As genetics technology proceeds, practices of genetic testing have become more heterogeneous: many different types of tests are finding their way to the public in different settings and for a variety of purposes. This diversification is relevant to the discourse on ethical, legal and societal issues (ELSI) surrounding genetic testing, which must evolve to encompass these differences. One important development is the rise of personal genome testing on the basis of genetic profiling: the testing of multiple genetic variants simultaneously for the prediction of common multifactorial diseases. Currently, an increasing number of companies are offering personal genome tests directly to consumers and are spurring ELSI-discussions, which stand in need of clarification. This paper presents a systematic approach to the ELSI-evaluation of personal genome testing for multifactorial diseases along the lines of its test characteristics.This paper addresses four test characteristics of personal genome testing: its being a non-targeted type of testing, its high analytical validity, low clinical validity and problematic clinical utility. These characteristics raise their own specific ELSI, for example: non-targeted genetic profiling poses serious problems for information provision and informed consent. Questions about the quantity and quality of the necessary information, as well as about moral responsibilities with regard to the provision of information are therefore becoming central themes within ELSI-discussions of personal genome testing. Further, the current low level of clinical validity of genetic profiles raises questions concerning societal risks and regulatory requirements, whereas simultaneously it causes traditional ELSI-issues of clinical genetics, such as psychological and health risks, discrimination, and stigmatization, to lose part of their relevance. Also, classic notions of clinical utility are challenged by the newer notion of 'personal utility.'Consideration of test characteristics is essential to any valuable discourse on the ELSI of personal genome testing for multifactorial diseases. Four key characteristics of the test - targeted/non-targeted testing, analytical validity, clinical validity and clinical utility - together determine the applicability and the relevance of ELSI to specific tests. The paper identifies and discusses four areas of interest for the ELSI-debate on personal genome testing: informational problems, risks, regulatory issues, and the notion of personal utility. In discussions on ethical, legal and societal issues (ELSI) surrounding genetic testing, there is no longer any single satisfying definition of what constitutes 'a genetic test'. Practices of genetic testing are becoming more and more heterogeneous, not only with regard to the setting and purpose of testing, but also with regard to the technical aspects of the tests themselves. Some of these technical differences between genetic tests are ethically significant or have implications for legal or societal issues. Therefore, a clear understanding of the relevant test characteristics of genetic tests is a necessity for any meaningful discussion of the ELSI surrounding genetic testing.Over the last decades, new technologies for genetic testing have been developed that differ in many respects from those used in traditional clinical genetic testing for monogenic diseases. One important development is the advent of personal genome testing on the basis of genetic profiling for the prediction of common multifactorial diseases. Multifactorial diseases, such as cardiovascular diseases , age-relWith this paper, we aim to clarify the relations between the more technical characteristics of a genetic test and the ELSI with which the test is associated. We believe that a thorough understanding of the technical characteristics of personal genome tests themselves forms a necessary basis for all further ELSI-discussions in the field. Our focus on the test characteristics implies that, in this paper, we will not be able to discuss other aspects that are relevant to ELSI-discussions, such as characteristics of the diseases tested for, or the settings in which tests are offered. Although there are moral differences, for example, between the offering of personal genome tests by private companies and the offering of the same tests by public health care systems, or between testing for diseases for which there are treatment options available and testing for diseases for which there are no such options, these differences are not the main subject of this paper. As personal genome tests are currently offered almost exclusively in a direct-to-consumer context, we take that context as the background to our discussion.First, we will introduce the practice of personal genome testing. In the second section, we will distinguish and briefly discuss the following four key test characteristics of genetic testing: from targeted to non-targeted testing, analytical validity, clinical validity and clinical utility. The third section of the paper discloses and discusses four major areas of implications of these test characteristics for the ELSI-debate.Personal genome testing for multifactorial diseases is conducted on the basis of genetic profiling. In a genetic profile, multiple genetic variants are combined that are associated with increased or decreased risks for a particular multifactorial disease. Presently, single nucleotide polymorphisms (SNPs) are used within genetic profiles . SNPs arFor almost a decade, companies have been offering genetic profiles based on SNPs directly to consumers via the Internet. Initially, personal genome testing companies marketed single profiles for specific health conditions, or a limited set of profiles for related diseases . Today, 1 Whether providers make use of genome-wide SNP-analysis or whole-genome or exome sequencing technology, however, the prediction of common multifactorial diseases and other complex traits will continue to be based upon multiple genetic variants, and thereby upon the construction of genetic profiles. In this respect, therefore, the scope of this paper is wide and encompasses all potential forms of personal genome testing based on genetic profiling: current and future, commercial and clinical forms, including sequencing technologies.Other direct-to-consumer companies are starting to offer genetic profiling on the basis of whole-genome sequencing technology: the analysis of all three billion base pairs.There are four key test characteristics relevant to the ELSI-debate to be discussed: from targeted to non-targeted testing, analytical validity, clinical validity and clinical utility , and negative if it is absent  . In the 4 such that most current "genomic assays have high sensitivity and specificity for measured genetic variants"  What has changed, however, is the data that the company uses to calculate genetic risk. In May, deCODEme added six new genetic variations to its algorithm for assessing its customers' risk of having a heart attack, on the back of new research ."7The probability of receiving contradictory results over time is quite high. A modeling study on genetic profiling for type 2 diabetes has shown that the update from one relatively strongly predictive SNP by an additional 17 less predictive SNPs, causes 34% of the study's population to switch risk categories either from above average risk to below average risk or vice versa . Due to In this section, the implications of the test characteristics of personal genome tests for the discourse on ELSI will be discussed. As shown in section II, 3, the test characteristic of analytical validity is no longer a topic of major concern in the ELSI-debate since the advent of a new generation of personal genome testing services. Therefore, it will not be discussed any further. The three remaining test characteristics (together) do lead to ELSI that are of importance to the current debate on personal genome testing in a direct-to-consumer context: informational problems, risks, regulatory issues and notions of utility.non-targeted genetic profiling: quantity, complexity, and fluidity of information. The informational problem is associated with the practice of non-targeted genetic profiling itself, whether within or outside the clinic, now or in the future.8The most important ELSI-issues in personal genome testing are related to information. Within the ELSI-debate, it has already been argued that priority ought to be given to informational problems; critics have stated, for instance, that both public and professional institutions ought to take up the responsibility to inform the general public, to raise awareness of the risks of direct-to-consumer genetic testing , and to Discussions of the difficulties surrounding the provision of genetic information are not new: in clinical genetic testing, patients are routinely offered extensive genetic counseling prior to consenting to undergo genetic testing. During counseling sessions, the patient receives detailed information about the disease, the genetic component thereof, the testing procedure, possible outcomes, therapeutic options, implications for reproductive choices and possibilities, consequences for the family, the communication of possible risks to relatives, social implications, privacy issues, potential adverse effects on employment and insurance, etc. . Ideallyexceptional quantities of information on dozens to over a hundred different diseases. It will be very arduous, if not impossible, to inform patients or consumers beforehand in detail on all relevant aspects for so many diseases without inducing information overload and therewith foregoing the actual aim of informed consent . Fal. Fal25].yndromes . Hypotheyndromes . Thus, wFinally, there are