Apis mellifera), whose complex genetic and reproductive characteristics are a challenge to model statistically. The aim of this project was to determine the genetic parameters of several traits important for Canadian beekeepers with a view to establishing a breeding program in a northern context. Our results show that the five traits studied  are all heritable. Furthermore, the genetic correlations between these traits are all positive or null, which means that these traits can be selected simultaneously in a honey bees genetic selection program. Our results are instrumental to the development of a selection index that can be used to improve the capacity of honey bees to thrive in northern conditions.Genetic selection is still little applied to honey bees (Apis mellifera), whose complex genetic and reproductive characteristics are a challenge to model statistically. Advances in informatics now enable creation of a statistical model consistent with honey bee genetics, and, consequently, genetic selection for this species. The aim of this project was to determine the genetic parameters of several traits important for Canadian beekeepers with a view to establishing a breeding program in a northern context. Our results show that the five traits measured  are heritable. Thus, the rate of V. destructor infestation has a high heritability (h2 = 0.44 ± 0.56), spring development and honey production have a medium heritability , and winter consumption and hygienic behavior have a low heritability . Furthermore, the genetic correlations between these traits are all positive or null, except between hygienic behavior and V. destructor infestation level. These genetic parameters will be instrumental to the development of a selection index that will be used to improve the capacity of honey bees to thrive in northern conditions.Genetic selection has led to spectacular advances in animal production in many domestic species. However, it is still little applied to honey bees ( Genetic selection has led to spectacular progress in the production of many animal species ,2,3. IndAs a result, there are still very few sustainable genetic selection programs for honey bees worldwide ,12. To oVarroa destructor and Nosema spp., shortage of flower resources, pesticide exposure, or stresses associated with pollination services; are causing important winter colonies losses in Canada [The BLUP-Animal Model is a promising tool for integration in a Canadian genetic selection program to improve honey bee production, health status and hardiness, which are greatly prized by Canadian beekeepers. Indeed, while some Canadian beekeeping operations are in temperate regions, most are in cold climates, where the honey flow season is short (June–August) and the wintering time is long (October–April) ,17,18,19n Canada ,25,26. Tn Canada . This len Canada ,28,29. GP corresponds to the phenotypic value of the animal, G to the genetic value of the individual (random effects), N to the identified effects of the environment (fixed or random effects depending on the choice of modeling) and E to the residual effects (random effects). Additionally,A is the additive value and I represents the interaction effects . Thus,Before colonies can be selected from breeding values, it is important to estimate the genetic parameters of the traits of interest for selection  to ensure efficacy of the genetic program . With thResolving this equation not only predicts the genetic value of an animal for selection but also indirectly allows the phenotypic variance of the population to be broken down to obtain the variance of the genetic origin of a trait in a population. We can then estimate the genetic parameters of traits such as heritability, or genetic correlations between traits ,32.h2 < 1) that evaluates the proportion of the phenotypic variation of a trait that is of genetic origin in a specific population . This index is used to predict the promptness of response of a selected trait. A higher value of the index is associated to higher transmission of the trait from one generation to the next, thus, the response to selection will be faster [r < 1). This index measures the intensity of the link between two quantitative traits [r > 0), opposite (r < 0) or independent (r = 0) [The heritability genetic parameter is represented by an index  . Most of (r = 0) .Varroa destructor infestation, spring development, honey production, winter consumption, and hygienic behavior).To our knowledge, little information is available in the technical or scientific literature on the use of various genetic parameters in North American honey bee breeding programs. The aim of our study is to estimate the genetic parameters (heritability and genetic correlations) of several valuable traits of interest to our beekeeping industry (https://buckfast.dk/index.php/en/). After evaluating the performance of these colonies, 7 were selected for queen production and 10 for drone production to together produce 100 queens  who would constitute the first generation of the breeding program (year 2011): The entire resulting breeding program is therefore based on this common ancestral stock.Our study was conducted in honey bee colonies at the Centre de recherche en sciences animales de Deschambault (CRSAD) Québec, Canada . The CRSAD has housed a breeding program based on phenotypes since 2010. The selection program started with 26 mother colonies. The queens of eleven of these colonies were from local breeders in Quebec or elsewhere in Canada with European-derived stock, as well as 15 colonies of Buckfast lines imported from Denmark  were measured at the colony level . Colonies kept for breeding (queen-producing colonies and drone-producing colonies) were therefore selected based on their phenotypic performance only, hence, the males produced by entire group of drone-producing colonies are not related. Our pedigree database has information on the genealogical background of all the colonies in our breeding program. For each queen in the pedigree database, we identified her mother, the entire group of fathers involved in the fertilization of her mother, the entire group of fathers involved in her fertilization, her sisters and her daughters (if she was selected for breeding) . A uniquIn total, our pedigree database contains the genealogical information of 604 colonies across 8 generations from 2010 to 2017 (performance measures from 2011 to 2018). Each generation includes about a hundred colonies and is composed of 7 to 14 different queen-producing colonies, depending on the year. From each of these selected colonies we produced 10 to 12 sister-queens at the start of the generation .Each year, about 20 colonies with the best overall performance were selected for breeding. Of about 20 colonies selected, between 7 to 14 were used for queen production, the mothers-of-queens, and the other 10 were used for drone production, the fathers-of-workers . The choYoung queens were produced from the selected mother lines (10 to 12 per generation) using the Doolittle queen rearing method . One dayAll queens were identified by marking the back of their thorax with a queen marking pen  and clipping half of the right wing to prevent them from swarming. Colonies were inspected regularly (every 15 days from early June to late August) to destroy royal cells. However, if a queen swarmed or died before or during the performance measurement period, the colony was excluded from the breeding program and the performance measured after the event was no longer included in the databases. These precautions ensured the pedigree of the queens and the validity of data ,41.Varroa destructor. The varroa treatment alternates every second year between Thymovar® and Apivar®. In mid-November, all double nuclei are overwintered in an environmentally controlled room (4 ± 1 °C and 40–50% RH). The following spring (early May), the queens and their colonies are transferred in Langstroth 10-frame hives  and equally distributed between 4 different apiaries, so that sister-queens of all the lines are present in all apiaries [The generation interval of our breeding program is two years. Each year, colonies are evaluated on their performance traits  and, in July of the second year, best performing colonies are selected for the next generation of queens/lines . The queapiaries . These aEach year, problematic colonies  were removed from the selection program and all data from these colonies was removed from the pedigree and performance databases. Of all the colonies produced at each generation, 50 to 70 colonies were integrated in our databases each year from 2010 to 2018.The drone-producing colonies (fathers-of-queens or fathers-of-workers) were selected according to their global performance, but the kinship relationship between these selected colonies were not considered, thus statistical modelling of the pedigree of males was not considered in our model. Integrating the male pedigree into our statistical model would have altered the accuracy of traits heritability estimations. The statistical modelling of our selection scheme is therefore was made only on the female side.V. destructor infestation level; hardiness, as reflected in winter weight loss and spring development.The traits studied were chosen taking into account the requirements and challenges reported by the Canadian beekeeping industry ,43,44. TColonies are provided with honey supers, which are placed above a queen excluder. Each colony has at least two supers during honey flow, each with frames of drawn comb. Honey weight gain is obtained by weighing honey supers when added and removed from a colony or by placing the entire colony (brood chamber and honey supers) on a platform scale .Hygienic behavior: To perform the test, the colony must be opened, and a comb is selected that contains a solid patch of sealed worker brood at the pupal stage, with pupae having pink or purple eye color. Two PVC tubes (2″ inside diameter) are pressed down to the midrib of the comb. The number of empty cells (“misses”) are counted in each tube. Liquid nitrogen is then applied, 300 mL/tube, to “freeze kill” the brood. Frames are marked and returned to the colony. After 24 h, the number of cells that remain capped or partially removed are counted. The total number of cells removed are counted and yield a percentage of hygienic behavior ,46. ThisV. destructor infestation level was assessed by the natural mite fall method using sticky boards placed on the bottom boards of hives [of hives ,48. MiteWinter weight loss: This reflects the consumption of sucrose syrup during winter. It is calculated as the difference in colony weight before vs. after wintering (November vs. April). This measure was taken during the second wintering, when colonies were wintered in one brood chamber (not for the 4 frame