason, genetic variance is often inferred based on covariance among more distant relatives, which reduces the power of the analysis. This simulation study shows that genome-wide identity-by-descent sharing among close relatives can be used to quantify additive genetic variance solely from within-family variation using data on extremely small family samples.Identity-by-descent relationships among full-sibs were simulated assuming a genome size similar to that of humans (effective number of loci ~80). Genetic variance was estimated from phenotypic data assuming that genomic identity-by-descent relationships could be accurately re-created using information from genome-wide markers. The results were compared with standard pedigree-based genetic analysis.For a polygenic trait and a given number of phenotypes, the most accurate estimates of genetic variance were based on data from a single large full-sib family only. Compared with classical pedigree-based analysis, the proposed method is more robust to selection among parents and for confounding of environmental and genetic effects. Furthermore, in some cases, satisfactory results can be achieved even with less ideal data structures, i.e., for selectively genotyped data and for traits for which the genetic variance is largely under the control of a few major genes.Estimation of genetic variance using genomic identity-by-descent relationships is especially useful for studies aiming at estimating additive genetic variance of highly fecund species, using data from small populations with limited pedigree information and/or few available parents, i.e., parents originating from non-pedigreed or even wild populations. Estimates of additive genetic variance are commonly based on data from large pedigreed populations incorporating all known relationship information. Additive genetic relationships can be defined as twice the identity-by-descent (IBD) probability of two randomly drawn alleles, which can be estimated from pedigree data. The advantages of these pedigree-based analyses are that they do not require any knowledge about the genetic architecture of the traits and that the additive relationships are easily inferred from a known pedigree. However, these methods also have some major limitations. First, such analyses ignore relationships beyond those included in the known pedigree. Second, the assumed relationships are expected relationships  rather than actual relationships. In fact, the pedigree relationship is exact only under an infinitesimal model , i.e., aDue to linkage between loci within the same chromosome, parents tend to pass on long segments of DNA to their offspring. Hence, the "effective" number of segregating loci within a full-sib family will be much lower than the corresponding number for the whole population, even for species with a larger genome. For example, recent reports have indicated that the effective number of segregating loci among full-sib pairs in humans is only about 80 ,5. When en) within a family . Effective loci are defined as the number of independently segregating "loci" that would yield the same standard deviation of the proportion of genome shared among full-sibs as observed in real genomic data from human sib pairs [i, the IBD relationship of two full-sibs was therefore defined as 0 if none of the paternal and maternal "alleles" (haplotype blocks) were IBD, 0.5 if either their paternal or maternal "alleles" were IBD and 1 if both their paternal and maternal "alleles" were IBD. The actual relationship between two full-sibs was then defined as the average relationship across all "effective loci" . An example of the distribution of actual relationships in a large simulated full-sib family is shown in Figure e n= 80), the actual relationship matrix cannot be of full rank for large size families, which introduces numerical problems in data simulation and analysis. Therefore, the relationship matrix was forced to be positive definite by adding a small positive value (10-3) to each diagonal element variance structure).The IBD relationships were simulated so that they closely resembled the relationships estimated with real data for humans, and thus they were typical of species with relatively large genomes. Variation in IBD sharing was simulated using a model with 80 "effective loci"  and individuals 200-1000) with data . Hence, 1200 different datasets were generated and analyzed.Phenotypes were generated using the following model:μ is the overall mean, G is the actual IBD relationship matrix, I is an identity matrix of appropriate size, the Z-matrices are appropriate incidence matrices and G was set up over all "effective loci", while for Scenario 3, only the first "effective locus" was used to calculate G. The breeding values in a were then generated as:where L is a lower triangular Cholesky decomposition of G, and z is a vector of standard normal deviates of length N . This assumes that (1) genetic variance is evenly distributed across the genome, and (2) gene effects are normally distributed, or that the aggregated effect of many genes, i.e., the breeding values, are approximately normally distributed, even when individual gene effects are not . It is also assumes that the different founder alleles at an "effective locus" have unique allelic effects, because an "effective locus" contains many genes and thus contains a unique combination of alleles at these genes. All datasets were generated using the MATLAB® software http://www.mathworks.com.where The data sets were analyzed with the general linear modelXβ includes fixed effects of each family, or in absence of common environmental family effects, the overall mean only. If fitted, common environmental family effects were included as fixed effects due to the fact that the number of families included was very small (and the number of observations per family large) and thus the associated variance was difficult to estimate.where G was calculated over all genomic regions, i.e. it was assumed that the inheritance of the DNA segments from parents to offspring could be accurately traced using marker information, and genetic variance was evenly distributed over genome segments (irrespective of the simulation scenario). The genomic IBD animal model is equivalent to a genome-wide gametic model, i.e., a model in which the original gametes received from the sire and dam and their associated actual relationships are reconstructed using genomic data [In this model, the additive genetic effects were assumed: mic data . The relA is the numerator relationship matrix (inferred from the pedigree). Furthermore, since the classical model uses only between-family variation to estimate additive genetic effect variance, environmental effects common to full-sibs could not be included in the model for this simple data structure (irrespective