Association mapping using abundant single nucleotide polymorphisms is a powerful tool for identifying disease susceptibility genes for complex traits and exploring possible genetic diversity. Genotyping large numbers of SNPs individually is performed routinely but is cost prohibitive for large-scale genetic studies. DNA pooling is a reliable and cost-saving alternative genotyping method. However, no software has been developed for complete pooled-DNA analyses, including data standardization, allele frequency estimation, and single/multipoint DNA pooling association tests. This motivated the development of the software, 'PDA' , to analyze pooled DNA data.® language, but it can also be executed on a Windows system without installing the MATLAB®. PDA provides estimates of the coefficient of preferential amplification and allele frequency. PDA considers an extended single-point association test, which can compare allele frequencies between two DNA pools constructed under different experimental conditions. Moreover, PDA also provides novel chromosome-wide multipoint association tests based on p-value combinations and a sliding-window concept. This new multipoint testing procedure overcomes a computational bottleneck of conventional haplotype-oriented multipoint methods in DNA pooling analyses and can handle data sets having a large pool size and/or large numbers of polymorphic markers. All of the PDA functions are illustrated in the four bona fide examples.We develop the software, PDA, for the analysis of pooled-DNA data. PDA is originally implemented with the MATLAB.PDA is simple to operate and does not require that users have a strong statistical background. The software is available at  The millions of single nucleotide polymorphisms (SNPs) now available are ideal for association analyses that identify important genetic variants in populations as well as genes predisposed to diseases involving complex traits ,2. AlthoDNA pooling is highly efficient. Many researchers have investigated the performance of DNA pools while estimating allele frequency and have measured the impact of pooling on association test results. The results show that allele frequencies can be estimated accurately and precisely using DNA pools after considering coefficient of preferential amplification (CPA) ,12; moreAlthough many single-point pooled DNA association tests have been developed, multipoint analysis still presents a challenge due to the large numbers of genotypic combinations in DNA pools. The difficulty increases substantially with the pool size and/or the number of SNPs involved. Several of the recently proposed advanced multipoint estimations and tests have been haplotype oriented -17; neve® software platform that is adapted to the Windows systems (MS Windows® 98/ME and MS Windows® NT/2000/XP/2003). For MATLAB® users, PDA can be run with a graphical user-friendly interface where users merely click the checkboxes to carry out data analysis. The PDA user interface is shown in Figure ® system, we used the MATLAB® compiler to generate standalone executables of PDA, which can be deployed on machines without installing the MATLAB®. The guide to the installation and initialization of PDA on Windows is illustrated in Appendix A  of CPA estimates, allele frequency estimates, single-point pooled DNA association test and multipoint pooled DNA association test.Item 1. Input/Output directory: The directories of input and output files must be specified. PDA will read data from the assigned input directory and automatically save outputs in the output directory. The format of input and output is illustrated in Appendices C and D , users check the box 'Two groups' and determine whether to carry out association tests after calculating estimates for CPA and allele frequency. PDA provides the flexibility of equal or unequal CPA statistical inference that the user may choose as needed. Check 'Yes' for equal CPA inference or 'No' for unequal CPA inference.Item 3. Data type for CPA estimation: Two types of data are acceptable. The first type is peak intensity data from genotyping experiments. The second type is raw CPA/heterozygote ratio from empirical studies or databases. If peak intensity data are inputted, then users should provide the number of pairs of peak intensities for each locus.Item 4. Calculation of the bootstrapped s.e. of the CPA estimate: Bootstrapping is a resampling technique used to estimate the s.e. of CPA. Users can determine whether s.e. is to be calculated. If users want to calculate the bootstrapped s.e. then they should check 'Yes' and assign a number of bootstrap replications between 10 and 1000. A larger number of bootstrap replications will take longer to calculate but yields a more reliable estimate.Item 5. Estimation of adjusted allele frequency: Users can check 'Yes' to calculate the adjusted allele frequencies or 'No' to omit the calculation.σE, if an association test is conducted.Item 6. Single-point pooled DNA association test: Users can carry out association tests only for the analysis of a two-group study. Because the test statistic of association tests depends on experimental error, users must assign a proper value for the experimental standard error, Item 7. Multipoint pooled DNA association test: Users can carry out association tests only for the analysis of a two-group study. If they check 'Yes', they must answer seven options to conduct this test. The seven options are as follows. (1) Data type for the association test. Two types of data are acceptable: peak intensity data or raw p-values from previous single-point association tests. (2) Map information. Users can check 'Yes' to provide information on marker positions for the latter graph demonstration of multipoint p-values or check 'No' to ignore the inter-marker distances. (3) Weight function. Users can choose to assign equal weights to all marker loci by checking 'Equal weight' or provide a set of weights by checking 'User-specified weight'. (4) Threshold value of truncation. PDA provides a function to truncate insignificant p-values in the analysis. The value is between 0 and 1, and p-values greater than the threshold will be excluded from the analysis. (5) Number of Monte Carlo simulations. Users must provide a suitable number of simulations between 500 and 10000. A large number of simulations increase the accuracy of the empirical p-value estimation, but a longer computational time may be required. (6) Window size, defined as the number of markers in a window prior to p-value truncation. Users should specify a suitable number of markers in a window according to the attributes of their data. Window size must be = 2, with the upper limit being the total number of SNPs in the study. (7) SWEPT statistics. PDA provides three statistics for multipoint association tests; i.e., multiplicative, additive and minimum p-value statistics.The statistical theory is introduced in the next section.We developed PDA based on a four-stage procedure, which combines the concept of a three-stage DNA pooling experiment  with theA and a, CPA represents the relative magnitude of the averaged amplified intensities of the different alleles and is defined mathematically as κ = μA/μa, where μA and μa are the average peak intensities of alleles A and a. The parameters can be estimated from heterozygous individuals who provide a standard for a 50:50 ratio for a pair of peak intensities of two heterozygous alleles. When κ = 1, there is no preferential amplification; when κ > 1, the first allele is more likely to be amplified than the second; when κ < 1, the second allele is more likely to be amplified than the first. PDA provides three discrete estimates for the CPA: arithmetic mean adjustment nheter denote the number of heterozygous individuals and {j), j), j = 1,...,nheter} is the pair of peak intensities of heterozygous individuals derived from individual genotypings. The mathematical formulas of the three CPA estimators are presented as follows:Firstly, PDA provides estimates for the CPA, which affects allele frequency estimation and association testing in a pooled DNA study. For a diallelic SNP with alleles d errors . Let nhewhere A is estimated by hA and ha denote the peak intensity of alleles A and a in a DNA pool  is the usual indicator that takes the value of 1 if event A is true; otherwise, it takes the value of 0. The non-negative wij is a standardized weight of the p-value, vj, in the ith window (i.e. the weight satisfies the requirement that the weights in the window sum to one). The standardized weight is calculated by dividing the original weight by the sum of all original weights in the window under the given original weights. The multiplicative SWEPT statistic is a sliding-window extension of the truncated product method [where t method , and thet method . The thiZMin = minj = i,...,i+k-1{vj}, i = 1,...,N + 1 - k.The minimum SWEPT statistic extended the technique of taking the minimum score, which has good performances in test power and type 1 error and has been used broadly in genetic studies ,23.There are other efficient p-value combinations, such as the rank truncated product method , which mV with a correlation matrix ∑ from an independent p-value vector V0, based on the following correlation-invariant transformationTo assess the statistical significance of the SWEPT in each window, PDA applied a Monte-Carlo procedure recommended in  to calcuV = 1 - Φ(C-1Φ-1(1 - V0)),C is a lower triangular matrix satisfying the Cholesky decomposition, ∑ = CCT. We estimated the correlation matrix ∑ using an autocorrelation function of p-values. We recalculated the SWEPT statistics based on the generated p-value vector, V. The previous procedure was repeated B times to yield {Zb), b = 1,...,B}. Hence, the empirical p-value of the ith window with window size k can be calculated as the following:where Φ(.) is the cumulative distribution of a standard normal random variable and Z* is the corresponding SWEPT value based on real data. The SWEPT offers several advantages over conventional DNA pooling analyses. (1) SWEPT can work well even in cases where only p-value data are available; hence, it can analyze data from different study designs and is applicable to meta-analysis. Because SWEPT allows a p-value truncation, it also handles data containing unpublished insignificant p-values. (2) The SWEPT statistics make adjustments for preferential amplification, a critical aspect that has never been considered before in pooled DNA multipoint analyses. (3) The simplicity of the SWEPT statistics lowers processing time and significantly reduces the computational complexity. (4) The SNPs involved in multipoint analyses can be determined conveniently once the window size has been determined, thereby avoiding the common perplexity of selecting SNPs in haplotype-oriented or other multipoint analyses. (5) SWEPT is comprehensive in that it covers conventional single-point test statistics and can be applied to the analysis of individual genotyping data, although this aspect is not the primary concern of PDA.where We give four examples to illustrate functions of PDA: (1) One-group allele frequency estimation. (2) Two-group single-point DNA pooling studies. (3) Two-group multipoint association test based on peak intensity data. (4) Two-group multipoint analysis based on p-value using PDA. Throughout this paper, we set the host name of working directory to be 'C:\Program Files\MATLAB71\PDA'. All input data files for these four examples are available with software PDA and saved in the example directory, 'C:\Program Files\MATLAB71\PDA\Example'.We used the six SNP data published in our previous paper  to illusst column shows the SNP number. The 2nd column shows the SNP name. The 3rd column shows the number of heterozygous individuals. Three discrete adjustments , and the s.e. is 0.051. After applying CPA