at least two perceived societal risks involved in personal genome testing of low clinical validity: indirect economic risks, and loss of public confidence in genetics research and applications. Firstly, on the basis of personal genome test results, consumers may turn to their physicians for advice, follow-up research or medication. As the clinical validity of test results are uncertain, most of the follow-up will be unnecessary while it does drive up the costs of public health care 65]. Emp. Emp65].12 In spite of widespread concern among ELSI-researchers, we therefore think that the fears of discrimination and stigmatization are not justified in the context of present-day personal genome testing.13 We agree, however, that the risks of discrimination and stigmatization will again be of relevance to the ELSI-debate if personal genome testing gains sufficient clinical validity and becomes capable of reliably discriminating between individuals with high and low risks of developing multifactorial diseases.Although personal genome testing implies its own potential societal risks, there is at least one such risk that has always been paramount to the ELSI-debate on clinical genetic testing, but that lacks ground in the context of genetic profiling for multifactorial diseases: the risk of discrimination or stigmatization. Genetic profiles of limited clinical validity will not be of interest to insurance companies or employers due to their limited utility for the purposes of risk stratification . Indeed,A third main ELSI associated with personal genome testing for multifactorial diseases is the legal and societal issue of regulation. The regulatory issue entails the ethical question whether it is morally justifiable to offer genetic tests of limited clinical validity and utility to the public, and if so, on what conditions. Within the regulatory debate, there are roughly two perspectives: the first emphasizes the value of protection and the second that of autonomy and consumer choice. From within the first perspective, there have been calls for enhanced government oversight and regulation -72, wherinformation at all. Personal genome tests are considered to be flawed as medical tests, or even as informational products , and, and78],15 Thus, the notion of personal utility of such tests could be questioned, as it may be based not so much upon considered valuations of consumers, but rather upon misconceptions that could partly be rebutted through the provision of information. In the absence of any clinical validity, we think that personal approaches to clinical utility, especially in the context of testing for disease risks, are unjustified.Critics have warned against the emphasis on the recreational value of personal genome testing: genetic tests cannot and must not be said to be purely (or even primarily) recreational when in fact they inform on (among other things) risks for serious medical conditions . They haOn the other hand, in the presence of sufficient, reasonable or increasing levels of clinical validity, we believe that a personal approach toward utility may indeed be sensible: consumers may wish to decide for themselves whether informative non-targeted genetic profiling is valuable for them, and in what way. The high standards of clinical utility that are used within public health care evaluations need not be identical to the standards applicable to individual consumer valuations of personal utility. For example, consumers may find personal utility in knowing their genetic risk for Alzheimer's disease, despite the absence of preventive options. This issue deserves further elaboration, which is beyond the scope of the present article.For a well-informed and meaningful discourse on the ethical, legal and societal issues (ELSI) of present-day personal genome testing for multifactorial diseases, it is important to clarify the relevant test characteristics of personal genome tests. Test characteristics that are most essential to the current ELSI-debate are the following: non-targeted testing, high analytical validity, limited clinical validity, debatable clinical or personal utility, and the quantity, complexity and fluidity of the generated personal genetic risk information.Non-targeted personal genome testing yields a vast amount of information that is complex and probabilistic, sometimes for a dozen to over a hundred multifactorial diseases simultaneously. Further, test outcomes may change over time as providers include additional genetic variants in their algorithms. Quantity, complexity, and fluidity of genetic information together pose urgent problems with regard to the provision of information and informed consent. Providers of personal genome testing are facing these informational problems at several moments within the testing process: pre-test informed consent, post-test delivery of test results, and post-test dealing with future  findings and changing interpretations. There is a pressing need for well thought-out models for valid informed consent and information provision in the context of a lot of complex and fluid information in non-targeted personal genome testing.Since personal genome testing is increasingly based on highly accurate and reliable genome-wide SNP-scanning technology and performed in high-quality laboratories, the test characteristic analytical validity has moved away from the centre of ELSI-discussions. Current debates are focused rather on the clinical validity and utility of genetic profiles for multifactorial diseases, which vary strongly, but are likely to increase given time. Awareness of the currently limited clinical validity is at the basis of both conservative and liberal stances within the regulatory discourse: it is used either as an argument in favor of stricter regulation, or as an argument against it.The notion of clinical utility is challenged by personal approaches towards the significance and usefulness of genetic information. It is far from impossible that consumers continue to attribute personal utility to genetic information and pursue the acquisition of their genomic data even after having been informed about the current clinical limitations of genetic profiling for multifactorial diseases. Standards of clinical utility that are used for public health evaluations, however, need not be identical to those used for individual valuations of utility.As a consequence of their limited clinical validity, present-day personal genome tests for multifactorial diseases have a much lower potential for adverse psychological effects than do clinical genetic tests for monogenic diseases. Neither do they imply as many health risks, or societal risks, such as discrimination, stigmatization and misuse of genetic information by insurance companies or employers. This holds true only on the condition that the general public as well as other stake-holding parties are sufficiently informed to understand the limitations to the clinical validity and utility of genetic profiling for multifactorial diseases, and are willing to act accordingly. In the future, as genetic profiles will attain more discriminative ability, both traditional psychological risks and concurrent health and societal risks will again be of concern to the discourse on ELSI.The applicability and the relevance of ELSI-issues to the discourse on personal genome testing will fluctuate with the analytical and clinical validity of genetic profiles, with their clinical utility and with their being targeted or non-targeted. Thus, consideration of test characteristics is indispensible to any valuable ELSI-debate on personal genome testing for multifactorial diseases.The authors declare that they have no competing interests.All authors have contributed substantially to the conception and design of the manuscript. EB has drafted the manuscript. Both MS and AJ have critically revised it. All authors have read and approved the final manuscript.http://www.knome.com/ 1. To the knowledge of the authors, the company Knome (pronounced as 'know me') is the only direct-to-consumer provider of whole-genome sequencing that offers (among other tests) risk profiling for multifactorial diseases. See: http://www.navigenics.com or Pathway Genomics at http://www.pathway.com 2. See for example Navigenics at http://www.23andme.com . Currently, the company 23andme offers risk profiles for 195 diseases and other phenotypic traits, but the number of traits tested for increases monthly. The company also provides ancestry and carrier status information. While most other direct-to-consumer personal genome testing companies are currently offering scans for no more than a few dozen diseases, they are likely to expand their services in the future rather than restrict them.3. See http://www.knome.com . With the Clinical Laboratory Improvement Amendment (CLIA) of 1988, the US government has set quality standards for all laboratory tests, ensuring their accuracy, reliability, and timeliness.4. Most US-based companies collaborate with CLIA-certified laboratories, see for example https://www.23andme.com/you/faqwin/dataaccuracy/. 5. The company 23andme states that it makes use of a chip which demonstrates over 99.9% reproducibility: "This means that if [the laboratory] ran the same DNA a second time on a new chip, more than 99.9% of the data would be the same compared to data from the first run" https://www.23andme.com/health/Age-related-Macular-Degeneration/techreport/, http://www.arcticdx.com/ ). Relatively strong predictive abilities such as those in genetic profiling for AMD are far from typical for common multifactorial diseases.6. There are other diseases that are more promising for predictive genetic profiling: SNPs have been found to be associated with almost 3-fold risks for age-related macular degeneration (AMD). Genetic profiles for AMD have already been made available online .10. See for example 23andme at https://www.23andme.com/health/Colorectal-Cancer/ 11. See 12. Respondents were US cancer genetics professionals involved in highly predictive genetic testing for familial cancer syndromes. Respondents may be assumed to consider the risk of genetic discrimination to be lower in case of genetic profiling in the context of multifactorial diseases.13. It should be noted that misuse and abuse of genetic risk information by employers or insurance companies cannot be excluded completely, for employers and insurance companies are susceptible to misinterpretation of genetic test results, and to overstatement of their significance.http://www.pathway.com .14. See for example Pathway Genomics at 15. These studies have been conducted in the context of genetic testing for highly predictive single-gene makers, such as BRCA1 and BRCA2. Women had been informed, for example, that genetic test results are often inconclusive and that they are of unclear significance in the absence of a family history of breast cancer ,70. As tThe pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1472-6939/12/11/prepub


