Lipid phenotypes were simulated in the Framingham Heart Study using several measured variables including causal genes (based on a 500 k SNP panel), smoking, dietary intake, and medication usage. We report a sensitivity analysis of how medication use  affects association inferences. Associations were performed for LDL-c under seven different correction schemes: 1) ignore medication use entirely (no correction) and adjust for covariates; 2) delete medicated subjects then adjust for covariates; 3) include medication use (Yes/No) as a covariate during covariate adjustments; 4) correct raw values using clinical trials information then adjust for covariates; 5) correct raw values using the actual simulation protocol ("truth") then adjust for covariates; and 6-7) over-corrections  then adjust for covariates. Results indicate that failure to properly correct for medication usage can profoundly affect the heritability, and hence the association results. The empirical results yielded one genome-wide significant locus on chromosome 22 (RS2294207), consistent with the simulation protocol. This signal was detected under all schemes that corrected the raw values , but was not detected under the first three adjustment schemes . In summary, we confirm that failure to  Schemes 6 and 7: we applied two over-adjustments by adding arbitrary constants (X + 75 mg/dl for Scheme 6 and X + 100 mg/dl for Scheme 7).Our analysis plan consisted of seven different correction schemes , smoking (yes/no), and diet (intake per day). Additionally, medication use (yes/no) was a covariate for Scheme 3. In summary, a given measure was regressed on these covariates, separately in males and females, using in a stepwise procedure. Only significant terms (5%) were retained to compute the residual from this regression. The residual was standardized to a mean of zero and a variance of one. The entire correction and adjustment procedure was carried out separately for each of Visits 1 and 3 data, leading to seven Visit 1 and seven Visit 3 variables, as outlined in Table Each of the (raw or corrected) LDL-c traits was then adjusted for several covariates using a stepwise regression method. The covariates included a polynomial in age  effects is compared to another having both the environmental and a polygenic heritability (e + g) component. The likelihood comparison tests the null hypothesis that the heritability is zero. The association analysis is equivalent to a non-TDT family-based method that contrasts the environmental plus polygenic model (e + g) with one in which the environmental, polygenic and association component is included (e + g + a). The comparison tests the hypothesis that the association component is zero.Variance-components models as parameterized in the computer programs QTDT  and MerlFor covariate adjustments, age was the primary predictor, accounting for up to 10% of the variability. Additional covariates of smoking and diet did not substantially alter the estimates.Heritabilities are in Table The purpose of this study was to perform a sensitivity analysis to determine how medication use, or rather how we adjust for medication use, affects genetic inferences in association analyses when the "true" answer is known. As expected, our results confirm that appropriate adjustments provide more accurate results. However, an important message is that some commonly used methods to correct for treatment may yield misleading results.The "true" results were derived by pre-correcting the raw LDL-c values using the simulation protocol and then performing typical covariate adjustments. A series of analyses were performed under a variety of correction scenarios and compared to the truth. As expected, ignoring medication (Scheme 1) use biased the results for Visit 3 data . However, two commonly used adjustment methods also were biased . While both of these methods slightly improved the signals as compared with doing nothing (heritabilities increased from 25% to ~30%) they did not yield "true" results. And, in fact, the association signal was completely missed under all three of these schemes. Removing medicated individuals decreases both the sample size and the power, but even more importantly these are the very individuals who provide the best evidence for genetic effects - the original upper tail! The alternative method that used medication as a binomial covariate (Scheme 3) assumes that the original unmedicated values of both the treated and untreated groups are similar. However, this is not the case  because only those individuals with sufficiently high original values are treated. In other words, this method tries to equalize the means across treated and untreated groups, when in fact the means by definition cannot be the same.For the remaining schemes we found that any of the methods that attempted to recover the original (untreated) levels prior to covariate adjustment worked fairly well in terms of finding the "true" answer. In this dataset, the clinical trials method (Scheme 4) yielded a slight under-adjustment, while adding arbitrary values of 75 mg/dl (Scheme 6) or 100 mg/dl (Scheme 7) led to slight over-adjustments, as shown in the magnitudes of the heritabilities and the association signals. This strongly suggests that independent data such as that from the clinical trials literature can be used to provide correction factors that, while they are not exact, seem to be close enough to recover signals that otherwise would have been missed. However, in some situations more accurate approaches may be required. For example, information on not only the mean (as used here) but also on the variation is available in the clinical trials literature which could be used to account for individual variation in treatment effectiveness. In addition to genes, "environmental" factors such as compliance levels or longitudinal information about treatment also may influence individual variation. Although beyond the scope of the current study, all of this information could be incorporated using a mixed effects model, through multiple imputation, or through a regression that takes into account censored information such as the Tobit model see ,3)..3]).In the last two to three years, GWA studies have been making great strides in identifying common polymorphisms associated with disease risk. For example, over 50 novel loci have now been detected that modify the risk for type 2 diabetes and cardiovascular disease  and overIn summary, these results show that medication use can have a significant effect on genetic analysis. Careful consideration in how one corrects for treatment is important in the case of association studies in which very small effects are expected as compared to the relatively large degree of noise that is produced when values are modified due to treatment effects.GAW: Genetic Analysis Workshop; GWA: Genome-wide association; LDL (LDL-c): Low-density lipoprotein-cholesterol; QTDT: Quantitative transmission-disequilibrium test; SNP: Single-nucleotide polymorphism; TDT: Transmission disequilibrium testThe authors declare that they have no competing interests.CCG carried out quality control analyses of the genotypes. GS extracted the data and set up genotype and map files in the appropriate formats. YJS performed association analyses. DCR participated in the design and coordination of the study. TKR conceived the study, derived all medication correction phenotypes, performed final formatting of analysis datasets, collated all results, and drafted the manuscript. All coauthors approved the final manuscript.


The Framingham Heart Study is a well known longitudinal cohort study. In recent years, the community-based Framingham Heart Study has embarked on genome-wide association studies. In this paper, we present a Framingham Heart Study genome-wide analysis for fasting triglycerides trait in the Genetic Analysis Workshop16 Problem 2 using multivariate adaptive splines for the analysis of longitudinal data (MASAL). With MASAL, we are able to perform analysis of genome-wide data with longitudinal phenotypes and covariates, making it possible to identify genes, gene-gene, and gene-environment (including time) interactions associated with the trait of interest. We conducted a permutation test to assess the associations between MASAL selected markers and triglycerides trait and report significant gene-gene and gene-environment interaction effects on the trait of interest. Current advances in genotyping technologies, such as the Affymetrix 500 k GeneChip, make genome-wide association studies (GWAS) feasible for identifying common variants that underlie complex traits. Some of the recent genetic variants discovered by GWAS include age-related macular degeneration (AMD) ,2, inflaPreviously, several approaches ,6 were pIn our study, we use the multivariate adaptive splines for analysis of longitudinal data (MASAL) presented by Zhang  to analyyij, tij, and xk, ij denote the response variable, time-dependent covariate, and kth non-time-dependent-covariates  for the ith subject at the jth exam, where j = 1, ..., T, i = 1, ..., n, k = 1, ..., p; n is the number of study subjects and Ti is the number of exams for the ith subject. In MASAL, we consider the following nonparametric model:We present a brief review of MASAL and refer to Zhang ,10 for tf is an unknown smooth function and εij is the error term.where Based on a set of observations, MASAL selects a model using a forward step from the following class of functions:βm is the regression coefficient and Bm(x) is a special basis function of the p + 1 covariates x =  , and M is the number of terms. Specifically, Bm(x) is either one of (xk - τ)+ and xk or their product , and a+ = max for any number a and τ is called a knot.where yi, and Wi is the within-subject covariance matrix for i = 1, ..., n. After the forward step, all knots are