adjustments, the accurate allele frequency estimate is about 0.64 and s.e. is 0.06. Three different adjustments yield similar results. In this example, there is a serious overestimation of allele frequency if the CPA adjustment is ignored.In Table In this example, we analyze the data set from our previous project that compared the allele distributions of three main Taiwan subgroups in the human major histocompatibility complex (MHC) region. We selected two subgroups (Hakka and Han groups) and 4 SNPs for the illustrations. The operation procedures are illustrated in Appendix F See .The results are shown in Tables Table X based on the three different CPA adjustments were conducted. The s.e. of experimental error was set to be 0.02 according to our previous study [In Table us study . For exaIn our previous project, these four SNPs were also genotyped individually and the allele-based association test based on individual genotyping data yielded the exact p-values for these four SNPs are 0.00795, 0.00006, 0.52346 and 0.23972 respectively. The conclusions are consistent with the results from the adjusted association tests and demonstrate the importance of CPA adjustment.In this example, we illustrate a multipoint analysis, an important utility of PDA. We analyzed 10 SNPs from our MHC study to screen for potential candidate regions that could distinguish Hakka and Han groups. The operation procedures are illustrated in Appendix F See .The results are shown in Tables Table In our previous project, these ten SNPs were also genotyped individually, and the allele-based association test based on individual genotyping data yielded exact p-values for these ten SNPs: 0.0216, 0.0052, 0.0115, 0.6859, 0.0232, 0.9440, 0.1628, 0.4468, 0.4082 and 0.9443. However, the previous single-point pooled DNA test only identified SNP 6419. In this case, the important SNPs, 6421 and 6422, were not identified by the single-point association tests; however, the two SNPs are included in the region from SNPs 6421 to 6424, which was identified by a multipoint analysis based on a sliding window with size 5.In this example, we illustrate the implementation of p-value analysis using PDA. To conduct multipoint association tests, we used the same 10 SNPs as in Example 3, based on the p-value derived from a single-point pooled DNA association test with unbiased adjustment of CPA. The operation procedures are illustrated in Appendix F See .Because we only implemented the p-value of each SNP, the procedures for the CPA estimate, allele frequency estimate and single-point association test cannot be considered in the analysis. Only multipoint association tests can be conducted.First, PDA shows the input information for the analysis in this example, as follows: p-value data were used; no map information was provided; user-specified weights were used; the threshold value of truncation was 1; the number of Monte Carlo simulations was 10000; the size of each window was 5; the SWEPT statistic was calculated using the additive model. The results are summarized in Table CPA estimation is based on peak intensity data of heterozygous individuals. Data of heterozygous individuals in a pilot study may not be available occasionally. Public accessible CPA databases for SNPs provide important information ,26. PDA PDA provides an extended single-point association test allowing for different CPAs between two comparative groups. This test reduces to the conventional test in  if the eHaplotype-scoring  and locuWeights for different SNPs in each window may affect the significance of a multipoint association test. If there is no prior knowledge in this regard, then equal weights can be employed. The other strategy is to consider weights according to genetic/physical or linkage disequilibrium maps of SNPs . Using iThe sliding window procedure emphasizes a local effect, which assumes the neighboring SNPs provide sufficient information for the window of interest and that other SNPs outside the window do not impact the inference of the window once SNPs within the window have been considered. A small proportion of SNPs is considered each time, making the sliding-window approach a convenient and practical procedure for chromosome-wide studies once the window size is determined. A sliding-window size of 5 for the selection of genetic markers for association tests with individual genotyping data was suggested in , but thePDA provides simultaneous analyses of the CPA adjustment, adjusted allele frequency estimate and single/multipoint DNA pooling association tests that are usually essential for complete DNA pooling studies. All of the PDA functions are illustrated in the four bona fide examples contained in the program. PDA is simple to operate and does not require that users have a strong statistical background..PDA software can be downloaded from the web site: Project name: DNA pooling projectProject home page: Operating system: MS Windows®Programming language: MATLAB®Other requirements: NoLicense: PDA licenseAny restrictions to use by non-academics: On request and citationPDA: Pooled DNA analyzerCPA: Coefficient of preferential amplificationSWEPT: Sliding-window empirical p-value testHCY conceived the statistical methods and experimental designs and prepared the manuscript. CCP programmed the software. CYL and CSJF contributed to the discussion and preparation of the manuscript. All authors have approved the final manuscript.Appendix A – Installation and initialization of PDAClick here for fileAppendix B – Description of working directoriesClick here for fileAppendix C – Data input formatClick here for fileAppendix D – Results output formatClick here for file®Appendix E – Execution of PDA without MATLABClick here for fileAppendix F – Operation procedures in examplesClick here for file


Five measures included in the operational criteria of the metabolic syndrome by the Adult Treatment Panel III were examined. A method for estimating sibling recurrence risk with correction for complete ascertainment was used to estimate the numerator, and the prevalence in the whole cohort was used as the denominator of λs.The purpose of this study was to estimate both cross-sectional sibling recurrence risk ratio (λs was found in terms of different time-points for the cross-sectional definition, the times of fulfilling the criterion for lifetime definition, and different components. Obesity and hyperglycemia had the highest cross-sectional λs of the five components. Both components also had the largest slopes in the linear trend of the lifetime λs. However, the magnitudes of the lifetime λs were similar to that of the mean cross-sectional λs, which were <2. The results of nonparametric linkage analysis showed only suggestive evidence of linkage between one marker and lifetime diagnosis of low high-density lipoprotein cholesterol and metabolic syndrome, respectively.Considerable variability in the λs of the metabolic syndrome and its components varies substantially across time, and the λs of lifetime diagnosis was not necessarily larger than that of a cross-sectional diagnosis. The magnitude of λs does not predict well the maximum LOD score of linkage analysis.The λ The metabolic syndrome is a clustering of diabetic and cardiovascular risk factors, including obesity, hypertriglyceridemia, low high-density lipoprotein (HDL) cholesterol, hypertension, and glucose intolerance . It preds, is a commonly used index [s is proportional to the power of an affected-relative-pair genetic linkage analysis [s is influenced by many factors, including the ascertainment process and overreporting [s by itself does not provide a reliable parameter for estimating the statistical power of a proposed linkage study. Nevertheless, within a range of low magnitudes of genetic effect, which probably is true for many susceptibility genes of complex diseases, the relationship between λs and genotype relative risk is relatively predictable regardless of underlying genetic models [s with correction for ascertainment [s can serve as an initial assessment of the genetic effect conferred to a trait under study.In assessing the familial aggregation of a disease or physiological trait, the sibling recurrence risk ratio, λed index . Under aanalysis . Recent eporting , as welleporting . Thus, λc models . Furthertainment . Therefos has been estimated exclusively on the basis of cross-sectional assessment of phenotypes. When a disease has a variable age at onset and the phenotype is repeatedly measured, it is not known whether the λs is stable over time. In this study, we estimated both cross-sectional λs and lifetime λs for the metabolic syndrome and its individual components over time among sibships in a prospectively followed-up cohort. We hypothesized that a component with a higher λs is more likely to be the central genetic contribution to the syndrome. The distributions of various definitions of λs over time for the metabolic syndrome and its individual components were compared. In addition, genome-wide scans were performed for both cross-sectional and lifetime phenotype definitions for the metabolic syndrome and its individual components.So far λParticipants were from Cohort 1 and Cohort 2 of the Framingham Heart Study data set provided by the Genetic Analysis Workshop 13 (GAW13). In Cohort 1, 1231 individuals  with a mean baseline age of 41.5 years (ranging from 29.0 to 62.0) were included. Cohort 2 included 1672 individuals  with a mean baseline age of 32.7 years (ranging from 5.0 to 64.0). Some participants in the two cohorts were from the same pedigree, and in total 330 pedigrees were identified. Among these pedigrees, 1702 individuals were further genotyped for genomic scanning.2. The cut-off points of hypertension in the Framingham Heart Study data set were slightly more stringent than those recommended in the ATP III criteria (≥ 130 / ≥ 85 mm Hg). The specific criteria adopted in this study were as follows: 1) Blood pressure: systolic blood pressure (SBP) ≥ 140 mm Hg or diastolic blood pressure (DBP) ≥ 90 mm Hg, or under anti-hypertension treatment; 2) Obesity: BMI > 30 kg/m2; 3) Fasting glucose: ≥ 110 mg/dl; 4) Triglycerides: ≥ 150 mg/dl; and 5) HDL cholesterol: < 40 mg/dl (men) or < 50 mg/dl (women). The diagnosis of the metabolic syndrome was made when three or more of the criteria were met. Phenotypic data provided in the GAW13 included those for the first 40 years of follow-up in the Cohort 1 (Exams 1 through 21) and those of the first 20 years of follow-up in the Cohort 2 (Exams 1 through 5).In this study, we considered five measures that are included in the operational criteria for the metabolic syndrome as defined by the Adult Treatment Panel III (ATP III) . Becauses) is defined as sibling recurrence risk (Ks) divided by the population prevalence (Kp). Since the original pedigrees were not ascertained via any particular phenotype considered in the present study, the determination of proband status is problematic. Under this circumstance, we considered every diseased individual as a proband and employed the method for estimating sibling recurrence risk (Ks) when proband status is unknown as proposed by Olson and Cordell [Sibling recurrence risk ratio (λ Cordell :ns(a) is the number of sibships of size s with a affecteds. This estimator of Ks has been shown to be unbiased and consistent when the ascertainment is complete [s, the population prevalence, was estimated from the whole cohort. Since the age range of Cohort 2 was wide, we limited the analysis of Cohort 2 to those aged between 30 and 60 years old.where complete , which is for each exam and then calculated the standard deviation and the range of the estimated λs across the exams. Furthermore, we defined a lifetime λs,e≥t as the condition that an individual had ever had the episode fulfilling the criterion at least t times during the whole study period. As