The availability of high-density single nucleotide polymorphisms (SNPs) data has made the human genetic association studies possible to identify common and rare variants underlying complex diseases in a genome-wide scale. A handful of novel genetic variants have been identified, which gives much hope and prospects for the future of genetic association studies. In this process, statistical and computational methods play key roles, among which information-based association tests have gained large popularity. This paper is intended to give a comprehensive review of the current literature in genetic association analysis casted in the framework of information theory. We focus our review on the following topics: (1) information theoretic approaches in genetic linkage and association studies; (2) entropy-based strategies for optimal SNP subset selection; and (3) the usage of theoretic information criteria in gene clustering and gene regulatory network construction. Following the completion of human HapMap project and the development of high-throughput genotyping biotechnologies, SNPs have become the primary source to unravel the genetic roots of complex diseases which has led to the flourishing of vast amount of SNP data analysis methodologies . Geneticenotypes . Howeverenotypes . The estenotypes , 8, 9. Despite great successes achieved by GWAS so far, only a small proportion of the heritability of complex traits has been explained , 11. OneWith the development of next-generation sequencing technology, more dense SNP markers are generated, raising the hope for the discovery of rare variants associated with complex diseases. However, current bottleneck in genetic association studies is not limited by genotyping technology, but rather limited by high computational cost and the lack of efficient and powerful statistical methods. The underlying gene action mode is never clear to us, leaving us great difficulty in choosing right models to fit the data. The information theory, which initially emerged in the 1940s  to quant2X with a set of possible values {1x, 2x, nx}, and corresponding probability distribution p(ix) = P (X = ix), i = 1, 2, ..., n, entropy of X is defined asWe present some key concepts of information theory in this section. Information theory quantifies the uncertainty of a random variable by its entropy section . Given aX and the entropy is maximized when all the possible values of X are equiprobable. When using the base 2 logarithm (b = 2), the entropy is measured in bits. The higher the entropy, the higher the uncertainty we may predict the outcome of X and Y is defined analogously as, p = P refers to the joint distribution of the two variables. An important property of entropy is subadditivity: H ≤ H(X) + H (Y), equality holds if and only if X and Y are statistically independent. Joint entropy of two discrete random variables Mutual information is defined as the information common to both random variables,I  provides the information of X contained in Y, or vice versa. I = 0 when X and Y are statistically independent. The uncertainty remaining in r.v. X given the knowledge of r.v. Y is defined as the conditional entropyY given theknowledge of r.v. X is measured by conditional entropy I = H(X), i.e. the mutual information of a r.v. X with itself indicates the decrement in the uncertainty of r.v. X given the information of itself. Thus entropy is also referred to as self-information. Fig.(1) shows a Venn diagram of I , H (X), H (Y). The left and right ellipses indicate the magnitude of the marginal entropies H (X), H(Y), respectively, and their overlap corresponds to the magnitude of the mutual information I .X and genotypes at marker B Y as, then p(·), q(·) denote corresponding genotype frequencies. The marginal entropy of marker A and marker B are then defined byAs an illustrative example of entropy in genetic studies, let’s consider two bi-allelic SNP markers. Suppose alleles of the two SNPs (SNP A and SNP B) are A, a and B, b, respectively, so there are three genotypes at the two markers. Denote genotypes at marker A as 33.1One of the major issues of single-SNP based analysis is little power in detecting variants with small/moderate effect size or SNPs with small minor allele frequency (MAF). Entropy takes the nonlinear transformation of allele frequencies. When testing for allele frequency difference between cases and controls, this nonlinear transformation can potentially increase the testing power , 26. Wit3.22 test for GWAS via the nonlinear transformation of haplotype frequencies. Furthermore, Zhao [Haplotypes are combinations of marker alleles that are closely linked on the same chromosome and are inclined to inherit as a unit. They are essential to decipher the genome-wide human LD pattern. In the association study of complex disease gene mapping, the haplotype-based approaches are statistically more appealing than those single locus based approaches in certain scenarios. Haplotype association analysis can be conducted under a population-based or family-based design . Considere, Zhao  systematre, Zhao  are equi3.3Genes are the functional units in most living organisms. The sequence information and function of genes are highly consistent across diverse populations, which makes the gene-based association study more robust in terms of replication compared to the single SNP-based analysis . MoreoveCui ’s method can be easily extended into other scenarios. For example, one can consider multiple SNP variants that are physically located close to each other on a chromosome band, or one can extend the gene idea to a set of genes or pathways and assess the association from a gene-set or pathway level. The gene-set or pathway information can be pre-defined, such as from the KEGG pathway  or Gene 3.4In population-based association studies, all samples cannot be guaranteed to come from a homogeneous population. When individuals are sampled from different populations and if different populations have different disease allele frequencies, false positive or false negative could be observed if the population substructure is not properly taken care of in the analysis . This pr3.5Quantitative trait loci (QTL) mapping has been a powerful tool in elucidating the genetic basis of complex traits . In a ty3.6It is commonly recognized that gene-gene (G×G) interaction or epistasis, and gene-environment (G×E) interaction play key roles in determining the risk of complex human diseases , 38. Thep and (1 - p) are the proportions of cases and controls in a sample, respectively. Then the disease entropy condition on a SNP can be calculated by adding the sub-entropy corresponding to all the possible genotypes. This approach not only can discover the G×G interaction effects by a package called ESNP2-S as most of the other method did on epistasis, but also can identify the optimal model from all two-locus genetic interaction models. For a case-only design, Kang [From an information content perspective, Moore  developegn, Kang  developeGET to test gene-environment interaction, and iEGT to test the genotype and environment interaction. They showed that the information theoretic approach is more powerful than the logistic regression approach. They found that p-values calculated from GET are smaller than those obtained by iEGT in general, which is due to the difference in the amount of information used by GET and iEGT. This evaluation provides a new framework for the study of G×E interaction and could be extended to a genome-wide association analysis. Fan [For the study of G×E interaction, Wu  investigsis. Fan  proposed3.7X1 and X2 with regard to the phenotype Z is defined as, I and I are zero, then the relations between X1 and X2 are defined as conditional independence and activity independence, respectively [I and I are equal. Schneidman [X1; X2 ... Xn} and a phenotype Z as,iiS = {X1, X2 ... Xn} and ∩iiS = ø. The generalization from bivariate synergy to multivariate synergy is consistent with the intuitive idea that synergy is the extra information conveyed to the phenotype by comparing the information contribution between the entire set and the maximum information partition of the entire set. To identify a set of variants in a gene, a gene set or a pathway associated with complex diseases, one needs to model the joint effect of those variants as a system. Synergy analysis provides a novel strategy for this purpose. The bivariate synergy between two SNPs hneidman  defined ectively . Note thhneidman  systemathneidman  later onAnastassiou  elaboratThe synergy analysis opens an