found and each corresponding basis function will be treated as if it is a given predictor. In the backward step, based on generalized cross-validation (GCV), we delete one least significant term from the large model at a time. The final model we select is the one that yields the smallest WLSl is the WLS of a reduced model with l terms and λ  sum of squared residuals: ly λ = 4 ,11) is tα, and it grows the model by adding terms that minimize the WLS in the forward step, and then it selects a final model by deleting one least significant term at a time in the backward step. In general, the final MASAL model isIn GWAS, we use MASAL to establish the relationship between a trait of interest and genomic markers as well as other non-genetic covariates. MASAL starts with a model that contains only the intercept a where β = 0, where β = '. The Wald statistic can be written asTo access associations between the selected SNPs and the trait of interest, we define a Wald statistic to test whether W The permutation test is done by randomly assigning the phenotype while keeping the set of genotypes intact for each individual and then performing the GWA analysis using MASAL. It is noteworthy that non-genetic covariates go together with the phenotype.where We perform GWA analyses of TG trait with MASAL. We consider the genotype at every SNP as a covariate in the model in addition to sex and age variables. MASAL has the option of setting the maximum order of interactions in the model. We set it to three in our analyses because it is difficult to interpret interactions higher than the third order. We first use MASAL to perform GWA analyses in the Offspring Cohort, in which the repeated TG values and the familial correlations are properly accounted for in the analysis. Next, we perform GWA analyses with MASAL in the Original Cohort in which the subjects are considered to be independent, whereas the longitudinal trait values are considered to be correlated. Then, we examine significant SNPs, SNP-SNP, and SNP-covariate interactions in the two generation data sets analyzed and compare the level of concordance of significant associations in the two samples.In the Offspring Cohort, some pedigrees have more than 100 subjects, which cannot be treated as independent individuals. These families with repeated traits induce a large covariance matrix in MASAL and thus markedly limit its efficiency. Therefore, in the analyses of Offspring Cohort, we split all pedigrees into sibship units according to the information from the Original Cohort. We obtained 1,767 sibship units, for which each sibship unit consists of one set of siblings and their spouses from each nuclear family. All subjects included in our study have all TG trait values  and genotypes. In the Original Cohort, we used 146 individuals who have all TG trait values (Exams 7 and 11) and genotypes. All of these subjects were genotyped for 488,146 SNPs.p-value for testing Hardy-Weinberg equilibrium <0.001. Thus, there were a total of 294,434 SNPs remaining in our analysis.We applied MASAL to analyze the FHS 500 k SNP data set (GAW 16 Problem 2). Before the analysis, TG level values were log-transformed to approximate a normal distribution, although there is no such limitation when using MASAL. Furthermore, in order to minimize false-positive associations due to rarer SNPs and genotyping artifact, we limited our analyses to SNPs with minor allele frequency ≥ 10% and the In the analyses of Offspring Cohort, the fitted model given by MASAL ist is the age of exam, s is the indicator for sex, and cth chromosome. The value of corresponding Wald statistic is 325.73. Similarly, for the analyses of original cohort, the fitted model given by MASAL iswhere and the value of the corresponding Wald statistic is 359.575.p-values of the two tests. However, the permutation study based on the entire genome is extremely time-consuming. Thus, instead of using the entire genome, we randomly selected a series of subsets of SNPs in 500 increments  of 500 k SNPs to characterize the pattern of the empirical distribution of W. Based on each subset, we used the above permutation procedure to establish the empirical distribution of Wand calculated 93%, 95%, 97%, and 99% quantiles. In Figure W based on 10,000 randomly selected SNPs.We attempted to establish the null distributions of the two Wald statistics by using a permutation procedure and calculating the p-value of the Offspring Cohort based association test is less than 0.001, which suggests strong associations between the SNPs selected by the final model and TG trait. In contrast, the adjusted p-value of the Original Cohort based association test is 0.056, which indicates marginally significant associations between the selected SNPs and TG trait. Our results confirm that MASAL can properly take account of the familial correlations in the GWA analysis. MASAL identified 13 significant SNPs for the Offspring Cohort and 6 SNPs for the Original Cohort. Table MLXIPL on chromosome 7 and TG; and Kathiresan et al. [TBL2 and MLXIPL) 8, and 19 to be associated with TG. Our models do not include SNPs in those regions. Further investigation is warranted to confirm our findings. Although MASAL is a unique approach to accommodating correlated phenotypes in genetic studies, its potential has not been fully explored. It is beyond the scope of this short article to compare MASAL with other methods, but it is a highly worthy effort to thorough study the utility of MASAL for longitudinal genetic data.The adjusted In this report, we proposed a testing procedure to perform GWAS for longitudinal data, using a nonparametric regression method (MASAL) presented by Zhang . In contWe used a permutation procedure to establish the null distribution of the Wald statistic and then estimated the significance level. However, the computation time was lengthy, especially for the large pedigree and large number of exams of each subject. Theoretical studies exploring the asymptotic distribution of the involved statistic would be useful.AMD: Age-related macular degeneration; FBAT: Family-based association test; FHS: Framingham Heart Study; GAW: Genetic Analysis Workshop; GCV: Generalized cross-validation; GWAS: Genome-wide association studies; MASAL: Multivariate adaptive splines for the analysis of longitudinal data; SNP: Single-nucleotide polymorphism; TG: Triglyceride.The authors declare that they have no competing interests.WZ and HZ designed the study and carried out the data analysis. WZ, KC, and HZ drafted the manuscript. KC, XC, MZ, and MW participated in data analysis.All authors read and approved the final manuscript.


Quantitative trait locus (QTL) analysis is a powerful tool for mapping genes for complex traits in mice, but its utility is limited by poor resolution. A promising mapping approach is association analysis in outbred stocks or different inbred strains. As a proof of concept for the association approach, we applied whole-genome association analysis to hepatic gene expression traits in an outbred mouse population, the MF1 stock, and replicated expression QTL (eQTL) identified in previous studies of F2 intercross mice. We found that the mapping resolution of these eQTL was significantly greater in the outbred population. Through an example, we also showed how this precise mapping can be used to resolve previously identified loci (in intercross studies), which affect many different transcript levels (known as eQTL “hotspots”), into distinct regions. Our results also highlight the importance of correcting for population structure in whole-genome association studies in the outbred stock. In rodents, as in humans, traits such as obesity or diabetes are under the influence of many genes spread throughout the genome. Using linkage analysis, the locations of the major contributing genes can be mapped only to very large regions of chromosomes, usually encompassing hundreds of genes. This has made it difficult to identify the underlying genes and mutations. Another approach, analogous to genome-wide association in human populations, is to use association analyses among outbred stocks of mice. In this proof-of-principle article, we make use of common variations that locally perturb gene expression to demonstrate the greatly improved mapping resolution of association in mice. Our results indicate that association analyses in mice are a powerful approach to the dissection of complex traits and their underlying molecular networks. Quantitative trait locus (QTL) analysis has been the primary tool for geneticists to study complex genetic traits in experimental organisms. However, while such QTL mapping has great power to identify loci controlling the traits, resolution of mapping is usually quite low and as a result few candidate genes have been successfully identified using this approach. The use of molecular phenotypes, in particular gene expression levels, as quantitative traits for mapping, coupled with the ability to measure thousands of such traits simultaneously, has added a tremendous spark to the field of complex trait genetics. The integration of expression QTL (eQTL) with complex clinical traits using statistical modeling has allowed the identification of genes and pathways involved in a variety of complex traits. Some of the recent successes of this integrative approach have been identification of causal genes underlying the QTL for clinically relevant trait Despite these successes, such integrative genomic approaches using F2 populations suffer from the same limitation that has hindered the success of the traditional physiological trait QTL mapping, namely lack of resolution in mapping A total of 110 outbred MF1 mice were studied for whole genome transcript levels in liver and subjected to genotyping using the Affymetrix 5K Mouse Chip. From the 5024 SNPs on this array, 1813 SNPs  had a minor allele frequency of 5% or greater, and were used for the analyses described below. The average and median distance between neighboring SNPs for these 1813 markers were 1.38 Mb and 0.57 Mb respectively. To determine the percent coverage of the genome by these 1813 SNPs, we used the average distance of the adjacent markers which had an r squared value of 0.9 or higher. This analysis showed that on average adjacent markers will have such a high LD when within 1.03Mb. Based on this, approximately 3000 informative evenly spaced markers are needed to allow coverage of the entire genome. This calculation is based on the assumption that the haplotype blocks in this population are all about the same size. However, as we show below, this is not true for some regions of the genome, and thus the estimate is very approximate. Given the non-uniform distribution of the 1813 markers across the genome the genetic coverage for these set of markers is 72% of the genome. This means that in the whole genome association analysis described in this report we would expect to miss 28% of the signals.