Canis latrans) pairs over two successive litters to determine whether among‐litter differences in behavior  and hormones  corresponded with parental plasticity in habituation. Thus, we explicitly test the hypothesis that accumulating experiences of anthropogenic disturbance reduces parental fear across reproductive bouts, which should have disparate phenotypic consequences for first‐ and second‐litter offspring. To quantify risk‐taking behavior, we used foraging assays from 5–15 weeks of age with a human observer present as a proxy for human disturbance. At 5, 10, and 15 weeks of age, we collected shaved hair to quantify pup hormone levels. We then used a quantitative genetic approach to estimate heritability, repeatability, and between‐trait correlations. We found that parents were riskier  with their second versus first litters, supporting our prediction that parents become increasingly habituated over time. Second‐litter pups were also less risk‐averse than their first‐litter siblings. Heritability for all traits did not differ from zero (0.001–0.018); however, we found moderate support for repeatability in all observed traits (r = 0.085–0.421). Lastly, we found evidence of positive phenotypic and cohort correlations among pup traits, implying that cohort identity  contributes to the development of phenotypic syndromes in coyote pups. Our results suggest that parental habituation may be an ecological cue for offspring to reduce their fear response, thus emphasizing the role of parental plasticity in shaping their pups’ behavioral and hormonal responses toward humans.A fundamental tenet of maternal effects assumes that maternal variance over time should have discordant consequences for offspring traits across litters. Yet, seldom are parents observed across multiple reproductive bouts, with few studies considering anthropogenic disturbances as an ecological driver of maternal effects. We observed captive coyote ( Our fouesources , both ofesources , underscFrom our questions, we make several predictions Figure . First, 22.1X ± SD]). Pups were born in March and April of both years and observed from 5–15 weeks of age. This age was selected because pup emergence from natal dens becomes more frequent, pups are progressively weaned by their mothers, and pups refine their social skills and conspecific communication  – National Wildlife Research Center's (NWRC) Predator Research Facility, in Millville, UT. Eight breeding pairs  and their offspring were observed in 2011 and 2013, and all pairs were nulliparous  before the study  to assess risk‐taking behavior in coyote parents and pups, as seen in previous work  as a binary response (yes/no) over a 7‐min period. Thus, riskier individuals, by definition, ate at food piles more frequently over development than others. We chose a 7‐min observation period because in preliminary feeding observations, this was the maximum amount of time for coyotes within a pen to consume all food provided, regardless of whether few individuals monopolized food rations or if all animals ate.2.3Hair has quickly become a viable alternative to quantify individual hormone levels, particularly because hair concentrations represent an accumulated hormonal average over a period of months to years, rather than days  received the odor cues four times over a 20‐day period, whereas control pairs  received water as a delivery control , individual identity , maternal identity , and cohort identity  as random effects  for the G priors (random effects) and the residual variance fixed to one  for the R priors, similar to previous studies  with Markov chain Monte Carlo (MCMC) estimation , and output was robust to slight changes to these priors.To determine whether offspring traits differed as a function of parental reproductive bout, we used univariate animal models with litter  included as a fixed effect, as well as all other fixed effect variables found in parental models . Pup risk‐taking was fit with a categorical error structure with residual variance fixed to 1, comparable with models for parental risk. Endocrine variables, in contrast, were fit with a Gaussian distribution after confirmation of normality using Levene tests implemented from the “Rcmdr” R package  was partitioned into additive genetic , permanent environment , maternal , and cohort  variance parameters by fitting the model with the random terms of “animal”, “ID”, “dam”, and “Litter ID”. Thus, PV = A V+ PEV + MV + C V+ RV, in which RV accounted for the residual variance  in the model , which included the distribution‐specific variance term (π2/3) of a binomial model with a logit link , maternal effects (m2 = MV/P V+ π2/3), and cohort effects (C = CV/P V+ π2/3) were similarly estimated  divided by the phenotypic variance, r = IV/(PV + π2/3). Again, we included the distribution‐specific variance term (π2/3) of a binomial model with a logit link function. Variance parameters for pup endocrine traits were estimated similarly without the binomial‐specific variance term.To investigate whether pup traits  were repeatable, as well as determine the relative weight of our variance components on trait repeatability, we combined parental and pup data  into a single analysis to effectively estimate variance components in our sample population. In addition, we used our previous univariate models from the first two aims to estimate quantitative genetic components within each age class . The total phenotypic variance (PCOV) from the multivariate model were partitioned into additive genetic (ACOV), maternal (MCOV), cohort (CCOV), and residual (RCOV) covariance components. To calculate correlations, we divided the respective covariance estimate for a pair of traits by the square root of the product of variances. For instance, the genetic correlation among pup risk‐taking and cortisol, where “1” is equal to risk‐taking and “2” is equal to cortisol, was calculated as: Ar = COVA/√(VA(1)*VA(2)); whereas the maternal correlation was calculated as: Rr = COVM/√(MV(1) * MV(2)))  = ACOV M+ COV C+ COV R+ COV) by the square root of the sum of all variances  = (A(1)V M(1)+ V C(1)+ V + VR(1))*(A(2)V M(2)+ V C(2)+ V + VR(2)). Hence, phenotypic correlations among traits were estimated as: Pr = COVP/√).To determine whether offspring traits were correlated, we used a multivariate animal model that contained all fixed and random effects in previous univariate models to estimate genetic, maternal, cohort, and phenotypic correlations among offspring traits. Before analysis, we binned trait data according to the hair hormone survey window, then proceeded to analyze each window separately. In other words, risk‐taking behavioral data were partitioned into a single row with shaved hair samples collected at 10 and 15 weeks of age, respectively. These periods corresponded with ecologically‐relevant developmental periods established in the literature , heritability (h2), and all correlations  were derived from animal models as the mode of the posterior distribution with accompanying 95% credibility intervals  in parentheses. Bayesian estimates were considered statistically significant when the credibility intervals do not overlap zero  versus the first reproductive bout (mean ± SE: 3.6 ± 1.2 pups). Because of the potential confounding relationship between parental parity and litter size, we compared model fit between null animal models containing all previous fixed effects, and alternative models that additionally included the interaction term between litter year and litter size. We then selected the model with the lowest deviance information criterion (DIC) value, in which the optimal model had a ΔDIC = 0  in 2011 and second litters  in 2013  died of unknown causes at 6 to 7 weeks of age. Thus, we observed n = 72 pups up to 5 weeks of age, n = 68 pups up to 10 weeks, and n = 60 pups up to 15 weeks for a grand total of n = 1763 observations of pups and n = 790 observations of parents over a 2‐year span. Model selection results can be found in Supporting information Appendix Coyote pairs gave birth to their first litters  and fathers  were riskier with their second litters than with their first , we found evidence of a positive phenotypic correlation among risk‐taking behavior and cortisol Figure . We did 4The combination of prior experiences paired with current environmental context induce parental plasticity over multiple reproductive bouts (Plaistow et al., We provide evidence to suggest that second‐litter pups had lower testosterone at 5 weeks of age Figure , Table 2Evaluating the genetic and environmental sources of variance in personality and endocrine function is integral to understanding the importance of such effects on evolution (Dingemanse & Araya‐Ajoy, Capreolus capreolus, Debeffe et al., Phodopus sungorus, Kanda, Louon, & Straley, Amatitlania siquia, Mazué, Dechaume‐Moncharmont, & Godin, Trachemys scripta, Carter, Paitz, McGhee, & Bowden, Our results provide evidence for the emergence of personality and phenotypic syndromes in early life history, as similarly shown in other mammals (European roe deer, The expression of personality traits can occasionally differ amongst wild and captive individuals (Mason et al., To conclude, the relationship among parent and offspring can dictate how future generations will navigate predicted ecological conditions (Badyaev & Uller, The authors do not have any conflict of interest to declare.C.J.S. conceived the research, conducted field work, analyzed data, created figures, and wrote the manuscript. J.K.Y. conceived the research, conducted field work, provided comments on the manuscript, and provided funding. E.V.L. conceived the research and provided comments on the manuscript. R.M.S. conceived the research, assisted in endocrine lab work, provided comments on the manuscript and provided funding. J.M.M. conceived the research and provided comments on the manuscript.https://doi.org/10.5061/dryad.pr00820.The datasets used for these analyses, as well as the R script used to perform the analyses, are deposited in the Dryad Digital Repository as  Click here for additional data file.