nucs winter the first year). Colonies were weighed individually using a platform scale .2  was used to convert the area to obtain the number of immature workers honey bees [Spring development: Colony strength was evaluated by measuring the area occupied by immature worker honey bees (eggs + larvae + capped brood) in colonies early June. This was done by measuring width and length of the brood surface area on each side of every brood frame. The rectangular surface obtained was multiplied by 0.8 to compensate for the elliptic shape of the brood pattern. These values were added to calculate the total brood surface in each colony. A factor of 25 worker cells per 6.25 cmney bees ,50.All relevant information on the performance measurements was compiled using another database, performance database. This database contains the performance measurements associated with each colony, the apiary, the year in which the measurements were taken and the unique identifiers of each queen belonging to the breeding program (defined in V. destructor infestation level trait have been transformed using the logarithm function). A kinship relationship matrix was established with the SAS INBREED procedure to obtain the relationship coefficients between all colonies of the breeding program since 2010. Multiple father ids are recorded in the mating of a queen, so this information was not included in our analysis. Variance components of phenotype traits were estimated with the Residual Maximum Likelihood (REML) Animal Model using the SAS MIXED procedure, with the LDATA option. This allows the integration of relation coefficients and their data set within the genetic relationship matrix. The univariate model used for our data analysis is as follow:ijkP = performance trait  for colony k; µ = general mean of the population for this phenotype; iyear = fixed effect of the year; japiary = fixed effect of the apiary; year × (ij)apiary = fixed effect of interaction between year and apiary; kcolony = random genetic direct effect of colony k assumed distributed ~N, with A equal to the relationship matrix constructed using SAS INBRED procedure; and ijke = residual associated with the measurement. Once the variance analysis was calculated with the model described above, the heritability index for each trait was then obtained by the following formula:Statistical tests were conducted using SAS software  at the 0.05 level of significance. The databases containing colony pedigree and associated performance were merged using the SAS MERGE procedure. Data were previously sorted according to the queen’s unique identifier with the SAS SORT procedure. All dependent variables were tested for normality using the Skewness and Kurtosis test, and a Box-Cox power transformation was used when necessary to meet the normality assumptions of the model 2Genetic correlations have been estimated from the Pearson correlations between breeding values from the univariate models for each trait with positive heritability using the SAS CORR procedure. Bivariate or multivariate models have not been used for these estimations.Distributions of phenotypic data measured from 2010 to 2018  showed ah2) for traits measured between 2010 and 2018 ranged from 0.11 to 0.44 (V. destructor infestation level trait showed high heritability (h2 = 0.44 ± 0.56). Heritability values for spring development and honey production showed medium heritability  and the winter weight loss and hygienic behavior showed low heritability  [Estimates of heritability ( to 0.44 . All the ± 0.13) .r = 0.09, n = 604, p < 0.05), spring development  and winter weight loss . This is also the case between spring development and winter weight loss  and honey production . Hygienic behavior was negatively correlated with V. destructor infestation level . Honey production was not correlated to V. destructor infestation level  and winter weight loss , just as the V. destructor infestation level was not correlated to winter weight loss  and spring development .The genetic correlations between traits  show thah2 > 0), the genetic correlations between these traits are estimated to determine their degree of genetic relationship in the population. Information on the genetic parameters of beekeeping traits as well as their associated economic values will then be taken into consideration for the establishment of the selection index, according to the relative value will be attributed to them by the breeder. Otherwise, if the trait is not heritable (h2 = 0), it cannot be improved by selection using the current model and data. However, other mechanisms can be considered to enhance this trait, for example, improving colony nutrition, colony management, and apiary site management or choice [The aim of this project was to study the genetic parameters  of several beekeeping traits that are important for Canadian honey bee breeding programs. To achieve this aim, we first studied the distribution of data for each trait and then, we measured the variance additive value of these traits in CRSAD breeding program colonies using the REML approach. The variance additive value for each trait was then used to calculate the heritability index for each trait, as shown in the flow chart in r choice ,34.In this study, the genetic parameters of the selected traits allowed us to affirm it is possible to improve zootechnical performance for these traits through a selection program. Indeed, all traits studied showed genetic variance in the population, meaning that these traits have the potential to be genetically improved. Also, all these traits have low to high heritability, which means that genes associated with these traits are transmitted from one generation to the next and that the trait will improve with selection ,33. In ah2 = 0.30 ± 0.14 and h2 = 0.20 ± 0.13, considered medium). Heritability for honey production and spring development were similar to the estimated heritability indexes for these same traits in other honey bee populations in Europe, with heritability ranging from 0.23 to 0.45 for the honey production trait [h2 = 0.18 ± 0.13, considered as low). However, even if our heritability value for the hygienic behavior trait is lower than that estimated by Facchini et al. [h2 = 0.37 ± 0.25), by Guarna et al. [h2 = 0.57) or by the USDA Honey Bee Lab at Baton-Rouge, Lousiana (h2 = 0.65 ± 0.61) [V. destructor infestation level trait will improve quickly  and winter weight loss will improve slowly . Unfortunately, we were unable to compare the heritability for these two traits because, to our knowledge, there are no heritability estimates using variance analysis and the same methodology for data collection of phenotypic measurement.Thus, spring development and honey production traits will improve relatively rapidly  , this prV. destructor infestation level (2015 to 2017) and hygienic behavior (2015) resulted in large standard errors. In the future, we plan to continue evaluation of the performance traits using a measurement rather than a classification system . This yields phenotypic measurements with good accuracy and allows a robust estimation of the genetic value [Integrating the male pedigree into our statistical model would have altered the accuracy of traits heritability estimations. The statistical modelling of our selection scheme is therefore based on the female side, which could have underestimated some results. Furthermore, three years of missing data for the ic value ,36,63. Bic value .r = 0.50, n = 604, p < 0.001), which seems obvious because it has been shown that the larger the honey bee population in spring, the greater the number of bees available for honey harvesting and the greater the honey production [r = 0.25, n = 437, p < 0.001).Estimating genetic correlations between heritable traits allows to anticipate whether selection on one trait will have collateral impacts on other traits which needs to be considered in multi traits selection. In our study, some traits were positively correlated, meaning that both traits will improve when selecting only one trait, and some traits had no correlation, meaning that the traits are genetically independent and therefore the selection of one will not have no impact on the other. Several of the genetic correlations calculated are consistent with the literature. For example, spring development was genetically correlated with honey production . However, this is a desirable result since we seek to obtain colonies with the highest levels of hygienic behavior and the lowest varroa mite population [V. destructor infestation level. Given these results, future development should look at multivariate models for the prediction of breeding values to integrate the genetic correlations between traits.One negative genetic correlation between the different traits was estimated, between hygienic behavior and pulation . This neV. destructor infestation level). In conclusion, we have shown that the five traits studied are heritable and independent of each other or favorably correlated. Therefore, selection for these traits will lead to genetic improvement and for optimize response should be performed simultaneously with a selection index in the context of a selection program. In the future, the statistical modelling and kinship relationship matrix constructed in this study will allow us to pursue an effective breeding program, to continue the development of the genetic evaluation program and to study the genetic parameters of other important traits such as gentleness, suppressed mite reproduction [The objective of this study was to estimate genetic parameters with a view to improve their zootechnical performance through a genetic selection program. Traits were chosen for this study based on their importance in beekeeping in northern climates, as Canadian beekeepers increasingly favor a honey bee with high productivity (high honey production), good hardiness (rapid spring development and low winter weight loss) and that is also healthy (high hygienic behavior and low oduction , and pol