−16). These results suggest that, as with Affymetrix arrays Gene expression measurements were performed on RNA isolated from liver using Illumina's mouse whole genome expression BeadChip (MouseRef-8-v1 Expression BeadChip) see . We appl2 were calculated among all the SNPs, and the average r2 measures for different ranges were used to look at the LD structure in the population. Visual inspection of the LD between markers within the same chromosome revealed a complex LD structure in the population . Despite this low r2, inspection of the distribution of the chi-squared statistics p-values for expected r2 in absence of LD indicated significant LD between certain pairs. These observations were consistent with our visual inspection of LD maps suggesting the existence of a complex relationship pattern among different loci, presumably due to population substructure within the MF1 stock.We first computed the degree of linkage disequilibrium in the population of 110 MF1 mice. Pair-wise rpulation . In partTo further investigate this we performed hierarchical clustering of mice based on the kinship matrix which we derived from the overall correlation of genotypes between pairs of mice. The results revealed clear evidence of familial clustering  indicaticis) and distant (trans) eQTL. Before correction, regression analysis of gene expression on markers revealed a total of 812 significant associations . From these, 444 (55%) were local and 368 (45%) were distant eQTL. After correcting for population structure, there were about two thirds as many significant associations as found originally . This result suggested that about one third of the associations found in the absence of correction were false positives due to the relatedness of the mice. The reduction in significant associations among local and distant eQTL, however, was not the same. For local eQTL, there were 18% fewer associations after correction (366 vs 444), but for the distant eQTL there was a 70% reduction in the number of associations found (112 vs 368). We also examined the results at 5% and 10% FDR thresholds (In order to correct for population structure we used Efficient Mixed Model Association (EMMA). The underlying statistical algorithm for performing such correction has recently been published resholds  and as wresholds .One of the limitations of using outbred stock for mapping complex traits has been the statistical power issue and the need to include large number of mice in the study cis eQTL  and 701 were classified as distant or trans eQTL ; on Chromosome 7 we found 6 local eQTL including the enhancer binding protein CCAAT/enhancer binding protein alpha (Cebpa), lipolysis stimulated lipoprotein receptor (Lisch7), peptidase D (Pep4), coiled-coil domain containing 123 (Ccdc123 or 2610507L03Rik), androgen regulated gene RP2 (Nudt19 or D7Rp2e), a Rho GTPase binding protein rhophilin 2 (Rhpn2); and in the most enriched hotspot on Chromosome 4 we only found one local eQTL methylthioadenosine phosphorylase (Mtap).We also searched for the presence of co-localizing distant eQTL (eQTL “hotspots”). To do this, the entire genome was divided into 2 Mb bins  and the number of significant distant eQTL were counted in each bin. Plotting of the eQTL frequencies at various genomic regions indicated a non-random distribution of the mapping locations . Several−/− and C3H/HeJ.ApoE−/− parental strains (herein referred to as the BxH cross) Since MF1s offer a higher mapping resolution, this resource may be used to fine map eQTL previously identified in other crosses. The limitation is that not all the eQTL found previously will be present in the MF1 population. Here we sought to compare the mapping results in the MF1 population by empirically estimating the fraction of eQTL detected in this study compared to what was found in a previously published cross from our laboratory. For this comparison we used the previously reported eQTL study of the liver tissue for the F2 intercross population generated between C57BL/6J.ApoETtf2, Insig2, Frzb, and Pparg) between BxH and MF1 populations. As shown, the MF1 data (grey curve) provided much narrower candidate region than the BxH data (black curve). As expected for any local eQTL, the candidate regions in both studies encompassed the genomic region for the gene itself and peaked directly over the physical location of the gene. However, the QTLs in the BxH cross encompassed a much broader region than the association results in MF1 data. These results indicate that the MF1 population yields a much better resolution for eQTL mapping than a traditional cross.Previous studies suggest that outbred stocks offer a high resolution mapping resource, but these studies did not have prior knowledge of the location of the causal variant for the trait trans-acting) eQTL. Several eQTL studies with intercross or RI strain mice have observed that many distant eQTL map to the same location on a chromosome, giving rise to what is known as distant eQTL hotspots. Here we illustrate how the MF1 whole genome association analysis can be used to resolve such co-localization of distant eQTL. We had previously detected a distant eQTL hotspot in the middle region of chromosome 6. A total of 96 unique genes (99 probes) had a distant eQTL at this locus. Based on the assumption that the variation in the expression of the causal gene underlying this hotspot mediates its effects, we identified 8 local eQTL  as primary candidate genes for this hotspot. Thus, judging by the F2 data alone, any one of these 8 local eQTL could be regulating any of the 96 distant eQTL. We turned to the MF1 data and asked, first, if we could resolve the co-localization of the local eQTL on the chromosome 6 locus and, second, if we could resolve the co-localization of the distant eQTL in this region. From the 8 local eQTL in the BxH F2 cross, 3 of them  were also observed in the MF1 population. Cidec mapped to two markers, rs13478971 and rs13478971, at 112.1 and 111.2 Mb, respectively, which are ∼1 and ∼2 Mb away from where the physical location of the Cidec gene (113.3 Mb). These were the closest markers to the physical location of Cidec. These results suggested the presence of either a distal regulatory element for this gene or the presence of a closely linked regulatory gene for Cidec on Chromosome 6. Genotyping with denser markers closer to Cidec location might correctly position the highest peak above the gene itself. Interestingly, the Bcl2l13 transcript levels, in addition to mapping to the nearest marker to the physical location of the gene at 121.6 Mb, also showed significant association with markers located at 114.2, 114.3, and 114.7 Mb, suggesting the possible presence of an additional distal regulatory locus near the local eQTL for this gene. After resolving the three local eQTL, we examined the distant eQTL which mapped to this locus in the BxH cross and asked which of these distant eQTL mapped to any of these three local eQTL on Chromosome 6. From the 96 distant eQTL colocalizing to the chromosome 6 locus in the BxH study, 14 were replicated in the MF1 population . Judging by the location of the association peak markers, despite the co-localization in the BxH, these 14 eQTL mapped to varying loci within the chromosome 6 region. In particular, 3 of these genes  mapped over the Pparg locus, another 2 genes  mapped over the Cidec locus, and 9 genes had the most significant associations with markers at other than the three local eQTL loci  at the genome wide significance level of 2.43e-05 p-value (10% FDR). The candidate region for this association spanned a 3.6 Mb region (from rs13478570 at 142.8 Mb to rs13478583 at 146.4 Mb) with the peak marker at 144 Mb (rs13478573). To fine map this region each of the 110 animals were genotyped for an additional 5 markers by the PCR-ARMS technique The amount of resolution achieved in our mapping study was limited to the density of the markers used (average marker density 1.37 Mb). Next, we asked whether typing more SNPs in a region would enhance the mapping resolution. For this, we focused on the distal locus of chromosome 5 where the results of the whole genome association had identified 3 distant eQTL  S3-12 and 13 other previously identified chromosome 6 locus distant eQTL. At the genome wide cutoff value, only 2 of these genes were significantly associated with markers on chromosome 6 locus, but with the nominal p-value of 0.01, 12 additional genes also showed evidence of significant association to this locus. The use of less stringent criteria for local QTL studies has also been implemented in other reports and shown to correctly rediscover and validate previously identified QTLs The number of significant associations observed at various false discovery thresholds in this study  were