Chinese Simmental beef cattle are the most economically important cattle breed in China. Estimated breeding values for growth, carcass, and meat quality traits are commonly used as selection criteria in animal breeding. The objective of this study was to evaluate the accuracy of alternative statistical methods for the estimation of genomic breeding values. Analyses of the accuracy of genomic best linear unbiased prediction (GBLUP), BayesB, and elastic net (EN) were performed with an Illumina BovineHD BeadChip on 1,217 animals by applying 5-fold cross-validation. Overall, the accuracies ranged from 0.17 to 0.296 for ten traits, and the heritability estimates ranged from 0.36 to 0.63. The EN  model provided the most accurate prediction, which was also slightly higher (0.2–2%) than that of GBLUP for most traits, such as average daily weight gain (ADG) and carcass weight (CW). BayesB was less accurate for each trait than were EN  and GBLUP. These findings indicate the importance of using an appropriate variable selection method for the genomic selection of traits and suggest the influence of the genetic architecture of the traits we analyzed. Chinese Simmental beef cattle play an important role in the Chinese beef industry because of their high adaptability and the rapid growth of their young with sufficient feeding. The beef is also well marbled and tender, contributing to its marketability. The primary objective of beef production is to improve beef quality and yield grades. Estimated breeding values of economical traits, such as carcass weight (CW), eye muscle area (EMA), and marbling score (MS), are generally considered to be the major selection criteria in beef cattle breeding. The average daily weight gain (ADG) is also important as an indicator of growth.In China, small holders manage a large proportion of young bulls, in some cases keeping fewer than 100 cattle. Considering this situation, these bulls cannot be managed well. The bulls often lack phenotypic or pedigree records due to the absence of unified management. In this context, an alternative to traditional genetic evaluations is genomic selection (GS), which was originally introduced by [The methods for genomic prediction can be classified into two groups: linear and nonlinear methods. The commonly used linear methods include unbiased and biased prediction models. Unbiased prediction models include genomic best linear unbiased prediction (GBLUP) and ridge regression best linear unbiased prediction (RR-BLUP) , while bPrevious studies have reported the performance of genomic prediction methods for various traits in different breeds. These studies suggested that statistical methods may perform differently for different traits because of the large differences in the genetic architecture of complex traits . When feThe aim of this study was to provide a scientific basis for the application of GS to the Chinese Simmental population. A comparative evaluation of the performance of multiple methods is essential to identify those best suited to GS for Chinese Simmental cattle. Here, we evaluated the relative performance of five methods for GS. We expected to identify suitable methods for estimating the predictive accuracy for various growth, carcass and meat quality traits of economic relevance using the BovineHD SNP array in the Chinese Simmental cattle population.All animals used in the study were treated following the guidelines established by the Council of China Animal Welfare. The research was undertaken with the approval of the Committee on the Ethics of Animal Experiments of the Chinese Academy of Agricultural Sciences (CAAS) . The use of animals and private land in this study was approved by the respective owners.2), CL (cm), hind leg length , and MS. The ADG was the rate of weight gain per day over the fattening duration. LW was measured before slaughter after fasting for 24 h. CW was the remaining cold carcass measured after slaughter and bloodletting by eliminating the hide, head, feet, tail, entrails and gut fill. The total BNW was created to sum the weights of the carcass bones. EMA was measured at the 12th and 13th rib interface 48 h postmortem via the dot grid method. TW, SW and CL were measured directly from carcass anatomy. MS was visually scored on a seven-point scale depending on the degree of marbling on the cut surface of the rib eye. The statistics used for each trait to estimate variance components are presented in Our resource population included 1,225 Chinese Simmental cattle born between 2008 and 2014 in Ulgai, Xilingol League and Inner Mongolia of China. After weaning, these cattle were moved to Jinweifuren Co., Ltd. for fattening, with all animals sharing uniform feeding and management conditions. More detailed descriptions of the breeding and management have been described previously . The catβ is a vector of fixed effects , X is the design matrix of relevant observations to the corresponding fixed effects, and y* is the random residual. The residual y* was subsequently used in the prediction models.The fixed effects were used to adjust the phenotypic values of traits of interest before the analysis:P-value for Hardy-Weinberg equilibrium test (HWE) higher than 10−5. The samples with more than 10% missing genotypes were excluded, which resulted in a final dataset of 1,217 cattle with 608,696 SNPs. Additionally, the imputation for sporadic missing alleles was performed using Beagle 3.3.2 [Genotypes were generated with the Illumina BovineHD BeadChip. Quality control of genotypes was conducted through an iterative process using the following SNP selection criteria: call rate (CR) higher than 0.95, minor allele frequency (MAF) higher than 0.05 and le 3.3.2 .y* is the vector of the corrected phenotype, μ is the overall mean, and I is an identity matrix. For GBLUP, Z is an incidence matrix for individual effects. γ is the vector of breeding values assumed to follow a multivariate normal distribution K is calculated following Zk is the vector of SNP genotypes for n individuals at locus k for k = 1, …, m, where m is the number of markers and n is the number of individuals. SNP genotypes are represented as 1, 0, or -1 to denote a diploid genotype value of 11, 12, or 22, respectively. For BayesB, Z is a genotype matrix corresponding to γ. kth SNP effect, π) and absence (with probability π) of the marker k.GBLUP and BayesB: GEBVs were calculated based on the following equation:=1mZkZkT , and d=1y* is treated as the corrected phenotype and is described by the following multiple regression model:y* is the vector of the corrected phenotype. Z is the matrix of genotype codes for SNPs with -1, 0, and 1. γ is the vector of SNP effects. Zk is the vector of SNP genotypes for n individuals at locus k for k = 1, …, m, where m is the number of markers and n is the number of individuals. γk is the SNP effects for marker k. The ordinary least squares (OLS) estimates the parameter γ by minimizing the residual sum of squares:Elastic net method: The LASSO  penalty The RR  penalty The EN  penalty The lambda in Eqs , 5) and and5) anAll analyses were conducted using R language; GBLUP was obtained by solving the mixed model equations (MMEs). The BGLR package was used to conduct BayesB. RR, LASSO, and EN were conducted in the glmnet R package. For BayesB, the MCMC was run for 50,000 iterations with a burn in of 2,500 and thinning of 10. These numbers of iterations were sufficient in that increasing the number did not change the results. An MCMC sampler with shorter chain and burn-in  was tested, but the results were discarded. To identify the best performance of the EN method, we used penalty weights of α = 0.001, 0.01, 0.05, 0.1, 0.4, 0.7, and 1.y1 and y2 are vectors of trait 1 and 2, respectively; X1 and X2 are incidence matrices for fixed effects; b1 and b2 are the vctors of the fixed effects; Z1 and Z2 are incidence matrices relating the phenotypic observations to vectors of the polygenic effects for two traits; and e1 and e2 are random residuals for two traits.Genomic heritability, genetic correlation, and phenotypic correlation were analyzed using the ASReml v3.0 software package . An animFor GBLUP, the GEBVs of all genotyped individuals were predicted by solving the MMEs. For BayesB and EN methods, the GEBV was calculated by adding all the marker effects estimated from the training population. The formula was Two validation procedures were considered for assessing the predictive accuracy of different methods in this article. We mainly used 5-fold CV procedures. The other method is a generation validation procedure. Because phenotype records were not available for all genotyped animals for all traits, the number of animals in the training and test datasets differed among traits. However, the test populations were the same for each method. In 5-fold CV procedures, the population (for each trait) was randomly split into five parts of approximately equal size. Then, the analysis was performed using each subset of the data as the validation sample and the other 4 subsets as the training population. For EMA, the numbers of animals in training and test populations were 893 and 224, respectively. For the other traits, the sizes of the training and validation populations depended on the traits.y* and predicted GEBV on the test subgroup using the formula In generation validation procedures, the dataset was divided into training  and test  subgroups, which contained 1135 and 82 animals, respectively. Three traits were measured in this part, including ADG, CW and MS. The accuracy of the genomic predictions was measured as the Pearson correlation between tability .The estimates of heritability for ten traits are presented in Cv). The average accuracy of the GEBV for the ten traits was approximately 0.25, ranging from 0.18 to 0.30. Among the three methods, the lowest GEBV accuracy was found in BayesB for each trait. The prediction method yielding the highest accuracy differed among traits. GBLUP had the highest accuracy for traits EMA, CL, and MS, whereas EN had the highest accuracy for traits ADG, LW, BNW, SW, TW, and HLL. The comparable performances of the GBLUP, BayesB and EN methods suggested that the accuracy of the EN method (α = 0.001) was slightly better (0.1 to 2.5%) than that of the GBLUP and BayesB methods for six traits. GBLUP usually performed well, whereas the BayesB method performed poorly. The Cvs were similar among the methods, except for trait BNW, which was higher (+0.07) in the EN method than that of the other two methods. Additionally, we computed the accuracy of the GEBV using the RR-BLUP method, and the results were the same as those using GBLUP for each trait; therefore, the results are not shown in this study. We also fixed π = 0.99 and 0.999 in the BayesB method for the CW, ADG, EMA, and MS traits  and GBLUP outperformed BayesB. Compared with those of the GBLUP and BayesB methods, EN (α = 0.001) predictions had higher accuracy for most traits. LASSO had the lowest accuracy for most traits. In a previous study on genomic prediction, most cases showed that the accuracy of GBLUP outperformed that of the Bayesian method with real data, with the opposite trend for simulation data ,20.EN, LASSO and RR are penalized least-squares methods. We used the glmnet/R package to perform EN, LASSO and RR analyses because glmnet/R is extremely fast for use with our dataset. In glmnet/R, the optimal value of the tuning parameter λ, which controls the degree of shrinkage, can be obtained through CV. The λ value producing the minMSE is