t increases, the person is supposed to have higher probability of carry the genetic susceptibility of the phenotype. In assessing the lifetime λs for the metabolic syndrome, any particular criterion was considered fulfilled if the individual had ever any measures above the threshold during lifetime, and a positive diagnosis was given if three or more criteria were met in this way.We estimated cross-sectional λFor each component and the metabolic syndrome per se, we then performed nonparametric linkage analyses for the pedigrees using GENEHUNTER  with thes in both cohorts, there were substantial variations across exams for hyperglycemia and obesity, while the values of λs for hypertension and low HDL were relatively stable  to 0.68 (hyperglycemia). For the lifetime λs in Cohort 1, a linear trend of increase with the number of episode was noted for obesity and hyperglycemia. The lifetime λs of low HDL had a sudden drop at λs,e≥3, probably due to small number of participants meeting the threshold, whereas the three lifetime λs of hypertension were very similar to one another.For the cross-sectional λe Figure . Meanwhis for individual components of the metabolic syndrome was that for hyperglycemia, followed by obesity . Indeed, a linear trend in lifetime λs was noted in the majority of the components of the metabolic syndrome, especially hyperglycemia. However, the value of the lifetime λs,e≥3 was not necessarily greater than the mean cross-sectional λs of its counterpart, with some even lower than the latter . This may further call into question the use of cross-sectional λs.With our definitions of lifetime λs of the components of the metabolic syndrome, obesity and hyperglycemia stand out as the highest components for both Cohort 1 and Cohort 2. Both components also had the largest slope in the linear trend of the lifetime λs. Thus, both obesity and hyperglycemia might be the central components of the metabolic syndrome in terms of familiality. This is consistent with the findings in twin analysis [Comparing the one-time λanalysis  of the ms (its mean not greater than 2) found in this study implies that very large samples of affected relative pairs are needed for the detection of definitive genetic linkage [s and the number of markers with LOD score ≥ 1.5, the widely varying number of affected individuals for each component might account in part for this. Nevertheless, the repeated appearances of some markers across different traits indicate that these traits might indeed share certain common genetic susceptibility.Despite the evidence supporting the familial aggregation of the metabolic syndrome and its components, the modest to moderate magnitude of the λ linkage ,7. This


CYP11B1/CYP11B2 genes.Using the genome-wide screening data of the Framingham Heart Study  we have quantified the underlying genetic diversity through high-dimensional genetic feature vectors and constructed a genetic vector space for the analysis of population substructure. Adaptive clustering procedures led to three major subgroups that were regarded as being related to "biological" ethnicity and that included more than 70% of the subjects. Based on these subgroups we addressed the question of ethnicity-related and ethnicity-independent risk factors for coronary heart disease (CHD). To this end, we relied upon hypertension as an endophenotype of CHD and applied a multivariate sib-pair method in order to search for oligogenic marker configurations for which the sib-sib similarities deviated from the parent-offspring similarities. Indeed, the latter similarities are always "0.5" irrespective of the affection status of parents and offspring. Loci with significant contributions to the oligogenic marker configuration constituted a CHD-specific genetic vector space. We found several ethnicity-independent signals. One signal on chromosome 8 may relate to the  Coronary heart disease (CHD) is one of the most common illnesses in the Western world. As with most late-onset diseases, empirical evidence from numerous studies suggests that CHD is caused by an interplay between an unspecific genetic vulnerability and environmental factors. So far, attempts to identify specific genes have not yet been successful while a series of environmental risk factors – such as overweight, cholesterol, smoking, and alcohol consumption – have been found to be associated with hypertension and cardiovascular sequelae ,2. MarkeGenetic vector spaces are spanned implicitly by a set of genetic feature vectors, where a genetic feature vector comprises a set of scalar variables. The scalar variables can include genotype measures, quantitative scores on the phenotype level, and environmental details. The intrinsic regularities inherent in a set of genetic feature vectors is revealed by systematically evaluating the distances d between any pair of vectors xi,xj (0 ≤ d < ∞) or the respective similarities  = 1/ [d + 1] for i, j = 1,2,..., n). Similarity measures are better suited for structural analyses of empirically derived vector spaces because the similarity coefficients s are "normalized" such that 0 ≤ s ≤ 1. There exists a variety of different distance and similarity measures. One distinguishes between metric and nonmetric measures depending on whether or not the "triangular" criteriaA vector space is a well-established, universal concept for the analysis of multivariate data. d ≤ d + d and s × s ≤ [s + s] × se1 ≥ e2 ≥ e3 ≥ ... ≥ en  ≥1 >en+1 ≥ ... associated with the "eigenvectors" v1, v2, v3,..., where n denotes the number of significant eigenvalues that have been extracted. The significant eigenvectors constitute the dimensionality of the vector space, whereas the vector space's orthogonal complement, associated with the insignificant eigenvalues, is eliminated. Conventionally, eigenvalues e ≥ 1 are regarded as "significant". The amount of variance explained by the vector space can serve as a goodness-of-fit estimate.are met. In the case of metric distances the underlying vector space can be constructed from a set of vectors ("measurements") by means of a principal component analysis (PCA). The PCA axes are aligned in the direction of the vectors' largest variations identified through the largest eigenvalues of the covariance matrix. Specifically, the PCA yields a rank order of eigenvalues metric vector space can be constructed from the rank order of nonmetric similarities in such a way that the rank order of the resulting metric distances is identical with the rank order of the original nonmetric similarities. Of particular interest are oligogenic vector spaces spanned by n ≥ 8 uncorrelated, highly polymorphic microsatellites because such vector spaces provide a powerful means for the analysis of population admixture, thus clearing the way for a solution of the problem of genomic control  of the genotypes aiaj and arranging the F(ij)(κ) /2) in descending order, such thatIn empirical studies the question arises as to how to construct genotypic feature vectors that constitute a problem-specific vector space. One possible approach is the use of uncorrelated, sufficiently heterogeneous microsatellites. The heterogeneity of a microsatellite  can be quantified through the microsatellite's multitude of different allele combinations observed in a given population. For a polymorphism with (ij)(1) ≥ F(ij)(2) ≥ F(ij)(3) ≥ ... ≥ F(ij)n(n+1)/2,Fh = h as the number m for which the sum of the F(ij)(k)  in descending order becomes greater or equal to a prespecified percentage s, where s is typically in the range of 95–99%:we can define a heterogeneity coefficient An exponential/logarithmic transformation may be used to compensate for the non-normal distribution of the heterogeneity coefficients in standard marker sets if compatible ethnicity markers have to be selected for cross-comparisons between studies.Our study comprised 394 nuclear families with 1328 subjects from the community-based Framingham sample. Participants aged 29 to 62 years were followed up to 52 years with up to 21 repeated assessments. Of the nuclear families, 48 included sibships with both parents, 142 with one parent, and 204 sibships without parents. On the phenotype level, blood pressure, hypertensive treatment, tobacco and alcohol consumption, total cholesterol, fasting HDL cholesterol, fasting triglycerides, and fasting glucose were examined. With respect to blood pressure, 136 sib pairs (34.5%) were concordant for normal values, 183 sib pairs (46.4%) discordant, and 75 sib pairs (19.0%) concordant for hypertension. The subjects contributed a 20-ml blood sample from which DNA was extracted and genotyped for 397 microsatellite markers (modified Weber9 marker set).h, we assembled 20-dimensional feature vectors in order to derive a genetic vector space for the representation of the subjects as multidimensional points. Adaptive cluster analysis led to three major subgroups that were regarded as being related to biological ethnicity and included more than 70% of the subjects . Each segment si was systematically combined with each segment sj, thus yielding n(n - 1)/2 feature vectors of length 20 that allowed us to detect interactions between any two marker loci . Interactions were deemed to be present if the joint effect of two markers deviated from the sum of their single effects. We then determined the distribution of parent-offspring similarities together with the distribution of the between-sib similarities of affected, unaffected, and discordant sib pairs. Subsequently, our signal detection algorithm looked for significant differences between the parent-offspring similarities and the sib-sib similarities whose mean value was expected to deviate in affected sib pairs from the parent-offspring value if the feature vector included markers close to vulnerability or protection genes. Those loci that contributed significantly to deviations in the expected values were included in the oligogenic configuration that constituted our CHD-specific genetic vector space. We found several ethnicity-independent signals. One signal on chromosome 8  may relate to the CYP11B1/CYP11B2 genes at 142.3 cM .Using hypertension as endophenotype of CHD and treating the genome as a single entity, we subdivided the genetic regions, implicitly defined by the 397 marker loci, into Our quantitative concept of vulnerability and protection factors assumes etiologic and phenotypic heterogeneity in such a way that only a certain proportion of the affected sib pairs exhibit an elevated SBD/SBS score at a certain locus within an oligogenic configuration. Thus, each locus of the configuration is regarded as being, by itself, neither necessary nor sufficient for developing the phenotype. There also exist subsets of affected sib pairs with a significant genetic dissimilarity at a locus of the configuration as well. Since a "dissimilarity locus" interacts with at least one of the vulnerability loci, it is likely that it modifies the genetic risk of the phenotype. We therefore conjecture that affected siblings who are dissimilar on the genotype level also exhibit differences on the phenotype level, perhaps, in terms of onset and severity of illness.As to the genetic analysis of complex traits, oligogenic approaches to quantifying genetic diversity complement standard linkage and association methods by following a genotype-to-phenotype research strategy. This has the advantage that multilocus variations in the genome sequence can be correlated with specific quantitative conditions on the phenotype level, that is, not only with the presence or absence of the disease but also with lifestyle and environmental details. Such a viewpoint appears to be of particular importance in the case of CHD, where environmental factors such as smoking, alcohol consumption, obesity, and comorbid personality traits modify both the risk of and the prognosis for the disease.