alternative framework for the evaluation of G×G interaction. Under the gene-based association analysis , all va4SNPs are the most abundant genetic variants in human genome. The emergence of the high-throughput genotyping technology has made the fast genome-wide SNP association analysis possible. The genotyping cost is no longer prohibitively high with the advancement of the high throughput genotyping techniques. Nevertheless, serious multiple testing problems are triggered when all recognized SNPs are genotyped. Recent studies have revealed that SNPs in the human genome are distributed as block like patterns , 48. SNPrd step into consideration. For example, those summarized in (Table 1) of Weale [Halldórsson  commenteof Weale  select t2r, the coefficient of determination [X as a random variable of haplotype and jX as a random variable of allele at the jth SNP, Liu and Lin [H(S) and H(X) are the joint entropy of the subset and the complete set, respectively; and E is the relative entropy. It has been shown that ER successfully captures the pattern of complete LD. In contrast to the above two measures which estimate the LD pattern across a whole chromosome region, Zhang [Commonly used criteria for evaluating tSNP subsets within a haplotype block includes: the ratio of the haplotype diversity explained by the subset, and the association between the complete set and the tagging set, which is measured by mination . The secmination  first timination  based onmination  claimed  and Lin  proposedn, Zhang  developeIn addition to the above mentioned tSNP selection methods using information theory, several other works have also shown the benefit of using entropy in tSNP selection. Hampe  fused thAlthough informative SNP selection based on haplotype block structure has been intensively studied, Halldórsson  argued t5Information theory has also been applied in a number of other applications in genetics and genomics study. For gene clustering analysis, Zhou  proposedFor network inference, Margolin  designed6Last decades has visualized the great success in genome-wide association studies with common SNP markers. Association analysis in an information content framework has achieved large success and caught researchers’ attention in the past. Its applications in a variety of areas in genetic and genomic studies have led to the advancement of our knowledge about the genetic machinery of complex diseases. With the next wave of genetic association studies in gene-sets or pathway levels, more novel variants are expected to be discovered and insightful interpretation of disease signals could be reached. However, how to deal with the every-increasing dimension of SNP variants and extend them to the gene- and/or pathway-based analysis represents daunting challenges. We have recently developed an efficient approach in selecting potential SNP variants and further incorporated them into a gene- or pathway-based analysis . This seThe recent breakthrough in next-generation sequencing technology has made it possible to identify even more dense SNPs with a much larger range of MAF, especially small MAFs. These so called rare variants stimulate another wave of association analysis aimed to detect genome-wide rare variants contributing to common diseases . With th


TDT), multivariable test, combined TDT, and a likelihood ratio test based on a random-effects model. We find that the likelihood ratio test is most powerful in a wide range of settings and there is no negative impact to its power performance when common variants are also included in the analysis. If deleterious and protective variants are simultaneously analyzed, the likelihood ratio test was generally insensitive to the effect directionality, unless the effects are extremely inconsistent in one direction.Despite the success of genome-wide association studies (GWASs) in detecting common variants  many suggested that rare variants also contribute to the genetic architecture of diseases. Recently, researchers demonstrated that rare variants can show a strong stratification which may not be corrected by using existing methods. In this paper, we focus on a case-parents study and consider methods for testing group-wise association between multiple rare (and common) variants in a gene region and a disease. All tests depend on the numbers of transmitted mutant alleles from parents to their diseased children across variants and hence they are robust to the effect of population stratification. We use extensive simulation studies to compare the performance of four competing tests: the largest single-variant transmission disequilibrium test ( In human medical genetics, one often hypothesizes that genetic susceptibility to common diseases such as diabetes is mainly due to the alleles that have moderate frequencies in the population. However, there is increasing evidence showing that there is extreme allelic and locus heterogeneity and multiple rare variants underlie susceptibility to such diseases. In detecting associations with common variants, two approaches are often used. One approach is the single-variant test with family-wise error rate controlled by a multiple testing correction . Another approach is to use a multivariable method for testing all variants simultaneously. However, application of either approach with multiple rare variants involves multiplicity  and data sparseness, which will reduce power. To solve this problem when multiple rare variants are expected to jointly influence disease risk, an often-used approach is to group the variants according to some identified function and to test the combined effect of multiple rare variants. This motivates several collapsing methods, with or without weighting, to enrich association signals and at the same time reduce degrees of freedom . In the case of detecting association with multiple rare variants, we show these methods continue to be robust.Collapsing methods were mainly designed for detecting rare variants in case-control studies. However, recently, Mathieson and McVean TDTTDT based on combining the numbers of transmitted mutant alleles across all variants from parents, called combined TDT  holds. Thus The same probability but not transmitting the mutant allele is given byThus, given the heterozygous parent with respect to the Under the null association, this probability equals to 1/2. In this paper, we focus our discussion on the dominant genetic model  mutant alleles, usually minor alleles, across all variants to enrich the signals of mutation and simultaneously reduce the degrees of freedom. Several versions of the combining method also can be considered in this regard. However, one simple and yet more powerful version is to combine the numbers of transmitted and non-transmitted mutant alleles across all variants by defining Under the null hypothesis, the asymptotic distribution of jth variant, we note that At the 0: We note that To infer parameter To evaluate the performance of four tests: In all simulations, we randomly selected Given the disease risks for all functional variants, haplotype frequencies, and disease prevalence rate equal to 1%, we determined the disease status of each offspring and sampled families until a pre-specified number of case-parents trios (500–2000) was reached. In order to evaluate type I error rate of each test under the null hypothesis, we considered regions with , with TDT is defined to enrich the association signals across variants, while the single-variant and multivariable tests are not. The likelihood ratio test based on a random-effect model is a novel test derived from the use of a completely different concept. It is designed to test zero proportion of functional variants in a gene region. Other tests were designed to compare the genotype frequencies of the “case” and “control” populations. We have compared the performance of the four tests: Although the tests considered depend only on the transmitted and non-transmitted genotype vectors, they are defined in different ways or use different concepts. The combined In our computation of p-values, a permutation method based on random permutation of the transmitted and non-transmitted genotype vectors within each family is used; Shi et al. A number of studies have shown that alleles with a wide range of frequencies are involved in disease etiology The disadvantage of the usual collapsing methods is that they are not powerful when the effects are not all in the same direction. We also have investigated the properties of the competing tests when the effects are in different directions. In general, if the proportion of the variants that are protective is increased, then the power of the tests is decreased. However, if the number of family trios is increased, then the power loss can be reduced. TDT is based on the equal weighting strategy. Madsen and Browning The definition of the combined TDT (Sun et al. Note that the family study requires genotypes of the diseased individuals and their parents. Unfortunately, parental genotypes may not be always available. Several methods had been proposed to solve the issue of missing parental genotype, for example, using EM algorithm with the assumption that genotype missing is random conditional on the known genotypes of other family members; see the discussion in Chen and Cheng In this paper, we report only the simulation results under the multiplicative genetic model. The unreported power under the dominant genetic models only differs slightly but basic conclusions remain unchanged. One major reason is that the subject cases have only very small chance of being homozygous for the mutant allele, because of low MAF. For the same reason, we also find that all competing tests have very small power in detecting rare variants under the recessive genetic model.