deemed the optimal tuning parameter in training sets. To avoid the overfitting problem in training sets, some scientists advise using minMSE+1SE to obtain the optimal λ . We testRR assumes that many predictors all have nonzero coefficients and that they obey a normal distribution . SpecifiThe GBLUP method, which assumes a normal distribution of marker effects with an equal variance and that all SNPs have a nonnull effect, performed better than the BayesB method for each trait. Many studies show that the GBLUP model performs well for most polygenic traits in livestock ,24. The In the comparison of the EN with BayesB and GBLUP, the EN model was superior to the BayesB and GBLUP for most traits. The computational speed of EN was between those of GBLUP and BayesB. EN is a machine learning method that uses training sets to learn parameters in models and then predict the test sets with these models. Unlike GBLUP, which only outperformed in traits controlled by large SNPs with small effects, EN performed well in two models  if a suitable alpha was set. Since the EN method provided higher prediction accuracy for six of ten traits and because of its adjustability, it was a good choice for further GS in Chinese Simmental beef cattle.GS in Chinese Simmental beef cattle has been evaluated since 2008, but to date, GS has been not implemented in practical breeding. The reference population has been continuously updated, although with a limited increase in number. Many factors hamper the development of GS in China. A comparison of our results from the CV procedure with those in other beef cattle  shows thIn contrast, when generation validation was used, the accuracies of three traits using three methods were slightly higher. This might be caused by the larger training dataset and smaller test dataset. Few reports are available about the traits TW, SW, BNW, CL, and HLL. However, clear correlations between traits were observed, and TW and SW are also of economic interest. These observations may indicate that the genetic architecture of these traits is different among breeds, which could cause the different characteristics in different breeds of cattle. We suggest that ADG, CW, EMA and MS traits are suitable for the infinitesimal model, as these traits have small effects distributed on many loci.Ne = 103) was consistent with that reported by [N ≈ 973). Another influence factor is CV strategy to assess the predictive ability of different methods. There are already a few studies that show that a stronger genetic relationship between the training and test datasets results in the highest accuracy for GEBV [The accuracy of genomic predictions is influenced by the density of the SNP panel, the heritability of the trait, the training population size and the effective population size . In thisorted by . The estfor GEBV ,29. The The performance of the statistical methods used depended on the trait analyzed. The results showed that GBLUP and EN were clearly superior to BayesB for each trait, particularly LW and CL. When an appropriate alpha was set in the EN model, the results were better than those of GBLUP for most traits, with the exception of traits EMA, CL and MS. EN and GBLUP methods could be equally recommended for the implementation of GS for carcass traits in Chinese Simmental cattle. In addition, the results also suggest that the genetic architecture underlying the ADG, CW, EMA and MS traits are similar and are controlled by many SNPs. Overall, our results can be used as a reference for implementing genomic prediction in Chinese Simmental beef cattle.S1 Fig(PNG)Click here for additional data file.S1 Table(DOCX)Click here for additional data file.


Here, we present a case where a rare combination of common SNPs in one individual results in an extremely high polygenic height score that is correlated with an extreme phenotype. While polygenic scores are not clinically significant in the average case, our findings suggest that for extreme phenotypes, polygenic scores may be more successful for the prediction of individuals.Polygenic scores (or genetic risk scores) quantify the aggregate of small effects from many common genetic loci that have been associated with a trait through genome-wide association. Polygenic scores were first used successfully in schizophrenia and have since been applied to multiple phenotypes including multiple sclerosis, rheumatoid arthritis, and height. Because human height is an easily-measured and complex polygenic trait, polygenic height scores provide exciting insights into the predictability of aggregate common variant effect on the phenotype. Shawn Bradley is an extremely tall former professional basketball player from Brigham Young University and the National Basketball Association (NBA), measuring 2.29 meters (7′6 Polygenic, or genetic risk, scores are aggregate measurements of the effects of multiple common genetic loci that are associated with a trait. First used in schizophrenia , they haMost phenotypes  are complex and polygenic, and our understanding of the underlying biology is limited because of high data dimensionality and small sample sizes. Approximately 80% of adult height variation has been attributed to genetic factors –10, and ″) tall , measuring 2.29 m (7′ 6″) tall  and has The Cache County Study on Memory Health and Aging was initiated in 1994  and conshttp://adni.loni.usc.edu). ADNI is a large collaboration from several academic and private institutions, and subjects have been recruited from over 50 sites across the US and Canada. Currently, over 1500 adults (ages 55 to 90) participate, consisting of cognitively normal older individuals, people with early or late MCI, and people with early stage Alzheimer's disease. For up-to-date information, see http://www.adni-info.org.Whole genome sequences (WGS) from 809 individuals  were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database  from the Cache County study. All samples were sequenced using the Illumina HiSeq technology at an average of 30x coverage. We sequenced Mr. Bradley's exome using the Ion Torrent and the Ion Ampliseq Exome Kit at an average coverage of 30x. Sequence data from all studies were mapped to the human reference genome, version GrCh37 with BWA (Burrows-Wheeler Aligner) . We furtSNP data from the Alzheimer's Disease Genetics Consortium (ADGC) were used to examine patterns of linkage disequilibrium. The ADGC consists of 32 studies collected over two phases that include 16,000 cases and 17,000 controls. All subjects were self-reported as being of European American ancestry. More information about this dataset can be found in the study of Naj et al.  and the r2 = 0.8). We then extracted as many of the remaining SNPs as possible from Mr. Bradley's data, the ADNI samples, and Cache County samples. We calculated an additive polygenic height score [The GIANT Consortium reported 22,539 genome-wide significant SNPs associated with human height. We extracted these SNPs from the ADGC data and identified unique tag SNPs within each LD block to (1) estimate the number of unique signals in the GIANT data and (2) prevent counting the same signal more than once. We identified tag SNPs using default settings in Haploview  for eachht score  for eachTo estimate the number of SNPs needed to elevate Mr. Bradley's height score to the highest in the distribution, we performed a random selection of SNPs (bootstrap) at various SNP-set sizes ranging from 100 to 2000 SNPs, recalculating Mr. Bradley's height score and rank each time. We performed 1 million replicates for each SNP-set size and measured the range (minimum and maximum), first and third quartiles (25th and 75th percentiles), and the median for each SNP-set size.We also explored the difference in height scores between the observed distribution of height scores amongst the 1020 individuals from ADNI and Cache County compared to the null distribution, assuming no evolutionary constraints. We simulated genotypes and height scores across the extracted common SNPs for 20 billion individuals. Specifically, for each SNP, we randomly chose one of three possible genotypes and calculated the simulated individual's height score.Understanding whether Mr. Bradley's height is attributed to an increased proportion of heterozygous or homozygous genotypes associated with increased height could shed additional light on whether the SNP effects are additive or nonadditive . We tested for a difference between Mr. Bradley's genotype distribution and the average ADNI and Cache County genotype distribution using a goodness-of-fit test. Alleles with a positive effect size are associated with increased height, while alleles with a negative effect size are associated with decreased height. A significant difference that could indicate the effects on height are nonadditive, though more data from extremely tall individuals would be necessary to provide definitive evidence.We also tested whether height scores were correlated with actual height in 407 individuals from the ADNI and Cache County datasets for each individual with both height and genetic data available. We tested for a correlation between the two using Pearson's product moment correlation coefficient, which is calculated using the R statistical package .We tested whether a simple polygenic height score, calculated using SNPs that were statistically associated with human height in the GIANT consortium data , could aWe calculated height scores weighted by effect size  was 1.28 mm lower than the observed height score mean (0.98). The maximum simulated height score (8.37) was 1.95 mm lower than Mr. Bradley's (10.32).p = 1.28 × 10−24). Mr. Bradley has an increased proportion of homozygous genotypes for alleles with a positive effect  [r2 should be stronger than the correlation coefficient in our findings because of our smaller population size of 407 individuals of the ADNI and Cache County individuals as well as the fact that the GIANT consortium identified the 697 SNPs used for prediction directly from their population of 1914 individuals.Using available height data from the ADNI and Cache County data, we tested whether the height scores calculated using the 2910 SNPs were correlated with the self-reported heights (at age 18) for the 407 individuals for which we have both height and genetic data. We failed to detect significant correlation between the two . This appears to be driven by an increased proportion of homozygous genotypes for SNPs associated with increased height when compared to the average ADNI and Cache County genotype values. Despite this, his height score only predicted him to be 10.32 mm taller than average. This suggests that while Mr. Bradley's extreme polygenic score could accurately rank his height amongst 1020 individuals, it does not accurately predict his actual height measurement, demonstrating that there are significant factors unaccounted for. Similarly, and as expected, this model was not able to accurately predict actual heights among the 407 ADNI and Cache County individuals for which we had both height and genetic data. These results as well as Mr. Bradley's predicted height  suggest that other factors such as environmental factors , nonaddiHeight is a complex trait that may serve as an effective phenotype model for other complex traits and diseases because it is a noninvasive and easily-measured phenotype to study. By developing new models and studies to better understand all genetic contributors to an individual's height, researchers will be able to apply the methods to other complex data.