Historically, the majority of statistical association methods have been designed assuming availability of SNP-level information. However, modern genetic and sequencing data present new challenges to access and sharing of genotype-phenotype datasets, including cost of management, difficulties in consolidation of records across research groups, etc. These issues make methods based on SNP-level summary statistics particularly appealing. The most common form of combining statistics is a sum of SNP-level squared scores, possibly weighted, as in burden tests for rare variants. The overall significance of the resulting statistic is evaluated using its distribution under the null hypothesis. Here, we demonstrate that this basic approach can be substantially improved by decorrelating scores prior to their addition, resulting in remarkable power gains in situations that are most commonly encountered in practice; namely, under heterogeneity of effect sizes and diversity between pairwise LD. In these situations, the power of the traditional test, based on the added squared scores, quickly reaches a ceiling, as the number of variants increases. Thus, the traditional approach does not benefit from information potentially contained in any additional SNPs, while our decorrelation by orthogonal transformation (DOT) method yields steady gain in power. We present theoretical and computational analyses of both approaches, and reveal causes behind sometimes dramatic difference in their respective powers. We showcase DOT by analyzing breast cancer and cleft lip data, in which our method strengthened levels of previously reported associations and implied the possibility of multiple new alleles that jointly confer disease risk. Joint analysis of association between the outcome and a group of SNPs within a genetic region is increasingly recognized to complement single-SNP analysis and shed light on the underlying molecular mechanisms. However, the correlation among GWAS association results calls for specifically tailored statistical methods. Here we propose DOT  method that can efficiently combine evidence of association over different SNPs and genes within a pathway without access to the original genotypic data. DOT is fast, does not rely on a permutation algorithm, and is often dramatically more powerful than other popular methods, such as VEGAS and the recently proposed ACAT. We believe that DOT will become a useful addition to the toolbox of methods based on the summary statistics for the GWAS community. PLOS Computational Biology Methods paper.This is a During the recent years, genome-wide association studies (GWAS) uncovered a wealth of genetic susceptibility variants. The emergence of new statistical approaches for the analysis of GWAS have largely contributed to that success. The majority of these methods require access to individual-level data, yet methods that require only summary statistics have been developed as well. The rising popularity of summary-based methods for the analysis of genetic associations has been motivated by many factors, among which is convenience and availability of summary statistics and high statistical power that can often match the power of analysis based on individual records –3.Many types of association tests, including those originally developed for individual-level records, can be presented in terms of added summary statistics. For example, gene set analysis (GSA) tests or burden and overdispersion tests for rare variants , 4, 5, ci and j; Dij) is defined by the difference between the di-locus haplotype frequency, Pij, and the product of the frequencies of two alleles, Dij = Pij − pipj. Then, the correlation between a pair of SNPs is defined as Pij frequency is defined as the sum of frequencies of those haplotypes that carry both of the minor alleles for SNPs i and j. Similarly, pi allele frequency is the sum of haplotype frequencies that carry the minor allele of SNP i.When SNPs are coded as 0,1,2 values, reflecting the number of copies of the minor allele, the LD matrix of correlations can be obtained from SNP data as the sample correlation matrix. It can also be directly estimated from haplotype frequencies whenever those are available or reported. Specifically, the LD  of zero-mean MVN vectors, adding their squared values to obtain statistics TQj)( and computing the proportion of times when TQj). This general idea is straightforward and have been used in many contexts, including methods that utilize individual records . For insHere, we propose a new decorrelation-based method for combining single-SNP summary association statistics. We derive theoretical properties of our method and explore asymptotic power of both DOT and TQ type of statistics. To the best of our knowledge, we are the first ones to derive the asymptotic distributions of DOT and TQ under the alternative hypothesis. Our results show that decorrelation can provide surprisingly large power boost in biologically realistic scenarios. However, high statistical power is not the only advantage of the proposed framework. Once statistics are decorrelated, one can tap into a wealth of powerful methods developed for combining independent statistics. These methods, among others, include approaches that emphasize the strongest signals by combining the top-ranked results –16.L , while the proposed method steadily gains power under the same conditions. Specifically, the proposed decorrelation method gains power when the effect sizes and/or pairwise LD values become increasingly more heterogeneous. The reasons behind the respective behaviors of tests based on TQ and DOT are explored here theoretically and confirmed via simulations. We further derive power approximations that are useful for understanding power properties of the studied methods.Our theoretical analyses also reveal an unexpected result, showing that in many practical settings tests based on the statistic TQ do not gain power with the increase in ESR1), fibroblast growth factor receptor 2 (FGFR2), RAD51 homolog B (RAD51B), and TOX high mobility group box family member 3 (TOX3) genes, without access to raw genotype data. We first test for a joint association between SNPs in those four genes and breast cancer risk by decorrelating summary statistics based on the overall LD gene structure. We then describe how to follow up on the joint association results and identify one or more SNPs that drive joint association with disease risk. To further validate the utility of DOT, we also applied it to summary statistics of a recent GWAS of cleft lip with and without cleft palate. Both of our real data analyses confirmed previous associations and revealed new associations, suggesting new potential breast cancer and cleft lip SNP markers.To showcase our method, we evaluate associations between breast cancer susceptibility and SNPs in estrogen receptor alpha  and this difference of the two means caused the order of power to change, in agreement with our theoretical analysis. Powers now became 0.72 and 0.80, for TQ and DOT, respectively. In this case, empirical and theoretical powers matched to two digits. There is still difference in power at R = 0.2 (0.75 vs. 0.80), but of course, in the case R = 0, the two methods are identical. The power of DOT here is constant, and this reflects a special case, when only a single SNP has a non-zero effect size and, in addition, all correlations between SNPs are the same. We provide R software script which can not only reproduce these results, but is also capable of power analysis with larger correlation matrices, i.e., cases with multiple SNPs. Correlation matrices are generated as symmetric matrices of random numbers and then converted to positive definite ones using the package “Matrix” Reviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: In this paper, the authors present a new summary-statistics-based method for testing a group of common SNPs in aggregate for association to a phenotype. Unlike previous approaches, the authors' test statistic explicitly (and exactly) removes correlation between the individual SNPs' summary statistics.I generally like this paper and appreciate the authors' precision and rigor in deriving and presenting their method. Their theoretical results concerning the power of their test as well as others are also a valuable contribution. So I generally feel this is a very solid contribution to the field. In the long-term I would suggest that the authors consider applications of their framework beyond set-testing since my impression is that the growing number of highly significant associations between *individual* SNPs and phenotypes will eventually cause set-testing to decline as an approach in the common-variant realm. But this is beyond the scope of this paper and for now there remains a substantial community of users of set tests who could benefit from the approach described by the authors.Regarding the technical substance of the paper, I have the following major comments:- I'm unclear on the phenomenon whereby TQ tests don't experience an increase in power as more SNPs are added to the model, e.g., in Setting 1. Looking at the authors' model, in which the variance of the environmental noise, epsilon, is set at 1, it would seem that the more SNPs I add to the model with non-trivial effects, the more phenotypic variance is produced by the genetics. In the limit of infinite SNPs and constant-magnitude environmental noise then, the phenotype should be deterministically set by genotype. It would seem unintuitive that in this situation the TQ tests wouldn't have full power. What am I missing? Are the authors scaling something somewhere?- Relatedly, it would help if the authors included in their methods section more detailed descriptions of their simulation set ups especially including sample size and proportion of phenotypic variance explained by genotype for each simulation .- I don't know if the proportion of variance explained by genotype is high in the authors' simulations. But if it is, do they expect their results to generalize to settings where this is not the case? For real traits, any one set of tens to hundreds of contiguous SNPs typically only explains a very small proportion  of phenotypic variance, so I'd be interested to see if this is the case in the simulations here. Sometimes it's okay to simulate small sections of genome explaining high proportions of phenotypic variance as long as sample size is lowered in some corresponding way, but if this is the case here the authors should explain and perhaps use their theory to justify.- How do the authors expect their statistic to behave in the presence of near-perfect LD? It seems they don't regularize their LD matrix, which surprised me. I would be interested to see power results under a simulation setting where two SNPs, only one of which is causal and contains 75% of the causal signal in locus, have a) 99% correlation and b) 100% correlation.- For the simulations with real genotypes, how was the 100kb region on chromosome 17 chosen? Do the authors expect the simulation results to generalize to other regions of the genome as well? If they are unsure, is it computationally feasible to do simulations where random sets of contiguous SNPs are chosen from the whole genome?- How were the genes ESR1, FGFR2, RAD51B, and TOX3 chosen by the authors for demonstration of their method? Does this set include all the genes found in the Min et al paper to have association with breast cancer? Would it be possible to test a larger set of genes chosen more systematically so that readers can have a sense for whether the authors' approach should in general be preferred over other approaches? Or perhaps to test a few genes chosen by authors of other set testing methods papers?