Researchers may embark on a genome-wide association study before fully investigating candidate regions which have been reported to produce evidence to suggest that they harbour susceptibility loci. If the genome wide study had not been carried out then results which demonstrated only modest statistical significance from candidate regions would be judged to be of interest and would stimulate further investigation. However if hundreds of thousands of markers are typed then inevitably very large numbers of such results will occur by chance and those from candidate regions may attract no special attention.p value for each marker, calculated as LR = echiinv/2, and the posterior odds in favour of a true positive association are obtained. These odds can be used to rank the markers with a view to suggesting the regions in which further genotyping is indicated. We suggest that prior probabilities be specified such that a candidate marker significant at p = 0.01 and a routine marker significant at p = 0.00001 will yield similar values for the posterior odds. We show that this can be achieved by setting a value for prior probability of association to 0.1 for candidate markers and to 0.00018 for routine markers.An approach is proposed in which differential treatment is afforded to markers from candidate regions and from those that are routinely typed in the context of a genome wide scan. Different prior probabilities are assigned to the two types of marker. A likelihood ratio is derived from the reported p values to posterior odds is available from .It is essential that formal procedures be adopted in order to avoid modestly positively results from candidate regions being swamped by the huge number of nominally significant results which will be obtained when very many markers are genotyped. Software to carry out the conversion from  If one were to study a candidate gene on its own and obtain results of this nature one might declare them of interest and as supporting further investigation. However if a genome wide scan of 500 K markers is performed then one can expect 5,000 to be significant at p < 0.01 and the evidence supporting the candidate gene will look very unimpressive indeed.The ability to carry out so-called genome wide association studies using a standard panel of single nucleotide polymorphisms (SNPs) presents obvious difficulties. For many diseases it will be the case that particular genes have already been implicated as possibly or probably being involved. Typically, there will be some positive and some negative association studies, some groups will report positive results with particular markers or haplotypes while others will report negative results with those markers but positive results from some other markers nearby and a third set of groups will report positive results with a different, though related, phenotype. There may be general consensus that the gene is worthy of further investigation. In an ideal world groups possessing appropriate datasets would test additional polymorphisms within the gene in an attempt to provide some definitive answer as to whether or not it influences susceptibility. However, the world is not ideal and instead what may happen is that groups who have yet to fully investigate the gene may submit their datasets for genotyping with a standard, very large, set of SNPs which will include a few markers located in the gene in question along with a few hundred thousand which are not. The problem is that markers in the candidate gene may produce relatively modest evidence in favour of association, perhaps at the level of p value than markers producing apparently significant results by chance. If the expected significance, based on effect size and sample size, for a given truly associated marker is 0.01 then the p value actually obtained will probably be fairly close to this value. However among 500 K markers one may expect 50 or so to be significant at p < 0.0001 by chance and there is no reason to suppose that any of these will represent a true positive effect. If there are particular markers which are incorporated in the genome scan which do happen to be strongly associated with the disease then it is true that they may produce very highly significant p values. However our concern is that markers which are truly associated to some degree may yield only modest p values [We should emphasise at this point that a marker having a real, but modest, effect is not expected to produce a smaller p values  and thatp < 0.01 when one has the results for 5,000 more which are just as significant.We propose that formal methods should be used for treating markers in candidate regions which have been previously implicated in a different fashion from the many thousands of SNPs spread across the genome which will be routinely typed in the course of a genome scan. One approach which would be theoretically sound would be to declare in advance a set of genes, and hence markers, which one regarded as being worthy of special consideration and to go on to analyse them first, before giving any consideration to those routinely genotyped. One could go on to publish the results from these analyses before analysing the other markers. However we believe that in practice it would be difficult to implement this approach. Typically, when a genome scan is performed all markers are analysed at once. It would be difficult to persuade readers and reviewers that special attention should be paid to certain markers giving results at p value such that markers achieving that threshold will have less than the declared probability of being false positive. Guidelines have been proposed in which candidate genes and genome screens can be treated differently when the FPRP approach is applied in order to satisfy different criteria for a finding to be "considered noteworthy" [a priori of interest.One approach which is relevant in this context is to estimate the false positive report probability (FPRP) . This aieworthy" . Howevereworthy" . This usWe believe that the FPRP approach is not quite correct from a conceptual point of view when assessing the results of genome wide association studies. One reason is that such a study must always be regarded as an intermediate step. This is because the polymorphisms which are genotyped have been selected on the grounds of being appropriately spaced or because they tag other markers but they are not themselves expected to directly influence susceptibility to a disease. Rather, it is hoped that a genotyped marker showing association with the phenotype may be in linkage disequilibrium with a polymorphism which does influence susceptibility. Thus a positive finding should lead on to further genotyping of other polymorphisms in the region. In this situation, there is no point in fretting over what the probability is that one has a "noteworthy" finding. Once the scan has been performed one will inevitably want to go on to perform further genotyping and the question will be which regions appear the most promising to pursue. Thus the aim of the scan is to assist in the ranking of regions for further genotyping rather than to come up with any definitive answers as to whether particular regions are implicated or not. Once reasonable attempts have been made to type all available polymorphisms within a region then one can address the question of the overall strength of evidence in favour of association. One can then make a judgement as to whether it is worth proceeding to more demanding investigations such as functional studies. At this point one may feel that a method which explicitly aims to quantify the probability that a finding represents a true positive is of value. However we do not believe that such judgements are necessary when the question is not whether, but where, to perform further genotyping. Additional disadvantages of the FPRP approach are that one needs to declare the power of each test, that is the probability that a marker near a susceptibility locus will support association at a certain significance level, and that one must declare a plausible prior probability for each marker being truly associated. It is doubtful whether either of these values can be realistically quantified but the application of FPRP seems to yield an artificially concrete estimate of a probability which is in reality quite uncertain.p value obtained [An alternative approach using a Bayesian strategy has been proposed which uses the prior probability for a test to detect association in order to weight the obtained . The weiHere we propose a related method which pragmatically seeks to rank results in a comparative way rather than to come up with absolute judgements as to whether they reflect true or false positive results. It also explicitly aims to afford special treatment for already implicated candidate genes within the context of a genome wide association study. Although it utilises "prior probabilities" for association of markers within candidate genes and of routinely typed markers it is recognised that these probabilities are essentially arbitrary and they are used mainly as a way of distinguishing the two sets of markers. It does not require any assumptions about the genetic effect size or power to detect association.The procedure we arrive at consists of a number of stages. The first, and perhaps most problematic, step is to divide the markers into those which are candidate markers and those which are routinely genotyped. Candidate markers must either all be declared in advance or must be definable by some explicit rule. They must not have been already typed in the current sample.  The authors reporting the study must report the basis on which they have declared markers as candidates. An example of a rule for defining a candidate marker might be to say that one will include all markers within 200 kb of a gene which has shown evidence from either three studies at p < 0.01, two at 0.001 or one at 0.0001 and for which the ratio of negative to positive association studies does not exceed 2:1. We believe that such a rule might reasonably reflect how association studies are commonly interpreted but we emphasise that any rule can be chosen, as long as it is explicit. Of course, one could set up hierarchies of candidate regions, whereby some were regarded as more strongly supported than others, but given the difficulty in making firm judgements about the weight of evidence any particular study provides we doubt that such complex schemes would be justified.Methods section we would suggest prior probabilities of PPrCAND = 0.1 and PPrROUT = 0.00018 but obviously these values are to a large extent arbitrary.Next, appropriate values for the prior probabilities, and hence odds, of association for candidate and routinely typed markers should be chosen. From the argument set out in the p value we obtain for it we write:Finally, for each marker and the chiinv/2OPo = OPr* eCAND = PPrCAND/(1-PPrCAND) or OPrROUT = PPrROUT/(1-PPrROUT) as the prior odds depending on how we have categorised the marker and we can then go ahead to rank all results according to the posterior odds of association. If we wish to obtain the posterior probability rather than odds we can write ProbPo = OPo/(OPo+1), which for the example above is equal to 0.755 for either the candidate marker significant at 0.01 or the routine marker significant at 0.00001. However we again emphasise that we do not regard the posterior odds or probability as representing an absolute value but rather as acting as a means to rank the results from different markers in order to guide further investigation.We can use either OPrThe main emphasis of our proposal is that approaches are adopted which provide