Another set of simulations was run with parameters mdr = − 0.53. All models showed similar behavior for the first 20 years. Throughout the study, we observed a higher genetic gain in the direct than in the maternal effects and a smaller gain with a stronger negative covariance. In the long-term, however, only the infinitesimal model predicted sustainable linear genetic progress, while the finite locus models showed sublinear behavior and, after 100 years, only reached between 58% and 62% of the mean breeding values in the infinitesimal model. While the infinitesimal model suggested a reduction of genetic variance by 33% to 49% after 100 years, the finite locus models saw a more drastic loss of 76% to 92%. When designing sustainable breeding strategies, one should, therefore, not blindly trust the infinitesimal model as the predictions may be overly optimistic. Instead, the more conservative choice of the finite locus model should be favored.Stochastic simulation studies of animal breeding have mostly relied on either the infinitesimal genetic model or finite polygenic models. In this study, we investigated the long-term effects of the chosen model on honeybee breeding schemes. We implemented the infinitesimal model, as well as finite locus models, with 200 and 400 gene loci and simulated populations of 300 and 1000 colonies per year over the course of 100 years. The selection was of a directly and maternally influenced trait with maternal heritability of  A major concern in sustainable breeding and conservation programs is the preservation of genetic variance in the population –4. To esAlthough existing software, such as ADAM , can runEstimates of the long-term effects of animal breeding, especially the limits of selection possibilities, were introduced in , relatinSimulation studies have already examined the effects of selective breeding for multiple traits and found that breeding value estimation should consider the correlation between different traits , 29. TraSimulation studies that include indirect effects appear to be almost exclusively based on the infinitesimal model –39. It hStrategies used to address the problem of uncertain paternity in several breeding schemes were first developed in  and 43]43]. By iVarroa resistance, can be influenced maternally by the queen through her egg laying frequency or pheromone release, as well as directly by the workers . Maternal and direct effects in honeybees differ slightly from their respective counterparts in mammals in two aspects.The biology and current breeding schemes of honeybee show the aforementioned peculiarities to a heightened degree. They combine negatively correlated maternal (queen) and direct (worker) effects with an uncertain paternal descent and a strong need for the maintenance of genetic diversity, which is threatened by inbreeding depression , 48. ComThe direct effect is not attributed to an individual but to the collective of workers in a colony. It is, thus, often seen as their average effect .The maternal effect of a queen is not directed at the next generation of queens but at her worker group, an entity that is not directly involved in the selection process. Hence, selection for maternal effects does not suffer from the inaccessibility of information in the latest generation, which leads to reduced selection potential in maternally affected traits in other species .Nevertheless, the notion of maternal and direct effects in honeybees resembles corresponding concepts in other farm animals very closely and are regularly seen as their equivalents , 51, 52.Moreover, there is a specific current reason to investigate bee breeding. Because of a lack of selection, small honeybee races in Europe get increasingly replaced by two selected subspecies . New breAfter the establishment of a BLUP-based breeding value estimation for the honeybee , 58, a nHoneybee colonies were modelled as consisting of a single queen and her offspring (a group of non-reproducing worker bees). In addition to the diploid queens and workers, haploid drones were individually modelled. After being created, young queens immediately mated with 12 drones each. Afterward, the genetic information of the queen and the 12 drones was used to create a worker group and daughter queens. Drones received their genetic information from their respective dam queens alone.In accordance with common breeding practices  BQs were created without specified parents. These base population BQs were assumed to be unrelated. Similarly, in each of the first three years, four (resp. 10) pseudo sires were created without specified parents. Every pseudo sire comprised eight DPQs. No relationships among DPQs or between DPQs and BQs were assumed in the first three years. Beginning in the third year, two-year-old BQs were available, and their colonies had estimated breeding values. With the breeding value estimation following the theory developed in , queens mdr = −0.18, resp. mdr = −0.53. These relations are in broad accordance with the breeding parameters determined for several traits by various studies [The genetics of the bees were simulated according to three different models: (i) a finite locus model with 200 unlinked loci, (ii) a finite locus model with 400 unlinked loci, and (iii) the infinitesimal model. We will refer to the three models as FL200, FL400, and INF, respectively. For the simulations, we decided on a normally distributed trait with an initial direct (worker) additive genetic variance of  studies , 63–65. loped in . No domiWe refer to the matrix of initial genetic parameters asA fulfilled three purposes in the simulations.The matrix ΣIt defined the variance structure of breeding values in the base population.A was not updated for this purpose during the simulation, although the genetic parameters shifted due to drift and selection in the finite locus models)It described the assumed genetic covariances in the BLUP breeding value estimation. , and population sizes (300 BQs and 1000 BQs per year) by adding the genetic correlations as subscripts and the population sizes as superscripts to the genetic model , genetic correlations , respectively 400 (FL400), unlinked biallelic loci, each possessing a queen and a worker effect. In accordance with , we deciThe genetic setup for the base population was created as follows:a1 and a2 be the two possible alleles at a locus with frequencies p1 and p2, respectively. Let n be the total number of loci. Then the allele effects a1 and a2 were determined in the following steps:Let e following a distributionGenerate a random vector Obtain preliminary allele effectsA. In a concrete realization, however, the obtained additive genetic variance A. Therefore, after the preliminary allele effects were calculated for all loci, we applied a further post-correction step to obtain the allele effects and ensure that the additive genetic variance over all loci is ΣA.By this transformation, the expected value for the allele effect at each locus is zero, and the expected additive genetic variance generated by all loci is ΣE1 and E2 asCalculate W of a queen Q received as its breeding value the mean value of Q’s TBV and the average TBV of the drones that Q mated with, as shown in the following equation:The script for the creation of these possible alleles at the QTL was written in R  using thA, as explained in [B and a drone D to a daughter queen Q was realized asA. FB denotes the inbreeding coefficient of B. In vivo, exact estimations of relationships and inbreeding coefficients are difficult to obtain because the relatedness of two offspring of the same dam can vary between 0.25 and 0.75 [in silico, drones were simulated individually, so we could keep track of the exact relationships between siblings and, consequently, between all individuals. The calculation of relationships was performed as stated in [FB as described in [Q to a drone D was realized asA. The worker group W of a queen Q received its breeding value as in the finite locus models according to In the infinitesimal model (INF), all queens from the base population received their true breeding values as realizations of a normally distributed random vector with expectation ′ and variance Σained in . The inhained in  with necand 0.75 . Howevertated in . Thereforibed in  or [49, ribed in . The inhWhen all queens of a year were created, the BQs were assorted to the pseudo sires. BQs with the same dam were sent to the same mating station. The distribution of the sister groups to the pseudo sires was random. When a queen was assigned to a pseudo sire, 12 drones were produced, where the dam of each drone was chosen at random among the 8 DPQs of the mating station. The data of the drones  were henceforth stored together with the queen data.After the mating of the newly-born BQs, the one-year-old BQs were assigned a colony. The colony received true breeding values according to A as specified in Subsequently, a BLUP breeding value estimation was carried out using the BLUPF90 software  with theSimulations using the INF model were repeated 100 times to obtain reliable results. In the simulations with a finite number of loci, we created 10 sets of allele frequencies and allele effects for each of the finite locus settings. With each of these sets, simulations for both population sizes were repeated 100 times.A posteriori, we did an investigation on the repetitions needed for future simulations based on the variance of the outcomes within the respective repetitions. Following [s, and the tolerated error in the estimation for the mean value is e, then, with a minimum repetition number ofα that the calculated mean value differs from the true expectation by less than e. Here, z refers to the quantiles of the standard normal distribution.ollowing , we estiin vivo observations due to model inaccuracy are possible, we decided to allow for an error rate of up to 10%. We further investigated how many repetitions are needed to fall below this error rate with a confidence level of 99%.As the outcomes of the simulations depend on the chosen model and deviations between simulation results, and −0.18) and 1.17 (INF−0.53). In the long-term, model INF always predicted the highest genetic gain, followed by FL400 and FL200, in this order. The ratio between genetic gain in the PC under model INF and model FL200 ranged between 1.62 , and 1.74  after 100 years. After 20 years, this ratio only ranged between 1.013  and 1.038 . Genetic gain in model INF showed an almost linear increase, whereas models FL200 and FL400 showed clearly sublinear behavior after 20 to 25 years  and 6.58  after 100 years. After 20 years, this ratio only ranged between 1.07  and 1.13 .The development of the mean inbreeding coefficients of the breeding queens depended mainly on the population size. After 100 years, the average inbreeding coefficient in the small populations was between 0.37 Click here for additional data file.S2 AppendixSmall-scale study on the effect of linkage on the results of the simulations.(PDF)Click here for additional data file.S3 AppendixSmall-scale study on the selection response to direct and maternal genetic effects dependent on the ratio of additive genetic variances.(PDF)Click here for additional data file.