However, the proportion of variance accounted for in validation samples is much smaller . While such 2R values vastly exceed what has been previously reported using a reduced number of pre-selected markers (<0.10), given the heritability of the trait (∼0.80), substantial room for improvement remains.Despite rapid advances in genomic technology, our ability to account for phenotypic variation using genetic information remains limited for many traits. This has unfortunately resulted in limited application of genetic data towards preventive and personalized medicine, one of the primary impetuses of genome-wide association studies. Recently, a large proportion of the “missing heritability” for human height was statistically explained by modeling thousands of single nucleotide polymorphisms concurrently. However, it is currently unclear how gains in explained genetic variance will translate to the prediction of yet-to-be observed phenotypes. Using data from the Framingham Heart Study, we explore the genomic prediction of human height in training and validation samples while varying the statistical approach used, the number of SNPs included in the model, the validation scheme, and the number of subjects used to train the model. In our training datasets, we are able to explain a large proportion of the variation in height ( While previous genome-wide association studies have implicated numerous loci associated with complex traits, such loci typically account for a very small proportion of phenotypic variation. However, a recent study using height as a model trait has illustrated that common single nucleotide polymorphisms can explain a large amount of genetic variance when evaluated through whole-genome statistical models. However, it is unclear to what extent higher proportions of explained variance will translate into improved predictive accuracy in future populations. Here we evaluate the predictive ability of whole-genome models for human height while varying the modeling approach, the size of the training population, the validation design, and the number of SNPs. Our results suggest that whole-genome prediction models can yield higher accuracy than what is commonly attained by models based on a few selected SNPs; yet, given the heritability of the trait in question, there exists room for improving prediction accuracy. While gains in predictive accuracy are likely to be small based on more expansive genotyping, our results indicate that more substantial benefits are likely to be gained through larger training populations, as well through the inclusion of related individuals. Despite the success of GWAS in the discovery of many novel disease variants, the variants identified as being statistically significant typically account for minimal fractions of the genetic variance, even for highly heritable traits Typically, predictive models from GWAS are constructed using a small number of Single Nucleotide Polymorphisms (SNPs) that have been pre-selected using extremely low p-values derived from single-marker regressions. This approach is most sensible under the assumption that only a few loci affect the trait of interest; however, it performs poorly for complex traits Using a Whole Genome Prediction (WGP) method, the authors from the Yang Study estimated that common SNP variation (through Linkage Disequilibrium (LD) with causal polymorphisms) explained 45% of the phenotypic variance, thus accounting for more than 50% of the expected heritability of height, which is reported to be approximately 80% The results of the Yang Study are particularly exciting due to their implications for eventual application to preventive and personalized medicine. However, a remaining question is the extent to which WGP methods improve the prediction of yet-to-be observed phenotypes, given the distinction between proportion of variance accounted for (as a measure of goodness of fit) and predictive accuracy . HeritabH and GY, respectively, producing altogether three separate models. Goodness of fit was evaluated by means of the estimated residual variance and the proportion of variability accounted for by the fitted model in the training (TRN) dataset, Using data from the Framingham Heart Study As the number of markers increases, DIC decreases, indicating that information is continually being added to the model. This conforms with expectations under an infinitesimal model where the proportion of variance at Quantitative Trait Loci (QTLs) accounted for by regression on SNPs should increase with marker density Approach A- 10-fold cross-validation (CV) with assignment of individuals to folds at random. Because of the multiple generations present in the Framingham dataset, it is possible for children to be used to predict their parents in this design, which does not correspond to a standard prediction problem. To avoid this situation, we employed Approach B- using parents to predict children, we constructed a training dataset (TRN) with 1,493 parents and a testing dataset (TST) comprising offspring . Because of the structure of the data, the size of the training sample used in Approach B is much smaller than that used in Approach A. Theory and empirical evidence Approach C- randomly split the sample 10 times into TRN  and TST sets . Therefore, Approaches B and C differ in the way individuals were assigned to TRN and TST sets but not on the size of the TRN set. While approaches A and C allow for replicate datasets (10 in this study), Approach B is constrained to one replicate. As an aside, replicate datasets yielded highly similar To evaluate predictive ability, we used three different validation designs. chniques . Predicti is in the testing population and individuals j is in the training population, and zero otherwise. Using this score we classified individuals into four groups , and therefore larger LD spans, have reported high accuracy with much sparser coverage In all validation designs, we found that predictive ability increased with the number of SNPs, suggesting that a large number of SNPs are needed to capture genetic variance at QTLs. These results are similar to findings in the animal breeding literature for infinitesimal traits −3; suggesting that prediction accuracy may be harmed by including a large number of markers in a predictive model. However, an important difference between this study and ours is that in the former, marker effects were estimated using a fixed effects model while we use a Bayesian mixed model framework where all unknowns are modeled as random effects. Unlike the fixed effects approach, the Bayesian mixed model framework induces a shrinkage of estimates which, to some extent, controls over-fitting and seems to prevent a reduction in predictive ability in models with p≫n.A recent study H or GY where all markers are equally weighted.Importantly, we found no drastic differences between any of the statistical methods we considered. This is not surprising given that all three methods are based on an underlying additive model and that height likely conforms to an infinitesimal architecture. Moreover, these results are in agreement with findings reported in the animal breeding literature e, the genetic architecture of the trait, and the degree of propinquity between individuals whose phenotypic outcomes are to be predicted and those used to train the model. For nominally unrelated individuals under an infinitesimal model for a trait with 2h = 0.8, Goddard and Hayes 2×0.8 ), requires training samples of approximately 4,000 and 50,000 individuals, respectively. However, as our results illustrate, prediction accuracy can be increased substantially by using information from related individuals.Theoretical Simulation Clearly, there exists a redundancy between the information conveyed by the pedigree and that provided by markers. However, this redundancy is not complete and there may be benefits to incorporating pedigree and marker information in the model. For example, Vazquez et al. (2010) used data from US Holsteins to quantify the prediction accuracy using pedigree-based predictions, marker based WGP, and predictions combining pedigree and markers. The study confirmed the superiority of marker-based models (with a correlation of 0.42 for pedigree-based predictions and 0.649 for the marker-based predictions in CV) and found that, when more than 10,000 markers were available (for a Holstein sample), combining pedigree and molecular marker data was no better than using marker data only. This suggests that dense markers are able to capture genetic similarity due to recent family history as well as other sources of genetic similarity not described by pedigrees. Therefore, we speculate that the largely incomplete pedigrees of most humans will provide little to no additional information for the prediction of complex traits, especially given the high density of markers typically available.A pertinent question is whether a WGP model fitted to one population can be used to predict phenotypes in a distantly related population; this remains, so far, an un-answered question Population structure, admixture, or other population features can lead to spurious associations and affect prediction accuracy; therefore accounting for these features has been an important focus for GWAS analyses In conclusion, WGP methods provide a promising approach for the prediction of complex traits. The results of the Yang Study and those reported in this study both support this conclusion: they account for a larger proportion of the expected genetic variance and, as our study indicates, are able to predict yet-to-be observed phenotypes with greater success. Yet, it is apparent that predictive ability depends to a large part upon how many close relatives are included while training the model, and there is an apparent need for improving the accuracy of predictions of nominally unrelated individuals. Therefore, while whole-genome prediction of complex human traits can yield more accurate predictions than those based on models using a reduced number of markers, prediction of such traits remains difficult and significant room for improvement exists.http://www.ncbi.nlm.nih.gov/projects/gap/cgi-bin/study.cgi?study_id=phs000007.v3.p2. SNPs with call rates less than 90% and with a minor allele frequency (MAF) less than 3% were excluded. The remaining missing genotypes were imputed by sampling from a Binomial distribution using the empirical MAF estimate under the assumption of Hardy-Weinberg Equilibrium.Subjects were genotyped using the Affymetrix GeneChip Human Mapping 500K Array Set. For details on genotyping, see i, g = {ig} is a vector of genetic values, and In all models, age and sex-adjusted height of individual i,j computed from marker genotypes and H) and the Yang Study (GY) to generate G from the marker data. In method GY, relationships are standardized so that the average diagonal value equals one. In order to make the genetic variance parameters comparable, this standardization was also applied to GH by dividing the entries of G by the average diagonal value.All models were implemented in a Bayesian framework with inferences based on the posterior distribution of the unknowns given the data. Models differed in the number of markers used and the way they were incorporated into http://www.R-project.org). We specified the hyper-parameters in To estimate the remaining model parameters, we utilized a Bayesian approach by assigning prior distributions to thl marker. Marker effects were inferred using the Bayesian LASSO (BL) of Park and Casella H and GY in which all markers are equally weighted. The joint posterior distribution given by In a third model, genetic values were described as a linear regression on marker covariates: estimates Click here for additional data file.