- Do the authors think it would make sense to compare (either in simulation or in practice) to the gene-level test in de Leeuw 2016 PLOS Comp Bio since that method also provides a way to test the SNPs surrounding an individual gene for association while accounting for correlation between variants in order to boost power? Relatedly: ACAT seems to be a method intended primarily for testing of rare variants in sequence data; could it be that this makes it an inappropriate comparison point?- I liked the way the authors argued for their particular choice of pseudoinverse by suggesting that exchangeability of SNPs should be preserved by this operation. Kudos!I also have the following minor comments:- It seems that the claims about the scaling of power as a function of L are for fixed rho > 0, because when rho=0 the tests considered are equivalent. The authors may want to clarify this.- In the definition of r_ij on page 2, should there be a square-root in the denominator?- On page 3 there is a typo in "This general idea is straightforward and HAVE been used..." (emphasis mine)- What was the sample size of the breast cancer data set that the authors analyzed?- In Equations 4 and 5, rho_ij appears on both sides of the equations.- The derivation of the covariance matrix of the vector of summary statistics can be carried out without the delta method but under the assumption of Gaussian genotypes (which is justifiable for large sample size and MAF bounded away from zero). See Proposition 2 in the supplement of Reshef et al 2018 Nat Genet. The authors may wish to comment on whether these two derivations give different results and if so why not.- For the results in Table 6: 1) which set of genotypes were the phenotypes simulated from? 2) Which set of genotypes was used as the reference panel? The only genotypes I saw mentioned were 1000 Genomes, but two distinct sets of genotypes are required for the described analysis.Reviewer #2: Zaykin et al propose DOT, a new method for Gene Based Association Testing. There is demand for a gene (or set-based) method, so a method that improves upon previous methods would be of much interest and (with easy to use software) could become highly used. Zaykin perform many simulations to show that DOT has the potential to improve on a state-of-the-art method, VEGAS . They also have a real data example, but this is very limited. While I am not convinced from this draft alone, I believe that by including an extra simulation method, and a more convincing application, DOT could be a useful addition to the field.Major pointshttps://onlinelibrary.wiley.com/doi/pdf/10.1002/gepi.20219). Thus I require convincing this method is different to / an improvement on those.Reading the method , DOT appears similar to methods which first compute principal components for each gene (ie eigen decompose the snp snp correlation matrix), then regress the phenotype on these . But to my knowledge, the most common methods are SCAT, or magma, and my preferred is Fast-LMM-Set, so would ideally like at least one of these considered (or a statement with justification that these very similar to VEGAS)The application is very limited. While I appreciate there is justification for the choice, unfortunately it looks odd to consider only four handpicked loci, rather than perform a genome wide analysis.I believe you require odds ratios for the SNPs in table 8 Minor PointsI applaud the range of simulations, and also of considering situations where DOTS is not well-suitedI also like the insight into how DOT has the potential to gain power .In the simulations, it is hard to understand the effect sizes. Can you instead report in terms of heritability, ideally both (average) phenotypic variance explained by the gene/region, and (average) variance explained by most significant individual snpThe tables (and I think figures) require captions. In generally, these should give a full description , rather than relying on the user to parse through the main text.Good that a github page is provided with software Please provide a summary of run time for a decent sized analysis.Very Minor PointsIntro; It is important to distinguish situations ... I suggest you replace second "in which" with "from those" or something similarI would prefer if you provided more thresholds when testing the false positive rates It is good you can accommodate covariates, but is this feature used in application?Signed Doug SpeedReviewer #3: In this manuscript, the authors combined single-SNP summary statistics in order to conduct joint analysis of a set of SNPs without accessing original genotype-phenotype datasets. To develop efficient overall summary-statistic, the authors used a decorrelation trick To simplify the correlation structure of the the vector of the single-SNP summary-statistics. The later are correlated by construction. Thus, by rotating the this vector over the eigenvectors of its corresponding correlation matrix one can simplify its correlation structure. Although the decorrelation-trick of a response vector is not a new concept—it has been used for kinship matrix several times in linear mixed models in presence of familial data, e.g. FastLMM— the theoretical and analytical development of the DOT p-values in this manuscript is relevant, in the context summary-statistic association.Major and Minor Comments are dteailed in a PDF file attached to this review.**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Computational Biologydata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: YesReviewer #2: YesReviewer #3: Yes**********what does this mean?). If published, this will include your full peer review and any attached files.PLOS authors have the option to publish the peer review history of their article  and your responses to reviewer comments. If eligible, we will contact you to opt in or out[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).Important additional instructions are given below your reviewer comments.Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.Sincerely,Jennifer ListgartenAssociate EditorPLOS Computational BiologyThomas LengauerMethods EditorPLOS Computational Biology***********************ploscompbiol@plos.org immediately:A link appears below if there are any accompanying review attachments. If you believe any reviews to be missing, please contact [LINK]Reviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: Overall the authors have addressed my theoretical and methods-related concerns quite well in this revision.However, I still have serious reservations about the authors' analysis of real data, which analyses a very small set of genes that were not chosen systematically. I previously wrote: "Would it be possible to test a larger set of genes chosen more systematically so that readers can have a sense for whether the authors’ approach should in general be preferred over other approaches? Or perhaps to test a few genes chosen by authors of other set testing methods papers?"The authors did not perform this analysis, and so I still do not know whether their method is more powerful than existing methods beyond the very small set of genes they have analyzed.  My understanfing of what the authors have shown is that:a) DOT assigns lower p-values than other methods do to the 4 selected breast cancer genes. This seems weak to me first because lower p-values don't necessarily correspond to higher power , and second because these genes have already been prioritized by other methods, suggesting that their connection to breast cancer is not a new discovery enabled by DOT. For example, these genes seem from the text to harbor previously reported risk SNPs. Am I missing something?b) DOT can point at new SNPs associated with breast cancer and cleft lip at these known loci (Tables 10 and 12). But the authors also state (appropriately) that since these results don't come with p-values they should be interpreted with caution, and they also state that cannot conclude that these SNPs are causal but rather only additional proxy SNPs. So I'm unsure what we can confidently learn from these results.I personally don't find (a) or (b) to be strong reasons that practitioners should use DOT.Overall, I see two ways forward:1. The authors can carry out a systematic analysis of the performance of their method on real data. For example, they could run the method on a larger set of genes . This would allow the authors to say things like "in a systematic analysis, our method identified X genes to be in loci that are significantly associated with breast cancer, while competing methods identified only Y such genes." I think this would make a much stronger case for the use of this method. And if it's not true, then that is important for potential users to know even if it doesn't preclude publication of the paper.2. Alternatively, recognizing they have performed extensive revisions already, the authors can add a statement explaining that the genome-wide performance of their method is yet-uncharacterized and would be important to assess in future work.I suppose it would be okay to publish the paper in case 2, but my opinion is that I would be less excited about it. Not answering the central question of whether DOT is more powerful than other methods in practice on real data is not consistent with the otherwise high level of statistical rigor in this potentially interesting paper.Minor comments:- Just above Table 1, you have a typo: "the column labeled \\hat\\gamma provide the average noncentrality value" ("provide" should be "provides")- In the sentence “Different combinations of sample size, the number of causal SNPs, their individual effect sizes and LD patterns among them, resulted in total proportion of phenotypic variance explained...", whose addition I appreciate in this revision, sample size should not be enumerated as one of the parameters that affects the total proportion of phenotypic variance explained.- On page 10, you cite "Min et al. " but neither of refs. 27 or 28 has Min as the last name of a first author in your bibliography.- In your response to R1.1.6, you state that eqns 22 and 27 in Reshef et al. 2018 are derived under the null, but this is not true: Eq 22 defines the computation of summary statistics from data (regardless of model) and Equation 27 includes a parameter beta which can be non-zero. A question therefore remains about the relationship between your derivation and the derivation that assumes Gaussian genotypes. (Fine if you want to drop this issue.)Reviewer #2: The authors have made a careful response and I am happy with the changes.Reviewer #3: No addtional comments**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Computational Biologydata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: YesReviewer #2: YesReviewer #3: None**********what does this mean?). 