Z score statistics are categorized as the response variables. I analytically show that the regional association analysis methods developed within the framework of the classical multiple linear regression model with additive effects of genetic variants can be reformulated in terms of the new model without the loss of information. The results obtained from the regional association analysis utilizing the classical model and those derived using the proposed model are identical when SNP-by-SNP correlations and SNP-level statistics are estimated from the same genetic data.Here I propose a fundamentally new flexible model to reveal the association between a trait and a set of genetic variants in a genomic region/gene. This model was developed for the situation when original individual-level phenotype and genotype data are not available, but the researcher possesses the results of statistical analyses conducted on these data . The new model was analytically derived from the classical multiple linear regression model applied for the region-based association analysis of individual-level phenotype and genotype data by using the linear compression of data, where the SNP-by-SNP correlations are among the explanatory variables, and the summary  Unfortunately, the polymorphism of the loci identified to date can explain only a small fraction of the genetic variability of traits. This situation is typical of almost all complex traits2 and therefore the ‘missing heritability problem’ formulated several years ago5 remains the central issue of genetics.Over the past ten years, significant progress has been made in understanding human genetic variation and developing DNA reading technologies. Genome-wide association studies (GWAS) have emerged as a powerful tool for investigating the genetic architecture of complex traits. With the GWAS approach, a number of loci involved in the control of various complex traits, including diseases, have been identified. It turned out that most of the genetic variants associated with complex traits are located in noncoding regions of the genome, and their potential effects are associated with changes in the regulatory functions of the genome8. It is likely that these variants are located in the protein-coding regions and change the structure and function of the corresponding proteins. Recently, thanks to two scientific advances, it has become possible to identify rare genetic variants. One is large-scale exome sequencing, which allowed imputing a large number of missing genotypes using the reference data from ‘HapMap’9, ‘1000 Genomes’10, ‘HRC’11 and other projects. The other is а series of powerful statistical methods developed for regional association analysis (RAA) (for example16). The general principle of these methods is to simultaneously analyze all the rare genetic variants in a gene or a genome region (e.g. these belonging to a certain metabolic pathway). The family methods for RAA address a number of problems related to the low frequency of individual variants, multiple testing and interpretation of the results obtained, and increase the power of analysis17.One approach to finding missing heritability is by identifying rare genetic variants. Rare genetic variants with relatively large and therefore potentially recognizable effects are expected to contribute to almost all complex traits20, principal component analysis methods21 and functional data analysis methods26. The random-effects models are based on the principle of decomposition of a total trait variance into components. These methods compare the genotypic and phenotypic similarities of individuals, and the parameter of interest is the component of trait’s variance that is explained by the genomic region. Under random-effects models, the kernel smoothing technique has proved to be successful30. This technique is based on the formation of a kernel matrix using the kernel smoothing density functions to measure the genetic similarity between individuals in the region. Methods exist that combine collapsing and variance components approaches31. In these methods, the parameters of interest are both the variance and the mean of the effect sizes. All the above RAA methods work with genotypic and phenotypic data measured for each individual .For analysis of regional associations, several model-based methods that use different regression models of trait inheritance have been developed. The main difference between these models is the assumption concerning the type of genotypic effects (fixed or random effects). In fixed-effects models, the immediate parameters of interest are genetic effect sizes. Models as these are used in the classical method of complete multiple linear regression analysis and its variants, when the genetic effect sizes or their specific linear combinations are estimated. Examples are as follows: collapsing methods32. The Global Lipids Genetics Consortium genome-wide association studies33 and Genetics of Personality Consortium34). This makes it possible to increase the number of individuals involved in the analysis by up to several hundreds of thousands . Many effective approaches have been reported for gene-level meta-analysis of rare variants. The most popular ones are those based on burden19, SKAT and SKAT-O37 and also FLM tests40. It has been shown that meta-analysis significantly increases the power of GWAS42.To improve the accuracy of the analysis of specific traits, the results obtained from different samples can be combined using meta-analysis methods applied by various consortiums . They have low power because the size and direction of the effect of each genetic variant are not taken into account. More powerful RAA methods using summary statistics have been developed that reproduce complete multiple linear regression (MLR) method41, collapsing (Burden) and variance components methods (SKAT and SKAT-O)36. For these methods, region-based statistics have been reformulated to use SNP-by-SNP correlations and SNP-level Z score statistics as input data instead original genotype and phenotype data.Recently, due to an increased emphasis on reproducibility and data sharing promoted by some journals and funding agencies, the SNP-level summary statistics obtained from GWAS have become increasingly available. Meta-analysis results, which are freely accessible, are usually presented with the size and significance of the effect of each genetic variant. Several methods have been developed to perform the RAA using summary statistics. These methods manipulate the p-values calculated for genetic variants within the region  and functional linear models (FLM) can be reformatted for summary statistics.However, for the inheritance of traits, regression models using summary statistics have not been developed, although such models can allow us to define dependencies between SNP-by-SNP correlations and SNP-level Here I derive a common generalized model for combining dependent SNP-level summary statistics to perform a region-based association analysis between a single trait and a set of genetic variants of a genomic region. As data, the model utilises not only the summary SNP-level GWAS results for the trait of interest but also the SNP-by-SNP correlations estimated from the original genotype data or reference sample data. I analytically show that RAA methods developed for the classical linear regression model with additive effects of genetic variants can be reformulated within the new model without loss of information. The results  of regional association analysis obtained using individual-level data and the proposed models are completely identical, when correlations between genetic variants and summary statistics are calculated from the same genetic data. Moreover, I analytically show that the proposed model can be extended to combine summary statistics obtained for other objects (genomic regions or traits) but using the same non-object-related data.n unrelated individuals. For each individual, the phenotype and genotypes of m genetic variants (SNPs) in a genomic region are measured.For simplicity, suppose I have a sample of y is the (n × 1) known vector of continuous trait values; G is the (n × m) known matrix of SNP genotypes; en is the (n × 1) vector of n units; μ is the scalar intercept; W is the (m × m) diagonal matrix of weights assigned to SNPs (see Box 1); С is the (m × k) method-dependent matrix operator of the linear transformation of the weighted genotypes (see Box 2); ξn is the (n × 1) vector of random regression residuals, ξn is caused by an environmental factor and is supposed to be multi-normally distributed with a zero mean vector and the covariance matrix In is the identity matrix of order n; finally, β is the (k × 1) vector of regression coefficients measuring the effects of the WС-transformed genotypes on the trait.Consider the classical multiple linear regression model with additive effects, where the genotypes of the genetic variants are explanatory variables, and the phenotype is a response variable. This model can be written in a generalized form for the most popular model-based RAA methods asy in Model (1) follows a multi-normal distribution with a mean vector E(y) and a covariance matrix Cov(y) determined in accordance with the type of genotype effects (fixed or random effects). For the fixed-effects (FE) models, β is the vector of parameters of interest, concerning which the null and alternative hypotheses are formulated (H0: β = 0 against H1: β ≠ 0). For the random-effects (RE) models, E(y) = enμ and τ2 is the parameter of interest known as the trait variance component explained by the genomic region (H0: τ2 = 0 against H1: τ2 > 0).It is standardly assumed that the trait c threshold:w is the weight function and MAFi is the minor allelic frequency of the ith genetic variant.There are several weighting techniques. Since it is considered that rarer SNPs are more likely as causal variants with larger effect sizes, the SNP weights can be assigned inversely to their minor allelic frequency. One of such weighting techniques is simple thresholding, which filters genetic variants using the 30:B is the beta function with the pre-specified parameters a and b.Later, a continuous weighting technique was developed for kernel methods and it has become successful not only for the RE-model-based methods, but also for the FE-model-based methods. To calculate weights, the flexible beta density function has been proposed37. Using tools  the user can attempt to make a computational prediction of the functional impact of genetic variants and assign weights as the a priori probabilities of the functionality of the genetic variants.There is one more weighting technique using biologically functional information about genetic variantsC can be defined in various ways depending on the RAA method used. In general, C has some constraints. For the FE-model, to avoid over-parameterization and multicollinearity, C must be a full rank matrix with a limited size (m × k) under condition that n ≥ m ≥ k. Moreover, to improve the model’s performance, it is advisable that the columns of C be orthogonal vectors (i.e. СTC = Ik). For the RE-model, C must be such that СTC is able to be interpreted as a correlation matrix for effects β.The linearly transforming operator C is an (m × 1) vector of units. It provides summation across the (weighted) genotypes of all the genetic variants into one vector. In FLM-based methods, C is an (m × k) matrix of values of k pre-specified basis functions  at m relative SNP positions and serves for functional (continuous) smoothing of the weighted genotypes. In PCA-based methods, C is given as an (m × k) incomplete (truncated) matrix of eigenvectors obtained from the spectral