Recently there has been great interest in identifying rare variants associated with common diseases. We apply several collapsing-based and kernel-based single-gene association tests to Genetic Analysis Workshop 17 (GAW17) rare variant association data with unrelated individuals without knowledge of the simulation model. We also implement modified versions of these methods using additional information, such as minor allele frequency (MAF) and functional annotation. For each of four given traits provided in GAW17, we use the Bayesian mixed-effects model to estimate the phenotypic variance explained by the given environmental and genotypic data and to infer an individual-specific genetic effect to use directly in single-gene association tests. After obtaining information on the GAW17 simulation model, we compare the performance of all methods and examine the top genes identified by those methods. We find that collapsing-based methods with weights based on MAFs are sensitive to the “lower MAF, larger effect size” assumption, whereas kernel-based methods are more robust when this assumption is violated. In addition, many false-positive genes identified by multiple methods often contain variants with exactly the same genotype distribution as the causal variants used in the simulation model. When the sample size is much smaller than the number of rare variants, it is more likely that causal and noncausal variants will share the same or similar genotype distribution. This likely contributes to the low power and large number of false-positive results of all methods in detecting causal variants associated with disease in the GAW17 data set. To date, genome-wide association studies (GWAS) have been successful in unveiling many common single-nucleotide polymorphisms (SNPs) associated with common diseases, including type 1 and type 2 diabetes, rheumatoid arthritis, Crohn’s disease, and coronary heart disease . HoweverThe advent of next-generation sequencing technology has offered great opportunities for discovering novel rare variants in the human genome, associating these rare variants with diseases, and increasing our biological knowledge of disease etiology. In particular, as pointed out by Choi et al. , proteinIn the GAW17 mini-exome data set , most ofWe apply these methods to each of the genes in the GAW17 unrelated individuals data set to identify genes associated with the given traits , adjusting for the effects of environmental covariates . The results from these methods are compared. In addition, for each given trait, we use the Bayesian mixed-effects model to estimate the phenotypic variance that can be explained by the given environmental and genotypic data and to infer an individual-specific genetic effect to use directly in single-gene association tests.iX denote the vector of given environmental covariates such as Age and Sex, and let iY denote the vector of a quantitative or qualitative trait for individual i . Our general framework can be described as follows. For a binary trait,Let and for a quantitative trait,ikG is a vector of minor allele counts for SNPs within gene k for individual i. In this framework, h(·) represents the genetic effect, adjusting for the effects of covariates iX. Then our main focus is on hypothesis testing for h(·) = 0 for each gene k.where k are collapsed so that one genetic variable ikg is obtained from ikG using an indicator function for the presence of rare variants in this gene for each individual i. Morris and Zeggini [ikg, based on ikG, defined by the proportion of rare variants. In a groupwise association test procedure proposed by Madsen and Browning [ikg is defined through a weighted sum of the mutation counts based on their MAFs. As shown in Eqs. (1) and (2), we would like to take into account environmental covariates in our testing models; these covariates are not included in the testing procedures just described [ikg for each ikG and model h(ikG) as h(ikG) = β · ikg. Then association testing is reduced to testing for β = 0.The collapsing method was first introduced by Li and Leal  for dete Zeggini  extendedBrowning  a new geescribed ,9. ThereAs suggested by Li and Leal , markersnsβ = synβ = 0.Then we perform association testing for β and h are estimated by maximizing the penalized likelihood function:An alternative powerful multimarker association test is the kernel-based association test (KBAT) ,11. KBATλ is a tuning parameter. The representer theorem by Kimeldorf and Wahba [h(·) can be expressed as:where nd Wahba  shows thk. Then the estimates of β and α  can be easily obtained by plugging the h(G) obtained from Eq. (5) into the penalized likelihood function (Eq. (4)). For more details on the estimation, see Wu et al. [h(·) ~ N, where τ is a scalar and K is an n × n matrix whose th component is K. As a result, testing hypothesis h = 0 is simply reduced to testing τ = 0. For the hypothesis testing for τ = 0, a score test statistic proposed by Zhang and Lin [for a given kernel function u et al. . The rel and Lin  can be u and Lin . KBAT meK measures the genetic similarity between two individuals i and j on the basis of their genotypes at the SNPs in gene k. If:Note that a prespecified kernel function j does not influence i and j at the SNPs within gene k. If ikG = , where rikM denotes the genotype of individual i at SNP r in gene k, then a weighted IBS kernel can be defined by:then lkw is a weight based on lkq, the MAF of SNP l within gene k, and is defined by:where lkw is replaced by a constant, say, 1. The underlying idea behind the weighted IBS kernel is that similarity in rare alleles is more informative than similarity in common alleles for the trait similarity between two individuals so that the IBS kernel weights similarity in rarer alleles more.here. For an unweighted IBS kernel, We propose a Bayesian mixed-effects model to jointly analyze 200 simulation replicates. The main idea of our Bayesian mixed-effects model is to treat the genetic effect for each individual as a random effect and the environmental effect as a fixed effect. For disease status, we consider the logistic regression framework:and use the linear regression framework for Q1, Q2, and Q4, that is,ike ~ N, k = 1, …, 200, is the index for replicates and i = 1, …, 697 is the index for individuals. In both models, ig is the genetic effect of individual i and ig ~ N, in which βΣ is a diagonal matrix. The diagonal elements of βΣ, σ2 are further assigned noninformative inverse gamma distributions. For each trait, we fit the model using the Markov chain Monte Carlo algorithm.where During the first round of association tests for different traits, we noticed a dramatic difference in the number and magnitude of significantly associated genes and environmental variables. Therefore we suspect that the variance in different traits that can be explained by the provided genotype data and environmental components may vary.βΣ, σ2. We found that Q1 is affected by both given genotype data and environmental variables; in contrast, Q2 is mainly affected by genetic but not environmental variables, and Q4 is not affected by any given genotypic data  synonymous method, which examined only noise variables, also reported PIK2B in the top 10 gene list, indicating that some synonymous variants in PIK2B also contain association signals. Indeed, we found that a noncausal synonymous SNP (C8S886) in PIK2B had an identical genotype distribution with a causal SNP (C5S5156) in FLT4 .Table NOTCH2NL contains a SNP (C1S6297) that is identical with C18S2475 in PIK3C3. The false-positive genes PRH1, PRR4, and TAS2R48 are colocated on chromosome 12 and share SNP C12S717, which has the same genotype distribution as C7S5144, a causal variant for Q2. The false-positive gene SUSD2 contains the SNP C22S929, which is identical with causal variants C1S3181 in ELAVL4 (associated with Q2) and C6S5448 in VNN3 (associated with disease status). The false-positive gene KIT contains C4S1839, which is close to and identical with the causal variant C4S1873 in KDR. Some other commonly identified false-positive genes  share similar but not exactly the same genotype distributions with causal genes , and their genetic scores are highly correlated (p < 2.2 × 10−16).Some false-positive genes were often identified by multiple methods for similar reasons. For example, the false-positive gene In summary, there are many confounded signals in the GAW17 data set. We found 1,494 SNPs sharing exactly the same genotype distribution with at least one of the 160 causal SNPs. This posed a big challenge in the identification of causal genes, especially for traits with a large number of underlying causal variants, such as disease status. This may be a common problem in rare variant association studies because the sample size is usually much smaller than the number of variants. When most variants have extremely low MAFs, it is likely that their genotype distributions will coincide.After obtaining the simulation answers from the GAW17 meetings, we analyzed 200 simulated data sets and then counted how many causal genes and false-positive genes were identified by each method at different significance thresholds, and we plotted receiver operating characteristic (ROC) curves for all methods Figures . From thβ) of causal variants for Q1 and Q2 and found that correlation for Q1 (−0.17) is not significantly different from 0 (p = 0.3), whereas correlation for Q2 (−0.23) is only