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For an example in PLOS Biology see here: Reproducibility:http://journals.plos.org/ploscompbiol/s/submission-guidelines#loc-materials-and-methodsTo enhance the reproducibility of your results, PLOS recommends that you deposit laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see  6 Mar 2020Attachmentresponse_to_reviewers.pdfSubmitted filename: Click here for additional data file. 23 Mar 2020Dear Dr Zaykin,We are pleased to inform you that your manuscript 'DOT: Gene-set analysis by combining decorrelated association statistics' has been provisionally accepted for publication in PLOS Computational Biology.Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. 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All press must be co-ordinated with PLOS.Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology. Best regards,Jennifer ListgartenAssociate EditorPLOS Computational BiologyThomas LengauerMethods EditorPLOS Computational Biology***********************************************************Reviewer's Responses to QuestionsComments to the Authors:Please note here if the review is uploaded as an attachment.Reviewer #1: I thank the authors for their revision, and I am happy to recommend acceptance given the clarifications the authors made about their analysis of real data.Setting aside this one point of disagreement, I feel this is very high quality work and I commend the authors on their valuable contribution to the field.**********Have all data underlying the figures and results presented in the manuscript been provided?PLOS Computational Biologydata availability policy, and numerical data that underlies graphs or summary statistics should be provided in spreadsheet form as supporting information.Large-scale datasets should be made available via a public repository as described in the Reviewer #1: Yes**********what does this mean?). 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Mendelian randomization is a powerful tool for inferring the presence, or otherwise, of causal effects from observational data. However, the nature of genetic variants is such that pleiotropy remains a barrier to valid causal effect estimation. There are many options in the literature for pleiotropy robust methods when studying the effects of a single risk factor on an outcome. However, there are few pleiotropy robust methods in the multivariable setting, that is, when there are multiple risk factors of interest. In this article we introduce three methods which build on common approaches in the univariable setting: MVMR-Robust; MVMR-Median; and MVMR-Lasso. We discuss the properties of each of these methods and examine their performance in comparison to existing approaches in a simulation study. MVMR-Robust is shown to outperform existing outlier robust approaches when there are low levels of pleiotropy. MVMR-Lasso provides the best estimation in terms of mean squared error for moderate to high levels of pleiotropy, and can provide valid inference in a three sample setting. MVMR-Median performs well in terms of estimation across all scenarios considered, and provides valid inference up to a moderate level of pleiotropy. We demonstrate the methods in an applied example looking at the effects of intelligence, education and household income on the risk of Alzheimer’s disease. Three assumptions are required in order that a genetic variant is a valid instrument: it must be associated with the risk factor of interest; it must not be associated with any confounder of the risk factor-outcome relationship; and it must be independent of the outcome conditional on the risk factor and confounders.2Mendelian randomization is a technique for estimating the causal effect of a risk factor on an outcome using observational data.3 These features allow practitioners to harness publicly available summary data from genome wide association studies (GWAS). Two-sample approaches, where the genetic variant-risk factor and genetic variant-outcome associations are estimated in different samples, open up vast combinations of risk factor-outcome relationships to be studied.4 The major limitation in Mendelian randomization analyses is therefore the potential presence of pleiotropy, which is when genetic variants associate with traits other than the risk factor of interest. If any such trait provides an alternative causal pathway to the outcome not via the risk factor, then the corresponding genetic variants are invalid instruments and causal effect estimates may be biased.Genetic variants are good candidates for instrumental variables: they are naturally independent of many environmental factors which are common sources of confounding, and mitigate the potential for reverse causation. Furthermore, methods for Mendelian randomization have been developed which allow for combining many instruments in a single analysis, and which can also be used when only summary statistics of the associations between the genetic variants and traits are available.5 One motivation for its use is to account for pleiotropy in a univariable analysis via a set of measured covariates. It can be an important sensitivity analysis if there are known biological pathways linking the genetic variants and the outcome. If any such biological pathways bypass the risk factor, the genetic variants will not be independent of the outcome conditional on the risk factor and outcome, and the instrumental variables assumptions are violated. Another motivation is if there are a number of correlated traits with shared genetic predictors which are all hypothesized to have potential causal effects on the outcome. A multivariable model can distinguish between the direct effects of the risk factors on the outcome and the total effects inclusive of mediators.6 A genetic variant is a valid instrument for multivariable Mendelian randomization if: it is associated with at least one risk factor; it is independent of any confounder of each risk factor-outcome relationship; and it is independent of the outcome conditional on all risk factors and confounders. Causal pathways from a genetic variant to the outcome that do not pass via one or more of the risk factors are referred to as unmeasured pleiotropy . For the purposes of this article, we use the word pleiotropy to mean unmeasured pleiotropy.Multivariable Mendelian randomization fits multiple risk factors in a single model.7 Each method provides valid estimation of the causal effect under different sets of assumptions. Although these assumptions are, generally, untestable, an applied analysis will typically employ a range of methods. Consistency of results across various methods which rely on different assumptions gives strength of evidence to the findings.8 There are, however, few methods for pleiotropy robust multivariable Mendelian randomization.10 Valid estimation of causal effects, therefore, typically relies on the assumption that all causal pathways between the genetic variants and the outcome are accounted for via the measured risk factors.There are a number methods in the literature for univariable Mendelian randomization  which are robust to pleiotropy.In this article we propose a number of novel approaches to multivariable Mendelian randomization which provide robustness to different forms of pleiotropy. The methods are developed for use with summary level data, and so access to individual level data is not required. We examine the performance of the methods under various pleiotropic settings in a simulation study. We then demonstrate the methods in an applied analysis looking at the effects of intelligence, years of education and household income on the risk of Alzheimer’s disease.22.111 and Kang et al12 in the single risk factor case. For individual i, let iY be the outcome, iX1, …,iKX be K risk factors, Gi1, …,ipG be p genetic variants and iU represent confounders of the risk factor-outcome relationships. The data generating model is: Xikv and Yiv are independent error terms with mean zero. Note that the Xikv are not necessarily independent of each other, and so the risk factors may be correlated via the correlation between these error terms as well as their common association with iU . We assume that the genetic variants are independent of each other and independent of iU .We further assume that p > K, that each genetic variant is associated with at least one risk factor, and that the p × K matrix with th element Xjkβ is of full column rank. This latter assumption ensures that the genetic variant associations with the risk factors are not multicollinear and means that each risk factor is associated with at least one genetic variant. The parameters of interest which we aim to estimate are the kθ’s, which represent the causal effects of the kth risk factor on the outcome.We assume the following model, which is similar to a multivariable version of the models set out by Bowden et al2.2jth genetic variant, pleiotropy is caused by the jα term. Since the model allows for no direct association between the genetic variant and iU, jG is a valid instrument if at least one of βXj1, …,XjKβ are non-zero and jα = 0. Note that although the model suggests jα represents direct effects of the genetic variant on the outcome, it may also represent an association via an unmeasured trait, including via a confounder of the risk factor-outcome relationship.12The relationships between a single genetic variant, the risk factors, confounders and outcome in model jα = 0 for all j, then there is no pleiotropy and all genetic variants are valid instruments. When not all j’sα are zero, we consider two patterns of pleiotropy. The first is referred to as balanced pleiotropy, which is where the j’sα are distributed with mean zero. The second is referred to as directional pleiotropy, which is where the jα’s are distributed with mean not equal to zero. It may also be that most of the jα are equal to zero but a relatively small number of them are non-zero and possibly large in magnitude. We will refer to these non-zero j’sα as outliers.If 2.3jα = 0 for all j), the assumptions given in 12 the parameters α1, …,pα,kθ, …,KΘ are identifiable if there is a unique solution to j = 1, …,p, given k = 1, …,K, where ijG, iY, and ikX, respectively. Let K = 1 and consider all subsets of {1, …,p} which have the property that q, for all j in the subset. Kang et al12 showed that the model parameters are identified if and only if there is no such subset which contains invalid instruments which is as large as, or larger than, the subset which contains all the valid instruments. This is often referred to as the plurality valid assumption. Thus, a sufficient condition for identifiability is that more than half the