decomposition of the covariance matrix of the weighted genotypes. C serves to control the number of first principal components (k) involved in the analysis to cover 80–90% of total variance observed in the genomic region. In SKAT with a linear kernel, C is an identity matrix (k = m), and in SKAT-O with a linear kernel, C is a square root matrix of the matrix of correlations between genetic effects. It was introduced by Lee et al.37 as ρ is a pairwise correlation among the genetic effect coefficients, em is the vector of units and Im is the identity matrix of the m-th order.Here I consider several examples for the most popular RAA methods. In the Burden method, Z score statistics are calculated via the standardized  y and G. Centering can be achieved through pre-multiplying all the terms of Model (1) by an (n × n) projection matrix, S with diagonal elements σgi is the genotypic standard deviation of the ith variant (see Box 3). In this way, I obtain a new regression equationy, G and ξn in Model (1).For further notational convenience, I rewrite Model (1) into the standardized data format in accordance with H0, since GWAS SNP-level W, С, β and, as will be shown below, on the region-based test statistic, since centering the data is reflected only in the intercept μ (μ = 0), and scaling the data leads to only the formation of the matrix S.Note that the reformatting of Model (1) has no effect on β, and β.According to Model (2), the parameters of the distribution of S represents an (m × m) matrix obtained by diagonalization of the vector of genotypic standard deviations divided by the standard deviation of the trait. The matrix S is caused by the scaling of phenotypes and genotypes and allows the regression coefficients to remain the same as in Model (1). Diagonal elements of S can be expressed via GWAS SNP-level beta standard errors, se (βGWAS):i indicates the ith variant.The matrix In the FE-model-based methods, tests based on the F distribution test statistics are often used:n), the maximum number of independent columns of the predictor matrix R2) calculated asβ maximizing R2 is Statistic (3) depends on the sample size (β into (4) gives:The substitution of the estimate of k ≤ m ≤ n must be fulfilled, and the matrix product It followed that to avoid model over-parameterization and matrix non-invertibility problems in the FE-model, the condition Q, are commonly applied:K is an (n × n) linear kernel matrix determined as K expresses the between-individual genetic similarity caused by the genomic region. Under the null hypothesis of no association, the distribution of Q is approximated by the weighted sum of K denoted by eigen45. Using the spectral decomposition property that the nonzero eigenvalues of XTX are the same as the nonzero eigenvalues of XXT, I have:In the RE-model-based methods, tests based on a score test statistic, Thus, Model (2) is a linear regression model generalized for the popular RAA methods Table  that assZ score statistics and correlations between the genetic variants.In this section, I am developing a new model for regional association analysis, where SNP-level summary statistics and SNP-by-SNP correlations are used as input data. The new model follows from Model (2), for which the linear compression of data is applied. As a result of this compression, the individual-level phenotypic and genotypic data are converted to the corresponding summary m × n) compression matrix introduced here as I pre-multiply all the terms of Model (2) by an  SNP-by-SNP correlation matrix, therefore, N, where Two key points should be noted. First, the rank of the predictor matrix in Model (8) is the same as in Model (2), which means that the use of matrix Thus, I obtain a new linear regression model from Model (8):z is an (m × 1) vector of summary Z scores calculated in GWAS as Box 3). In fact, the matrices W serve for weighting the SNP genotypes to control their impact on a trait of interest , while the matrix C is method-dependent. C is set by the researcher in accordance with the selected gene-based method and serves for linear smoothing/compressing the SNP genotypes.Here m × k) matrix that provides weighting and smoothing/compressing of the genetic data.For better readability, Exp. (9) can be rewritten asz follows approximately a multivariate normal distribution as N, which was also shown in a work by Pasaniuc et al.47 and under the alternative hypothesis, z is distributed as Thus, within the framework of Model (10), under the null hypothesis of no association, the vector Like Model (2), Model (10) is based on the same standard assumptions as any linear regression with additive effects, namely: linearity and additivity of the relationship between dependent and independent variables, homoscedasticity and normality of the distribution of the regression residuals.F test statistic depends on the same parameters that describe the F test statistic (3) obtained from the original individual-level data: the sample size (n), the maximum number of independent columns of the predictor matrix R2 reformulated from (5) asFor RAA methods based on Model (10) with fixed effects, the It is obvious that by analogy with Q is calculated asKs is an (m × m) linear kernel matrix determined as Ks expresses genetic similarity between Q from (12) is approximated by a weighted sum of Ks:In RAA methods based on Model (10) with random effects, Q (12) and (6) formulated in dissimilar terms are identical. Obviously, the new model (10) appears as a generalized model for the RAA model-based methods (Table Note that the region-based test statistics Q 2 and (6)So, Model (10) using summary statistics is based on the same standard assumptions as any linear regression with additive effects. However, when introducing Model (2), I also assumed that the sample consists of unrelated individuals, and the trait analysed is continuous. These restricting assumptions were made only for the sake of simplicity. They can be dropped within the framework of the new model, since for combining the computed Z scores, it does not matter what type of the trait is analyzed and what the structure of the sample is. This information is already taken into account when estimating the Z scores. For Model (10), it is primarily important that each Z score follows asymptotically a standard normal distribution, and correlations between these Z scores were known under H0.48). Since the estimates β and R2 in the new model are equivalent to those in the classical model using individual-level data, the regularization term,RSS) is the same in both models, because RSS=1 − nR2. Then the objective function to be minimized with respect to β is:λ is the tuning parameter. Without regularizing, the beta-coefficients are estimated as To be able to exclude predictors from the model by shrinking their beta coefficients to zero, Model (10) can be expanded by using the lasso procedure, which is usually included for such purposes in the regression analysis  are, first, the Z score statistics measuring the associations of the same single trait with various SNPs, and, secondly, the correlations between the genotypes of these SNPs. GWAS SNP-level beta standard errors are optional input data for Model (10) because they are only required for the SNP weighting procedure.Thus, Model (10) is intended to combine a set of correlated SNP-level 47, i.e. when zi is the Z score calculated as ith SNP with the standardized genotype z denotes the vector of Z scores calculated at genetic objects, each of which is a genomic region (including a SNP and a gene), and U is the correlation matrix between these Z scores; the remaining variables are the same as in (10). In Model (16), the matrix of weights was removed, since it is unlikely that the researcher can formulate a hypothesis on the contributions of region-level Z score statistics to their combination. Such an interpretation (16) allows one to aggregate the region-level statistics, which were calculated for the same phenotype data (a trait or a set of traits), using the same RAA method, and then transformed into Z score statistics.It has been shown that the SNP-by-SNP correlations are asymptotically equal to the correlations between the Z scores at these SNPs under null dataZ scores calculated for these traits on the same genotype data49, namelyZ score statistics. One confirmation of this is the score-based method, metaUSAT50, developed to find an association between a single genetic variant with multiple traits when using summary statistics. This method can be rewritten in terms of the new model with random effects. Then the C matrix in Exp. (16) can be formed by analogy with the SKAT-O method (Box 2) as ρ is a pairwise correlation among the genetic effect coefficients.Moreover, since under null hypothesis the phenotypic correlations between traits are asymptotically equal to the correlations between Note that to determine the association between a gene and a single trait, metaUSAT can be applied using the new model with non-unit weights for genetic variants. Obviously, the new regression model (16) includes a submodel that underlies the metaUSAT method.Z scores. Thus, Model (16) is universal because it is suitable for addressing tasks requiring that Z score statistics of any level be combined.Finally, I claim that Model (16) is suitable for aggregating test statistics calculated by various methods  provided that the same original individual-level genotype and phenotype data were used and then, as usual, transformed into Z score statistics serves as a response variable.By using the technique that allows the linear compression of phenotype and genotype data to the level of statistics  without the loss of statistical power, I have derived a new model for finding associations between traits and genomic regions. This model represents a linear regression model of fixed or random effects, where the vector-columns of the SNP-by-SNP correlation matrix serves as explanatory variables, and the vector of the SNP-level summary Table The methods built on the new model have a number of important advantages. First, these methods do not require access to individual-level phenotypes and genotypes, and use the SNP-level summary statistics obtained from the GWAS meta-analysis and deposited in public databases as input data. Secondly, these methods are fast, because most of the calculations have already been performed. Finally, these methods are universal, because they can combine dependent Z scores without requiring information on the population structure of the sample or the trait type. This information has been taken into account when calculating the summary Z scores.In this work, I considered only the RAA methods that suggest an additive allele action in a genetic variant. For these methods, only the values of the SNP-level summary statistics and the correlations between genetic variants are required. Other methods that assume the dominant allele action can also be implemented under the new model using summary statistics. However, for such methods, higher-order SNP-by-SNP correlation matrices should be calculated. Noteworthy, the new model can serve as a basis for the development of new methods for conducting regional association analyses using SNP-level summary statistics and SNP-by-SNP correlations instead of individual measurements. Moreover, as has been shown in this work, the proposed model can be extended to combine Z score statistics derived from the test statistics with any level. In other words, the latter can be calculated for various genomic regions using the same individual phenotype data, for various traits using the same individual genotype data or by various methods using the same individual phenotype and genotype data.


