As the cost of sequencing decreases, the demand for association tests that use exhaustive DNA sequence information increases. One such association test is multivariate distance matrix regression (MDMR). We explore some of the features of MDMR using Genetic Analysis Workshop 17 simulated data in search of potential improvements in distance measures. We used genotype data from 697 unrelated individuals, in 200 replications, to test the power of MDMR to detect 13 trait Q2 causative genes based on the Euclidean distance metric. We also estimated the false-positive rate of MDMR using 508 control genes. In addition, we compared MDMR with Mantel’s test and collapsing analysis for rare variants. MDMR performed comparably well even with the Euclidean distance measure. High-throughput sequencing technology allows identification of new rare alleles in a human population, but the sparseness of these alleles in samples becomes an important obstacle to detecting the true effects of rare variants under a single-variant-based paradigm. The increasing number of identified genetic variants also requires more statistical tests, thereby aggravating the issue of low statistical power. Several methods, based on collapsing or grouping rare variants, have been developed to alleviate this problem. However, these methods arbitrarily define common and rare variants and treat them differently for analysis. This artificial classification may fail to capture important biological reality. For example, it has been shown that a common variant can act as a modifier of a rare variant’s effect ,2. It isMultivariate distance matrix regression (MDMR) can provide a flexible platform for a phenotypic association test of DNA sequence . In thisMDMR is based on the multivariate multiple regression model ,6,7 defiY is an n × p data matrix, in which n is the number of subjects and p is the number of unknown biologically relevant genetic variables; and X is an n × m model matrix, in which m is the number of predictor variables. β is an m × p matrix of beta coefficients and ε is an n × p residual matrix. Because genotype is regressed using a combination of traits and covariates under this model, m regressors can be phenotypes  and covariates . A pseudo-F statistic can be constructed to test the null hypothesis of β = 0:where where the projection matrix is:G matrix is Gower’s centered matrix, which can be calculated from an n × n distance matrix D. The G matrix replaces YY′ and can be calculated from any symmetric distance matrix allowing for nonmetric dissimilarity measures. The I matrix is the n × n identity matrix, and tr stands for the trace of matrix. To assess statistical significance of the pseudo-F statistic, one can use permutation tests.and the We calculated Euclidean distances using numerically coded genotypes of 13 Q2 risk genes for all possible pairs of the 697 unrelated individuals:a and b. Genotypes were coded as the number of minor alleles with no weighting of single-nucleotide polymorphisms (SNPs) was applied. For each gene and each Q2 simulation, we constructed a 697 × 697 genotypic distance matrix D and a 697 × 1 phenotype matrix X, which consists of the individual Q2 trait values, and used them to calculate a pseudo-F statistic under the regression model that includes the Q2 trait as the sole independent variable. Each of the 13 × 200 tests underwent 1,000 permutations in which the rows and columns of its raw genotype matrix  were shuffled at random. The empirical p-value was determined as the frequency of observing more extreme pseudo-F statistics in permutations than in the actual gene case. MDMRs were performed either using all variants within a gene or using only rare variants with minor allele frequency (MAF) less than 0.01. Similarly, we selected 508 noncausative  genes for Q2 and tested them using all 200 replications. We omitted a subset containing 125 genes from these 508 control genes for the rare-variant-only analyses because they contained no rare variants.where the Euclidean distance is defined as the L2 norm between two individual genotype vectors P-values were empirically determined using 1,000 permutations. We estimated the power and false-positive rates on the basis of the significance threshold value of 0.05 and compared them with the values from MDMR and collapsing analysis.The Mantel test measures the correlation between two distance matrices . In our Collapsing analysis is a simple regression analysis that uses a collapsed variable  into whiVNN1 was significantly (p < 0.05) associated with Q2 in 94% of the total replicates when MDMR using all variants was used. The Mantel test, however, discovered association in only 25% of VNN1 replicates and failed to find any risk gene with a detection rate greater than 50% regardless of the MAF-based SNP filtering.Table PDGFD and SIRT1, with power greater than 50%. VNN1 did not survive the cutoff this time, presumably because one of its causal SNPs was common and thus removed from the analysis. The common variants of PDGFD and SIRT1 were all noncausal, and removal of these variants may have enhanced performance of MDMR on these genes. PDGFD was also found by collapsing analysis using the cutoff value of 50%, along with SREBF1. The estimated power for the 12 Q2 risk genes was comparable between the Euclidean MDMR and collapsing analysis, whereas the Mantel test appeared to be less sensitive than the other methods.When variants with MAF > 0.01 were removed, MDMR identified two genes, Because MDMR is computationally intensive, we focused on only 508 control genes to compare the false-positive rates of all three methods. The false-positive rates of MDMR and collapsing analysis were similar and slightly inflated Figure . The ManAlthough the best performing method differed from gene to gene, the power and false-positive rate of MDMR, the Mantel test, and collapsing analysis were inclined to be positively correlated, implying that there is a general agreement in performance between these methods Figure . Causal MDMR is a statistical method to test the aggregate effect of genetic variants based on a pairwise genotypic distance matrix. We applied the Euclidean MDMR to the GAW17 simulated data to compute the power of the method for detecting causative Q2 genes. We estimated the power and false-positive rate of MDMR using 200 simulated replicates of the Q2 trait for 13 Q2 causative genes and 508 control genes, respectively. MDMR was compared with collapsing analysis and the Mantel test for its performance on rare variants. We observed that the Euclidean MDMR performs comparably to collapsing analysis. The Mantel test, another distance-based method, seemed to behave more conservatively than the other methods. Our study also suggests that MDMR will perform better than the Mantel test and collapsing analysis in some genetic settings, presumably depending on the number of causal and noncausal variants under interrogation, their respective effect sizes, and any dependencies among them. To dissect the effect of each individual factor on performance, we need to examine various rare variant analysis methods in more contrived, simulated settings that confine confounding variables.Originally designed for a region of genome, MDMR anticipates multiple causal SNPs with joint actions on the phenotype(s). Thus the unit of test can be easily reduced to a functional domain or a small set of adjacent SNPs  if we exThe Euclidean MDMR performed comparably to collapsing analysis to detect the Q2 causal genes. The Mantel test was less sensitive than these methods with a slightly reduced type I error rate. Potential progress can be made because the distance matrix appreciates genotypic dissimilarities relevant only to phenotypic dissimilarities.The authors declare that there are no competing interests.DC conceived of the study, performed the statistical analysis and drafted the manuscript. QZ developed the analytical pipeline for collapsing analysis. ATK, IBB and MAP contributed ideas to improve the design and the manuscript. All authors read and approved the final manuscript.