instruments are valid.If all genetic variants are valid instruments /2 genetic variants are valid instruments.It follows that in the multivariable case, assuming all genetic variant-risk factor association vectors are linearly independent, we require there to be no subset of {1, …,2.4kth risk factor and outcome, respectively, on the jth genetic variant. We have that Xjkε and Yjε are independent for all j. If Xjkε and xjlε are independent. Otherwise, the correlation between Xjkε and Xjlε depends on the correlation between the kth and lth risk factors. Finally, although in practice they are estimated from data, it is commonly assumed that j, k.We denote by Although the model assumes that the risk factors and outcome are continuous, categorical traits are possible, and in fact common in practice. In this case, the relevant genetic variant-trait associations are estimated by logistic regression  and represent the change in log odds ratio of the trait per extra effect allele in the genetic variant.Note that summary level estimates used for Mendelian randomization are typically unadjusted for other variants . In large samples, the differences between the unadjusted estimates and adjusted estimates  will be negligible.2.53 is not affected by genetic variant orientation, but methods which model pleiotropic effects, such as the MR-Egger method13 and the lasso-based approach of Rees et al,14 are.Each genetic variant can be coded in two ways, depending on which allele is chosen as the effect allele. The choice of effect allele is arbitrary, but will change the sign of the genetic variant-trait associations. Some Mendelian randomization methods may give different results depending on the orientation of the genetic variants. For example, in the single risk factor case, the inverse-variance weighted method9 suggest orientating the genetic variants such that each one has a positive association with the primary risk factor of interest. If there is no single primary risk factor of interest, or as an additional sensitivity analysis, the impact of the orientation may be assessed by repeating the analysis multiple times, re-orientating the genetic variants with respect to each risk factor.In univariable Mendelian randomization, it is conventional to orientate the genetic variants such that an additional copy of the effect allele has a positive association with the risk factor. In the multiple risk factor case, however, this may be done in multiple ways, since forcing positive associations with respect to one risk factor may change the sign of the associations with respect to the others. Rees etal2.613 At a ‘population’ level, the InSIDE assumption is that the αj’s are independent of each of Xj1β, …, XjKβ. In a finite sample, we require that the correlation between the sample estimates of the j and each of the Xj1β, …,XjKβ, for the given set of data, is equal to zero. This latter condition will rarely be true in practice, since there will typically be residual correlation due to random variation. If the former is true, however, then this correlation will tend to 0 as the number of instruments increases.An assumption that is often required for pleiotropy robust Mendelian randomization is the InSIDE assumption (instrument strength independent of direct effects).3We proceed to recall existing methods for multivariable Mendelian randomization in 3.115 fits the multiple linear regression model j = 1, …,p, where jε is normally distributed with mean zero and variance The multivariable inverse-variance weighted (MVMR-IVW) methodθ. If not all genetic variants are valid instruments, the estimator remains consistent if pleiotropy is balanced and InSIDE is met. Thus, it is sensitive to outliers and directional pleiotropy.If all genetic variants are valid instruments, 9 This method fits an intercept term in If some of the genetic variants are invalid and pleiotropy is directional, the causal effect can still be consistently estimated using the MVMR-Egger method.The MVMR-Egger estimator is robust to directional pleiotropy, even when all instruments are invalid. However, it relies on the InSIDE assumption for consistent estimation. Furthermore, it results in lower precision. A final drawback is that it may produce different results depending on the orientation of the genetic variants.16 has been proposed to handle the case where pleiotropy is balanced but there are outliers. Broadly speaking, the method performs a test based on a heterogeneity measure to identify outliers, which are then removed from the analysis. Although Verbanck et al16 has been proposed to handle the case where pleiotropy is balanced but there are outliers. Broadly speaking, the method performs a test based on a heterogeneity measure to identify outliers, which are then removed from the analysis. Although Verbanck et al16 describe the method for the single risk factor case, the authors have also produced a multivariable version, which is a straightforward extension. Specifically, the method computes inverse-variance weighted estimates by leaving out one genetic variant at a time. Letting jth genetic variant, it then computes the following quantity, termed the global observed residual sum of squares: The MR-PRESSO methodM) times: j = 1, …,P, are drawn the normal distribution with m = 1, …,M. Finally, for each j, an empirical P-value is computed as 1A(x) is the indicator function. If the jth empirical P-value,multiplied by the number of variants (in order to apply a Bonferroni correction), is greater than the chosen significance level , then the respective genetic variant is identified as an outlier. If there are no outliers identified, the estimate obtained is the same as MVMR-IVW. If true outliers are identified and removed, it is expected to reduce the bias and be more efficient than MVMR-IVW. However, the method is not expected to perform well when a large number of genetic variants are pleiotropic, particularly if the pleiotropy is directional.This is compared with an expected residual sum of squares, which is computed multiple . A method for performing robust regression in univariable Mendelian randomization is described in Rees et al,This method of robust regression provides robustness to outliers by effectively capping residuals of a certain magnitude. The approach is thus expected to be robust to pleiotropy when there are a relatively small number of invalid instruments. In this case it should be unbiased and more efficient than MVMR-IVW. However, it may not perform well if there are a relatively large number of invalid instruments.3.2.2θ by An alternative approach to robust regression is to use least absolute deviations regression. That is, we estimate 18 Since Least absolute deviations regression is a special case of quantile regression which estimates the 50th percent ile. Thus, K = 1, the estimator obtained using least absolute deviations regression is equivalent to the weighted median estimator for univariable Mendelian randomization proposed by Bowden et al19 with weights given by Similar to robust regression, least absolute deviations regression is less affected by outliers than least squares regression. It is not robust to influential observations, as robust regression is. However, it may be expected to perform better when the distribution of the 21 in a study which examined the effects of education on coronary heart disease.A disadvantage of least absolute deviations regression is that we lose the asymptotic theory of least squares estimation which leads to easy to compute and accurate standard errors for use, for example, in inference. Confidence intervals are typically produced using a rank inversion technique, or via resampling methods , ijG ~ Binomial , Xiv1, …, XiKv, Yiv, iw ~ N , independently. These parameter values give R2 statistics  of approximately 12%. Two sets of values for the causal effects were considered: in the first, θ1 = 0.2, θ2 = 0.1, θ3 = 0.3, θ4 = 0.4; in the second, θ1 = 0,θ2 = -0.1, θ3 = 0.1, θ4 = 0.2. Four scenarios were considered with different patterns of pleiotropy. For each scenario either 10%, 30%, 50% or 70% of genetic variants were invalid. jδ’s were set to zero and the jα’s corresponding to invalid instruments were generated from the N  distribution.Balanced pleiotropy and InSIDE assumption met: All jδ’s were set to zero and the jα’s corresponding to invalid instruments were generated from the N  distribution.Directional pleiotropy and InSIDE assumption met: All jα’s corresponding to the invalid instruments were generated from the N  distribution and the jδ’ s corresponding to the invalid instruments were generated from the Uniform  distribution.Directional pleiotropy and InSIDE assumption violated: The jα’s corresponding to the invalid instruments were generated from the N  distribution and the jδ’s corresponding to the invalid instruments were generated from the Uniform  distribution.Balanced pleiotropy and InSIDE assumption violated: The We conducted a simulation study to compare the performance of the methods described in the previous section under scenarios with different amounts and types of pleiotropy. We simulated from model 11 It should also be noted that the overall pleiotropic effect in Scenario 3 is directional because the jδ parameters have non-zero mean.In scenarios 3 and 4, the InSIDE assumption is violated because of the direct association of the genetic variants with the confounder. This creates a pleiotropic effect of the genetic variants on the outcome which is not independent of the effect of the genetic variants on the risk factors due to the association of the confounder with the risk factors.θk values, the simulations were replicated 1000 times. For each replication, the genetic variant-trait association estimates and their standard errors were computed from the individual level data using simple linear regression with an intercept. The causal effects were then estimated using the methods described in θk values, there is a true causal effect, and for the second set of values there is no causal effect). The mean, standard deviation of estimates, mean standard error and power/type I error rate, at the 0.05 significance level, are shown in For each scenario, level of pleiotropy, and set of All methods performed well in terms of bias when there was balanced pleiotropy. The MVMR-IVW and MVMR-Egger methods were biased when pleiotropy was directional, increasing as the proportion of pleiotropy increased. These methods were also less precise than all other methods, with the largest standard deviations of estimates, and were very low powered. In theory, MVMR-Egger should be robust to directional pleiotropy when InSIDE is met. However, there was a fair amount of bias from this method in Scenario 2 when these conditions are satisfied. Given that the conditional F statistics were fairly low, a possible explanation is that the bias is due to weak instruments, which this method is particularly susceptible to. Some evidence supporting this explanation was provided in a supplementary simulation study, where the instrument strength was increased and the bias from MVMR-Egger was reduced to close to zero. This supplementary simulation study is discussed in more detail later in this section.