There is considerable interest in estimating the causal effect of a range of maternal environmental exposures on offspring health-related outcomes. Previous attempts to do this using Mendelian randomization methodologies have been hampered by the paucity of epidemiological cohorts with large numbers of genotyped mother–offspring pairs.We describe a new statistical model that we have created which can be used to estimate the effect of maternal genotypes on offspring outcomes conditional on offspring genotype, using both individual-level and summary-results data, even when the extent of sample overlap is unknown.in utero exposures related to fetal growth restriction on future risk of disease in offspring. We illustrate our framework using examples related to offspring birthweight and cardiometabolic disease, although the general principles we espouse are relevant for many other offspring phenotypes.We describe how the estimates obtained from our method can subsequently be used in large-scale two-sample Mendelian randomization studies to investigate the causal effect of maternal environmental exposures on offspring outcomes. This includes studies that aim to assess the causal effect of We advocate for the establishment of large-scale international genetics consortia that are focused on the identification of maternal genetic effects and committed to the public sharing of genome-wide summary-results data from such efforts. This information will facilitate the application of powerful two-sample Mendelian randomization studies of maternal exposures and offspring outcomes. Statistical methods exist for partitioning genetic effects at single loci into maternal and offspring genetic components.Estimates of maternal genetic effects obtained from these methods can be used in large-scale two-sample Mendelian randomization studies to investigate the causal effect of maternal environmental exposures on offspring outcomes.in utero exposures related to fetal growth on future risk of disease in offspring. Two-sample Mendelian randomization studies can also be used to assess the causal effect of ,There is considerable interest in elucidating the causal effect of maternal environmental exposures on offspring outcomes. However, traditional observational epidemiological studies are prone to confounding, bias and reverse causality. Mendelian randomization (MR) is an epidemiological method that was developed to estimate the causal effect of environmental exposures on medically relevant outcomes.in utero exposures related to fetal growth on offspring outcomes using this method. We illustrate our methods using an example involving susceptibility to type 2 diabetes and offspring birthweight. Our methods have broad applicability to investigating hypotheses involving the Developmental Origins of Health and Disease (DOHaD).In this manuscript, we describe a statistical method based on structural equation modelling that we have recently developed to partition genetic effects on offspring phenotypes into maternal and offspring components.,,iY and ioSNP refer to the (offspring) phenotype and SNP genotype of the ith individual, imSNP is the genotype of the individual’s mother at the same locus, oβ and mβ the estimated offspring and maternal effects of the SNP, α is an intercept and iε a normally distributed residual term. However, a problem with this simple approach is that there is a paucity of cohorts worldwide with genome-wide association study (GWAS) data on both mothers and their offspring. Additionally, loci influencing complex traits in the offspring are typically of small effect and, since maternal and offspring genotypes are highly correlated, power is often low to definitively partition genetic effects into maternal and offspring components. For example, in a recent study of offspring birthweight, we initially attempted to partition the genetic effect at 58 autosomal genome-wide significant birthweight SNPs into maternal and offspring components using simple conditional linear regression in some of the largest genotyped cohorts of mother–offspring pairs that were available at the time . However, we were unable to resolve these effects definitively at nearly all of the birthweight loci.The resolution of genetic effects into maternal and offspring components is important because it not only provides meaningful insights into the mechanism through which genotypes exert their effects on offspring phenotypes, but it can also be leveraged for a number of informative downstream analyses including MR.,O) and their own genotype (SNP). The latent unobserved variables (in circles) represent the genotypes of the individual’s mother  and the genotype of the individual’s first-born offspring (GO). The total variance of the latent genotypes for the individual’s mother (GG) and offspring (GO) and for the observed SNP variable are all set to Φ , which is estimated from the data. According to quantitative genetics theory, the causal path between the individual’s own genotype and both their mother and offspring’s latent genotype is set to 0.5. The mβ and oβ path coefficients refer to maternal and offspring genetic effects on birthweight, respectively, and are equivalent to the coefficients estimated using the conditional linear model in ɛ and Oɛ, respectively, and the variance of both these terms is estimated in the SEM. The covariance between the error terms is denoted by ρ. Models can be fit by maximum likelihood using the OpenMx software package.IJE online, of this manuscript.Recently, we developed a new statistical model based on the mathematical technique of structural equation modelling, which has allowed us to estimate maternal and offspring effects on birthweight utilizing data from the large UK Biobank Study.Our approach is flexible in that the results from ‘singletons’ can be incorporated into the analysis  and the results from the SEM can be meta-analysed with results from conditional genetic association analyses of genotyped mother–offspring pairs. Importantly, our method can incorporate both individual-level genotype data (e.g. from the UK Biobank Study) and summary-level GWAS results data from other research groups and publicly available websites. The result is an extremely large, powerful, combined dataset for the estimation of maternal and offspring effects on birthweight. Additionally, when the model is applied across the genome (e.g. for locus-detection purposes), it can increase the power to detect variants that have opposing maternal and offspring effects on birthweight that may not otherwise be apparent.http://evansgroup.di.uq.edu.au/MGPC/) indicate that, whilst our approach is less powerful than similar-sized samples of genotyped mother–offspring pairs, the two study designs have similar power when the residual correlation between maternal and offspring phenotypes (ρ in N = 85 518), their own birthweight only (N = 178 980) or their offspring’s birthweight only (N = 93 842). In the third design, we combine the previous two data sources and conduct the SEM described in We have recently described the statistical properties of our approach, demonstrated that it yields unbiased estimates of maternal and offspring genetic effects on birthweight, gives similar answers to conditional genetic association analyses of genotyped mother–offspring pairs and has low sensitivity to random measurement error.pes ρ in  approachy-axis) and offspring (x-axis) genetic effects on birthweight at each of these loci. MTNR1B, EBF1, ACTL9) and at least six loci that exhibit maternal and offspring effects in opposite directions. Interestingly, the loci that manifest opposing effects through maternal and offspring genomes include the type 2 diabetes-associated loci HHEX-IDE, CDKAL1 and ADCY5. Our results are consistent not only with smaller and less powerful conditional regressions in genotyped mother–offspring pairs,,GCK gene.To further illustrate the utility of our method, we applied our SEM to 58 known autosomal SNPs for birthweight using data from the interim release of the UK Biobank Study.Whilst we have discussed our method within the context of offspring birthweight, there is no reason why the same or similar methods could not be applied to understand the genetic basis of other offspring phenotypes. Many offspring traits are thought to be influenced by maternal phenotypes,In order to illustrate this potential, Once genetic effects on offspring phenotypes have been partitioned into maternal and offspring components, the resulting estimates can be utilized in a variety of informative downstream analyses. In this section, we argue that the availability of maternal effect estimates on offspring phenotypes allows us to estimate the causal effect of maternal environmental exposures on offspring outcomes via two-sample MRin utero, but the method is also relevant for evaluating post-natal maternal influences on offspring phenotypes). Mendel’s Law of Segregation ensures that genetic variants segregate randomly and independently of environmental factors, whilst Mendel’s Law of Independent Assortment suggests that the genetic variants should also segregate independently of other traits, provided certain conditions are met. This means that genetic variants are less susceptible to confounding than the ‘traditional’ variables used in observational epidemiological studies.MR is an epidemiological method that uses genetic variants robustly associated with a modifiable environmental exposure of interest to estimate the causal relationship between the exposure and a trait or disease of interest.,,Although MR is a useful method for assessing causality, there are a number of assumptions and potential complications that should be borne in mind that we have discussed at length elsewhere.,et al.MR has recently been used to examine the causal effect of maternal environmental exposures on offspring outcomes. Examples include the effect of maternal cardiometabolic phenotypes,One potential way to increase the power of MR analyses involving maternal exposures and offspring outcomes is to utilize two-sample MR in potentially extremely large samples of individuals.,The two-sample MR procedure we advocate relies on the