Characterizing associations among multiple single-nucleotide polymorphisms (SNPs) within and across genes, and measures of disease progression or disease status will potentially offer new insight into disease etiology and disease progression. However, this presents a significant analytic challenge due to the existence of multiple potentially informative genetic loci, as well as environmental and demographic factors, and the generally uncharacterized and complex relationships among them. Latent variable modeling approaches offer a natural framework for analysis of data arising from these population-based genetic association investigations of complex diseases as they are well-suited to uncover simultaneous effects of multiple markers. In this manuscript we describe application and performance of two such latent variable methods, namely structural equation models (SEMs) and mixed effects models (MEMs), and highlight their theoretical overlap. The relative advantages of each paradigm are investigated through simulation studies and, finally, an application to data arising from a study of anti-retroviral-associated dyslipidemia in HIV-infected individuals is provided for illustration. The increased availability of data on single-nucleotide polymorphisms (SNPs) has led to heighten interest in understanding how this genetic information correlates with measures of disease progression. One analytic challenge plaguing these genotype-trait association studies is the potential for multiple SNPs to be implicated in complex diseases. In this manuscript, we describe applications and performance of two latent variable paradigms, namely structural equation models (SEMs) and mixed effects models (MEMs), for addressing this challenge.SEMs constitute a broad range of multivariate regression models that allow complex dependencies among multiple predictors and outcome variables and are widely used in economics, sociology and psychology   is invertible , given by J1 and J2, are not necessarily equal, these vectors need to be concatenated with vectors of 0 of appropriate length. That is, Eq.  with all 0 elements. In order for this SEM to simplify to the MEM with two levels of clustering, we additionally need to assume α=0 and where, without loss of generality, we assume K>2 genes, we make the following three assumptions for the SEM to reduce to the MEM: (i) SNPs, represented by Xi, are fixed, so that the model of Eq. ––6)–(7), xis is a binary SNP indicator, y, where we assume y increase while y=0) is 0.03 for both the SEM and MEM approaches.The results of fitting both SEMs and MEMs to these data are provided in B=0. In this case, our model can be written as:Second, we simulate data according to an SEM with two uncorrelated latent variables. That is, we let the data arise from Eqs.  and 8)––8)–9) w w9) with1 and α2 are set to 0 and we restrict y, 1, and ψ2 are assumed to vary as described in where The results of fitting the SEM and MEM to these data are given in 12=0.1 and ψ12=0.2, where in both cases ψ1=ψ2=0.4. These models correspond to correlations between latent variables of 0.25 and 0.5, respectively. The results are reported in Finally, we simulate data according to an SEM with two correlated latent variables, where the model is the same as described above for two uncorrelated latent variables, with the exception that it is assumed S=4 SNPs, each coded as binary indicators with s=1,2,3. A single continuous covariate y=0 has reduced power for detecting association. The type-1 error rates are 0.05 and 0.03 for the SEM and MEM approaches, respectively.Next we simulate data under an MEM with a single clustering variable, as described by Eq. . We agaiy is relatively small in all cases while the power for the MEM approach is reasonable (>90%) for Finally, we generate data according to a two-level clustering MEM, as described by Eq. . Here wen=626 HIV-infected participants enrolled in 5 AIDS Clinical Trials Group (ACTG) trials who agreed to genetic testing. A complete discussion of the study design and patient demographics is given in Foulkes et al. (n=532) with complete data, including known drug exposure histories, are included in analysis.In this section we apply the SEM and MEM frameworks to data arising from the New Works Concept Sheet (NWCS) 224 study, an investigation of genetic factors that contribute to anti-retroviral-associated dyslipidemia in HIV-1-infected individuals. This cross-sectional study is comprised of s et al. . Here wep=0.002 and p=0.003) in the SEM and p=0.02) in the MEM for both the single-gene and two-gene models. These effects are attenuated  in adjusted models and may represent spurious associations resulting from population-admixture, i.e. confounding by race/ethnicity and study site. Adjusted models also included PI exposure as a three-level factor – no current PI exposure; currently exposed to a non-RTV-containing PI regimen; and currently exposed to an RTV-containing PI regimen – gender, race/ethnicity and study.The results of fitting unadjusted models are reported in ui, is estimated by p-values based on a Wald test are given by p-values of “0” are less than 1×10−8. Recall the first element of Λ is fixed at 1 for identifiability. These results suggest variant alleles at rs2032582  described in y) relating the latent gene variable to the trait. In the MEM context, on the other hand, the test of association is based on a variance parameter . Notably, for y depends on Interestingly, our simulation studies reveal that the performance of the two modeling approaches is comparable under the SEMs in terms of power and type-1 error rates; however, when the data arise from an MEM, power for the SEM approach is lower than the corresponding power for detecting association using the MEM approach. Also of note, when data are simulated under the SEM model, the estimates of the nuisance parameter, σr SNPs in one gene and s SNPs in a second gene, the original formulation of the MEM approach involves A notable limitation of both the MEM and SEM approaches is their potential inability to handle a large number of SNPs. In the context of the MEM, the number of genotype groups can become unwieldy as the number of SNPs increases, as described by Foulkes et al. . InteresWe also presented the results of applying an alternative two-stage BN analysis approach to the NWCS224 data, as well as to randomly selected simulated data sets. In the real data example, we were unable to fit a regression tree with splits beyond the root node subject to the specified pruning criterion for one of the genes under study. In this case, the analysis reduced to fitting a BN to single SNP variables rather than metavariables as described by Malovini et al. . On the Finally, further research is needed to characterize the performance of both the SEM and MEM frameworks in the context of more complex structures of association and underlying genetic models. In a recent manuscript, we describe application of a mixture modeling approach that may be more appropriate than the single normal prior assumption on the random effects under a dominant or recessive genetic model (Au et al.,


The data set simulated for Genetic Analysis Workshop 17 was designed to mimic a subset of data that might be produced in a full exome screen for a complex disorder and related risk factors in order to permit workshop participants to investigate issues of study design and statistical genetic analysis. Real sequence data from the 1000 Genomes Project formed the basis for simulating a common disease trait with a prevalence of 30% and three related quantitative risk factors in a sample of 697 unrelated individuals and a second sample of 697 individuals in large, extended pedigrees. Called genotypes for 24,487 autosomal markers assigned to 3,205 genes and simulated affection status, quantitative traits, age, sex, pedigree relationships, and cigarette smoking were provided to workshop participants. The simulating model included both common and rare variants with minor allele frequencies ranging from 0.07% to 25.8% and a wide range of effect sizes for these variants. Genotype-smoking interaction effects were included for variants in one gene. Functional variants were concentrated in genes selected from specific biological pathways and were selected on the basis of the predicted deleteriousness of the coding change. For each sample, unrelated individuals and family, 200 replicates of the phenotypes were simulated. The state of the science for localization and identification of genes that influence common complex diseases has changed rapidly over the past 20 years. As laboratory costs continue to fall with the development of more efficient high-throughput techniques, the field is quickly proceeding toward studies that make use of genome-wide sequence data. There is as yet no consensus on optimal, or even appropriate, statistical genetic approaches for analyzing exome sequence data, and few investigators have had experience analyzing such data sets. This was the motivation for the Genetic Analysis Workshop 17 (GAW17) “mini-exome” data set. The GAW17 data set is a hybrid of simulated and real data. Real exome sequence data from the 1000 Genomes Project were used as the basis for simulating a common complex disease and related quantitative risk factors. Two different study designs were simulated, unrelated individuals and large families, each with the same sample size.http://www.1000genomes.org) is designed to survey genetic variation at the sequence level across multiple human