of whether they were present or not).This is the classical animal model, assuming For both models, variance components were estimated with restricted maximum likelihood methodology using the ASREML software package .The genomic IBD model assumes that all animals are genotyped with a sufficiently dense marker map covering the entire genome. However, in some studies, selective genotyping of phenotypically extreme (high/low) animals within each family may be used to save costs. This may be a useful approach for QTL (Quantitative Trait Loci) detection, but our aim was to evaluate whether such data could be used to estimate quantitative genetic variation as well. In these analyses, we assumed a single family with 200, 500 or 1000 full-sibs, and for which only individuals with phenotypes deviating more than one residual standard deviation from the mean were genotyped. However, because including only the genotyped  animals in the analysis would probably yield overestimated variance components, the non-genotyped animals were also included in the analysis. For the analyses, genomic IBD relationships among genotyped individuals were combined with pedigree relationships of non-genotyped individuals in a common relationship matrix ,9.The estimated heritabilities (across-replicate means and standard deviations) for the different structures under Scenarios 1 and 2 are presented in Figures When assuming no common environmental variance (Scenario 2), the pedigree-based analyses were also unbiased but they were less precise than the genomic IBD analyses Figure . As expeN = 1000) family. However, single-family estimates based on smaller samples (200 or 500) tended to be overestimated, and the precision of the estimates were reduced compared to that with full genotyping , estimation of heritability was still unbiased for both the genomic IBD and the pedigree-based (more than one family) methods Figure . With lan = 2/h2. However, using the genomic IBD approach, this formula is no longer valid. For a genomic IBD analysis, including only a single large family will maximize the precision of the predicted Mendelian sampling deviations of all family members, and there is also no need to separate genetic and environmental family effects. Furthermore, the number of families is necessarily lower than the number of individuals, and prediction of Mendelian sampling deviations is therefore more informative than prediction of family means with respect to genetic variance. Estimation of genetic variance from family means  is also sensitive to selection of data (both within and among families), and failure to account for this is expected to give downward biased estimates of the genetic variance. In populations undergoing artificial selection, parents of the phenotyped animals are usually selected, and unbiased genetic analysis requires that the selection history is properly included in the data, i.e., the analysis should involve data on multiple generations, which is not always available. However, selection among parents will have little impact on the within-family genetic variance (in absence of inbreeding). Since the genomic IBD model uses mainly within-family variation, it is expected to be more robust to selection among parents.This study shows that tracing genomic IBD relationships using genomic information has clear advantages, not only for prediction of individual breeding values  but alsoN = 1000), although there was a tendency towards overestimated heritability values and less precise results for smaller family sizes. This may be explained by the fact that only the phenotypically most extreme individuals are genotyped and only these are informative with respect to partitioning of residual and Mendelian sampling variances.Even with selective genotyping, the genomic IBD model could estimate genetic variance relatively precisely for large families . For such species, a sire gametic model may be more relevant. In such a model, the sires' gametes are reconstructed and the actual relationships between them estimated. Genetic variance can then be estimated from variation among the sires' gametes , rather than variation among individuals (sires' and dams' gametes). In this model, genetic variation due to the dams' gametes will be included in the random residual term and genetic variance may be estimated from samples of offspring .The proposed method can also be generalized to IBD tracing in more complex pedigrees, i.e., beyond a single generation, allowing information from various types of relatives to be exploited by linkage analysis e.g., . More diG matrix. For instance, if a fraction q of the genome is not covered by markers, the total variance will also be underestimated by a fraction q when the single full-sib family design is used. When the design contains several families, the between-family variances are quite accurately predicted, even if part of the genome is not covered by markers, which will recover some of the underestimation. The underestimation may be completely recovered by including a polygenic effect in the model, which has a covariance structure equal to the pedigree-based relationship matrix, requiring several families of data.The proposed method will underestimate the total genetic variance in cases where a fraction of the genome is not covered by the markers, i.e., if some of the "effective loci" are not accounted for in the Scenarios 1 and 2 in the current study assumed that genetic variance is evenly distributed over genomic regions, as assumed in the genomic BLUP (GBLUP) model . The maiThe proposed genomic IBD method is particularly relevant for quantitative genetic studies aiming at estimating additive genetic variance of highly fecund species, using data on populations with limited pedigree information and/or few available parents. For example, genetic variance may be estimated based on a few full-sib-families with parents sampled from the wild or from non-pedigreed domesticated populations. In principle, the genomic IBD model  requires only a single large family for proper and accurate estimation of heritability for quantitative traits. In contrast, classical pedigree-based estimation requires the establishment of a sizeable pedigreed population consisting of numerous full- and (preferably) half-sib families to produce estimates with acceptable accuracy. Furthermore, the proposed genomic IBD model is expected to be less affected by selection among parents and will facilitate the separation of genetic and non-genetic family effects .The authors declare that they have no competing interests.JØ was mainly responsible for conception and design of the study, data analysis and writing of the manuscript. THEM contributed to the design of the study and to writing and revision of the manuscript. Both authors read and approved the final manuscript.