assumption that the variants associated with the maternal exposure of interest are also associated with these same traits during the critical period of interest (e.g. during pregnancy) and also that it is this time point that is relevant in terms of influencing the offspring phenotype. The same assumptions are of course also made in one-sample MR studies using genotyped mother–offspring pairs—the critical difference being that, in the latter case, it may be possible to investigate these assumptions empirically.,In all the MR analyses we describe above, we assume that maternal genotypes act additively on the maternal exposure of interest; the absence of parent of origin effects (e.g. from genetic imprinting); genetic non-additivity (including genetic dominance and epistasis); no gene–gene interaction between maternal, paternal and/or offspring genotypes; and the absence of paternal genetic effects affecting the offspring outcome at the same loci used in the maternal–offspring analyses. Recent work from the deCODE group and othersFinally, we note that an alternative approach consisting of fitting a multivariate model that includes maternal genotype, offspring genotype, maternal exposure and offspring outcome  and (independently) increased future risk of disease through developmental changes.in utero environment is difficult to measure and investigators may settle with infant birthweight as a proxy of this. There are several problems with this methodology, including the fact that offspring birthweight is an imperfect measure of growth in utero as well as all the usual limitations of traditional observational epidemiological studies . We believe that SNPs in mothers that are robustly associated with offspring birthweight (i.e. after conditioning on offspring genotype) exert their effects through the in utero environment and consequently that these variants could be used to proxy intrauterine growth restriction. Therefore, it should be possible to use the principles of MR to investigate the causal effect of in utero exposures influencing fetal growth on a variety of offspring phenotypes. Specifically, we propose that the existence of birthweight-associated SNPs in the mother that also exert maternal genetic effects on other offspring outcomes is highly suggestive of mechanisms consistent with DOHaD.One exposure that is of interest to the epidemiological community is fetal growth and the in utero exposures influencing fetal growth and other offspring outcomes). In the different panels of this figure, we consider the relationship between SNPs in the mother that are associated with offspring birthweight  and cardiometabolic disease in the offspring. The black ‘X’ in the diagram represents the effect of conditioning the association analysis on either the offspring or maternal genotype and therefore blocking the path between the conditioned genotype and the other variables of interest.To illustrate why this is the case, in utero environment lead to reduced fetal growth and subsequently lower offspring birthweight and developmental compensations that result in increased risk of offspring cardiometabolic disease in later life. Under this scenario, SNPs in the maternal genome that exert maternal effects to reduce offspring birthweight will also be positively correlated with offspring cardiometabolic risk (after correcting for offspring genotype). In panel (B), lower offspring birthweight is causal for increased risk of cardiometabolic disease. Under this scenario, SNPs in the maternal genome that exert maternal effects to reduce offspring birthweight will also be positively associated with offspring cardiometabolic disease risk (after correcting for offspring genotype). We stress that, although this model is broadly consistent with DOHaD, most advocates of DOHaD would not believe that birthweight is directly causal for future cardiometabolic risk, but rather a marker of an adverse in utero environment, as in panel (A). Nevertheless, we believe it is important to consider this possibility, particularly as naïve MR analyses making similar assumptions have begun to appear in the literature.In panel (A), and consistently with DOHaD, SNPs in mothers that produce an adverse In panel (C), the correlation between SNPs in the mother and offspring cardiometabolic risk is driven by genetic pleiotropy in the offspring genome. Under this model, SNPs that exert maternal effects on offspring birthweight will not be correlated with offspring cardiometabolic risk after conditioning on offspring genotype. This model encompasses the possibility of cardiometabolic disease-associated SNPs in mothers directly affecting offspring birthweight and then being transmitted to their offspring, where they increase the risk of cardiometabolic disease in later life.in utero environment. Any postnatal environmental effects of these variants, if they exist, on offspring phenotypes are likely to be small in comparison to their in utero effects. We note that the availability of mature genotyped father–offspring pairs would provide a test of this assumption, since we would expect that paternal genotypes will not be associated with offspring cardiometabolic risk in the absence of postnatal environmental effects.Finally, in panel (D), SNPs that exert maternal effects on offspring birthweight also pleiotropically influence offspring cardiometabolic risk through the postnatal environment. Under this model, SNPs with maternal effects on birthweight will be associated with offspring cardiometabolic risk after conditioning on offspring genotype. In general, however, we think this last model is unlikely, since the primary effect of these variants is likely to be on birthweight though the Thus, the different models illustrated in In this section, we have used birthweight SNPs to proxy intrauterine growth restriction and have consequently made the assumption that variation across the distribution of birthweight is informative for inference regarding the validity of DOHaD. Whilst utilizing birthweight-associated SNPs represents a useful starting point for investigations, some DOHaD literature concerns the effect of more extreme maternal exposures (e.g. famine) on fetal growth restriction and subsequent adaptation.N = 234 154 individuals reporting their own birthweight and N = 210 423 mothers reporting their offspring’s birthweight). Full details on the sample, genotyping and phenotypes are included in the IJE online. A total of 403 SNPs identified in a recent GWAS meta-analysis of type 2 diabetesIJE online). We fit the SEM described in –5), so the SEM was unable to converge using this SNP and it was therefore excluded from subsequent MR analyses. Using the effect sizes for type 2 diabetes reported in Mahajan et al.,In order to illustrate the framework espoused in this manuscript, we examined the relationship between maternal and offspring susceptibility to type 2 diabetes and offspring birthweight in the UK Biobank studyThe results of our analyses are displayed in IJE online.Importantly, causal estimates obtained using the SEM-adjusted estimates of the maternal and offspring genetic effects were far greater in magnitude than causal estimates obtained using the uncorrected estimates from simple linear regression. Likewise, the strength of evidence against the null hypothesis of no association was far greater for the SEM-derived estimates. Our results suggest that failing to partition genetic effects into maternal and offspring genetic components has the potential to bias the results of MR analyses and lead to misleading interpretations of the data. R code illustrating the SEM and two-sample MR analyses we have performed are included in the ,We envisage that the major challenge in implementing the ideas outlined in this manuscript will be practical rather than technical. With some notable exceptions,,A logical place to start might be the establishment of working groups focused on perinatal phenotypes, where maternal genetic effects on offspring phenotypes are expected to be strongest. We have already begun to do this in the case of offspring birthweight within the EGG consortium, where maternal genotypes and offspring phenotypes in the large-scale UK Biobank Study and many other population-based birth cohorts of appreciable size are available.in utero exposures influencing fetal growth restriction on future risk of disease in offspring. The establishment of large-scale international genetics consortia geared to identifying maternal genetic effects will assist in facilitating these sorts of studies.Statistical methods now exist for estimating maternal genetic effects on offspring phenotypes. The estimates obtained in these studies can subsequently be used in large-scale two-sample MR studies to investigate the causal effect of maternal environmental exposures on offspring outcomes. This includes studies that aim to assess the causal effect of D.M.E. is supported by an NHMRC Senior Research Fellowship (GNT1137714) and this work was supported by project grants . G.H.M. has a PhD grant from the South-Eastern Norway Regional Health Authority (Grant number 2015008). D.A.L.’s contribution to this piece is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement , the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 733206 (LifeCycle) and the United States National Institutes of Health (NIH): National Institute of Diabetes and Digestive and Kidney Diseases (R01 DK10324). D.M.E. and D.A.L. work in a UK Medical Research Council (MRC) Unit that is supported by the University of Bristol and MRC (MC_UU_00011/6) and D.A.L. is a National Institute of Health Research Senior Investigator (NF-SI-0611–10196). N.M.W. is supported by an Australian National Health and Medical Research Council Early Career Fellowship (GNT1104818). This research has been conducted using the UK Biobank Resource (Reference 12703). Access to the UKBB study data was funded by a University of Queensland Early Career Researcher Grant (2014002959). No funders influenced the study design, data collection or interpretation of results. The views expressed in this paper are those of the authors.dyz019-Supplementary_DataClick here for additional data file.