lowPrevious genetical genomics studies reported the presence of genomic hotspots where large groups of eQTL collectively map to single loci in the genome It has been widely acknowledged that standard statistical tests that do not account for population structure or familial relatedness are prone to identify spurious associations Genetical genomics is becoming increasingly popular due to its promise to bridge the gap between the physiological traits and the genetic variations in the population. To fully take advantage of such an approach it is imperative to understand the nature of the association between the transcript level of the gene and the causal genetic variation. Molecular networks underlying the physiological traits cannot be properly constructed without the proper knowledge of the interaction among genes. We believe that the genetical genomics approach coupled with the high precision mapping offered by the MF1 outbred stock will significantly advance the potential for identifying regulatory genes for distant eQTL and provide the necessary components to build such biological networks.Female MF1 mice, approximately 4–6 weeks of age, were purchased from Harlan . These animals were fed Purina Chow  containing 4% fat until 19 weeks of age, and then fed a Western diet  containing 42% fat and 0.15% cholesterol for ∼14 weeks until they were sacrificed at 33 weeks of age. All mice were maintained on a 12h light/dark cycle. Mice were fasted for 5 hours before being euthanized.http://www.ncbi.nlm.nih.gov/SNP/index.html).For the initial genotyping, the Affymetrix GeneChip Mouse Mapping 5K SNP platform was utilized. The DNA used for the genotyping was isolated from the tail clips of each mouse using the Qiagen's DNeasy tissue kit (cat# 69506). Overall, a total of 5024 SNPs were genotyped. The genomic location of all the analyzed SNPs were based on the snpdb126  AACTATCCAAATGCACACTGAAGCCA, forward outer primer GTGCTATCTCTTCAGCCCAGAGTGATAT, reverse out primer GAGGAGCGAACCATTCTCTAAAAGTTGT, C-allele size = 133bp, T-allele size = 163 bp, outer-products = 242 bp; 2) rs33318740 forward inner primer  AAGATGCCGGCCCCAGATTGCCCTGTGA, reverse inner primer  GGGAGAAAGCTCCCTGCTTTGTCCAAACTC, forward outer primer GGAAGGTGAGGAGACAGGCTTCCGGCAG, reverse out primer CTTATGGCAAACCACCCTGCCCAGCAGA, A-allele size = 193 bp, G-allele size = 162 bp, outer-products = 297 bp; 3) rs33492148 forward inner primer  TTCTGTCTTAATTGAGCCCATATGAAAAT, reverse inner primer  GCACATTCTTCCAGACTCTGCATATC, forward outer primer ACTCTTGACAAAGAAGAATGCTTGCTTT, reverse out primer ATGTTTTGGCTAAGCACAATCCTACTCT, T-allele size = 194 bp, G-allele size = 172 bp, outer-products = 311 bp; 4) rs32348286 forward inner primer  AGGACTGCCACAGGCCAGCATCTCACA, reverse inner primer  AGTGTATCTATCAGGTGAATTCCAGTAGTC, forward outer primer AAGCCAAGCTGTCTCCAAGTCCTAGAAA, reverse out primer ACCACTTGAAGCCTGATTAAAATGTGCC, A-allele size = 213 bp, G-allele size = 175 bp, outer-products = 331 bp; 5) rs29524465 forward inner primer  CCTCTAATCTCCTGAGGATTGGAACA, reverse inner primer  TCATTTGGACACTAGAGCTTCTTCATTATC, forward outer primer TCGGAGAGACAGTTGTCTGTTAGGTTTA, reverse out primer GACAATGACGAAAAGACAAGTCACTTCT, T-allele size = 116 bp, G-allele size = 127 bp, outer-products = 187 bp; 6) rs33719947 forward inner primer  GTACATGTTCTTTTAAAATTATTAATCGAC, reverse inner primer  AGAAAAAGACACTCCTTGGAGCATGA, forward outer primer CTGTGATTTAAAGGCGTGCTAGTACTAC, reverse out primer GTGAGAGAGAGAGTCTGGGAATATTTCT, C-allele size = 177 bp, T-allele size = 157 bp, outer-products = 278 bp. Each of the 110 MF1 mice were genotyped for these markers by PCR and the products were separated by 4% agarose gel and visualized by ethidium bromide staining.Fine mapping on the distal region of Chromosome 5 was performed using the PCR-ARMS technique RNA extraction was performed on the liver tissue obtained from each animal at the time of sacrifice, using Qiagen's RNeasy kit (cat# 74104). For the gene expression measurements, Illumina's Mouse whole genome expression BeadChips (MouseRef-8-v1 Expression BeadChip) were used. All amplifications and hybridizations were performed according to Illumina's protocol by the Southern California Genome Consortium microarray core laboratory at UCLA. In brief, 100 ng of total RNA was first reverse transcribed to cDNA using Ambion cDNA synthesis kit AMIL1791 and then converted to cRNA and labeled with biotin. 800ng of biotinylated cRNA product is hybridized to prepared whole genome arrays and allowed to incubate overnight ( 16–20 hrs) at 55 degrees C. Arrays were washed then stained with Cy3 labeled streptavadin. Excess stain wass removed by washing and the arrays were dried and scanned on an Illumina BeadScan con-focal laser scanner.Data normalization was performed using the rank invariant method by the Bead Studio software. After normalization, all gene expression data were log2 transformed.http://mouse.perlegen.com/mouse/index.html). Any probe which contained a SNP which was polymorphic between the proposed ancestors of the MF1 mice  was excluded. This filtering step resulted in the exclusion of 2160 probes. The remaining 10013 probes were used as the starting set for the whole genome association analysis. The gene expression data are deposited to the Gene Expression Omnibus (http://www.ncbi.nlm.nih.gov/geo/) at NCBI. These data are deposited under the accession number GSE10280.To filter genes, we selected probes which met two criteria; 1) the probes exhibited a reliable signal and 2) the probes contained no annotated SNP within their sequence. The former was determined according to the Illumina Bead Studio output. The detection value is equal to 1-probability that a signal level is due to nonspecific hybridization. This value can be interpreted as the probability of seeing a certain signal level without specific probe-target hybridization. For filtering we excluded any probe which had a detection value of lower than 0.95 in greater than eleven (10%) or more animals. To select against the bias in hybridization due to probe design, as described by Walter et al LD and FDR were calculated using R software algorithms. To compute the pair-wise LD between markers, we used the LD function in the Genetics package, which includes output of the chi square p-values for marker independence which we used to test for LD between markers on different chromosomes. For the FDR calculation we used the q-value package in R We first computed the genetic similarity matrix between the individual mice as the fraction of shared alleles (identity-by-state) for each pairs, and visualized it with heatmap R package. A complex multi-leveled population structure and genetic relatedness is observed in the genetic similarity matrix.e2I. u is a vector of random variables accounting for the effect from genetic background. u follows a multivariate normal distribution with Var(u) = σg2K, where K is the genetic similarity matrix described above. The fraction of variance explained by genetic background is computed as previously suggested with tr(SKS)/(n-1+tr(SKS)), where S = I-J/n, and J is a square matrix consisting of ones from the REML estimate of H1. The likelihood ratio test is performed by comparing the maximum likelihood of two hypotheses. The likelihood difference 2*(1-l0l) asymptotically follows a 1:1 mixture of the chi-squared distribution with zero and one degree-of-freedom 0 = 1 We applied the following variance component test to estimate the variance explained by genetic background and assess the statistical significance.1:β≠0 against H0:β = 0 after estimating restricted maximum likelihood (REML) variance components as described To account for population structure and genetic relatedness in association mapping, we applied the following standard linear mixed model as previously suggested Figure S12 is shown in the insert.LD structure on Chromosome 2-20. The order of markers in each heat map follows the physical location of the marker along the chromosome with the most proximal at the bottom and the most distal marker on the top. The correspondence between color and r(0.23 MB DOC)Click here for additional data file.Figure S2Effect of familial structure on gene expression association. Panel (A) shows the inflation of false positives at a transcript represented by as the average log p values across all the markers (x-axis) and the correlation between a transcript and genetic relatedness (y-axis). Panel (B) shows this correspondence after correcting for genetic relatedness using a linear mixed model.(4.23 MB DOC)Click here for additional data file.Figure S3Power Analysis. Figures A–E show the power expected for various various genetic backgrounds. A is for no genetic background effect, B is for genetic background effect of 0.1, C is for genetic background effect of 0.2, D is for genetic background effect of 0.4, and E is for genetic background effect of 0.5. In each panel, the power is calculated for various p-value cutoffs . For each calculation, the minor allele frequency is assumed 0.3.(0.08 MB DOC)Click here for additional data file.Figure S4Validation of distant eQTL hotspots. In each of the figures A through D, the 110 mice were randomly split two groups (55 MF1 mice in each) and for each subset the number of distant eQTL counts were determined across the genome. The genome is represented as 1287 equally sized bins of 2 Mb. The gray line depicts the 0.05 genome wide significance for eQTL enrichment after Bonferroni correction .(0.08 MB DOC)Click here for additional data file.Figure S5Association results in MF1 data for 14 distal eQTLs co-localized in the BxH F2 intercross. The location of local eQTLs Pparg (grey), Cidec (red), and Bcl2l13 (black) is shown at the bottom of each figure.(0.13 MB DOC)Click here for additional data file.Table S1Cis and trans eQTL at 1%, 5%, 10%, and 25% FDR.(1.17 MB XLS)Click here for additional data file.