marginally significant (p = 0.05). Interestingly, the kernel method using a weighted IBS kernel did not suffer much power loss in Q1, although it also assigned more weight to rarer variants. It performed favorably and at least as good as two baseline methods (collapsing and weighted-sum) no matter whether the assumption was true or not. In real data, when we do not know whether rarer variants have larger effect sizes, the kernel-based method is preferable.Another pattern revealed by the ROC plots for Q1 and Q2 is that the weighted-sum and CMC methods that assign more weight to rarer variants performed worse than other methods for Q1 and comparable to other methods for Q2. This is probably because the “lower MAF, larger effect” assumption does not hold for Q1. We checked the correlation between MAF and effect size (A key contribution of our work is the application of the kernel-based method in the setting of association tests with rare variants. Originally this method was proposed in common variant association studies to enrich signals from multiple genotypic markers and to reduce the degrees of freedom in association tests. We found it suitable for rare variant association studies as well because single-marker tests using rare variants have low power as a result of the extremely low MAFs. To our knowledge, the kernel-based method has not been widely applied to rare variant association studies, and our systematic comparisons of this method with other well-established collapsing methods provide a better understanding of its behavior and potential use in rare variant association studies.βΣ, σ2, we can estimate the proportion of phenotypic variation that can be explained by environmental variables and given genotype data. Second, the posterior mean of ig is treated as a new response without environmental covariate effects and is directly used in association tests with genotypic data. It provides the basis for a more reliable comparison of different collapsing-based and kernel-based association methods by evaluating the result consistency across different replicates.Another novel contribution we make is the application of a Bayesian mixed-effects model. This procedure makes use of all 200 simulation replicates and serves two purposes. First, by comparing the posterior mean of We have two major conclusions. First, collapsing-based methods that assign more weight to rarer variants are sensitive to the “lower MAF, larger effect size” assumption, whereas kernel-based methods are more robust and suffer less power loss, even when the assumption is violated. Second, many false-positive genes identified by multiple methods often contain SNPs with exactly the same genotype distribution as the causal variants used in the simulation model. When sample size is much smaller than the number of rare variants, it is likely that the causal and noncausal variants will share the same or similar genotype distributions. This might lead to poor power and a large number of false-positive results for all methods in detecting causal disease-associated variants in the GAW17 data set.The authors declare that there are no competing interests.All authors participated in the design of the study. LL, WZ and JL performed statistical analysis and drafted the manuscript. JF and XZ proposed and implemented the Bayesian mixed-effects model. XY made valuable suggestions and helped to organize the study. HZ coordinated and helped to draft the manuscript. All authors read and approved the final manuscript.


Current methods have focused either on improving the detection of causative genetic variants or their precise genomic location separately.With the advance of genome sequencing technologies, phenotyping, rather than genotyping, is becoming the most expensive task when mapping genetic traits. The need for efficient selective phenotyping strategies, Here we recognize selective phenotyping as a Bayesian model discrimination problem and introduce SPARE (Selective Phenotyping Approach by Reduction of Entropy). Unlike previous methods, SPARE can integrate the information of previously phenotyped individuals, thereby enabling an efficient incremental strategy. The effective performance of SPARE is demonstrated on simulated data as well as on an experimental yeast dataset.Using entropy reduction as an objective criterion gives a natural way to tackle both issues of detection and localization simultaneously and to integrate intermediate phenotypic data. We foresee entropy-based strategies as a fruitful research direction for selective phenotyping. As genotyping costs are rapidly decreasing, often phenotyping becomes the limiting factor for mapping genetic traits. Recent techniques of genotyping have enabled access to complete or quasi-complete genotypes, from model organisms like yeast to human individuals -3. Moreous (QTL) .Existing methods for selective phenotying methods can be classified into 2 categories (See  for a mohttp://www.lmb.uni-muenchen.de/tresch/mastermind.html.Here we present SPARE (Selective Phenotyping Approach by Reduction of Entropy), a method that tackles both the issue of QTL effect detection power as well as the issue of locus resolution in a single framework. The problem of selective phenotyping is seen as an experimental design question in which the choice of samples must be made so that the outcome of their phenotyping allows, in expectation, optimal discrimination between the candidate genetic models. Here we follow a Bayesian mapping approach and the objective criterion is the minimization of the entropy of the a posteriori probability of the candidate models. We describe our method for single-causative locus models and fully genotyped populations with only two segregating genotypes, such as backcross, double-haploid, or recombinant inbred lines. An implementation of SPARE in the statistical programming language R  is availx corresponding to the samples S is denoted by yS is the realization of a random variable YS. If S consists of only one individual s, we shall write by abuse of notation yS instead of ys} {and xs instead of xs}{.In our setting, we have genotyped a set ys of an individual with genotype xs, given the QTL locus is λ,The building block of our mapping strategy is the probability of observing a phenotype θ is a suitable set of parameters for the respective model. Assuming observational independence given the genotype, the QTL and the parameters, the likelihood for the locus λ is obtained by summing over the parameters:Here, θ | λ,xS) in (2) will be discussed later in this section. Note that the individual observations in yS, conditioned only on λ and xS, are not independent in general.Specific parameterizations of (1) as well as the choice of appropriate priors Pr can for instance be chosen to be uniform.The locus prior H (Λ) as a measure of the uncertainty about the causative QTL. After having phenotyped a sample subset S, the remaining uncertainty about the QTL is:Similarly to the model discrimination method of Box and Hill , we consS without knowledge of the phenotypic outcome yS, so the best we can do is to minimize the expectation of H with respect to YS|xS. Given a maximum number n of individuals to be phenotyped, the task is to select a sample subset S| = n which minimizesHowever, we must make our selection P(ϵ = ±1) = w±1. The typical choices are either w1 = 1, w-1 = 0 for an a priori known genotype-phenotype association, as in typical linkage analysis studies, or Ys deterministically depends upon θ which consists only of the variable ϵ:Three specifications for elementary phenotype distributions (Equation 1) will be discussed here. Generally, let ϵ ∈ {±1} indicate which of the two genotypes is associated with the largest expected phenotypic value. Since this may be unknown, e is a predefined random variable with This scenario is called the deterministic case. In the presence of binary measurements with errors, to which we refer as the Bernoulli case, (9) is replaced by a Bernoulli distributionθ = . Equation (2) has to be evaluated for all loci λ. Therefore we make use of conjugate priors for θ to obtain a closed form solution of the integral. The conjugate prior is a Beta distributionwith parameters α±1,β±1. This fully determines the prior if we furthermore assume parameter independence, i.e.with appropriate hyperparameters Finally, we discuss the so-called Gaussian case for quantitative phenotypes, in which the probability (1) is defined asθ = . We choose conjugate Gaussian priorsHere, σ2, η±1, and ν2 are considered given and fixed. Again this completely specifies the prior under the assumption of parameter independence, Pr(θ) = Pr(ϵ)Pr(μ1)Pr(μ-1). For all three cases, closed form solutions of equation (2) are derived in the supplements , (5) and (6) hold with S ∪ T in place of S. The cost function is now defined given T and yT:We propose a sequential selection strategy that alternates between a selection step and a phenotyping step. This makes use of phenotypic data as they are gathered and is a cost-effective approach for situations where phenotyping is expensive. Let S| samples are selected and afterwards phenotyped. These selection/phenotyping rounds are iterated until the desired number of individuals is reached. In our applications, we have implemented the case where selection is incremental thus the integration above is done over a single dimension.By means of cost (14), |It was a discussion about