population groups. It includes individuals of European, East Asian, South Asian, West African, and American Indian ancestry. Three pilot projects for the 1000 Genomes Project were completed in 2010: low-fold genome-wide sequencing of 179 individuals, higher fold sequencing of two parent-child trios, and exonic sequencing in 697 individuals  to infer missing genotypes and haplotypic phase. In the family data set (described later), we used the program CHRSIM [The 1000 Genomes Project genotypes were not phased, and some genotypes were missing as a result of incomplete sequence coverage in some individuals. We used the program fastPHASE [m CHRSIM  to drop As noted, the GAW17 genotypes differ from the official 1000 Genomes Project called genotypes for the same individuals because of differing approaches to genotype calling, inclusion or exclusion of regions of low-fold coverage, and the inclusion of the imputed genotypes in the GAW17 data set. Imputed genotypes were not identified as such in the distributed data and were treated as equally “real” as called genotypes in the phenotype simulations. These choices were motivated by a focus on designing a data set that would be useful for developing methods related to gene localization, identification, and characterization, with the 1000 Genomes Project data primarily serving as a source of sequence data with realistic patterns of SNP distribution, allele frequency, population variation, and linkage disequilibrium. However, these decisions limited the utility of the GAW17 data for population genetic analyses or for examination of effects of genotype calling or data cleaning on gene finding.KDR gene that was designated as functional in the phenotype simulations which had identical genotypes to multiple nonfunctional SNPs. Relatively few of the variants were common; 74% had minor allele frequency (MAF) ≤ 0.01 and only 12.8% had MAF ≥ 0.05 ; marker locations were recorded as RefSeq36 base-pair coordinates. The 24,487 autosomal SNPs detected in genotype calling were, for purposes of the simulation, assigned to 3,205 genes based on the first intersection found of the marker location and the base-pair coordinates of all genes obtained from RefSeq36 annotations. SNPs that overlapped multiple genes were assigned to only one of those genes. There were 1 to 231 SNPs per gene . Of the SNPs, 9,433 (38.4%) were private variants, occurring once in the set of 697 unrelated individuals. Multiple private variants carried by the same individual resulted in SNPs with identical genotypes, including a SNP in the n = 90), Denver Chinese (n = 107), Han Chinese (n = 109), Japanese (n = 72), Luhya (n = 108), Tuscan (n = 66), and Yoruban (n = 112) population groups. The second sample configuration used in GAW17 simulations consisted of 697 individuals in 8 extended families with the genotypes for the 202 pedigree founders being taken from the 1000 Genomes Project data. The founders of the pedigrees were chosen at random from the unrelated individuals sample and included 12 CEPH, 18 Denver Chinese, 19 Han Chinese, 28 Japanese, 50 Luhya, 66 Tuscan, and 9 Yoruban samples. Because of a computational error, the genotypes of pedigree founders were merged incorrectly across files, resulting in an incomplete match between the genotypes of the pedigree founders and the corresponding individual from the unrelated individuals sample. This affected a small proportion of genotypes (7%) but impacted all pedigree founders. Approximately one-third of the SNPs were unaffected, and for the two-thirds that had substitutions, most had only one or two founders with altered genotypes. Pedigree configurations were adapted from the pedigrees used for simulated data in Genetic Analysis Workshops 10 and 12 [Two disparate sampling designs were used in the construction of the simulated data. One sample consisted of 697 unrelated individuals, each of whom corresponded to an individual from the 1000 Genomes Project data. The 1000 Genomes Project subjects whose data were used came from the CEPH . Assigned ages were matched across the family and unrelated individuals data sets and ranged from 16 to 91 years, with a mean of 41.8 years.For the family data set, fully informative markers were generated at each gene  and used to compute identical-by-descent (IBD) allele sharing at each gene location under the rationale that family-based data sets were likely to have previous short tandem repeat (STR) or high-density SNP genotyping that could be used to estimate the IBD allele sharing. These IBD matrices were provided as part of the GAW17 data set.A common disease, with a prevalence of 30%, was simulated along with three related quantitative risk factors, Q1, Q2, and Q4. Smoking status  was also simulated. Phenotype simulations were performed multiple times to generate 200 replicates of the unrelated individuals and pedigree data sets. Note that the genotype data remained constant across replicates, as did age, sex, and pedigree configuration.http://www.genome.jp/kegg/) and the proprietary software Ingenuity Pathways Analysis (IPA), version 8.7 (http://www.ingenuity.com). The vascular endothelial growth factor (VEGF) pathway was observed to have numerous genes with available typed SNPs and was therefore selected as the source of a subset of the functional loci for phenotype simulations. The IPA version of the VEGF signaling pathway was used as the core source because it included most of the genes in the KEGG VEGF signaling pathway as well as some additional upstream information, primarily relating to VEGF transcriptional control. The overlap between the two data sources was considered significant enough not to impede the investigations of any researchers limited to the freely available KEGG data set.Knowledge about biological pathways and statistical predictions regarding the potential deleteriousness of coding variants was used in designing the simulation model. Pathways for gene enrichment were selected from the publicly available Kyoto Encyclopedia of Genes and Genomes (KEGG) database . Effect sizes for coding variants within genes were assigned using PolyPhen and SIFT predictions of the likelihood that the variant would be deleterious. The functional variants included both rare and common alleles and a range of effect sizes, with most having small effects but a few having large effects that should be reliably detectable in most replicates of the data set. Some genes contained a single functional variant and others contained many. Population origin of the 1000 Genomes Project participants was not used in the phenotype simulations. In general, there was little disequilibrium between the functional variants Figures , 4, withQuantitative risk factors Q1, Q2, and Q4 were simulated as normally distributed phenotypes. Disease was simulated using a liability threshold model, and the top 30% of the distribution was declared affected. All SNP effects were additive on the quantitative trait or liability scale, with each copy of the minor allele increasing the mean trait value by an equal amount. Genotype by environment effects were simulated for Q1. Because genotype, age, and sex were held constant across replicates, the variation in phenotype across replicates came primarily from the residual polygenic and residual environmental components. The residual polygenic components were correlated between relatives, by definition, and also correlated between Q1, Q2, and latent liability. The residual environmental components were unique to each individual and were simulated to be weakly correlated between Q1, Q2, and latent liability.β column in the table provides the displacement in mean levels of Q1 for each copy of the minor allele. Q1 also had a residual heritability of 0.44, resulting from variants at loci not included in the current sequence data set. The residual genetic component of Q1 was correlated with the residual genetic components of Q2 and latent liability. There were also weaker environmental correlations between Q1 and Q2 and latent liability. Values of Q1 were higher in smokers, and there was genotype by smoking interaction for the effects of variants in the KDR gene on Q1. Effects of the KDR variants were 50% higher in smokers than in nonsmokers. . Q4 was lower in smokers, decreased with age, and was lower in females. Q4 was protective; that is, individuals with higher levels of Q4 had lower risk of disease.A normally distributed latent liability trait (not included in the distributed phenotype data) was simulated; it was influenced by 51 SNPs in 15 genes with 1–24 functional variants per gene (see Table Liability to disease = latent liability + Q1 + Q2 – Q4. (1)Using this formula, a quantitative liability score was calculated for each individual, and the top 30% of the distribution in each simulation replicate was declared affected. A consequence of this assignment was that each replicate had the same number of affected individuals, although the identity of these individuals varied across replicates. The effect sizes in Table The authors declare that there are no competing interests.LA, TDD, and JB designed the phenotypic trait model and the pedigree and unrelated individuals data sets. JMP aligned and processed sequence data. JCC and JEC provided input on incorporation of pathway and functional information and selection of functional variants. TDD conducted the gene dropping and phenotype simulations. JWK, TDD, LA, and JB performed error checking on the simulated data. All authors read and approved the final manuscript.