Rheumatoid arthritis (RA) is a complex, chronic inflammatory disease implicated to have several plausible candidate loci; however, these may not account for all the genetic variations underlying RA. Common disorders are hypothesized to be highly complex with interaction among genes and other risk factors playing a major role in the disease process. This complexity is further magnified because such interactions may be with or without a strong independent effect and are thus difficult to detect using traditional statistical methodologies. The main challenge to analyze such gene × gene and gene × environment interaction is attributed to a phenomenon referred to as the "curse of dimensionality." Several combinatorial methodologies have been proposed to tackle this analytical challenge. Because quantitative traits underlie complex phenotypes and contain more information on the trait variation within genotypes than qualitative dichotomy, analyzing quantitative traits correlated with the affection status is a more powerful tool for mapping such trait genes. Recently, a generalized multifactor dimensionality reduction method was proposed that allows for adjustment for discrete and quantitative traits and can be used to analyze qualitative and quantitative phenotypes in a population based study design.In this report, we evaluate the efficiency of the generalized multifactor dimensionality reduction statistical suite to decipher small interacting factors that contribute to RA disease pathogenesis. HLA-DRB1 and PTPN22, have been reported to be associated with disease  as covariates for classifying the data into high and low risk groups.An initial screen of data for quality control was performed for the markers selected for the current study. Hardy-Weinberg equilibrium (HWE) was estimated in the case, control, and combined groups using the Haploview program (version 3.32). To understand the degree of correlation between the SNPs, linkage disequilibrium (LD) was estimated using the Haploview program (version 3.32). The D' statistics for the same is presented in Figure n = 867).In the current study, we used the Genetic Analysis Workshop 16 (GAW16) RA case-control data set (Problem 1) comprising a total of 2062 sample , typed on the 550 k Illumina chip. To evaluate the efficiency of the GMDR algorithm to detect small epistatic interactions involved in RA pathogenesis, analysis was performed on chromosomes 1, 2, 5, and 6, which have shown strong positive association earlier with the phenotype . BecauseThe GMDR is a score-based algorithm based on the MDR framework. Briefly, the MDR uses a novel constructive induction algorithm to facilitate the detection of nonlinear interactions among multiple discrete genetic and/or environmental factors that are predictive of a discrete clinical endpoint . Multi-l1. The GMDR scoring method: The GMDR method uses the original MDR data reduction method, with the ratio of cases to control being replaced by scores in each cell to discriminate between high risk and low risk followed by determining classification accuracy and prediction error. A detailed description of the methodology can be found elsewhere . This geS = exp(y)/1+exp(y), where y is the standardized quantitative trait. In brief, this was done by computing the mean and standard deviation (SD) of the quantitative trait. Scores where then assigned by subtracting the mean from the individual's quantitative trait value and then dividing it by the SD.2. We formulated a detailed scoring methodology by using the expression In the current study we used GMDR algorithm to evaluate its efficiency in detecting gene-gene interactions in the complex RA phenotype. For this we used markers information from the GAW16 data set from regions that have been previously implicated in RA. Additional file While the MDR software is designed to classify individuals into high risk and low risk groups, GMDR is a score-based method in which the ratio of cases to control is replaced by scores in each cell to discriminate between high risk and low risk cells and then assessing classification accuracy and prediction error. GMDR was performed on the genotype data (cases only) from the GAW16 Problem 1 data set with the computed scores. The phenotype scores used in the analysis were generated using the built-in GMDR scoring method and a detailed scoring method (described above in the Methods section). Analysis was performed individually for the separate chromosomal regions. An exhaustive search was performed to identify all possible one- to five-locus models. We report the prediction accuracy and cross-validation consistency for the most significant models identified by GMDR, the results of which are summarized in Table Rheumatoid factor (RFUW) has been widely used as a screening test for patients with RA. RFUW is prognostically useful because it correlates with functional and radiographic outcomes in RA . More reIn this study we used the GMDR methodology to evaluate its efficiency to detect gene-gene interactions in putative regions for RA using the anti-CCP and RFUW (IgM) values as covariates. Three out of the four models predicted reached statistical significance Table . None ofanti-CCP: Anti-cyclic citrullinated peptide; CPM: Combinatorial partitioning method; GAW16: Genetic Analysis Workshop 16; GMDR: Generalized multifactor dimensionality reduction; HWE: Hardy-Weinberg equilibrium; IgM: Immunoglobulin M; LD: Linkage disequilibrium; MDR: Multifactor dimensionality reduction; RA: Rheumatoid arthritis; RFUW: Rheumatoid factor; RPM: Restricted partition method; SD: Standard deviation; SNP: Single-nucleotide polymorphism.The authors declare that they have no competing interests.OM carried out all statistical analysis, data interpretation, and drafted the manuscript. KRS and PA contributed in data cleaning and analysis. SG contributed in data analysis manuscript writing. All authors have read the paper and approve its contents.SNPs used in this study.Click here for file


FTO, PON1, and PFKP on body mass index.For the Framingham Heart Study (FHS) and simulated FHS (FHSsim) data, we tested for gene-gene interaction in quantitative traits employing a longitudinal nonparametric association test (LNPT) and, for comparison, a survival analysis. We report results for the Offspring Cohort by LNPT analysis and on all longitudinal cohorts by survival analysis with cohort effect adjustment. We verified that type I errors were not inflated. We compared the power of both methods to detect in FHSsim data two sets of gene pairs that interact for the trait coronary artery calcification. In FHS, we tested eight gene pairs from a list of candidate genes for interaction effects on body mass index. Both methods found evidence for pairwise non-additive effects of mutations in the genes  The Framingham Heart Study (FHS) cohorts and simulated FHS (FHSsim) data available for Genetic Analysis Workshop 16 (GAW16) provide longitudinal data on quantitative traits and include families. FHSsim data are based on the FHS individuals and family structures, replacing only the phenotypes by simulated values. For both sets of data, we compared two association approaches to test quantitative traits for gene-gene interaction. Both approaches use baseline and available follow-up measurements and require a set of independent individuals with phenotypes and genotypes. The methods are: 1) a longitudinal nonparametric association test for cohorts (LNPT) which we recently developed and 2) survival analysis with the Cox model. The LNPT is a rank-sum procedure based on the longitudinal trait measurements. Survival analysis models event times derived from the longitudinal traits as age at the first exam when the trait crossed a predefined threshold.Both approaches remain mathematically valid regardless of trait distribution. This is most useful: quantities with mixture distributions such as the simulated trait coronary artery calcification (CAC) remain non-normal despite blom or log transformation. Also, LNPT and survival analysis are invariant with respect to monotone transformations such as blom or log: they change neither the order (ranks) of the trait data nor the event times, given that the threshold is also transformed.We analyzed quantitative traits with a skewed distribution: CAC in the FHSsim data and body mass index (BMI) in the FHS data. We verified that the type I error of 5% was kept, applying our methods on the original, untransformed data. CAC is influenced by two separate sets of interacting gene pairs according to the GAW16 answers [The FHS data provide 227 independent individuals and 980 pedigrees. For association analysis, 1631 , 1243 Offspring (Off), and 155 Third Generation (Gen3) Cohort) unrelated individuals had available genotype and phenotype data that could be extracted, including founders and one random individual from all remaining multimember pedigrees. Similar selection procedures were applied to FHSsim.For FHS, Gen3 provides baseline values only. OC and Off provide four consecutive measurements per individual. The LNPT requires that individuals are followed up at the same intervals but tolerates missing trait values. We harmonized follow-up times by allocating five examination time points T = 1, ..., 5 . T = 2 is missing for Off, T = 5 is missing for OC. Longitudinal analysis by LNPT used the phenotypes from all time points when both cohorts were examined . Some individuals have incomplete longitudinal phenotypes. For FHSsim, each cohort  provides simulated baseline trait values and two follow-ups, each 10 years apart. All longitudinal phenotypes are complete.2 (FHS) or a value of CAC>230 . For censored individuals, time to censoring was age at the last available exam. Censoring occurred when individuals did not pass the threshold either for any of the longitudinal measurements or for an initial sequence of longitudinal measurements after which they are missing to follow up. We excluded individuals who have baseline trait values which are either missing or above the threshold.For survival analysis, time to event was defined as the earliest age at which the individual presented a BMI ≥ 25 kg/mFor analysis of CAC (FHSsim), we selected the two interacting SNP pairs from the GAW answers  (single-PFKP); rs1121980 and rs9930506 ; and rs854560  and rs6971091 .For gene-gene interaction analysis of BMI (FHS), we selected five candidate SNPs previously found to be associated with alteration of BMI . The SNPF between these subgroupsThe LNPT tests for association of longitudinal quantitative traits with respect to a set of influencing factors, which divide the cohort into subgroups. The LNPT tests the null hypothesis of no difference in trait distribution C is a contrast matrix and F = {t and the influencing factors (kls) of interest. For gene-gene interaction analysis, two factors k, l are used to code for SNP genotype at the two loci  and we stratified by sex s = {m, w}. The test statistic resembles structurally a heteroscedastic repeated measures ANOVA which is performed on the mid-ranks of the longitudinal traits [Yi can be arbitrarily dependent between exams/time points t of