Advances in genotyping technologies have provided breeders with access to the genotypic values of several thousand genetic markers in their breeding materials. Combined with phenotypic data, this information facilitates genomic selection. Although genomic selection can benefit breeders, it does not guarantee efficient genetic improvement. Indeed, multiple components of breeding schemes may affect the efficiency of genetic improvement and controlling all components may not be possible. In this study, we propose a new application of Bayesian optimisation for optimizing breeding schemes under specific constraints using computer simulation.Breeding schemes are simulated according to nine different parameters. Five of those parameters are considered constraints, and 4 can be optimised. Two optimisation methods are used to optimise those parameters, Bayesian optimisation and random optimisation.The results show that Bayesian optimisation indeed finds breeding scheme parametrisations that provide good breeding improvement with regard to the entire parameter space and outperforms random optimisation. Moreover, the results also show that the optimised parameter distributions differ according to breeder constraints.This study is one of the first to apply Bayesian optimisation to the design of breeding schemes while considering constraints. The presented approach has some limitations and should be considered as a first proof of concept that demonstrates the potential of Bayesian optimisation when applied to breeding schemes. Determining a general "rule of thumb" for breeding optimisation may be difficult and considering the specific constraints of each breeding campaign is important for finding an optimal breeding scheme. This information along with phenotype data, has allowed breeders to estimate the effects of these markers on the phenotypic traits of interest and to assess the genetic value of their breeding population. Thanks to prediction models, breeders can now perform genetic selection (GS), which involves selection of un-phenotyped individuals based on their genotype data .Although using genomic selection can help breeders, it does not guarantee efficient genetic improvement. Therefore, other factors need to be considered when designing breeding schemes. For example, selecting only those individuals with the highest predicted values  can be iOther factors influencing the performance of breeding schemes include the constraints faced by breeders, including straightforward examples such as the budget allowed for the breeding campaign, the initial population, the genotyping or phenotyping capabilities, and so on. Breeders should not neglect these constraints when optimizing a breeding scheme as the optimal decisions made under particular constraints  may differ under different constraints .Several examples of methods for optimizing breeding improvement have already been reported, including weighted genomic selection , optimalIn this paper, we introduce the Bayesian optimisation method . BayesiaIn this study, we present the use of Bayesian optimisation to optimise breeding schemes under specific constraints faced by breeders. We consider the function associated with the breeding scheme outcome as the objective function. However, realizing actualbreeding campaigns to evaluate this objective function would have required several years of experimentation. Therefore, to evaluate the objective function in a reasonable time, like most studies presented above , the bre22.1In this section, we explain the relationship between the breeding scheme and optimisation. For simplicity, we will only consider one phenotypic trait of interest here. In case of multiple traits, a function that maps the value of each trait to a selectionindex value  can be used for generalisation, and the selection index can be considered the target trait. After a breeding campaign, the mean genotypic values for this trait over the individuals in the final population can be expressed as:x is the value of all the parameters representing the breeding scheme, f (x) is the expected genotypic value of the population under the parameter values x and ϵ is the residual due to the randomness of the breeding process.Where X be the domain of all possible breeding schemes. A breeding scheme can be parameterised by a very large number of variables. In this paper, we consider the following parameters for representing the breeding scheme:Let X = where:p: cost for phenotyping one plotn: cost for generating and genotyping one new individual in a breeding populationgen: total number of selection cycles  in the breeding campaigninitop: genotypes of the initial population initi: selection intensity for the initial generation i: selection intensity for all later generations repB: part of the total budget allocated for phenotyping; the rest will be used for generating new individualsppheno: period of phenotyping experiments. Individuals will be phenotyped every ppheno generationsiniti and i, respectively. As explained in section 4.3.4, the number of individuals in the second generation is directly related to the selection intensity used on the initial population, and the size of the first generation may have an impact on the performance of the breeding scheme.This parametrization separates the selection intensity of the first and later generations