differential treatment to markers in candidate genes and those which are routinely genotyped within a genome wide association study. Although we do not think the differences are critically important, our procedure does involve a slightly different emphasis from the FPRP approach. In order to obtain a value for the FPRP one must declare the power of the test to detect association. We believe it is fair to say that in most circumstances, at least within the context of markers typed routinely in the context of a genome scan, one can say nothing about the minimum value for the true genetic effect. Yet if the power of the test is declared to be higher than it actually is then the FPRP value will be too low. Ultimately, if the true genetic effect is minimal and the power to detect it is close to zero then the actual probability that one has a false positive finding will be close to one – all positives will be false positives. Yet a value for the FPRP based on some over-optimistic assessment of power may be quite small. We believe that in practice the value calculated for the FPRP may well not reflect the true probability that a finding is a false positive.post hoc, is determined to best fit the data. We have emphasised throughout that we would not regard the posterior odds or probability arising from this approach as truly reflecting the actual chances that a locus is involved in susceptibility. We believe that the FPRP value is in more danger of such a literal interpretation. In order to avoid such a literal interpretation we suggest that in applying our approach it may be preferable to quote the posterior "odds in favour of association" rather than "probability of association". That said, we do offer another approach to interpreting the posterior odds or probability obtained from our procedure. We suggest that it might be explicitly recognised as representing a best case scenario. Thus, if we were obtain a posterior probability of association of 0.75 we might make a statement along the lines of "the probability that this result reflects a true positive association is no more than 0.75". Arguing along these lines, we could say that the FPRP approach seeks to declare an estimate for the probability that a result is a false positive whereas ours seeks to declare a minimum probability for a result being a false positive, with fewer prior assumptions. We emphasise that we do not regard this distinction as critical.The alternative approach which we suggest does not require that any value for the power be specified. Rather, our method explicitly use a "best case" scenario. Rather than take the likelihood for the observations conditional on a pre-specified alternative hypothesis, our approach considers the alternative hypothesis which provides the best fit to the data, the maximum likelihood hypothesis. The criticism of this approach would be that it does not represent a true Bayesian scenario in which explicit statements are made about the prior probabilities of alternative hypotheses and the associated likelihoods of the observed data. The advantage of the approach is that it is not sensitive to any user-defined values regarding the genetic model, linkage disequilibrium parameters, sample size and associated power. It always explicitly deals with the alternative hypothesis which, p values will be overshadowed by the huge numbers of genotypes already obtained. There is a sense that researchers submitting their painstakingly acquired datasets for genome-wide studies are sleep-walking into a desolate space from which nothing will emerge but a set of p values conforming pretty much to chance expectation. The approach we propose aims to mitigate some of the worst effects of this.What we do regard as being of critical importance is to have some formal, public system for treating results from candidate regions differently from routinely typed markers within the context of whole genome scans. Whether our method or FPRP or some related procedure is used, it is essential that markers from candidate regions can be treated as special cases. If this is not done then important findings from such markers will be swamped. Large samples will typically have been collected over a number of years at considerable effort. Once they have been genotyped for hundreds of thousands of markers they will be at risk as being regarded as "burned out". Any subsequent studies carried out on these samples yielding modest p values but which are not yet regarded as candidates. Once genome scans have been completed and their results reported it will become necessary to review findings retrospectively from markers which "become interesting" after the scan is complete.We note that our proposed procedure will do nothing to highlight the importance of true positive markers which yield modest Doubtless these issues will be hotly debated, but we hope that the scheme we propose may represent a useful contribution to this debate and we look forward to the issues being progressed further.p < 0.01 as being quite interesting and as supporting further investigation of this gene, for example by finding and typing additional polymorphisms within it. We realise that there are many theoretical problems with taking such an approach but we believe that it reasonably reflects the interpretation which in practice many investigators apply when assessing the results of association studies. We know that we expect 5 of the 500 K markers to be significant at p < 0.00001 purely by chance. It is also possible that some highly significant results from routinely typed markers may represent true genetic effects from hitherto unsuspected regions. We propose that we would like to achieve a scheme of ranking results such that a p value of 0.01 from a candidate marker will be considered similarly to a p value of 0.00001 from a routinely typed marker in terms of providing an indication of where further efforts may directed. We recognise that some genuinely associated markers may produce even smaller p values than this and they will receive favourable consideration whether or not they are recognised in advance as occurring in candidate genes.The situation we envisage is as follows. We have a dataset suitable for association studies containing perhaps several hundred cases and controls. We are about to submit the sample for a genome wide association scan using, say, 500 K SNPs. For the phenotype under consideration there exist a few genes for which there is quite strong, but not completely compelling, evidence for association. Alternatively, there may be genes which are very strong candidates for other reasons such as the results of expression studies or pathway analysis. If we were not performing the genome wide scan we would be typing additional polymorphisms in these genes. As it is, there are some 40–50 SNPs in these genes which will be typed in the genome wide study which have not already been typed. If we were typing these SNPs in isolation then we would take a result significant at All of this fits very comfortably into a Bayesian framework. Taking a relatively sceptical view we can say that if a gene has been the subject of a few positive and some negative association studies then the probability that that gene really does influence susceptibility and that a previously untyped polymorphism within it demonstrates association might be in the region of 0.1. Then we would say that the prior odds for a true positive association are 0.1/0.9. Then we could we carry out our genome wide association study and obtain the likelihood ratio for the results for a candidate polymorphism to be observed assuming association compared with the null hypothesis of no association. To apply a true Bayesian approach we would need to be able to calculate a Bayes factor consisting of the ratio of the cumulative probability distributions for the observed results under the alternative and null hypotheses. Instead, we propose a modified approach which uses instead the likelihood ratio obtained in the course of carrying out a likelihood ratio test for heterogeneity of allele frequencies between cases and controls. This test uses the two likelihoods maximised over allele frequencies under the alternative and null hypotheses. We use these maximised likelihoods as proxies for the distribution of likelihoods which might be obtained over the universe of alternative and null hypotheses and which would be needed in a true Bayesian approach. Multiplying this likelihood ratio by the prior odds provides an indication of the posterior odds for the polymorphism to be associated with the disease. We can do the same thing for a routinely typed polymorphism but assign it a lower prior probability of being truly associated. Writing OPr and OPo for prior and posterior odds and CAND and ROUT for a candidate or routine polymorphism, we have:CAND = OPrCAND*LRCANDOPoROUT = OPrROUT*LRROUTOPop value of 0.01 would yield similar posterior odds to a routine marker yielding a p value of 0.00001.Our task would then be to select suitable prior odds for association for a routinely typed marker such that a candidate marker with a likelihood ratio yielding a p value and the likelihood ratio. Although in practice the test for association used may consist of a Pearson chi-squared statistic derived from a 2 × 2 contingency table it is safe to assume that this will be very similar to a likelihood ratio statistic calculated as 2*ln(LR) formulated as test for heterogeneity of allele frequencies between cases and controls. In the context of a genome wide association study this obviates the need to go through all the genotypings recalculating likelihood ratios because we can instead use the p values output from the analysis and convert each into the associated likelihood ratio. We can write p = chidist to indicate the p value associated with a chi-squared statistic of x having d degrees of freedom and likewise we can write the inverse function x = chiinv to indicate the chi-squared statistic, x, which would yield the stated p value. We can use this to select appropriate values for the prior odds for candidate markers and routine markers by considering the p values which we wish to make equivalent to each other. The allele-wise test for association yields a chi-squared test with one degree of freedom, so that we can state that a significance value of p = 0.01 from a candidate marker is equivalent to a chi-squared of chiinv = 6.65 and a likelihood ratio of e6.65/2 = 27.8, while a significance value of p = 0.00001 from a routine marker is equivalent to a chi-squared of chiinv = 19.5 and a likelihood ratio of e19.5/2 = 17150. If our aim is to make both sets of results yield similar posterior odds for association, and if we assume prior odds for the candidate marker of 0.1/0.9 then we can write:It is helpful to realise that, if one considers allele-wise tests for association with biallelic markers, there is a simple relationship between the ROUT = OPrCAND * LRCAND/LRROUTOPr= 0.1/0.9 * 27.8/17150= 0.00018As in this case the odds are small they are practically equal to the probability, so using this scheme we would be declaring that a marker routinely genotyped as part of a genome wide association study has a prior probability of 0.00018, or approximately 1 in 5,000, of being truly associated. If one considers that there are 25,000 or so human genes and that several may be involved in the susceptibility to a particular disease then this estimate may not be terribly wide of the mark.p values to posterior odds is available from . It is provided in the form of an Excel spreadsheet and also as the C source and MSDOS executable for a program which will read in output from the PLINK program, such as might be available from a genome wide association study.Software to carry out the conversion from