VEGFC gene with estimated power not exceeding 0.35 (FAM-MDR), when FWER was kept under control. The considered population-based methods gave rise to highly inflated FWERs (up to 90% for PBAT screening).In the quest for the missing heritability of most complex diseases, rare variants have received increased attention. Advances in large-scale sequencing have led to a shift from the common disease/common variant hypothesis to the common disease/rare variant hypothesis or have at least reopened the debate about the relevance and importance of rare variants for gene discoveries. The investigation of modeling and testing approaches to identify significant disease/rare variant associations is in full motion. New methods to better deal with parameter estimation instabilities, convergence problems, or multiple testing corrections in the presence of rare variants or effect modifiers of rare variants are in their infancy. Using a recently developed semiparametric strategy to detect causal variants, we investigate the performance of the model-based multifactor dimensionality reduction (MB-MDR) technique in terms of power and family-wise error rate (FWER) control in the presence of rare variants, using population-based and family-based data (FAM-MDR). We compare family-based results obtained from MB-MDR analyses to screening findings from a quantitative trait Pedigree-based association test (PBAT). Population-based data were further examined using penalized regression models. We restrict attention to all available single-nucleotide polymorphisms on chromosome 4 and consider Q1 as the outcome of interest. The considered family-based methods identified marker C4S4935 in the  Analyzing the effects of genes and/or environmental factors on the development of complex diseases is a great challenge from both the statistical and the computational perspectives. Calle et al. ,2 recentKDR gene and C4S4935 in the VEGFC gene) are assumed to be additive on the quantitative trait scale, such that each copy of the minor allele increases or decreases the mean trait value by an equal amount. In addition, values of Q1 were simulated to be higher in smokers, and the listed variants in the KDR gene were involved in KDR-smoking interaction effects on the trait. There are 944 markers in 81 genes on chromosome 4. The sample size for both population- and family-based data is 697 with family data comprising 8 families with 202 founders and 3 offspring generations. The founders were randomly sampled from the unrelated individuals data set, and genotypes of offspring were sampled using Mendelian inheritance. It should be noted that genetic information is the same for all replicates; only phenotype and smoking status differ.The data provided by GAW17 include a subset of genes grouped according to pathways that had sequence data available in the 1000 Genomes Project. Effect sizes for coding variants within these genes were assigned using PolyPhen and SIFT predictions of the likelihood that the variant would be deleterious. Two hundred replicates were generated. Our analyses involve the quantitative trait Q1, which was simulated as a normally distributed phenotype. Furthermore, we restrict attention to the available single-nucleotide polymorphisms (SNPs) on chromosome 4 . All simulated singular SNP effects  to the association test (PBAT) screening -7  on each of the individual cells versus all other cells with the response variable, using a liberal p-value threshold of 0.1 [X in {H, L, O} on the outcome variable. Association with the trait is investigated by testing H versus L [t test. In the last step, permutation-based step-down max T adjusted p-values [P procedure [The MB-MDR technique for one dimension involves three steps. First, each marker’s genotype cells are assigned to one of three categories—high risk (H), low risk (L), or no evidence (O)—on the basis of the result of association tests  markers in the model, to be in line with PBAT screening. However, sometimes a few more markers were selected (maximum of 12). The covariates Sex, Age, and Smoke are fixed and unpenalized in the regression model. We repeated this analysis for each replicate to obtain a screening technique for the main effects. After this screening procedure, the selected markers were put in a linear regression model to test for association with Q1, again fixing Age, Sex, and Smoke in the model. P-values are Bonferroni corrected for the number of markers in the data.To select the 10 most interesting predictors for Q1, we also apply a least absolute shrinkage and selection operator (LASSO) penalized regression . We decrFollowing Li and Leal  for discT and 0.34 for min P) to find C4S1878, the marker with the largest MAF (0.16), but also elevated FWER estimates . With penalized regression, the highest power is achieved for markers C4S1884 (MAF = 0.02) and C4S1877 (MAF < 0.001), irrespective of whether a gene-collapsing method was adopted or not. However, these results are downplayed by the inflation of the corresponding FWER (>0.3). PBAT exhibits extremely high power (0.94) to detect C4S4935 (MAF < 0.001) but also has an extreme FWER of 0.895. When a correction is made for the presence of linkage, interestingly, PBAT’s power drops to 0 and its FWER drops to 0.015. On the other hand, the FAM-MDR approach has only limited power (0.18 for max T and 0.17 for min P) to detect C4S4935 but keeps the FWER under control.Additional file A graphical representation of the relation between error rates for nonfunctional markers and the corresponding markers’ MAFs is given in Figure T and 0.38 for min P) and C4S1884 (0.205 for max T and 0.155 for min P) on the KDR gene and for the collapsed variable obtained from the rare variants on the KDR gene (0.355 for max T and 0.47 for min P). For the FAM-MDR analysis for families, collapsing increases the power to detect the variant C4S4935 on the VEGFC gene (0.275 for max T and 0.345 for min P). FWER for unrelated individuals remains high, whereas for family data FWER is under control.Finally, collapsing rare variants increases the estimated power of the MB-MDR approach on unrelated individuals, both for the common variants C4S1878  is only 0.01; hence linkage disequilibrium can be ruled out as an explanation of the increased FWER. In contrast, rare effects seem to be the major cause of the observed elevated FWER estimates. This is further supported by the observation that 1 out of 200 simulated replicates gives an erroneous result among the markers with MAF > 0.1  and the creation of variables that exhibit larger amounts of information. For unrelated individuals, of the 944 markers, 199 have MAF ≥ 0.01 and 745 have MAF < 0.01, and they are collapsed into 72 gene-specific variables. For the families, of the 541 nonmonomorphic markers, 227 have MAF ≥ 0.01 and 314 have MAF < 0.01, and they are collapsed into 60 gene-specific variables. Surprisingly, FWER increased when the MB-MDR approach was applied to unrelated individuals. Notably, one of the drawbacks of adopting collapsing methods is that singular effects for rare variants cannot be distinguished from global gene effects.We compared several genetic association strategies to detect main effects, including the MB-MDR approach, PBAT screening, and penalized regression. Although none of the methods exhibited sufficient power to detect rare variants, remarkable differences were observed between these methods within and between study designs. At this point it is not clear whether these differences are due to the particular way the genetic effects were simulated in the family-based or population-based data or whether they are actually due to the methods themselves. However, most important, we postulate that the rarity of certain marker alleles hampers the validity of model assumptions and distributional properties of test statistics as well as assumptions underlying some commonly used measures to correct for multiple testing or to control false-positive rates.The authors declare that there are no competing interests.JMMJ, TC and KVS participated in the statistical analyses related to the MB-MDR technique. LDL and KVS participated in the analyses related to PBAT and penalized regression. FVL participated in optimizing the MB-MDR software for this project. AE participated in data handling and manipulation. All authors read and approved the final manuscript.Table 1 - Power to detect functional markers and FWER Power and FWER results are shown for the MB-MDR approach and penalized regression on unrelated individuals, and FAM-MDR and FBAT results are shown for family data, both on original and collapsed chromosome 4 data. Power values greater than 0.1 are indicated in bold, and FWER values greater than 0.1 are indicated in italic. a At least 10 markers are selected using penalized regression, with Sex, Age, and Smoke as the fixed covariates. bP-values are Bonferroni-corrected according to the total number of markers. Sex, Age, and Smoke are fixed covariates in the final model. c PBAT screening uses FBAT statistic to test the null hypothesis of no association in the presence of linkage.Click here for file


Risk prediction that capitalizes on emerging genetic findings holds great promise for improving public health and clinical care. However, recent risk prediction research has shown that predictive tests formed on existing common genetic loci, including those from genome-wide association studies, have lacked sufficient accuracy for clinical use. Because most rare variants on the genome have not yet been studied for their role in risk prediction, future disease prediction discoveries should shift toward a more comprehensive risk prediction strategy that takes into account both common and rare variants. We are proposing a collapsing receiver operating characteristic (CROC) approach for risk prediction research on both common and rare variants. The new approach is an extension of a previously developed forward ROC (FROC) approach, with additional procedures for handling rare variants. The approach was evaluated through the use of 533 single-nucleotide polymorphisms (SNPs) in 37 candidate genes from the Genetic Analysis Workshop 17 mini-exome data set. We found that a prediction model built on all SNPs gained more accuracy (AUC = 0.605) than one built on common variants alone (AUC = 0.585). We further evaluated the performance of two approaches by gradually reducing the number of common variants in the analysis. We found that the CROC method attained more accuracy than the FROC method when the number of common variants in the data decreased. In an extreme scenario, when there are only rare variants in the data, the CROC reached an AUC value of 0.603, whereas the FROC had an AUC value of 0.524. The completion of hundreds of genome-wide association studies has brought numerous novel disease susceptibility loci to light. Yet for many diseases the common variants that have been identified explain only a small proportion of disease heritability. Additional genetic variants, including rare variants