an optimal strategy in the famous recreational Mastermind game that led us to formula 14) and initiated our research. Indeed if we choose loci, genotypes, phenotypes and genotype-phenotype modeling in an abusive way, identifying the hidden target sequence of colors is tantamount to bringing the cost function down to zero. We refer the reader to the supplements for a precise description of the algorithm. There, we implement a SPARE variant that plays the Mastermind game (and does it considerably better than most of us). The Mastermind game has been recently proposed as an educational tool to train biology students to experimental design . Multipl and initstrong linkage scenario and the unbalanced allele scenario. For both scenarios, linkage mapping for a backcross experiment was simulated with three biological replicates of each of the two parental F0 haploid strains and 100 F1 haploid trains. The 100 offspring genotypes were simulated by drawing a number of recombination events given the chromosome length in cM. The positions of the recombination events were drawn uniformly randomly along the chromosome. One unique QTL was drawn from one of the marker positions. The Gaussian model was applied. The fixed parameters were set to σ2 = 1,η±1 = 3, ν2 = 4, w±1 = 0.5. The Gaussian means μ±1 were drawn from the distributions (13), and the phenotypic values for the parental and the F1 strains were drawn according to (12). In the strong linkage scenario, we assumed a single chromosome genotyped at 100 markers equally spaced by 1 cM. No particular filtering is applied on the progeny so that at every marker, the two parental alleles are equally likely to occur. In the unbalanced allele scenario, we assumed a single chromosome genotyped at 100 markers equally spaced by 50 cM. Two alleles have low frequencies (20% in expectation) in the population: the causative allele and another randomly drawn one. This is obtained by first generating a much larger F1 population and then taking a random subset of 100 individuals weighting the probability to choose each individual according to its genotype for the two low frequency alleles. This procedure generates a population in which linkage is respected and the two low frequency alleles have in expectation a 20% frequency.Simulations for two scenarios were done: the On these datasets, SPARE was initialized given the observation of the parental phenotypes. This setting matches the typical situation in which a cross is used to map a phenotypic difference observed between parental strains. SPARE, however, does not require initialization with parental phenotypes. We iterated the selection step until 30 out of the 100 F1 strains were included. For each selection size, we selected the samples by optimizing the MMA criterion using a two-step implementation  . Unbalanced allele frequencies are common in outbred populations. As in our simulation, unbalanced allele frequency can also occur in recombinant inbred line panels if combinations of alleles are lethal or lead to phenotypes incompatible with the assay of interest. The whole procedure was repeated 100 times for each scenario. In our formulation, performing QTL mapping amounts computing the posterior distribution over the candidate loci. We used the Kullback-Leibler distance to estimate how close the posterior distribution for a particular selection is to the posterior obtained when phenotypes of the whole population are taken into account. This measure allows us to compare the performance of different methods based on the selection they propose.Simulations for two scenarios were done. In the As expected SSBL performs better than MMA in the strong linkage scenario Figure , whereasSPARE can integrate the information of phenotyped individuals, giving SPARE an advantage over previous methods. This can be well illustrated in the case of the strong linkage scenario, and holds true for the unbalanced allele scenario. Figure PHO84. SPARE, SSBL and MMA were run on this dataset, restricting for computational efficiency reasons to 314 markers regularly spaced at 100 cM distance. Different prior parameters were tried. The optimal prior, which takes as pseudo-counts of the Beta distributions the counts observed in the total population of 184 segregants: q1 ~ Beta and q-1 ~ Beta, the uniform prior, and a strong prior but with an association to the locus closer to random (60%): q1 ~ Beta and q-1 ~ Beta. The genotype-phenotype association is known here from the parental phenotypes and we therefore set w1 = 1 and w-1 = 0. As for the simulations, the Kullback-Leibler distance to the posterior estimated on the whole population was taken as a measure of the accuracy of the mapping for any selection.We then applied SPARE to a linkage analysis dataset in yeast . This daSPARE outperformed SSBL and MMA when run with the optimal prior, identifying the causative locus after phenotyping 12 to 15 segregants Figure . HoweverWe introduced SPARE, a selective phenotyping method based on Bayesian model discrimination that tackles both resolution and detection issues simultaneously. The objective function we minimize is an expected entropy which favors both a balanced selection of genotypes at all loci as well as evenly distributed crossovers across the genome. This formulation provides a natural way to deal with the issue of resolution: loci in linkage are seen as alternative competing models. Interestingly, linkage between neighboring loci does not have to be explicitly modeled. Indeed, two loci in linkage lead to similar likelihood. Moreover, our approach is able to make use of prior phenotype information and thereby proposes selections that enable the rapid reduction of QTL intervals. Simulations showed that SPARE can provide, on average, a more efficient selection strategy than previous methods. Application to a linkage analysis dataset in yeast showed that SPARE can propose, with appropriate priors, selections that enable a faster mapping of the QTL.i.e. how 'high and well-peaked', the mapping is. It relies on a Bayesian framework.Even though SPARE is a Bayesian method, one should note that just like other selective phenotyping procedure, its application is independent of the method employed to perform QTL mapping, which can be frequentist. It is worth noting however that comparing QTL-mapping results such as LOD score curves for different selections is a not a well-defined task. Indeed, the objective of QTL mapping is not only to detect the causative locus, for example with a significant LOD score, but also to have a narrow confidence interval on its location. One contribution of the present study is to propose the entropy of the posterior to quantify how certain, SPARE requires a prior distribution of the trait given the causative locus. The application to the yeast dataset showed that sensitivity to this prior distribution mattered. SPARE could only give the best results when applied with the prior distribution close to the optimal one. This is currently a weak point of the method.i.e. those involving multiple causative loci. Second, a selection step that chooses more than one sample at a time is desirable as phenotyping individuals in small batches might be more cost-effective than for one individual at a time. However, both improvements are computationally prohibitive if our cost function is used without modification, because then the task turns into a combinatorial optimization problem. Instead, efficient approximations must be developed to cope with the explosion in the number of hypotheses (Equation 5), and to solve the integration step in high dimensional space (Equation 8 and 14). The analogy to the Mastermind game suggests directions to a possible solution. For instance Cotta and colleagues [Two extensions could broaden the applicability of the current framework: First, the restriction to the case of simple traits with a single causative locus may be dropped. One would like to detect complex traits, lleagues  do not clleagues . An analOur simulations also showed that random selections are actually performing reasonably well compared to standard methods in a linkage analysis approach, with balanced allele frequencies. While unexpected, this observation can be explained. Indeed, optimal selections are selections that are simultaneously balanced at each polymorphism and with evenly distributed break points along the genome. The natural process of recombination actually promotes this situation. Random draws thus typically show these desired features.Recently, the use of Fisher's information to evaluate experimental designs in the context of QTL mapping, including selective phenotyping has been discussed . In contSPARE is a Bayesian model discrimination approach to perform selective phenotyping. Using entropy reduction as an objective criterion enabled tackling both issues of detection and localization jointly for the first time and integrating intermediate phenotypic data. Altogether, entropy-based strategies appear as a promising family of approaches for the question of selective phenotyping.JG and AT conceived the research. JG, ME and AT devised and implemented the algorithm. JG and AT wrote the manuscript. All authors read and approved the final manuscript.SPARE source code. A zip-compressed directory containing R source code and example data. See file README.txt in there for more details.Click here for fileSupplementary Information.Click here for file