Family-based association testing is an important part of genetic epidemiology. Tests are available to include multiple siblings, unaffected offspring, and to adjust for environmental covariates. We explore a susceptibility residual method of adjustment for covariates.Through simulation, we show that environmental adjustments that down-weight persons who are "destined" to be affected decrease the power to detect genetic association. We used the residual adjusted method on the Framingham Heart Study offspring data, provided for Genetic Analysis Workshop 13, and got mixed results.When the genetic effect and environmental effects are independent, a susceptibility residual method of adjustment for environmental covariates reduces the power of the association test. Further study is necessary to determine if residual adjustment is appropriate in more complex disease models. T = Y - μ, adjusts Y, a dichotomous indicator of affection status, by μ, a covariate value based on a dichotomous trait, measured trait value , or age-dependent phenotypic value.Family-based association tests, such as Spielman's transmission disequilibrium test (TDT)  rely on T, such that T = Y - p, where p is the estimated predicted probability from the logistic model. In this context, we call T the susceptibility residual.To expand this covariate adjustment to a more general setting, we propose the use of a susceptibility score. For instance, we model affection status on the covariates of interest using logistic regression and then use the susceptibility residual as a measure of deviance from the predicted outcome. Genotypes are excluded from the regression because we are not adjusting for genetic effects. Our logistic model differs only slightly from the suggested calculation of A concern with the covariate adjustment method in FBAT is that persons with low deviance from the predicted outcome are heavily down-weighted. The assumption is that persons with low deviance were susceptible to affection regardless of genetic makeup and therefore should not contribute fully to the association test. In the extreme case in which the affection status is predicted perfectly by the covariate in FBAT, that individual contributes no information to the result. Through simulation we explore the power and type I error rates of the adjusted Horvath test compared to the unadjusted version of the same test. We vary the influence of the genetic component and the non-genetic (covariate) component from no influence to an increased odds-ratio of 4.95. This comparison of methods allows us to investigate the advantages and disadvantages of covariate adjustment. Briefly stated, we found that for those cases in which a moderately strong genetic influence is determinative, adjustment for environmental cofactors can reduce the power to detect association between the disease and the disease allele. We apply the susceptibility residual adjustment to the Framingham Heart Study offspring data, available for the Genetic Analysis Workshop 13 (GAW13), to observe the effects of adjustment on nonsimulated data.In each replicate 100 families were generated with two parents and 2 to 10 offspring, according to a truncated Poisson distribution (λ = 4), such that each family contained at least one affected offspring. Parental genotypes were determined by random selection assuming Hardy-Weinberg equilibrium and random inheritance was used for determining offspring genotypes. For these simulations, a bi-allelic marker was considered such that the disease allele has a population frequency of 0.15. Offspring exposure to the covariate was randomly determined using a population exposure rate of 0.20. Disease status was determined by the following logistic model:disease|x1 = x2 = 0) = 0.05 and β1, β2 are the regression coefficients corresponding to the indicator variables x1 and x2 for raised genetic and environmental risk, respectively. The β values were varied at intervals of 0.2 from 0.0, no risk, to 1.6, increased odds of 4.95, thus creating nine levels of the genetic effect and nine levels of the covariate effect. The probability of disease calculated by the above model was compared to a random number selected from a uniform  distribution. If the probability of disease onset is greater than the random number, then the individual in question is labeled as affected.where α ≈ -2.9, such that Pr per day for at least one of the five exams.The genotypes of interest were determined using a genome scan with FBAT on a random selection of nuclear families. We selected markers GATA48G07A 10q25.3) and GGAA5D10 10q26.13) on chromosome 10, and GGAA7D11 (17q11.2) on chromosome 17 because each showed a significant association to hypertension using FBAT. Also, literature suggests association for neighboring regions on chromosome 10 [.13 on chq25.3 andp < 0.05. The marker was tested on 100 random selections of nuclear families from the full pedigree data. Thus correlation between families within a pedigree was avoided. The program default minimum of 10 informative families for testing was maintained in these FBAT runs.To match the methods of the simulation, FBAT was run using a dominant model of inheritance and the bi-allelic testing mode. Significance of an allele was determined by p < 0.05 for the disease allele). Figure p < 0.05, with Benjamini-Hochberg correction for multiple comparisons). The type I error rate of the model was determined at varying covariate effect levels, using no genetic effect. Figure p = 0.71). Thus we find no change in type I error rates from the covariate adjustment.The power of the FBAT procedure was determined for each parameter set by testing 1000 simulated data sets, 100 families each, and recording the frequency of significant results . The increased information in the adjusted test allowed it to run on all 100 random samples (median N = 23). Allele 362 is over-transmitted and shows 65% significance in the unadjusted method and only 42% significance in the adjusted method, suggesting reduced power in the adjusted method.In Table In Table In Table Results for the negative control markers show low false-positive rates for both methods, data not shown. At marker GATA88F09, the negative control on chromosome 10, the unadjusted test showed no significant results  and the adjusted test showed a low number of significant results  on the five alleles tested. At marker 217YD10, the negative control on chromosome 17, the unadjusted test showed no significant results  for all but one of the seven alleles tested . The adjusted test at marker 217YD10 showed a low number of significant results  for the 11 alleles tested.In simulated data the covariate adjustment procedures in FBAT demonstrated the expected loss in power attributable to the "discounting" of information for affected persons who had a high prior environmental risk of disease. This phenomenon of "discounting" has the paradoxical attribute that more definitive information about the environmental contributors to a trait may actually cause more difficulty in detecting a genetic signal.Interpretation of the comparison between the adjusted and unadjusted versions of the FBAT test when applied to real data is difficult primarily because the suggested loss of power was not uniform across marker loci. In particular, at the three markers a total of eight alleles were detected by one or both of the tests, of which four were detected more frequently with the unadjusted test and three were detected more frequently with the adjusted test. All four of the alleles detected more frequently by the unadjusted test had increased transmission. Yet only two of the three alleles detected more frequently by the adjusted test were under-transmitted and allele 350 on marker GATA48G07A, detected in 100% of the tests for both methods, is over-transmitted. Also, no pattern arises between allele frequency and the frequency of detection by either test method, unadjusted or adjusted.The negative controls suggest that the false-positive rates for both tests are low; however, the spurious 30% false-positive rate presumed for one allele on marker 217YD10 (chromosome 10) for the unadjusted test lends doubt to the results for alleles showing significant association at low frequencies. In particular the significance of allele 286 on marker GGAA7D11 (Table The mixed results, regarding power loss in the susceptibility residual adjusted test, from the three markers of interest could reflect variation in the extent of genetic determination of the trait or possibly interaction between genetic and environmental factors that is not specified in our model. Further simulations, accounting for variable influence of different alleles at a marker and gene × environment interaction, would be needed to explain this set of results more satisfactorily.When the genetic effect and environmental effects are independent, a susceptibility residual method of adjustment for environmental covariates reduces the power of the association test. Results of the method comparisons in the Framingham data are not conclusive and further study is necessary to determine if susceptibility residual adjustment is appropriate in more complex disease models.Our results should not be taken as indication that the FBAT adjustment should be avoided. Further simulations, involving correlation between genetic and covariate influences, are being explored in an effort to find situations in which the susceptibility residual is beneficial for the adjustment of covariate effects. Likewise, the reader should note that we have explored only one method of covariate adjustment for family-based association testing. We hope that our efforts in this matter will encourage others to evaluate covariate adjustment measures, since we believe that accounting for nongenetic influences is important in modeling complex diseases.