and gene-gene or gene-environment interactions, remain uncovered. Among these, great attention has been given to the rare variants. Current genome-wide association studies include only single-nucleotide polymorphisms (SNPs) with a minor allele frequency (MAF) greater than 5% ,2. WithiThe uncovered rare variants, particularly those that are yet to be identified by future whole-genome sequencing studies, can be combined with the known common variants and clinical risk factors for more accurate disease prediction. However, few approaches are available for assessing the combined effect of both common and rare variants in early disease prediction. Statistical approaches, such as the collapsing approach  and the The collapsing approach first combines all rare variants into a single common variant and then analyzes it with other common variants using multivariate test statistics. Although this approach was originally proposed for genetic association studies, the idea can be used for genetic risk prediction research as well. We here develop a collapsing receiver operating characteristic (CROC) approach for risk prediction research that considers both common variants and rare variants. The new approach is an extension of the previously proposed forward receiver operating characteristic (FROC) approach, which was developed using optimal features of the likelihood ratio rule ,9. A mulThe receiver operating characteristic (ROC) curve is commonly used in genetic risk prediction research to evaluate the accuracy of a risk prediction model. The ROC curve plots the sensitivity of a prediction model against its specificity by continuously changing the cutoff points over the whole range of possible outcomes. When the ROC curve is formed on the likelihood ratio (LR), which is defined as the ratio of the frequency of a particular test outcome in case subjects to that in control subjects, it attains the maximum performance at each cutoff point. The corresponding one-dimensional summary accuracy index, the area under the ROC curve (AUC), is also the highest among that of all approaches . Based oK loci, which consist of kM multilocus genotypes, we calculate the likelihood ratio The FROC approach uses a computationally efficient algorithm, the forward selection algorithm, to search a large number of genetic predictors for an optimal risk prediction model. The forward selection algorithm starts with a null model and gradually adds new predictors to the model to improve its accuracy. In each step, the forward selection algorithm searches all loci for a locus that most significantly improves the prediction model. The whole selection process continues until adding a new predictor no longer increases the prediction accuracy. A series of prediction models, with different levels of model complexity, are thus obtained. Among these models, the first model with only one predictor is the simplest model, and it has the lowest AUC value; the last model, which has the largest number of predictors, is the most complex model, and it has the highest AUC value. Although the more complex models tend to have an estimated classification accuracy that is higher, they are more likely to overfit the data when there are more risk groups. We therefore used 10-fold cross-validation to identify the best model with the appropriate complexity, and we used that as our final model. Assuming that the best model selects D and kM multilocus genotype is assigned an LR value. Based on the ranks of the LR values, we can form an optimal ROC curve and calculate its corresponding AUC value:where DN and ψ is given by:where The FROC approach was proposed for risk prediction research on common variants. When both common and rare variants exist, the FROC approach hardly selects rare variants because of their low frequency, which could lead to low accuracy of the prediction model. To deal with both common and rare variants, we have extended the FROC approach and are introducing the CROC approach here. The CROC adopts a multistage collapsing procedure to collapse rare variants into pseudo-common variants and then uses the forward selection algorithm of the FROC approach to search both common and pseudo-common variants for the best prediction model.k − 1) rare variants are selected in the previous (k − 1) steps and then collapsed into a pseudo-common variant. The pseudo-common variant is defined by:The development of the multistage collapsing procedure is based on the ideas of Li and Leal . Similark, we search the remaining rare variants for one locus that increases the AUC value most significantly and collapse it into the pseudo-common variant. The procedure keeps collapsing new rare variants into the pseudo-common variant until the AUC value stops increasing. A pseudo-common variant is thus formed.The accuracy of the pseudo-common variant is then measured by its AUC value. In step We repeat the collapsing procedure on the remaining rare variants and generate a set of pseudo-common variants. The multistage collapsing procedure stops when there are no rare variants left in the data. One of the advantages of using a multistage collapsing procedure instead of the original collapsing procedure is that it could potentially consider bidirectional effects. The forward selection algorithm is then used to search both common variants and pseudo-common variants for an optimal risk prediction model. Because the pseudo-common variants have a higher frequency, they are more likely to be selected by the forward selection algorithm, which could result in increased accuracy of the prediction model.We evaluated the performance of the CROC approach using the simulated Genetic Analysis Workshop 17 (GAW17) mini-exome sequencing data. The data are composed of 697 individuals, in which 209 individuals make up the case group. Thirty-seven candidate genes were selected based on the simulation results provided. There are 533 SNPs in the candidate genes, including both disease susceptibility and noise loci. The MAFs of these SNPs range from 0.00072 to 0.45122. Among those, 400 SNPs are rare variants (MAF < 0.01). Using the GAW17 data, we investigated whether the accuracy of the risk prediction model could be improved by considering rare variants in the analysis. An additional analysis was also conducted to compare the performance of the CROC and FROC approaches.We started the analysis by forming a risk prediction model on all common variants. To assess the accuracy improvement by adding rare variants, we also built a risk prediction model using all genetic variants. In all, 200 replicates were used for the analysis. Risk prediction models were built based on the first 100 replicates, using both the CROC and FROC approaches; they were then evaluated on the remaining 100 replicates. The reason for evaluating the model on a separate replicate is to ensure that the rare variants are evaluated. Because of the small sample size of the data set, rare variants are commonly carried by one individual or by a small number of individuals. If we split the data into training and testing data sets, then rare variants present in the training data set will likely be absent in the testing data set and therefore cannot be validated. Using two replicates ensures the presence of the same rare variants in both data sets. However, we should note that the estimation might be biased because of the potential correlation between two replicates.The accuracy of the risk prediction models was averaged over 100 replicates and is summarized in Table From Table Next-generation sequencing technology is anticipated to be a powerful tool for uncovering novel genetic variants associated with complex disease, particularly for those variants with a low frequency. The rare variants identified through future whole-genome sequencing studies, if confirmed to have functional importance, may provide novel insights into underlying pathological and etiological processes. Yet, even if they are merely predictive and without functional importance, these rare variants can still be harnessed into clinical translational research applications. By incorporating these rare variants into current risk prediction models, we can predict disease outcomes more accurately.We developed a CROC approach for future risk prediction research on sequencing data. The approach extends the previously developed FROC approach to deal with both common and rare variants by using a multistage collapsing procedure. The idea of collapsing was originally introduced in genetic association studies to deal with both common and rare variants. We have now integrated those ideas into the CROC approach for risk prediction research on sequencing data. By applying the approach on the simulated mini-exome sequencing data, we demonstrated the advantage of using both common and rare variants in risk prediction research. However, in this application, limited improvement was gained when additional rare variants were combined with common variants. This may be because rare variants account for only a small proportion of phenotype variation. However, in a different scenario, rare variants might contribute significantly to phenotype variation. Therefore we artificially decreased the influence of common variants. When the number of common variants was decreased, we found that significant improvement could be attained by considering additional rare variants. The accuracy of the risk prediction models can be further improved by considering environment risk predictors and gene-environment interactions. We ran an additional analysis including the environment risk predictors and found that the prediction accuracy was significantly improved (data not shown).We considered only candidate genes in our analysis. Current risk prediction studies commonly adopt this strategy. However, for high-dimensional risk prediction research using millions of SNPs, variable selection becomes important. Although a forward selection algorithm is incorporated into the CROC approach, it could still be subject to false positives when dealing with whole-genome sequencing data. More sophisticated selection algorithms will be needed to deal with a large number of common and rare variants.We have developed a CROC approach for risk prediction analysis on sequencing data. By applying this new approach to the simulated GAW17 mini-exome sequencing data, we have illustrated that current risk prediction models built on common variants can be further improved by considering additional rare variants. In addition, we compared the CROC approach with the existing FROC approach. The CROC approach outperformed the FROC approach, especially when a large proportion of the considered variants were rare.The authors declare that they have no competing interests.CW developed the statistical methods, conducted the analysis and drafted the manuscript. QL conceived of the study, developed the method, participated in its design and coordination and helped to draft the manuscript. Both authors read and approved the final manuscript.