θ. vector by comparing the mean squared error defined as the mean, across all replications, of θj. As shown in MVMR-Robust outperformed MVMR-PRESSO in all scenarios up to 50% pleiotropy with lower bias, more precision and correct type I error rates. Notably, MVMR-Robust had type I error rates at or below the significance level across all scenarios, even with 70% pleiotropy. MVMR-PRESSO had low bias at the lower level of pleiotropy, but did not perform well with moderate or high amounts. MVMR-Lasso was generally the most precise estimate: it had similar mean squared error to MVMR-Robust at 10% pleiotropy, but retained its performance in this regard at the higher levels of pleiotropy also. Similarly, it had comparable power to MVMR-Robust at 10% pleiotropy, but did much better as the proportion of pleiotropy increased. As expected, MVMR-Lasso had inflated type I error rates. MVMR-Median had comparable bias to MVMR-Lasso across all levels of pleiotropy. It was less precise and lower powered than MVMR-Lasso, but had type I error rates closer to the significance level. In terms of mean squared error, MVMR-Median was bettered uniformly across all scenarios only by MVMR-Lasso. As a further analysis, in p = 20, with the distribution of the Xjkβ parameters adjusted to retain similar 2R values), where the risk factors were correlated by setting cor  = 0.5 for all k ≠ l, and where the genetic variant-trait associations were all estimated from the same sample (one sample Mendelian randomization). For brevity, supplementary analyses were performed over Scenarios 1-3 and for the 10%-50% pleiotropy cases. The results are shown in The simulations were repeated for the cases where there were fewer instruments  with each risk factor separately suggest that both intelligence and years of education have a protective effect on Alzheimer’s disease. However, when both risk factors are included in a multivariable model, using MVMR-IVW, the effect of years of education, independent of intelligence, shifts toward the null. The implication is that years of education only has a causal effect on the odds ratio of Alzheimer’s disease via its effect on intelligence. Here, we reconsider this example and include household income as an extra risk factor.In this section we consider an applied example looking at the causal effects of intelligence, years of education and household income on Alzheimer’s disease. The effects of intelligence and years of education on health outcomes have been studied by Davies etal31 and Okbay et al,32 respectively. By clumping the combined list of genetic variants which associate with each risk factor at the genome wide significance level, Davies et al29 arrived at a list of 219 independent genetic variants to be used as instruments in multivariable Mendelian randomization analyses. We obtained the associations between these genetic variants and household income from the UK Biobank . Note that household income is an ordinal categorical variable, and so the genetic variant associations represent the increase in log odds of being in a higher income category per extra effect allele. Genetic variant associations with Alzheimer’s disease were obtained from the GWAS of Lambert etal.33 In total, 213 of the genetic variants used by Davies etal29 were available in both of the household income and Alzheimer’s disease datasets, and we used these as instruments in our analysis.Genetic variant associations with intelligence and years of education are taken from the GWAS of Hill et alNote that the genetic variant associations with both intelligence and years of education were all in the same direction, and so they were orientated in our analysis to be all positive with respect to these traits.yj±σ for each genetic variant. The plot provides a way of visualizing heterogeneity in the multivariable setting, similar to the scatterplots of 34We assessed instrument strength in the univariable setting using the mean F statistic across the genetic variants, and in the multivariable setting using the conditional F statistic see . When ta30 The estimated log causal odds ratio of Alzheimer’s disease per one standard deviation increase in intelligence is -4.20 (95% CI -0.57 to -0.27), and per one standard deviation increase in years of education is -0.59 (95% CI -0.83 to -0.36). The estimated log causal odds ratio of Alzheimer’s disease per unit increase in log odds ratio for a higher household income bracket is -0.60 (95% CI -0.89 to -0.31). Using the MVMR-IVW model, the estimates of the log causal odds ratio from both years of education and household income attenuated to the null, with 95% confidence intervals overlapping zero. The multivariable model however still suggests a protective effect from intelligence, with an estimated odds ratio of -0.47 (95% CI -0.86 to -0.07).The pleiotropy robust multivariable methods gave results which were broadly consistent with the MVMR-IVW results see . The MVMThe consistency of the findings give strength to the assertion that intelligence has a causally protective effect on Alzheimer’s disease, conditional on years of education and household income. However, there is no evidence of a direct effect of years of education or household income on Alzheimer’s disease. There are two potential explanations for why years of education and household income would appear to have an effect in the univariable analysis but not in the multivariable analysis. One is that these risk factors affect Alzheimer’s disease but only via their association with intelligence. That is, that intelligence is a mediator of the effect of these risk factors on Alzheimer’s disease. The other explanation is that the genetic variants that affect education and household income have pleiotropic effects, potentially via intelligence. It is also possible that both are true .6In this article we have presented methods for performing multivariable Mendelian randomization which are robust to pleiotropy. Existing methods either allow for invalidity at the cost of low precision and the InSIDE assumption (MVMR-Egger), orwere developed for the casewhere pleiotropy is balanced and there are a relatively small number of outliers (MVMR-PRESSO). We have considered methods which can handle higher proportions of invalidity and directional pleiotropy.When there is evidence of relatively few invalid instruments, MVMR-Robust was shown to outperform MVMR-PRESSO in all scenarios with pleiotropy up to 50%. MVMR-Lasso, another method which aims to identify and down weight outliers, performed best overall in terms of mean squared error, even when half of the genetic variants were in valid instruments and pleiotropy was directional. Although type I error rates were inflated,this can be mitigated when a three sample approach is possible. MVMR-Median was shown to perform almost as well as MVMR-Lasso in terms of mean squared error, and retained correct type I error rates at higher levels of pleiotropy. Although all methods showed considerable bias in the settings with high levels of directional pleiotropy , our newly proposed methods still out performed existing methods across the various metrics considered in these extreme scenarios. Furthermore, MVMR-Robust retains correct type I error rates even with very high levels of pleiotropy. As demonstrated in the applied example,a plot of the residuals versus fitted values from the MVMR-IVW method can be used to visualize potential outliers and pleiotropy, and to help determine the most appropriate choice of robust method.37Although our initial model assumes no causal pathways between the risk factors, the methods are applicable if there are such pathways, as demonstrated by the supplementary simulation analyses. The only difference in this case is in the interpretation of the causal parameters. In the presence of mediation, the estimates represent the direct effects of the risk factors on the outcome, excluding any indirect effects via the mediators. There is existing literature which examines mediation analysis in the Mendelian randomization setting.22A potential extension to MVMR-Lasso is to consider different penalty terms for each genetic variant. Although different penalty weighting schemes did not significantly affect our simulation results, this is an interesting area to consider for future research, for example if the association estimates have considerably different levels of precision. A potential extension could be along the lines of the adaptive lasso approach of Windmeijer et al.38The work has some limitations in the modeling assumptions made, in particular of the linearity and homogeneity  of the effects of the risk factors on the outcome. Furthermore, although we can handle non-continuous traits via the use of logistic regression to produce summary statistics, this may cause bias in the causal effect estimates due to the non-collapsibility of the odds ratio. Nonetheless, violations of these assumptions tend to attenuate causal effect estimates toward the null.Xjkσ = 0 for all j, k. This is a common assumption in Mendelian randomization analyses, and is justified by the very large sample sizes that these associations are typically estimated in, in contrast to the genetic variant-outcome associations which may be estimated using a relatively small number of cases vs controls. Provided the genetic variants strongly predict each risk factor, conditional on all the other risk factors, this assumption will have little influence on the analysis. Otherwise, the resultsmay be subject to weak instrument bias. In practice, weak instrument bias is mitigated by selecting a set of genetic variants which associatewith the risk factors according to some threshold related to, for example, F statistics or P-values. Furthermore, the approach of Sanderson et al10 provides a sensitivity analysis in case there is still a suspicion of weak instruments, as demonstrated in our applied example. Although in univariable Mendelian randomization measurement error will bias causal effect estimates toward the null,39 this will not necessarily be the case in the multivariable setting. Assessing the impact of measurement error in the multiple risk factor case, and how to account for this, is an active area of research.Another limitation is the assumption of no measurement error of the genetic variant-risk factor associations, equivalent to assuming In summary, the methods we have presented provide new ways for performing Mendelian randomization with multiple risk factors which are robust to different forms of pleiotropy. Each has advantages when applied to specific scenarios. Together with MVMR-Egger, these methods provide a suite of sensitivity analyses for multivariable Mendelian randomization.7https://github.com/aj-grant/robust-mvmr. Existing Mendelian randomization methods were implemented using the following R packages: Mendelian Randomization40 , MRPRESSO16 (MVMR-PRESSO) and MVMR6 (MVMR-Q(het)). Existing packages which were used in the implementation of the newly proposed methods include: robustbase41 (MVMR-Robust), quantreg42 (MVMR-Median) and glmnet43 (MVMR-Lasso).R code for performing the methods described in this article, and for reproducing the simulation results, can be found at Supplementary File