the same individual i while they are assumed to be independent between individuals. Individuals should be followed up at the same time intervals. The LNPT yields a set of adjusted p-values for tests of average effects of the loci, sex, and number of exam and for tests of all interactions. The LNPT can handle missing values for the longitudinal phenotype. The test statistic is computed using the existing phenotype measurements when values are missing. The LNPT can model covariates as additional factors. For all tested interactions, each stratum must contain a minimum of ten observations per exam to ensure consistent estimates of the test statistic.where l traits , estimatk, l. Gene-gene interaction analysis was restricted to the large Offspring sample (N = 1243), to individuals with no cholesterol treatment (79% of Off sample). For BMI (FHS), we restricted baseline age to the interval 25-46 yr to avoid effects of BMI-related mortality (manifest as stagnating median BMI for old age). We adjusted for age by using a two-level factor, testing only age-averaged interaction effects. Levels are assigned according to baseline age with median baseline age as break point.Prior to genetic analysis, we tested for a cohort effect  by using factors for cohort type and for sex, omitting the two genetic factors Survival analysis evaluates the time to the occurrence of an event of interest and its dependence on particular characteristics such as sex, genotype, and cohort. We employ the Cox proportional hazard model and its extension for time-varying factors , which aA significant main effect of cohort type and a significant interaction of cohort type with sex were detected in FHS BMI data (LNPT) and FHSsim CAC data (LNPT and Cox model). We report gene-gene interaction analysis on the Offspring Cohort (LNPT) and on all longitudinal cohorts (Cox model with cohort effect adjustment).p-values ≤ 0.05 were estimated for the interaction tests for SNP pairs rs854560 and rs1121980 (FHS) and τ1, τ2 (FHSsim) after permutation of the assignment between longitudinal phenotypes and individuals. Permutation destroys associations but retains the distributional properties of the trait.Rates of false positives  for 1, τ2 and τ3, τ4 are known to contribute to the analyzed trait CAC [p ≤ 0.05) of the interaction test on the 200 replicates. The LNPT detected both interactions with 100% power in the largest ascertainable sample (n = 856 on average) and with 99%  and 100%  for sample size n = 400. The Cox model had a power of 94%  and 100%  for the largest ascertainable sample  but only 76%  and 97%  for sample size n = 400. The Cox model did benefit from cohort effect adjustment: SNP interaction τ1, τ2 was found with 94% power after adjustment instead of 69% with no adjustment (n = 808). For CAC, the LNPT is more powerful compared with survival analysis. Estimates of type I error at the α = 5% level were 4.76% (LNPT) and 4.90% (Cox).The tested SNP interactions τpcohort = 0.0001, pcohort × sex = 0.03). It is due to men . The Cox model yields threshold-dependent results  detecting no interaction between cohort and sex. In the remainder of this paper, we report results on the Cox model with threshold BMI>25 (yielding more events) and cohort-effect adjustment.A significant cohort effect was detected by LNPT (FTO (rs1121980 and rs9930506), PON1 (rs854560), and PFKP (rs6602024). p-Values for gene-gene interaction on BMI are given in Table p ≤ 5%). For SNPs rs6602024 and rs6971091, LNPT analysis distinguished only between carriers and non-carriers of the rare minor allele. For SNP rs6602024, minor allele homozygotes were excluded from survival analysis. Survival analysis computes a single interaction p-value for each SNP pair by the likelihood-ratio test comparing the interaction model against the null model of no gene-gene interaction. In contrast, the LNPT yields a set of p-values for the interaction tests that are adjusted to the α = 5%-level. In the present data, gene-gene interaction is detected as effect on time-course  and candidate SNPs for BMI tend to be rare, yielding small two-locus genotype subgroups. For the SNP-pair with smallest subgroup size (rs854560 and rs6602024), the likelihood-ratio test performed for survival analysis was observed to yield significance although simultaneously, the computed confidence intervals for the HR ratios were not significant but included very high HRs.Transformation from follow-up data to event-time data can be viewed as reduction of information content. It requires a priori a well reasoned threshold value. Results of survival analysis depend on the choice of this threshold. We illustrated this for the detection of the cohort effect . For some scenarios, it may well be possible that longitudinal data can not be analyzed by survival analysis. For example, if a particular two-locus genotype would efficiently protect against overweight, one could face the problem that the threshold either provides insufficient contrast or does not yield sufficient numbers of events for analysis for the two-locus genotypes of interest. For BMI, an established threshold value of 25 kg/mSurvival analysis has the benefit that it yields hazard ratios as readily interpretable estimates of effect size. The LNPT tests for a much more general feature, namely differences between whole trait distributions without making assumptions about an underlying model. Consequently, it primarily provides a tool for testing for group differences but does not offer readily interpretable estimates of effect size. When a set of factors are chosen, by default the LNPT also tests their interactions, with the consequence that interactions with weak marginal effects are less likely to be missed.CAC has a strongly skewed distribution. For parametric approaches, this would cause loss of power or inflated rates of false positives. Neither is the case for the LNPT and survival analysis. We have shown that they both keep the rate of false positives at the 5% level while they successfully detected gene-gene interactions for simulated and real phenotypes with good power. The LNPT was found to be more powerful compared with survival analysis for detection of a cohort effect in BMI (FHS) and more powerful for detection of gene-gene interaction, particularly for CAC (FHSsim). The LNPT is a longitudinal approach. In its ability to incorporate loss to follow up, the LNPT is better than survival analysis: the LNPT uses all remaining trait measurements, whereas for survival analysis these individuals are likely to become censored or even must be excluded in case of missing baseline value.For high-dimensional data such as genome-wide studies, sophisticated boosting techniques exist for survival analysis . They yiBMI: Body mass index; CAC: Coronary artery calcification; FHS: Framingham Heart Study; FHSsim: Framingham Heart Study simulated data; GAW16: Genetic Analysis Workshop 16; Gen3: Third Generation Cohort; HR: Hazard ratio; LNPT: Longitudinal nonparametric association test; MAF: Minor-allele frequency; OC: Original Cohort; Off: Offspring Cohort; SNP: Single-nucleotide polymorphismThe authors declare that they have no competing interests.DM designed and coordinated the study, carried out statistical analyses, and drafted the manuscript. YB carried out statistical analyses. JPL participated in data preparation. DM and HB conceived the study. All authors contributed to the writing of the manuscript. All authors read and approved the final manuscript.


Clinical heterogeneity of a disease may reflect an underlying genetic heterogeneity, which may hinder the detection of trait loci. Consequently, many statistical methods have been developed that allow for the detection of linkage and/or association signals in the presence of heterogeneity.This report describes the work of two parallel investigations into similar approaches to ordered subset analysis, based on an observed covariate, in the framework of family-based association analysis using Genetic Analysis Workshop 15 simulated data.With an appropriate choice of covariate, both approaches allow detection of two loci that are undetectable by the classical transmission-disequilibrium test. For a third locus, detectable by the classical transmission-disequilibrium test, a substantial increase of power of detection is shown. For several diseases, one can suspect that clinical or environmental heterogeneity, expressed through covariates such that the age at onset (AAO) of the disease, its severity, or a measure of exposure to an environmental factor, is associated with genetic heterogeneity. This heterogeneity may hinder the detection of trait loci.The ordered subset analysis (OSA)  was desiThe performance of OSA for detecting gene × environment interaction has been assessed , showingHerein, we introduce two slightly different approaches, inspired by OSA and by Spielman's transmission-disequilibrium test (TDT) , for famThe OSA-TDT approach is very similar to Hauser's approach, and aims at detecting linkage and association of a marker with the disease in a covariate-ordered subset of families. The order TDT (OTDT) approach  is slightly different, and aims at the detection of genetic heterogeneity between the first and the last families of the ordering.We used all 100 replicates of 1500 nuclear families. The ascertainment was based on the presence of at least two affected sibs. The true value of each covariate, including age at onset, severity, and IgM level, is given for all affected individuals. In each family, we chose as the index case the affected child who first appeared in the file; the studies were performed on trio families including the parent and the index.The covariates used were AAO, disease severity, and a quantitative phenotype (IgM level). AAO depends on a latent variable that weighed the hazard, latent severity, and an independent random effect equally. Hazard and latent severity both depend on genotypes at different loci: all trait loci except G and H are involved in the hazard, whereas severity is determined by G and H only. In addition, Locus F influences disease risk directly through the IgM level. Thus, the IgM level is best conceptualized as an endophenotype.For the OSA-TDT, both severity and IgM level were used separately as covariates, in ascending and descending ordering, in data from each of the 100 replicates. For the IgM covariate, we used a reduced data set, consisting of the first 250 families . For the severity covariate, we used both the reduced data set (250 families) and the full data set (1500 families). We selected two single-nucleotide polymorphisms (SNPs) nearest to Locus F for assessing power when IgM level was used as a covariate and two SNPs nearest Locus G and H when severity was used as covariate. In addition, we selected a group of SNPs unlinked with any simulated disease loci to demonstrate that the approach maintains the proper type I error rate. Additional details regarding selected SNPs can be found in Tables For the OTDT, we used only one covariate, namely the AAO, and on each replicate we used the full data set. We performed the classical TDT and the OTDT, focusing