Genetic mapping has been used as a tool to study the genetic architecture of complex traits by localizing their underlying quantitative trait loci (QTLs). Statistical methods for genetic mapping rely on a key assumption, that is, traits obey a parametric distribution. However, in practice real data may not perfectly follow the specified distribution.Here, we derive a robust statistical approach for QTL mapping that accommodates a certain degree of misspecification of the true model by incorporating integrated square errors into the genetic mapping framework. A hypothesis testing is formulated by defining a new test statistics - energy difference.Simulation studies were performed to investigate the statistical properties of this approach and compare these properties with those from traditional maximum likelihood and non-parametric QTL mapping approaches. Lastly, analyses of real examples were conducted to demonstrate the usefulness and utilization of the new approach in a practical genetic setting. Genetic mapping of quantitative trait loci, or QTLs, plays prominent roles in understanding the genetic basis of many phenotypic variations -4. Depen2 mapping population of 500 mice derived from two inbred lines . Based on the error (3), we havewhere ij ωis the conditional probability of QTL genotype j given the marker genotype of progeny i.where Thus, the estimator of the parameters ise Eis approximated twice, one by the LLN and the other by the calculation of φ(ie). However, if the QTL is scanned at a marker position, only the approximation by the LLN is needed because no mixture density is used in this situation. The energy function at the marker position em Eis expressed asIn practice, the genomic location of a QTL is estimated by scanning positions across the genome. When the QTL is assumed to exist between the two markers, the k is the number of progeny in the marker genotype group k and where Nk  has a different probability of linking with the QTL genotypes, the phenotype density is marker-dependent. The density for marker genotype k is expressed asUnlike the error density, the phenotype data density contains a mixture of density functions each corresponding to a different QTL genotype. Also, because each marker genotype group ij ωis the conditional probability of QTL genotype j given the marker genotype k. From Eq. (2), the energy function for marker genotype k iswhere Notice thatThus, we havekφ(iy) = kφ(i y| jm), which is not in a mixture form. The When a QTL is assumed to be at a marker position, dE) asTo combine the information from all nine marker genotypes, we take a weighted sum of the marker energy functions to calculate an overall energy function for phenotype data (orh(x) is a monotone increasing function with respect to x. The reason for choosing such an h function is that the more progenies in one marker group, the better approximation accuracy achieved by the LLN, and the more weights should be put on this group. To determine the exact form of h, we need to use the formulation for eL2E. Because at marker positions, the eL2E and pL2E approaches use exactly the same information for the derivation of energy function, they should agree at those positions. Therefore, a comparison between the energy functions in (5) and (7) for eL2E and pL2E at marker positions suggests that k) = Nk/N hh.We performed Monte Carlo simulation studies to examine the statistical properties of the L2E approach performs better than the eL2E. Therefore, for the simulation studies and real-data analysis, the pL2E method will be used. This is reasonable because the derivation of eL2E involves two approximations but pL2E involves only one by the LLN. For the ease of notation, hereinafter L2E means pL2E.By scanning the simulated chromosome with a step size of 2 cM from the left end to the right end, the ED values were calculated and smoothed. Figure 2E method, under different sample sizes  and heritabilities . In each case, 100 replicates were run to evaluate the consistency and efficiency of the mapping methods. First, we consider simulation scenarios where error distributions in model (1) are normally distributed without any outlier data; i.e., the normal distribution is the true model. Because the simulation results from different sample sizes display similar patterns, here we only show the result for N = 400, which is tabulated in Table 2E and traditional ML methods obtained consistent estimators and similar standard errors for the genetic effect parameters. However, the ML estimators have better efficiency as evidenced by smaller MSEs. This is expected because under the true model the MLE is asymptotically efficient.Additional simulations were performed to examine the statistical properties of the Lt-distributions as error terms. In addition to different combinations of sample size and heritability, we also changed the degrees of freedom (df) of the t-distributed errors. When df is high , where the t-distribution approximate a normal distribution, the two methods perform similarly , where the t-distribution has much heavier tails than the normal distribution, the MLE method failed to give correct parameter estimates and yielded much larger standard errors. In the contrast, the L2E maintained the correct estimates with smaller standard errors. This demonstrates the robustness of the L2E method against model misspecification.Second, we simulated scenarios where error distributions in model (1) are non-normal, using a ly Table . However2E model with both ML and NP approaches. The outliers were generated from another normal density on the upper tail of a mixture density. Different percentages of noise points  were considered. The main results are shown in Table 2E estimators are consistent but the ML estimators become biased towards the direction of the outliers. As the outliers move further away from the true density (from 45 to 55), the ML estimators perform significantly worse, but the L2E estimators stay consistent with very little impact. (2) As the heritability becomes smaller and smaller, the difference between the two methods becomes less. This is because the variation of the mixture density increases with decreasing heritability and, thus, the relative positions of the outliers become closer. This is consistent with the point (1) (3) L2E and NP methods show similar robustness to the outliers. However, the L2E method maintains the interpretability of a parametric model and gives accurate estimates of genetic effects. Overall, the simulation results demonstrate that the L2E method is preferred to the MLE and NP methods when the true model is misspecified or non-ignorable outliers exist.Third, we simulated experiments where data contains outlier data points. Because NP mapping is popular for traits with outliers , we comp2 mice  derived from two inbreed strains, the Large (LG/J) and Small (SM/J). This map has a total map distance of 1780 cM (in 19 linkage groups) and an average interval length of 23 cM. The F2 progeny was measured for body mass at 10 weekly intervals starting at age 7 days. The raw weights were corrected for the effects of each covariant due to dam, litter size at birth, parity, and sex [Vaughn et al.  construc and sex .2E and ML methods were applied to map this trait. The profiles of the two test statistics, energy difference (ED) and likelihood ratio statistic (LRS) across the whole mice genome is shown in Figure Our analysis here focuses on identifying QTLs that may affect the body mass growth rate from ages 5 weeks to 10 weeks, which is defined as body mass ratio between week 10 and week 5. On the right side of the empirical density of this trait . Through this dissection the genetic architecture behind the quantitative traits can be elucidated, which provides a sound basis for future trait improvement. To better utilize the genomic data, considerable attention has been paid to develop powerful analytic methodologies that can increase the power, precision, and resolution of QTL mapping (8-16). Currently, almost all the QTL mapping methods proposed so far assume a parametric  distribution density of a trait. However, there is an increasing recognition of the limitation for the parametric assumption, given that in practice the true distribution of a trait is never known.2E, which may allow the fitted model to be different from the true model. We derived two different implementation of the L2E method into the mapping framework and show how they are connected. The simulation studies suggest that the pL2E method works better than eL2E method and were used for our further analyses. Additional simulation studies were performed to test the statistical behaviour of the L2E-based mapping approach. The L2E method is more robust in the model choice at a cost of lower efficiency. For a "perfect" data, the ML performs better than the L2E. However, when the data contains noises, the L2E outperforms the ML. The relative efficiency of the L2E increases with increasing percentage of noises. In practice, it would be unrealistic for us to know the true model underlying the data, but it can be almost assured that no data is perfect. Thus, a better strategy is that the L2E method can first used to explore the data, with results compared with the MLE method.In this article, we propose a QTL mapping methodology based on the principle of L2E method would be an excellent addition to the current toolbox of the QTL mapping.This work is our first attempt to incorporate the principle