Arabidopsis, which is more than was detected by the other methods.In genome-wide association studies, linear mixed models (LMMs) have been widely used to explore the molecular mechanism of complex traits. However, typical association approaches suffer from several important drawbacks: estimation of variance components in LMMs with large scale individuals is computationally slow; single-locus model is unsatisfactory to handle complex confounding and causes loss of statistical power. To address these issues, we propose an efficient two-stage method based on hybrid of restricted and penalized maximum likelihood, named HRePML. Firstly, we performed restricted maximum likelihood (REML) on single-locus LMM to remove unrelated markers, where spectral decomposition on covariance matrix was used to fast estimate variance components. Secondly, we carried out penalized maximum likelihood (PML) on multi-locus LMM for markers with reasonably large effects. To validate the effectiveness of HRePML, we conducted a series of simulation studies and real data analyses. As a result, our method always had the highest average statistical power compared with multi-locus mixed-model (MLMM), fixed and random model circulating probability unification (FarmCPU), and genome-wide efficient mixed model association (GEMMA). More importantly, HRePML can provide higher accuracy estimation of marker effects. HRePML also identifies 41 previous reported genes associated with development traits in  One strategy to solve this problem is to use more information beyond the p-value. For example, Xu, et al. [Arabidopsis genome-wide association boosting, which can identify weak association signals by integrating co-functional gene network information. Apart from this, the linear mixed model (LMM) has become a widely used methodology due to its capability in controlling for population stratification and the inclusion of related individuals [Genome-wide association studies (GWAS) can advance our understanding of molecular mechanism of complex traits ,2,3,4. T, et al.  proposed, et al.  presenteividuals . Howeverividuals ,10,11,12Lanczos derivative-free REML and Lanczos first-order Monte Carlo REML to further improve the computing speed. However, these existing methods for REML variance components estimation are mainly aimed at single-locus LMM, which is not effective enough to handle a complex genetic structure.With regards to this, several approaches based on sparse matrix operations have been developed to improve the calculating speed ,14. LippSeveral classical multivariate selection methods have good performances in association analyses when the number of SNPs is not far more than that of individuals, including Lasso and its derivatives ,18,19, pIn this study, we developed an efficiently hybrid method HRePML to perform GWAS, which takes full advantage of REML and PML. Under the linear mixed model framework, we firstly conducted single-locus marker scanning using REML and then carried out multi-locus marker selection based on the reduced subset, taking genetic relatedness and population stratification into account. We used pure C++ language to implement HRePML and overcome one key issue in the programming limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method . In ordeArabidopsis thaliana [A publicly available dataset of thaliana  is used A standard linear mixed model for association mapping can be expressed asenvalues ,37,38.y becomesTransform Equation (1) by left-multiplying QT  and geneIn order to get the estimation Limited-memory BFGS (L-BFGS) ,39 is an library  providesp-value corresponding to P cutoff is adopted.The Wald test is used to conduct a hypothesis test on each marker effect ith All the markers passing the REML step are placed into a multi-locus model  as(12)yPenalized maximum likelihood (PML) is a fast approach to use numerical optimization to obtain the maximum posteriori estimates when the penalty function is a probability density on the parameters ,22,42. LLet Then, the logarithm of penalized likelihood function can be defined asThe parameter vector Via For the nuisance parameter vectors, Set the initial values for Arabidopsis thaliana dataset [k among 1000 repetitions, k in the ith repeat, and kth QTN’s true effect. Four simulation experiments were designed to validate our new method HRePML. In the first simulation study, our goal was to explore the new method’s performance on the statistical power, mean square error (MSE), and running time. We generated a set of genotype data consisting of 500 individuals and 10,000 markers, which was based on the  dataset . Eight qting QTN ,43,44. MIn the second simulation study, we aimed at exploring the influence of polygenic background on HRePML. The polygenic effects were introduced with multivariate normal distribution In the third simulation study, our goal was to investigate the influence of the sample size on the running time and statistical power. The sample size was set to 500, 1000, 2000, and 4000, respectively. Meanwhile, the number of markers was fixed at 10,000. In the fourth simulation study, our aim was to investigate the impact on running time as the number of markers increased. The number of markers was set to 10,000, 50,000, 100,000, and 200,000, respectively. At the same time, the sample size was fixed at 500. In these two simulation studies, the repeat times were set to 100. The position and heritability of each QTN were set to the same as those of first simulation study. Their parameters are listed in We compared the statistical properties of the new HRePML method with those of the multi-locus mixed-model (MLMM) , fixed aTo further validate the performance of HRePML, an additive polygenic effect was involved in the second simulation study. The dataset was the same with that used in the first simulation study, except that polygenic effect was added to the phenotype. The same trend in statistical power was observed, and the average powers of HRePML, MLMM, FarmCPU, and GEMMA were 62.45%, 57.08%, 55.18%, and 40.46%, respectively, which showed that HRePML was still powerful and robust under polygenic interference B. As farIn the third simulation study, we investigated the effect of the sample size on the statistical power of HRePML. There were four datasets consisting of 500, 1000, 2000, and 4000 individuals, respectively, and 10,000 SNP markers, with 100 replicates. Eight true QTNs were set in each replicate. Then, each dataset could be regarded as having 1,000,000 SNPs and 800 true QTNs in total. The average powers of sample sizes 500, 1000, 2000, and 4000 were 59.88%, 75.75%, 82.00%, and 91.13%, respectively. Clearly, the statistical power improved as the sample size increased. The results demonstrated that the statistical power could be more than 80% for QTN, with heritability equal or greater than 0.03 when the sample size reached 1000. However, for QTN with very small heritability (0.01), the required sample size was at least 4000, and then, the statistical power could exceed 60% .® Core™ i5-7300HQ CPU 2.50 GHz, Memory 8.00 GB, Houston, TX, USA). In the first simulation consisting of 1000 repetitions, the total running time of HRePML, MLMM, FarmCPU, and GEMMA were 3.1419, 22.7274, 4.6653, and 2.4186 h, respectively. Compared with the other two multi-locus MLMM and FarmCPU methods, HRePML was the most computationally efficient, which was only slightly slower than the single-locus GEMMA method. In particular, HRePML achieved about seven times faster than the popular multi-locus method MLMM , the compiling command is: g++ -I/path/liblbfgs-1.10/include HRePML-Linux.cpp -llapack -lblas -llbfgs -o output, where it needs to include the C++ library of limited-memory BFGS. If Math Kernel Library (MKL) has been installed for Intel CPU users, the following compiling command is recommended: g++ -I/path/liblbfgs-1.10/include HRePML-Linux.cpp -lmkl_gf_lp64 -lmkl_sequential -lmkl_core -llbfgs -o output. In order to save the results, the HRePML program requires two output files, which are the results file and the computational time file. After compilation, the execution command is: ./output Genotype.csv Phenotype.csv Kinship.csv Fixed.csv Results.csv Time.csv. Then, the results and running time are output into Results.csv and Time.csv files, respectively.In this paper, we proposed a new fast multi-locus method HRePML for GWAS, which is based on a restricted and penalized maximum likelihood function. HRePML can take genetic relatedness and population stratification into account under the linear mixed model. In addition, we implemented the algorithm in pure C++ language and provide Windows and Linux platform versions for the researcher’s choice.p value to determine the final identified SNPs. Using LOD equal to 3.0 as the threshold, many real data analyses show that it is feasible to improve the statistical power [The new method adopts a two-stage approach to conduct multi-locus GWAS, which is widely used to improve the computational efficiency when hundreds of thousands of SNPs appear. The core idea is to conduct an initial screening of the marginal effects of all SNPs and select the ones with reasonably large effects for the next phase of the multi-locus test. Recently, multi-locus GWAS methods have become more and more popular, such as, MLMM , FarmCPUal power ,43,44. HHRePML can be flexibly applied to animal and human GWAS, not limited to plant research. Genetic architecture is more complex, and the genome is much larger in animals and human beings than that in plants. One important issue is allelic heterogeneity, which cannot be effectively handled by traditional single-locus methods. More importantly, genetic heterogeneity can lead to a noncausative marker being a better descriptor of the phenotype than a causative one . AnotherArabidopsis real data analysis showed that the results have relatively low consistency among HRePML, MLMM, FarmCPU, and GEMMA , HRePML can compute more efficiently when handling large individuals and millions of markers.