on the six non-HLA trait loci  that were given as candidates.Both of our methods use the principle of Spielman's TDT . This tev in the range of the values of the covariate divides the sample in two subsamples: the first subset of families of the ordering, up to the place where the value v is crossed, and the second subset, consisting of the last families of the ordering, starting from this very place. The smallest p-value obtained in any single ordered subset is the intermediate test statistic used for selecting the optimal value of v. Overall significance of the test is assessed via permutation. The permutation test is performed by repeating the process previously described in many  randomly ordered data sets. The minimum p-value from each randomly ordered data set is selected for inclusion in the null distribution against which the p-value from the covariate ordered subset is compared. The empirical p-value, representing the significance of covariate inclusion, is the proportion of p-values from the null distribution smaller than the p-value from the actual ordered subset. Analyses are repeated starting at both the highest and lowest values.The OSA-TDT method is an application of OSA to Spielman's TDT. In the testing procedure, families are ranked in order based on the value of the quantitative covariate. The association test is performed on the initial subset, starting at either the highest or lowest values. Pedigrees are added sequentially, one at a time, until the entire sample is analyzed. At each sequential addition of cases, the association test is repeated. A given value 0 against the alternative hypothesis H1, defined as follows:The permutation statistic is computed to test the null hypothesis H0: There is no value of v for which there is evidence of linkage and association in the ordered subsample.H1: For some v, there is evidence of linkage and association in the ordered subsample.HM1 at the locus M in the low and high subsample, designated here by τ and ρ. Our null hypothesis states that there is no difference between τ and ρ. We briefly describe the construction of our test, which is a log-likelihood ratio test.The OTDT is based on a very similar ordered-based division in subsamples, but the null and alternative hypothesis test are different. In the OTDT we are testing the specific hypothesis regarding the transmission rates of a given allele c of the covariate is used to form two subsamples of families: S(c) (and S'(c)), where S(c) is the subset of families that has an associated covariate value less than c, and S'(c) is the subset that has an associated covariate value greater or equal to c. The general model has three parameters: a parameter c in the range of the values of the covariate, and two parameters τ and ρ in  that are the transmission rate of M1in S(c) and S'(c). The null and alternate hypothesis can be stated as follows:A value 0: For every value of c, τ = ρ.H1: There exists c such that τ and ρ are distinct.Hnc 1,and nc 2,and n'c 1,and n'c 2,be the total number of transmissions of alleles M1 and M2 in the sample S(c) and S'(c), respectively. We let n1 = nc 1,+ n'c 1,and n2 = nc 2,+ n'c2,. These are the number of transmissions of M1 and M2 in the whole sample of families.Let c, τ and ρ can be written as maxc, τ, ρ Z. Similarly, under the null hypothesis, the maximum is maxc, τ Z. Our log-likelihood ratio is now Ω = log. Globally, the rate of transmission of one particular allele at these loci is significantly lower in families of patients with an AAO less than 31 years compared to that in families of patient with an AAO greater than 31 years.However, the role of the Loci G and H is not detected by the TDT, but is detected by the OTDT in all replicates with a permuted Both methods, the OTDT and OSA-TDT, allowed detection of the involvement of Loci G and H in the simulated disease, using the AAO or the severity as a covariate. Indeed, we know from the answers that these loci are related to the severity, which is itself one of the elements used to determine the AAO. These loci are not related to the affection status, which is why they are undetectable by a classical TDT.Locus F was also detected by the OSA-TDT method when families were conditioned on IgM level; this locus is indeed related to the IgM level, which directly influences the simulated RA risk; this influence explains why this locus is already detectable by a classical TDT. Nevertheless, we demonstrated dramatic increases in power when using small data sets. That is, the OSA-TDT method maintains very good power in conditions where the power of the TDT is low.The two ordering TDT methods allowed detection of two loci which are undetectable by the classical TDT, being unrelated to the affections status, but rather moderating its expression.For a third locus, detectable by the classical TDT, a remarkable increase in the power for detection has been shown, thanks to the inclusion of an intermediate phenotype (the IgM level).These ordering methods allow dealing with the particular situation in which a given covariate is suspected of reflecting heterogeneity of the disease. A distinct approach for testing heterogeneity, the predivided sample test , requireIt appears that these ordering TDT methods are promising for detection of loci modulating the expression of a disease, or the value of an endophenotype.The author(s) declare that they have no competing interests.


Genet. Epidemiol. 33:406–418, 2009. © 2009 Wiley-Liss, Inc.Odds ratios or other effect sizes estimated from genome scans are upwardly biased, because only the top-ranking associations are reported, and moreover only if they reach a defined level of significance. No unbiased estimate exists based on data selected in this fashion, but replication studies are routinely performed that allow unbiased estimation of the effect sizes. Estimation based on replication data alone is inefficient in the sense that the initial scan could, in principle, contribute information on the effect size. We propose an unbiased estimator combining information from both the initial scan and the replication study, which is more efficient than that based just on the replication. Specifically, we adjust the standard combined estimate to allow for selection by rank and significance in the initial scan. Our approach explicitly allows for multiple associations arising from a scan, and is robust to mis-specification of a significance threshold. We require replication data to be available but argue that, in most applications, estimates of effect sizes are only useful when associations have been replicated. We illustrate our approach on some recently completed scans and explore its efficiency by simulation.  Genomewide scans are becoming increasingly popular as a tool for estimating the association between large numbers of genetic variants, usually single-nucleotide polymorphisms (SNPs), and many common, complex diseases . AlthougP-values, then the expected value of an estimator, conditional it being the most significant, is again greater than its unconditional expectation. Ranking bias applies not only to the most significant SNP, but also to the second most significant, third most and so on, and is present even when there is no selection by significance. This is similar to the “winner's curse” effect  ≠ μ(1) [for details on the exact distribution of X(1) see (1), but it is often overlooked on account of its large variance. For this reason a weighted average of the first- and secondstage data is sometimes taken, giving an estimator of the form(1) of the formX(2), the second best performing treatment in stage 1. The two-stage estimation procedure is necessitated by the fact that no unbiased estimate for μ(1) exists based only on (finite) stage 1 data—see ith ranked stage 1 effect, and also to allow unequal variances among all stage 1 and stage 2 estimates. The estimate for μi)( in this case isith ranked effect is corrected by taking into account information from the (i – 1)th and (i + 1)th ranked stage 1 effects. In this formula X(0) and Xk+1)X. Let iZ = Xi)+ϒi are then sufficient and complete statistics for μ1,…, μk. The joint distribution of ϒi and X1,…,kX given Q, f is then transformed into f and Without loss of generality we assume that the event i)′. This means that, given the ranking in stage 1, the estimator is unbiased for the corresponding effects and has minimum variance among all such unbiased estimators.which enables the density P-value threshold critp. For a one-sided Wald-type test, we can make this extension by conditioning instead on the eventWe now propose some modifications to this formula in order to apply the same estimation procedure to a genome scan followed by replication. Instead of the magnitude of the point estimates determining the rank order, it is more common in the genomewide setting to rank SNPs according to the statistical significance of their effects, and to restrict attention to those passing an initial Q* as opposed to Q changes the limits of integration for iX and iY, with the result that s,tW in (4) becomesThis leads to slight changes in the proof since conditioning on Xk+1)(/σk+1)1,( = Φ−1(1 – pcrit). This expression was noted in P-value threshold. If SNPs that confer either an increased or decreased disease risk are of equal interest, as is usually the case, then the rank order of significance should be based on two-sided P-values. This now requires conditioning on the eventFor (8) to work generally we define In whereXk+1)(|/σk+1)1,(=Φ−1(1 – pcrit/2).and |We shall work with this last form in the remainder of the article. Note that this estimator may be easily calculated in standard software, using only the summary estimates obtained in the two stages.P-value threshold. Our intention is to compare the usefulness of this estimate to alternatives based on two-stage data. To do this we will calculate the bias and mean squared error of each estimator, to assess their bias-variance trade-offs. However, a subtle facet of this estimation problem is that the true effect of the ith ranked SNP, μi)( = iX) and can be evaluated to a high degree of accuracy by Monte-Carlo simulation.These expressions are the weighted average bias and mean squared error over the distribution of the random variable μThis point of view also holds that the usual formula for the standard error of the stage 2 estimate is inaccurate when applied to estimating the top-ranking effects in stage 1. If a fixed SNP is specified in advance, then the usual standard error does reflect the sampling variation in stage 2, but if the SNP is regarded as random then the weighted average (13) is more appropriate. This forces a distinction between whether a SNP is regarded as definitively associated after stage 1, in which case the fixed effect view is reasonable, or whether stage 2 is regarded as an integral part of the discovery process, suggesting the random effect view. Both views are justifiable in real contexts, but our view is that if we will combine estimates from both stages, then we should allow for variation across both stages even when basing an estimate on stage 2 data only. Although this