u is the target of selection and that a larger u is preferred. As breeders may not have control over the first five parameters or over ϵ , the best breeding scheme that considers the constraints can be expressed as follows:Here, we assume that Z= and g:z↦f with p, n, gen, initop are the respective values taken by the variables ℬ, 𝒞p, 𝒞n, 𝒩gen, init𝒫op under the breeder’s constraints.where To optimise this objective function, we used the Bayesian optimisation method using Gaussian process regression at the estimation step, and the value of the objective function was returned by the breeding simulation algorithm.2.2Bayesian optimisation consists of a three-step cycle, starting with some observed values of the objective function:Bayesian analysis step: Calculation of the posterior distribution of the objective function using all the observed values of the objective function (i.e. the training data).Sampling step: Selection of the following sampling points that maximize the acquisition function. The acquisition function is a computationally easy function that can be evaluated at any point of the research space using the posterior distribution of the objective function, as well as return how much evaluating the objective function at this point would help yield the global maximum.Evaluation step: Evaluation of the objective function at the previously selected sampling point.These steps are repeated until the stopping criterion is satisfied.g (z) over parameter space Z. The Gaussian process can be fully specified by its mean function m(z)=E[g(z)],∀z∈Z and its covariance function, also called a kernel, k,∀z,z′∈Z2  and a constant liar strategy  = L, ∀k < j+1 .Phenotyping: To phenotype the nPt ≠ 0 we fit a prediction model using a ridge regression  algorithm , the Euclidean distance of vectors of the marker genotype scores between individuals A and B, weighted by the estimated effect of each marker according to the latest prediction model. A mating table is generated associating ∀s ∈{1, n{St−1} Inds with sInd+1 and tnSInd with Ind1. If 2 ≤ tSn ≤ 3, a mating table is generated associating all individuals together, and if tSn = 1, the selected individual is self-fertilized. The mating table, composed of tCn crosses, is used to generate a total of tnewn offspring. To calculate the number of progenies for each crosses, we uniformly allocate the total number of offspring to each crosses. If there are any remaining progenies, some crosses are randomly selected to generate one more individual. The new individuals are then generated using a crossing simulation algorithm (cf. 2.3.3).is small compared to other possible sequences, with dist  interactions. The model used to simulate the phenotypes was as follows:sry is the phenotype of individual s repetition r, sG is a vector of the genotype of individual s encoded with an allele dose  for the reference alleles, β is the genetic effect of the reference alleles, and where 2.3.3To generate the single nucleotide polymorphism (SNP) genotypes of new individuals from those of the parents, we simulated gametogenesis, one from each parent:For each pair of chromosomes:con for each chromosome in a Poisson distribution of rate chrλ, which is the length of the chromosome in the unit of Morgan: con ~ Pois(chrλ)1. We draw the number of crossing-overs nco ≠ 0, we draw the positions of crossing-over independently in a uniform distribution along the length of the chromosome in Morgan. We then include the sampled positions kCOpos } of the crossing-overs so that COposk < COposk+1  with COpos0 = 0 and COposnco+1=λchr.2. When X be the 2 × lsnpn matrix representing the genotype of the parent for the current pair of chromosomes l. Each row represents one chromosome of the pair. Let Y be the vector of length lsnpn representing the recombined genotype of the gamete for the chromosome l. We set  =  or  with probability Y is then calculated as:3. Let hpos is the position of marker h.where The genotypes of the offspring were obtained by merging two gametes from their parents.2.3.4B, CP, Cn, ngen, iinit, i,Brep, phenop.As mentioned in the section 4.3.1, we used a different parametrisation for the breeding simulation than those used for the optimisation problem. In this section, we explain the derivation of the parameters of the simulation from the constraints and optimised parameters The breeding process detailed in 4.3.1 uses the following parameters:tPn the number of phenotyping in the t-th generation.tSn the number of selected individuals in the t-th generation.tnewn the number of new individuals to create in the t-th generation.tSn, the number of selected individuals in the generation t-th, is calculated as the number of individuals in the t-th generation multiplied by the selection intensity, initi for the initial population and i for the later, and rounded to the nearest strictly positive integer.The value of tPn and tnewn, we first needed to calculate the total number of phenotyping plots available for all generations totPn and the total number of new individuals created during the entire breeding campaign totnewn according to the phenotyping and new individual generation costs (pC and nC) as well as the total budget B.To calculate totnewn is calculated by the budget allocated to the generation of new individuals B × repB divided by the cost of creating a new individual