of the integrated square errors into the genetic mapping framework. There are many areas that can be explored in the future, such as how to apply this principle to examine the gene-gene interaction or gene-environment interactions. The L2E mapping, automatically manipulates data points that are apparently outliers by giving them less weight in parameter estimation, and therefore yields more accurate estimation of QTL locations and effects. In the case where the data cleaning is not possible or very hard to do so, our new method could be a very beneficial choice. Simulation studies showed that in the presence of outliers, L2E method outperforms the traditional MLE and non-parametric methods in terms of both accuracy and efficacy of the parameter estimations. A real data analysis of the mice body mass data also demonstrates the usefulness and utilization of the new approach in a practical genetic setting. We strongly encourage researchers to explore both the L2E and MLE mapping procedures in practice.In this article, we derive a robust approach for genetic mapping of complex traits by incorporating the principal of the integrated square error into the general mapping framework. This approached, called the LSW carried out the analysis, prepared and drafted the manuscript. GF participated in the design of the study. YC initiated the project design. ZW participated in the design of the study. RW initiated and established the overall project design, prepared and drafted the manuscript. All authors read and approved the final manuscript.


Does achievement independent of ability or previous attainment provide a purer measure of the added value of school? In a study of 4000 pairs of 12-year-old twins in the UK, we measured achievement with year-long teacher assessments as well as tests. Raw achievement shows moderate heritability (about 50%) and modest shared environmental influences (25%). Unexpectedly, we show that for indices of the added value of school, genetic influences remain moderate (around 50%), and the shared (school) environment is less important (about 12%). The pervasiveness of genetic influence in how and how much children learn is compatible with an active view of learning in which children create their own educational experiences in part on the basis of their genetic propensities. A measure of academic achievement uncorrelated with ability or previous attainment is an appealing construct, relevant for several current controversies and goals in education Twin studies provide a useful indication of the relative contributions of genetic and environmental factors on individual differences in measured traits Behavioral genetics and the twin method are concerned with the genetic and environmental influences on individual differences in complex traits. Such complex traits are the outcome of multiple genetic factors alongside  multiple environmental factors throughout development. The focus is on variation in genes and environments in relation to variation in outcomes; as such the statistics of behavioral genetics concern population (or sample) level effects, rather than case studies of particular individuals' genetic and environmental profiles.Previous twin studies on school performance have indicated moderate heritability around 40–60% The evaluation of schools has increased steadily since the early 1990s, with the ultimate aim of encouraging competition and thereby driving improvements in education. The concept of ‘added value’ was introduced to the UK school system in 2002 to overcome the issues of using ‘raw’ school attainment as a measure of school performance, because raw achievement is largely an index of the calibre of pupil intake, rather than any indication of the school's performance “The inclusion of the previous level of attainment is therefore intended to capture the effects on the current level of knowledge of all historical inputs, including inherited endowments such as innate ability as well as family background and schooling.” (page 209).Education has been slow to accept the role of genetic influences on educational outcomes Analyses of the determinants of added value suggest that both school-level and pupil-level characteristics are important The sampling frame for the present study was the Twins Early Development Study (TEDS), a study of twins born in England and Wales in 1994, 1995, and 1996 The twins' academic performance was assessed throughout the school year by their teachers, using the assessment materials of the National Curriculum for England and Wales (NC), the core academic curriculum developed by the Qualifications and Curriculum Authority (QCA) For the study at age 12, the NC Teacher Assessments at Key Stage 3 were used, which are designed for children aged 11–14 years old. When the twins were 12 years old, teachers assessed three broad areas of achievement: English ; mathematics ; and science . Teachers rate performance from levels 1 to 8. By the end of Key Stage 3 at age 14, children are expected to reach levels 5/6. These judgments were not made specifically for the present study, but rather form the continuing assessment of each child that ultimately leads to the final NC Teacher Assessment score submitted to the QCA at the end of the key stage, which summarizes the child's academic achievement during that period. Reminders of the NC criteria used to select the appropriate attainment level were provided as part of the questionnaire.http://curriculum.qca.org.uk.The teacher-rated achievement scales are highly correlated, with an average intercorrelation of .83. A factor analysis indicated that the first principal component accounted for 89% of the variance. For this reason, we calculated a composite achievement score as a mean of the three achievement domains: English, mathematics and science. There were 2824 complete pairs of twins with this NC achievement composite; 1039 monozygotic  pairs; and 1785 dizygotic  pairs. Further details about these measures have been published previously To complement our teacher ratings of school achievement we also collected test data on reading and math performance at 12 years. We created a web-based cognitive battery to allow collection of test data from the whole TEDS sample, which is spread across the UK. We have shown that our web-based cognitive test battery is a reliable and valid method for collecting cognitive data on children as young as 10 years old rc. The PIATrc assesses literal comprehension of sentences. The sentences were presented individually on the computer screen. Children were required to read each sentence and were then shown four pictures. They had to select, using the mouse, the picture that best matched the sentence they had read. All children started with the same items, but an adaptive algorithm modified item order and test discontinuation depending on the performance of the participant. The internet-based adaptation of the PIATrc contained the same practice items, test items and instructions as the original published test.Three measures of reading ability were used at 12 years: two measures of reading comprehension and a measure of reading fluency. The twins completed an adaptation of the reading comprehension subtest of the Peabody Individual Achievement Test rc, we assessed reading comprehension at age 12 using the GOAL Formative Assessment in Literacy for Key Stage 3 As well as the PIATReading fluency was assessed using an adaptation of the Woodcock-Johnson III Reading Fluency Test In order to assess mathematics, we developed an internet-based battery that included questions from three different components of mathematics. The items were based on the National Foundation for Educational Research 5-14 Mathematics Series, which is linked closely to curriculum requirements in the UK and the English Numeracy Strategy The test achievement scales were moderately correlated, with an average intercorrelation of .49. A factor analysis indicated that the first principal component accounted for 58% of the variance. We calculated a composite achievement score as the mean of the six achievement web measures. There were 5142 pairs of twins with this web achievement composite; 1892 MZ; and 3250 DZ.g) score as the mean of the four tests.As well as tests of achievement, we also included ability tests in our web-based cognitive battery at 12 years. The twins were tested on two verbal tests, WISC-III-PI Multiple Choice Information  and Vocabulary Multiple Choice subtests rc) and mathematics (NferNelson) from our 10-year web-battery When we correct for previous achievement at 10 years we use the same NC teacher reports  and tests . To create the ability-corrected achievement scores, both teacher ratings and test-based scores, we used a standard regression onto general cognitive ability, and saved the standardized residuals. The correlations between Because twins are perfectly correlated for age and same-sex twins are correlated perfectly for sex, variation associated with age or sex would contribute to the correlation between twins. That is, data uncorrected for age and sex would inflate twin correlations. For this reason, and as is standard in twin analysis, all measures were also corrected for age and sex effects using a regression procedure Twin intraclass correlations were calculated, which index the proportion of total variance due to between-pair variance g-free achievement” (or achievement adjusted for previous achievement) than residualizing on g, is multivariate genetic