With the improvement of genotyping technologies and the exponentially growing number of available markers, case-control genome-wide association studies promise to be a key tool for investigation of complex diseases. However new analytical methods have to be developed to face the problems induced by this data scale-up, such as statistical multiple testing, data quality control and computational tractability.p-values are assigned to genomic regions termed bins, which are defined from a gene-biased partitioning of the genome, and the false-discovery rate is estimated. We have applied this algorithm to data coming from three genome-wide association studies of Multiple Sclerosis.We present a novel method to analyze genome-wide association studies results. The algorithm is based on a Bayesian model that integrates genotyping errors and genomic structure dependencies. The method practically overcomes the scale-up problems and permits to identify new putative regions statistically associated with the disease. The last years have shown a tremendous increase in the number of markers available for association studies. Previous studies were dealing either with the whole genome at a very low resolution (for instance 5 264 microsatellites in ) or withFirstly, the multiple testing problem seems to cause association studies ability to detect associations to decrease as the number of markers increases. The classical analysis strategy, based on an association test for each marker , encountMethods like False Discovery Rate (FDR)  computatbins) on another basis, so as to limit the multiple-testing problem and make it independent of the number of markers. Combining these two arguments leads to choose one bin for each gene, and to create "desert" bins in large unannotated regions. It allows to associate a list of genes with a test, which simplifies the analysis of results. The drawbacks are (i) that it makes more difficult the study of these "deserts", however the goal is here to maximize, not the chance of finding an association, but the chance of elucidating a mechanism of a complex disease given the current knowledge (ii) that a bin might contain several haplotype blocks, resulting in a dilution of the association signal if only one block is associated. Reciprocally, neighbor bins are not independent because they may share a haplotype block. However, with the classical strategy, correlated neighbor SNPs would also be tested separately.Secondly, a genetic association of a given SNP is a statistical feature and does not explain by itself a phenotype. To biologically interpret an associated marker, its haplotype block should first be delimited. Then, the association can be refined by fine-scale genotyping technologies or ideally by full resequencing. This eventually allows to identify functional mutations. Most of the time, these mutations impact relatively close genes. This is a first argument to bias association analysis towards genes. Moreover, even if haplotype blocks are unreachable, DNA might be cut into distinct regions  and the error rate (i.e. errors left in the data) is difficult to adjust. Obtaining unbiased statistical results is then conditioned to good pre-processing filters. Indeed spurious markers must be eliminated and missing data correctly managed.Thirdly, genome-wide genotyping data are obtained by high-throughput experiments which encompass limitations requiring careful statistical methodology. Especially, with i) is not enough for good asymptotic approximations and (ii) should be considered with care given possible high error rate.In addition, for most of SNPs used in this study, some genotypes are held by less than few percents of patients, which, given the usual collection size of a few hundreds, . It identifies putatively associated bins, containing genes previously described to be linked to MS .Three association studies dealing with Multiple Sclerosis (MS) in three independent collections have been realized. Around 600 patients have been recruited for each study, half of them as cases affected by the disease, half of them as controls Table . Genotypv). Indices are noted in lower case (k), ranging from 1 to the corresponding upper case letter (K). Unless needed, this range of indices  is omitted. The number of different values is noted # is noted P , m ∈  , and a genotype value for each marker i is represented by this vector:Stochastic variables are noted with a round letter , errors on heterozygotic genotypes are more frequent. It can be detected through the deviation of a SNP from the Hardy-Weinberg equilibrium, which basically states that, noting P(a) = P(aa) + P(Aa)/2 and P(A) = P(AA) + P(Aa)/2:Due to i) if the number of missing genotypes is higher than 5% because the genotyping process quality was low for this SNP, (ii) if the minimum allele frequency in controls MAF = min(P(a), P(A)) is lower than 1%, because the SNP holds no information, or (iii) if the probability that the SNP follows the Hardy-Weinberg equilibrium in controls is lower than 0.02.Therefore, the following pre-processing filters are applied: SNPs are discarded  each individual subset of the study is a realization of the conditional distributions On the contrary, due to the experimental design, estimations of A general way to assess the association of a bin b is to estimate whether However, as only We have chosen likelihood ratio LR as a statistic to estimate the "distance" between estimations of As all patients are considered to be independently chosen, the LR of the set of patients available is:p-value πb needs to be computed. This is theoretically achieved by enumerating all possible outcomes Db(σ) of the experiment that lead to the observed data Db(σ0) (σ is a enumeration parameter to be defined. The following notation simplification is done: Db(σ0) = Db). Then the probability p(Db (σ)) of each outcome assuming that p-value is:To assess estimation errors due to randomness and sample size, the probability that p-values is based on permutations: possible outcomes are obtained through patient phenotype permutations σ and σ0 is the identity permutation. The probability of each permutation is uniform. The denominator of equation (8) is constant with respect to such permutations, therefore it is omitted. Sampling this space is possible: random permutations of the phenotypes are drawn and used to compute a LR. This is a Monte-Carlo procedure, for which we propose an optimized implementation that guarantees the precision required for FDR estimation:In this article, estimation of b, compute LR for new permutations of phenotypes until the number of permutations realized Nb satisfies the following equation, noting p-value:For each bin θ and γ/δ control the quality of the method: θ is an upper bound of the threshold that is expected to be used to select bins. γ/δ controls the error due to the randomness of the process: Assuming that two consecutive p-values πb1 <πb2 ≈ θ are sufficiently spaced (probability ps = eδ-), c = cdf(γ) . In this article, B = 11 264, θ = 0.001, δ = 1 and γ = Nb = 507 003, ps = 0.37 and c = 0.92.To address multiple testing, the method uses an FDR estimation defined as in :πb ≤ θ. p-values below. The ratio is therefore an estimation of the proportion of false negatives in the set of bins with a p-value below θ. Because we want to analyze thoroughly the FDR for around the 10 bins with the lowest p-values, the FDR is not controlled at a specified threshold as in . Assuming that sharing of haplotype block by neighbor bins is the only source of correlation between tests, the positive correlation seems reasonable. Indeed, if the p-value of a not associated bin decreases, the p-values of bins sharing the same haplotype block are more than likely to decrease too. The uniform distribution is less obvious, because the number of possible contingency tables is finite so that even the null distribution is not uniform. However, the sample size is one to two order of magnitude higher than in other applications of FDR to discrete data in which the problem is acute [This estimation relies on two main hypothesis:  p-is acute .Correlation between markers induced by LD is modelled with an inhomogeneous hidden Markov chain of order 1. Indeed, as a rough approximation, for each marker, most information is found on its first neighbor on each direction of DNA. In a directed graphical model, independence assumptions consist in:Finally, this assumptions also allow to obtain correct estimations because corresponding contingency tables are sufficiently filled. They implies that contingency tables are computed for 2 SNPs :An error model Figure  is introSince ∅ is the number of patients with either m is the number of cells. To obtain more regular estimates, a constant is added to all cell counts. It is a Dirichlet prior on parameters. This constant is chosen to be C = α0α0 is the chosen error rate and Where nOn the other hand, given the previously developed structure of errors, the following model of β is estimated for each marker through the resolution of the non-linear system drawn from the preceding model. The maximum error rate α0 is estimated during external comparison of Affy. technology and other technologies. In this study, the error rate is chosen to be α0 = 0.05. The error rate is α = min/)~OJb. Some aa from its left neighbor and AA from its right neighbor, the merging of this two inferences would results in a contradiction and thus a low resulting likelihood. On the contrary, the approximated likelihood does not detect this contradiction and is falsely increased. This likelihood is named thereafter "two-marker" likelihood.This equation considers information coming from two neighbor markers together. Compared to the full model, information flow is limited to pair of markers. The likelihood could be falsely increased in this extreme situation: suppose that a missing genotype is inferred Simplifying further leads to consider markers one by one. There is no model of linkage disequilibrium anymore, but noise is reduced as cells are better filled. This likelihood is named thereafter "naive likelihood" because it corresponds to a naive Bayesian model:A, B, C (Table ABC), considering the collection of origins as a co-variable. The overall computation time is about 10 days on a single processor.The method has been applied to each of the three collections  C Table  as well A (resp. B and C), out of 112 463 SNP, 84 430 (resp. 93 548 and 86 652) SNP remains. If all SNP satisfied the Hardy-Weinberg equilibrium, 2 249 SNP are expected to be discarded. 9 422 were for collection A. It can be explained (i) by artifacts of DM calling algorithm which has a higher error rate on heterozygotic genotypes (ii) by deviations from the assumptions underlying this theoretical equilibrium. The bin partioning algorithm divides the genome into 19 556 gene bins and 1 993 desert bins. Out of these 21 549 bins, only 11 264 (52%) contain one SNP or more after pre-processing in at least one collection and are considered for further analysis. Before pre-processing, out of 12 512 SNP with one bin or more, 2 781 have only one SNP, and 2 188 bins 10 SNP or more. The maximum is 210.The pre-processing filters discard around 20% of SNP: for collection p-values computed using the two-marker L2 or the naive L3 likelihood for the three collection design. Two-marker FDR remains below naive FDR until a p-value level of 0.01 and both increase slowly towards 1. FDR against the number of selected SNP plots are detailed by collection in Figure p-value. The oscillations are less important for the three collection design, maybe because of the three time increase of sample size. With a FDR threshold of 5%, only between 2 and 6 bins are selected depending on the collections and likelihood considered , and handles the multiple testing problem through FDR estimation while staying computationally tractable. The method has been applied to analyze three association studies in Multiple Sclerosis.The FDR threshold is chosen according to the desired application. To conduct expensive further experiments with putatively associated genes, a very low rate of false-positives is required. A FDR threshold of 5% seems reasonable. On the contrary, if one wants to minimize the false-negative rate, a FDR of 50% is acceptable.i) to assess the algorithm and evaluate the different parameters and design and (ii) to identify genes potentially associated to Multiple Sclerosis. We have evidenced that the three collection design outperforms the one-study design in terms of expected number of true-positives, despite differences between the studied collections, especially on the severity of the disease. Furthermore, with this three collection design, the two-marker likelihood L2 seems to be more efficient thanks to the additional information used. With this configuration, a FDR threshold of 5% gives 6 associated bins. Four of them are located in the MHC region, known to be linked to Multiple Sclerosis [OR2T2 and OR4A47. The biological meaning of such association is unclear but the extended MHC regions contain many other olfactory genes [Applying the method to experimental genome-wide association data on three collections permits (clerosis . It is ary genes  and olfary genes . At FDR The authors declare that they have no competing interests.NO, FK and JW conceived and designed the model. NO, KF, ML and GM wrote the analysis tool. NO, FK and JW wrote the manuscript.