distinction may seem technical, it will be important for illustrating our claim that the UMVCUE estimator is more efficient than that based only on the stage 2 data.In 2007, the Welcome Trust Case Control Consortium published genome scans of seven common diseases . About 2P-value threshold of 2.25 × 10−5, which is the P-value for the Wald test of the least significant SNP taken forward. This was certainly not the selection criterion used in the original study, but it suffices for our illustration.The stage 1 estimate is consistently higher than the stage 2 estimate and the UMVCUE, reflecting the upward bias. The MLE is similar to the combined odds ratios reported by P = 0.01) so we exclude this from our illustration. In P-value threshold was now taken at 4.9 × 10−5. The general pattern is the same in that the MLE and ZP show bias, whereas the UMVCUE is similar to the stage 2 estimate. In two cases, rs4958847 and rs10801047, the stage 2 estimates are actually higher than stage 1, which reflects sampling variation. The UMVCUE allows for this whereas for rs10801047 ZP has adjusted the MLE back toward the stage 1 estimate. Also the UMVCUE for the 2nd ranking SNP rs9292777 is higher than both the stage 1 and stage 2 estimates. This reflects a negative correction to the MLE, occurring because the stage 1 estimate shows less bias than expected given its ranking.P-value threshold, leading to differences between our results and those of P-value threshold of 2.25×10−5, whereas P-value threshold of 5 × 10−7.For comparison we also report the bias reduced estimates proposed by pcrit at stage 1, makes no correction for the two most significant stage 1 SNPs found by P-values so far away from pcrit that selection is judged to have no biasing effect. Conversely, the largest correction is made to the SNPs that are least significant, and therefore closer to pcrit. On the other hand, the adjustments made by our UMVCUE are not only uniformly greater but also take the rank into account, leading to a less obvious relationship between the rank and the degree of adjustment.To gain some insight into the nature of these estimators, we show in A) we conditioned on detecting a fixed set of SNPs, so we consider genome scans that detect these and only these SNPs. This just involves sampling the log odds ratios from their normal distributions, truncated to allow for the P-value threshold. For each disease, we took the UMVCUE estimates and stage 2 allele frequencies from To evaluate the bias and MSE of the proposed estimators, as defined in (12) and (13), we need to repeatedly simulate multiple effects from a genome scan. We took two approaches to doing this using the T1D and CD results as a model. In the first simulation  we allow for a genomewide set of SNPs and conditioned only on the number of SNPs detected. This was identified as a difficult computational problem , but if The ZP estimator performs well in terms of MSE, having in most cases the minimum of the four methods. Its bias is reduced compared to the unadjusted MLE, but not always by much, particularly for the top-ranking SNPs. The UMVCUE is unbiased as predicted, but its MSE tends to be higher than that of ZP, and sometimes of the unadjusted MLE too. This represents a bias-variance trade-off, but we know that the UMVCUE has minimum variance among unbiased estimators that condition on the stage 1 ranking. We consider this further in the section “Discussion”.We noted earlier that an analytic confidence interval is not known for the UMVCUE: the Rao-Blackwell theorem ensures but does not quantify the minimum variance property. However, since the UMVCUE is unbiased we can regard the MSE as a variance from which confidence intervals can be derived. If we simulate as above from the estimated effect sizes, then the variance gives a parametric bootstrap Wald-type interval subject to the assumptions and conditioning used in the simulation. Our first simulation approach has weaker assumptions about effect size and allele frequency, but stronger conditioning, while our second has stronger assumptions but weaker conditioning. An alternative simulation approach is to assume that all SNPs have equal frequency and therefore the same standard error , which wut proof ; but altWe have seen that the gain in efficiency of our UMVCUE over the stage 2 estimate depends on the sample size in stage 2, more precisely the relative size of stage 2 compared to stage 1. As stage 2 gets larger, our approach gives smaller gains in efficiency, although it is always more ef ficient. In a sense our examples from WTCCC were not ideal, because that study was designed as a proof of concept with a smaller sample size than might normally be attained, and the participating groups had access to a larger number of samples that became the de facto replication data even if they were not originally intended as such. While this may yet become a common model, we also considered a design with 20,000 cases and controls in stage 1 and 2,000 of each in stage 2, a possible scenario for future studies as genotyping costs decrease and samples are shared among consortia, such as for meta-analysis. Using simulation method A, conditioning on detecting a fixed set of SNPs, i)1,1,( and σi+1)1,, was approximately 0.75. In a simulation study we replaced the actual value of σ2,1 by 1,(1), 10σ1,(1)) to demonstrate how the MLE and UMVCUE estimates change. At each value of (1) close to the stage 2 estimate. By the same token, if the true value of μ(1) is close to the stage 2 estimate, the larger variance of the UMVCUE allows it to attain values close to the MLE without incurring bias. Note that if μ(1) is indeed close to the stage 2 estimate, then unbiasedness implies that the UMVCUE will sometimes be less than the true value, so will not be close to the MLE for every sample. Although the UMVCUE and MLE can become arbitrarily close as stage 1 gets larger, this will not be true for every sample drawn at a fixed and finite stage 1 size.The relative sizes of stages 1 and 2 also affect the value of the UMVCUE P-value threshold affects only P-value threshold.Because our adjustment to the ordinary MLE depends only on the two flanking stage 1 estimates, any P-value thresholds. It is clear that the estimate is fairly robust to this threshold so long as it is not grossly mis-specified.In Correcting for selection bias in genome scans is problematical because no unbiased estimator exists for the scan data alone. We have proposed a two-stage estimator that is unbiased while including the data from a genome scan, and moreover accounts for selection bias by both significance and rank. Our approach explicitly allows for multiple associations arising in a scan, which is important because of the different degrees of ranking bias induced, and also the possibility, as we have seen, of negative bias in some circumstances. We have shown that this estimator gives improved precision compared to estimation based only on replication data, with the greatest improvements when the replication sample size is smaller than the initial scan.are available. All approaches, including ours, become much more accurate than those based only on the first-stage data. Indeed, we found that the bias and mean squared error of the method of We argue that in many of the applications for which estimation is needed, associations should already have been replicated so that the two-stage design will usually be available. For example, it seems premature to estimate the total attributable risk for the detected loci until those loci have been confirmed. Of course, we are not advocating collection of a second sample only for unbiased estimation, and estimation from the scan data alone will sometimes be necessary, such as in setting the sample size for a replication. One-stage bias-reduced estimators have been proposed , which uNevertheless, if only initial scan data are available, we note that our approach can be incorporated into a single-stage analysis. Our focus in this article has been on unbiased point estimation of SNP effects. However, in the section “Replication Sample Size” we saw that while the two-stage UMVCUE for a top-ranking stage 1 SNP tended toward the biased stage 1 estimate as the stage 2 precision was set close to zero, its bootstrapped confidence interval width increased to keep the estimate nominally unbiased. This suggests the possibility that when only heavily selected stage 1 data are available, by postulating an unbiased estimate with a large variance (in the manner of a vague prior), the resulting UMVCUE confidence interval could also be used for hypothesis testing, to help distinguish null effects from true effects of small magnitude. It would be interesting to assess the performance of this ad hoc proposal relative to other more established multiple testing strategies.We considered a two-stage design consisting of a genome scan and a replication study, but two-stage genome scans can also be performed . We did Our estimator has the minimum variance among unbiased estimators that condition on the ordering in stage 1, but this holds only marginally for each estimate and not jointly. That is, each of P-values from a Wald test, but the current test of choice is the Cochran- Armitage test of trend, which is the score test from a logistic regression model [T be a statistic on which SNPs are ranked, and suppose it can be computed from an estimate of the odds ratio X and some nuisance parameters θ. Write T = T and let T−1 be the solution of T = x. Then equation (11) becomesWe give a formula based on ranking by on model . The Walon model . Let T b2 tests from the 2 × 2 allelic contingency tables inferred from the reported summaries [We applied this approach to Pearson χummaries . Here thOur results show that while our estimator is unbiased, it can have greater mean square error than the two alternatives we considered for this design: the unadjusted standard combined estimate, and the significance-corrected estimator of We have assumed that all effect estimates are independent. This is approximately true for the studied examples since replication was attempted for one SNP in each genomic region, with two exceptions (5p13 and 5q33 in CD) in which the replicated SNPs were in weak linkage disequilibrium (LD). However, each SNP was selected for being the most significant in its local region, so that some ranking bias is present in the stage 1 estimates that is not accounted for by our approach.We do not expect this to be a serious problem since the ranking bias decreases as LD increases, so that when our failure to account for LD is most acute, the resulting bias is minimal. Nevertheless, a precise unbiased estimator that takes into account known or estimated LD structure is an open problem for further work.Finally we note that our approach extends trivially to a meta-analysis of several studies following an initial scan, by treating the combined estimate from the replication studies as our stage 2 and then combining it with the initial scan to obtain an overall unbiased estimate. R code to implement the method proposed in this article is available from the authors on request.