analysis, which estimates the extent to which genetic and environmental factors that affect one trait also affect another trait. Rather than removing the g covariance from the variance of achievement scores prior to analysis, multivariate genetic analysis considers all of the variance of achievement scores as well as all of the variance of g, and it decomposes the covariance between them into genetic and environmental components of covariance A more elegant way to address the question of “SD) for the measures at 10 and 12 years are presented in The means and standard deviations  environments? Shared environment can be estimated by subtracting the above estimates of heritability from the MZ twin correlation. Shared environment was actually Structural equation maximum likelihood model-fitting analyses were performed in the program Mx lower when achievement was corrected for g: .30 versus .21 for teacher-rated achievement, a non-significant difference, and .20 versus .06 for tested achievement, a significant difference. And similar results were found for achievement corrected for previous achievement: .30 versus .12 for teacher-rated achievement (non-significant difference), and .20 versus .08 for test achievement (significant difference). Non-shared environmental influences were significantly greater when achievement was corrected for either g or for previous achievement.As indicated by the pattern of MZ and DZ twin correlations, shared environment was 2, C2 and E2 estimates indicate the genetic, shared environmental and non-shared environmental influences on achievement that are independent of those influences on g. For teacher-rated achievement these g-free estimates are .32, .21 and .17 respectively, all statistically significant. The total g-independent variance of teacher-rated achievement is .70 . If we re-standardize the g-independent genetic and environmental components of variance on the basis of this total g-independent variance of .70, the resulting ACE estimates are similar to the univariate model-fitting results reported in Results from the bivariate Cholesky decomposition model for general cognitive ability and 12-year achievement are shown in A similar pattern is found for genetic and shared environmental influences on test data of achievement Table 4.2 estimate of .29 indicates that 29% of the total variance of teacher-rated achievement at age 12 can be attributed to genetic influences independent of teacher-rated achievement at age 10. Similar to the results for g-free achievement, we can re-standardize the influences on 12-year achievement that are independent of those on 10-year achievement . For teacher-rated achievement, the total variance of 12-year achievement that is independent of 10-year achievement is .58 (i.e. .29+ .12+ .17 = .58), so the re-standardized heritability of 12-year achievement independent of 10-year achievement is 50% (.29/.58 = .50). The shared environmental estimate (.12) is not significant, indicating that shared environment independent of 10-year achievement does not contribute significantly to 12-year achievement; the re-standardized shared environmental component is 21% (.12/.58 = .21). Results for test scores of achievement at ages 10 and 12 , were highly similar, with twin correlations of 0.52 and 0.35 for MZ and DZ twins for teacher-rated achievement, and 0.47 and 0.27 for MZ and DZ twins for test performance.The purpose of this study was to test the hypotheses that added value indices of school achievement are less influenced by genetic factors and more influenced by shared (school) environmental factors than raw achievement scores. We find that neither nature nor nurture support these predictions about added value.g are linked genetically g-free achievement scores, as well as the first report of genetic and environmental influences on current achievement that is independent of previous achievement. For teacher assessments, heritability for g-free achievement is 0.49, which is almost as great as the heritability of 0.52 for achievement uncorrected for g. For test scores, g-free achievement was also significantly and substantially heritable (0.40) although non-significantly less heritable than achievement uncorrected for g (0.55).Although other genetic studies have shown that achievement and g-free achievement scores. The heritability estimates for 12-year achievement independent of 10-year achievement are 0.47 for teacher assessments and 0.46 for test scores. In other words, current achievement independent of previous achievement shows just as much genetic influence as raw achievement. A more elegant way to address this question uses multivariate genetic analysis, which comes to the same conclusion  twins, because the only factor that makes members of an identical twin pair different from one another is the non-shared environment. Therefore it is possible to correlate the differences in achievement scores within pairs of identical twins with differences in environmental exposure to determine whether such environmental exposures account for this non-shared environmental influence Despite the reasonableness of assuming – or hoping – that a measure of achievement independent of previous attainment (either g or previous school achievement) could be constructed that would be a pure measure of the school or learning environment, about half of the variance of corrected-school achievement is due to genetic differences between children. This does not mean that the enterprise has failed, or is hopeless. There is no doubt that these indices of knowledge gain are useful at an individual level. However, averaged out at the school level, they do not necessarily provide a pure indication of the school's effectiveness g or by previous achievement, which makes it a useful step towards assessing the value that schools add to children's achievement. What is new from our study is the finding that what is left is still heritable. In addition, we have shown that contrary to expectations, indices of added value show only minimal shared environmental contributions.Even though corrected-school achievement is not free of genetic influence, it is at least free of genetically-driven attainment as indexed by either g, personality, interests, attitudes, motivation, and even psychopathology and health, all of which are at least moderately heritable gg, achievement scores are still substantially heritable.Because half of the variance of achievement is not heritable, it would seem theoretically possible to create a better index of schools' added value by correcting for other genetically influenced traits. What are these other genetically influenced factors that affect children's achievement at school? Possible candidates include specific cognitive abilities not tapped by The concept of added value is not used in all educational systems; however, these results are relevant to any educational system because they highlight the dynamic influence of genes and environments on school-related processes. For example, it seems a reasonable assumption that genetic influences are the same throughout development because children's DNA sequences do not change. However, these results suggest that new genetic influences can come into play even between 10 and 12 years of age because there are genetic influences specific to 12-year achievement that did not influence achievement at 10 years. In contrast, unlike DNA sequence, we know that environments change during development – even the classroom a child finds themselves in changes from year to year, but what these results suggest is that we should be focusing on changes in non-shared (as opposed to shared) environments. That is, environmental influences that affect added value are unique to individuals.not follow is the nihilistic notion that if achievement shows genetic influence there is nothing that can be done about it. Heritability does not imply immutability how and how much they learn. This way of thinking is compatible with the current trend towards personalizing education by optimizing children's learning If we take seriously these findings that demonstrate that all aspects of achievement are suffused with genetic influence, we must conclude that it may not be possible to devise a measure of achievement that reflects only environmental influences such as the added value of schools. What follows from such a conclusion? What should instruction , this genetic perspective on learning suggests a return to the original meaning of education . That is, instead of a model of instruction in which children are the passive recipients of knowledge, a genetically sensitive approach to education suggests an active view of learning in which children select, modify and create their own education in part on the basis of their genetic propensities. In genetics, such processes are called genotype-environment correlationMore generally, instead of thinking about education as Genotype-environment correlation offers new ways of thinking about the environmental interface in which genotypes become phenotypes. Specifically, achievement independent of ability may be just as heritable as achievement including ability because achievement is as much a function of genetically-driven appetites as of aptitudes. Genotype-environment correlation is a way in which children add value to their own environments. Effective, responsive education systems may well be ones that increase the magnitude of such correlations between nature and nurture.