Although identification of cryptic population stratification is necessary for case/control association analyses, it is also vital for linkage analyses and family-based association tests when founder genotypes are missing. However, including related individuals in an analysis such as EIGENSTRAT can result in bias; using only founders or one individual per pedigree results in loss of data and inaccurate estimates of stratification. We examine a generalization of principal-component analyses to allow for the inclusion of related individuals by down-weighting the significance of individual comparisons. At the heart of all genetic case/control association analyses lies estimation of allele frequencies. For linkage analyses and pedigree-based association analyses, allele frequency estimates are used when a parental genotype is missing. Because cryptic population stratification results in misestimates of allele frequency, this can lead to false positives for any type of analysis with missing founder genotypes ,2. CurreM × N matrix . An eigenvalue decomposition is then performed on the N × N correlation matrix and population membership is inferred from the eigenvectors. One determines how many natural ethnicities are present by examining the sizes of the eigenvalues using a graphical scree analysis or a numeric approach .However, the nature of the eigenvalue decomposition introduces problems when individuals are related. Because biologically related individuals are already genetically correlated, this can bias the decomposition, especially in the presence of a large number of related individuals (such as in large pedigrees). Using only unrelated individuals limits the analysis to either the founders or a sampling of unrelated individuals. Although the founders provide all of the genetic variation present in the subsequent generations and therefore represent all available information, using randomly sampled unrelated individuals results in a loss of information.To allow for the analysis of related individuals, we will apply a recently developed method of linear dimensionality reduction . This mew, we can instead maximizeIf we instead consider a system of weights N × N matrix such thatproviding a weighted version of PCA. In particular, we define the Laplacian to be an One then performs an eigenvalue decomposition on the matrixThe use of the Laplacian causes an important change in the process. The PCA is normally computed on the matrixN × N symmetric matrix of the pairwise genetic covariance between subjects. However, since the Laplacian is an N × N matrix placed within the covariance calculation, we then produce an M × M symmetric matrix of the weighted pairwise covariance between markers. With dense SNP genotyping, we typically see more markers than individuals by several orders of magnitude, and this computation would be nearly intractable. However, since the Laplacian matrix is positive semidefinite, we can compute a Cholesky decomposition such thatan And thus,LetWe then see thatBut then we can compute the eigenvalue decomposition ofYT and the second set of eigenvectors  [This is much more manageable. Further, the eigenvalues for these two are the same and the eigenvectors of the original formulation are simply the product of ization) .For our analyses, we use a weight based on work by McPeek and colleagues . In partN individuals, let K be the N × N kinship matrix and let 1 denote a column vector of length N of 1 values. Then the vectorFor S children without genotyped parents, each child is given weight 2/(S+1). Note that as the number of typed children increase, the sum of the weights tends toward 2 - precisely the number of founders of the pedigree. This generalizes to any sized pedigree; namely, the total of the weights cannot be larger than the number of founders, since the founders were the only source of genetic material in the pedigree.provides the best linear weights to compute allele frequencies for related individuals. In a fully typed pedigree, each founder is given a weight "1" and all other individuals are given weight "0." In any pedigree with a single typed individual, that individual is given weight "1." In the simple case of a nuclear pedigree with L from the weight matrixFor pairwise weights between two individuals, we use the product of the individual weights. In particular, we derive We test this method compared with the standard EIGENSTRAT method using the Framingham Heart Study data. After cleaning, the Framingham Heart Study data consists of 1180 pedigrees, including 418 singletons. The remaining 762 pedigrees have an average of 8.3 genotyped individuals, including 9 pedigrees with more than 50 genotyped individuals. The best standard of comparison would be an analysis using all founder genotypes, but because not all founder genotypes are available, we apply an algorithm to identify the maximal set of unrelated individuals. We consider the resulting population membership as the "gold standard." We also consider the set of singletons and one individual chosen at random from each pedigree. Finally, we consider the full sample with all related individuals using the standard EIGENSTRAT method and our novel method. We then assess to what degree including related individuals influences the standard method and how well the novel method reproduces the "gold standard." We also examine the total weight of all the genotypes as a measure of how much information is used.We used the full 50 k marker set but kept only autosomal SNPs with a minor allele frequency greater than 0.05 and a genotyping rate greater than 99%, for a total of 31,068 SNPs. We dropped individuals with more than 5% missing genotypes, for a total of 6757 individuals.We considered five data sets for PCA: MaxUnrel, maximum number of unrelateds, our gold standard; Singletons, individuals without genotyped relatives; One per, one individual per pedigree chosen at random; Full, all individuals without weighting; and Weighted, all individuals with weighted PCA as described above. Table Figure Figure We propose the use of weighted PCA implemented through the presence of a Laplacian matrix to allow detection of stratification in related individuals. Our results indicate the methodology developed by McPeek and colleagues to compute allele frequencies in related individuals can be extended to detection of ethnic stratification. This method uses all available genotypic data, with an effective sample size that approaches the number of founders in the pedigrees. This exceeds other methods of selecting unrelated individuals. Furthermore, we see evidence of bias and outliers when using small subsets of individuals. Using too few individuals for stratification may also artificially inflate evidence of stratification. It does appear that the presence of related individuals in a very large sample seems to have little effect on the stratification analysis, but this might not hold in other circumstances. Furthermore, this method has only been tested on a European American sample with a single principal component (probably identifying a continuous population spread such as northern to southern European). Because the Framingham data does not have any obvious discrete clusters, this method still must be tested in a more diverse population.GWAS: Genome-wide association study; MCMC: Markov-chain Monte Carlo; PCA: Principal component analysis.The authors declare that they have no competing interests.ALH developed the method and performed analyses. RC and BKS assisted in design and methodology. CHJ assisted in statistics and data management. All authors read and approved the final manuscript.


Identifying quantitative trait loci (QTL) for both additive and epistatic effects raises the statistical issue of selecting variables from a large number of candidates using a small number of observations. Missing trait and/or marker values prevent one from directly applying the classical model selection criteria such as Akaike's information criterion (AIC) and Bayesian information criterion (BIC).p small n" data. Specifically, sparseness and possible asymmetry of the significant coefficients are dealt with by developing a Gibbs sampling algorithm to stochastically search through low-dimensional subspaces for significant variables. The superior performance of the approach is demonstrated via simulation study. We also applied it to real QTL mapping datasets.We propose a two-step Bayesian variable selection method which deals with the sparse parameter space and the small sample size issues. The regression coefficient priors are flexible enough to incorporate the characteristic of "large The two-step procedure coupled with Bayesian classification offers flexibility in modeling "large p small n" data, especially for the sparse and asymmetric parameter space. This approach can be extended to other settings characterized by high dimension and low sample size. With the advent of high-throughput biotechnologies to genotype dense molecular markers throughout the genome, statistical methodologies are crucial in understanding the genetic architecture of complex traits, and in locating genes underlying important traits. Since the pioneering statistical work by Lander and Botstein , much efŻak et al.  or even a fixed value  is usually chosen for the mixing weight. However, when n, and tackle the aforementioned issues. We then construct a two-step Bayesian variable selection approach for model (1) in the case that n ≪ (pβ + pγ). In the first step, we employ a restrictive prior for each of the coefficients in model (1) in order to enforce stochastic filtering of the large number of candidate variables. This prior also allows flexibility for the possible different numbers and/or scales of positive and negative coefficients (see .that is, expecting at most  Gaffney  and Yi e Gaffney , among oYn be the column vector including the trait values of all strains under investigation, let Xi be the vector of all marker values of the i-th strain and Zi be the vector of all epistatic candidate values of the i-th strain. Denote the marginal distribution of A as [A], and the conditional distribution of A given B as [A|B]. With data  and the prior specification in Section 3.1, we have the likelihood function, that is, the joint distribution function of the data , the parameters , Let Xn can be specified based on the available linkage map information [βj. Similarly, the conditional distribution of γj. The priors of the hyperparameters, θβ+, θγ+, φβ+, φγ+, θβ-, θγ-, φβ- and φγ-, are specified to be as noninformative as possible.The distribution of ormation . The conμ is set as the mean of the observed trait values. Then, with individuals having fully observed trait values, each component of β and γ is initially estimated using recursive univariate regression. Other parameters, β, and similarly, the initial values for γ. For example, we can initialize β, and then use max{#{j : βj > 2σβ+}, 1}/pβ to initialize wβ+.Since the specified priors are conditionally conjugate, Bayesian variable selection can be implemented with a Gibbs sampling algorithm. We initialize the algorithm by imputing missing genotypic values based on the observed genotypes and linkage information. The initial value of Xi,-j be Xi excluding the j-th component, and define βj -and γj -similarly. Based on the likelihood function in (4), the Gibbs sampler can be developed by recursively drawing the missing genotypic values, the missing trait values, and the model parameters from their full conditional posterior distributions as follows.Let Sample missing values: Sample each missing genotypic value Xij from its full conditional posterior distribution,Yi from its full conditional posterior distribution [Yi|Xi, μ, β, γ, and then sample each missing trait value μ: Sample Sample μ from its full conditional posterior distribution,β and γ: Sample Sample each βj and γj from their full conditional posterior distributions,where Sample wβ+, wγ+, wβ-, and wγ-: These parameters can be sampled from the conditional posterior distributions,βj > 0 : 1 ≤ j ≤ pβ} and βj < 0 : 1 ≤ j ≤ pβ}; γj > 0 : 1 ≤ j ≤ pγ} and γj < 0 : 1 ≤ j ≤ pγ}.where Sample , and : With conditionally conjugate priors, the posterior for all variance parameters are still inverse gamma distributions. Specifically,j-th additive effect, the posterior probabilities wβ j+ = P  and wβ j- = P . With the full conditional posterior distribution of βj and all the notations in the APPENDIX, we haveFor each variable in model (1), one pair of parameters is used to select the corresponding variable. They are, for the wβ j+ and wβ j- can be estimated with the Markov chains of wβ j+ and wβ j- are less than 0.5, the median of the posterior distribution of βj is zero. Similarly, the posterior probabilities wγ j+ = P  and wγ j- = P  can be estimated with the Markov chains of Therefore, the two parameters j-th additive effect based on max{wβ j+, wβ j-}, and rank the j-th epistatic effect based on max{wγ j+, wγ j-}. The top cpβ additive effects, and the top cpγ epistatic effects are selected, respectively. At the second step, we select variables out of those selected ccWe propose to select variables twice under the above Bayesian framework for model (1). At the first step, we use a restrictive prior for each coefficient to ensure an identifiable Bayesian model and enforce to stochastically search for an optimal low-dimensional parameter subspace. We then rank the H0 : βj = 0 vs. H1: βj ≠ 0 on the basis of the Bayes factor, which was defined asFollowing Jeffreys ,50, we tπ (βj = 0) and π (βj ≠ 0) are the a priori probabilities, and the last equality follows the fact that π (βj = 0) = π (βj ≠ 0) at the second step of our Bayesian Classification. As suggested by Jeffreys [B10 (βj) with value between 1 and H0; a B10 (βj) with value from H0; a B10 (βj) with value from 10 to 100 provides "strong" evidence against H0; and a B10 (βj) with value larger than 100 provides "decisive" evidence against H0. Similarly, we can test the hypothesis H0: γj = 0 vs. H1: j ≠ 0 using the following Bayes factorwhere Jeffreys , a B10 (® code and did the simulation study. MZ analyzed the real data. MTW participated in conceptual development, writing, reviewing and editing the manuscript. All authors read and approved the final manuscript.MZ and DZ both conceived the study and developed the method. DZ wrote the MATLABFully Conditional Posterior Distribution of βjj = 1, ⋯, pβ, the fully conditional posterior distribution of βj isFor each where the updated parameter values are