nC, rounded to the nearest integer.totPn is calculated by the remaining budget B − totnewn ×nC divided by the cost of phenotyping one plot pC, rounded down to the nearest integer.Beff = nnewtot × Cn + nPtot × CP may differ from the given total budget constraint B. However, this difference is smaller than PC which should usually be relatively small with respect to B.Owing to rounding operations, the effective budget used tPn, the number of phenotyping plots in the t-th generation. First the generations which will include phenotyping trials are identified. These are the first generation t = 1 and then every ppheno generation . All the other generation will have Ptn = 0. Consequently, the total number of generations with non-zero phenotyping trials is equal to genn/ppheno rounded up to the nearest integer. For thosegenerations, we equally allocate the total number of phenotyping plots available for the breeding campaign. If this number is not an integer, as many generations as the remaining plot are selected and those generations will be allocated one more phenotyping plot. Moreover, if P1n< 3, its value is increased to 3 to avoid errors during creation of the prediction model.We can now calculate new1n was set to the number of crosses proceeding at this generation C1n. Then the remaining number of new individuals to be created in the later generations were allocated equally. If this number is not an integer, as many generations as theremaining new individuals to be created are randomly selected and those generations will generate one more individual.Finally, we calculate the number of new individuals to create at each generation. The first generation was homozygous, and therefore, any individuals derived from the same pairs of parents would have the same genotype; as a result, 2.3.5https://github.com/ut-biomet/bayesianOptimizationForBreeding). The simulation algorithm developed for this study was integrated in the R package “breedSimulatR” (“mlrMBO” (renv.lock (https://github.com/ut-biomet/bayesianOptimizationForBreeding/renv.lock).All calculations were performed using R programming language (version 4.0.2) . A reposimulatR” , the Bay“mlrMBO” . An exha2.4To test our optimisation method, we ran several optimisations following the algorithm detailed in the “Materials and methods” section with different parameterisations.2.4.1Glycine Max) with a total of 4,776,813 SNP markers on 20 pairs of chromosomes. To make the simulations faster, the genotypes of the initial population consisted of a smaller subset of snpn = 3000SNPs that were randomly selected. For simplicity, we arbitrarily set the chromosome length to 1 Morgan (i.e. 100 cM) and calculated the linkage map positions based on physical positions, assuming a linear relationship between the two types of positions. This led to an average of one crossing over for each chromosome during the gametogenesis simulation.The genotypes of the initial population were created based on the whole-genome sequences of the accessions of soybean mini-core collections provided by the National Agricultural Research Organisation, Japan  Kajiya-[4.4.1]. β on the phenotypic traits were determined as follows. First, we selected a subset of qtnn=1000 SNPs in the pool of all available snpn SNPs in the genotypes of the initial population. These markers have non-null effects and are known as quantitative trait nucleotides (QTNs). Let mβ be the effect of the QTN m, for all m∈{1, n{qtn}}, mβ was drawn according to the formula mβ = ma × I with ma following an exponential distribution, ma ~ Exp(λ = 1) and I was equal to +1 or −1 with probability sn − nqtnn remaining markers were set to 0.The true genetic effects of the SNPs eσ was calculated according to the specific heritability for the initial population The residual variance of the simulated phenotypes tG is the genotype of individuals from generation t, ty is the phenotype of individuals from generation t, and beta is the marker effect. We can thus deduce that:where Var(tβG) and t = 0. Thus, The value of 2.4.2genn = 5 and genn = 10. For simplicity, both the costs for phenotyping one plot pC and for generating and phenotyping, one new individual nC were set as one and two total budgets were tested: B = genn × 200pC and B = genn × 600pC.We considered the constraints related to the total number of selection cycles in the breeding program and the total budget for the program. To test the proposed optimisation method on various breeding schemes, we used setups with a total number of selection cycles for breeding campaigns of All possible combinations of these parameterisations were tested, and yielded eight different scenarios:2.4.3tolfilter = 10-3The initial training data for Bayesian optimisation are a set of five points randomly sampled in a Latin hypercube ; we set The random optimisation algorithm does not require training data; however, to obtain the same number of objective function evaluations as those with Bayesian optimisation, we began its first iteration with five points randomly selected in the parameter space.itern = 50 iterations of q = 8 parallelly sampled points at each iteration. After each optimisation run, evaluationn = 32 breeding schemes were repeated using the parameters returned by the optimisation run. The results from these optimisations were used to compare the behaviour of Bayesian optimisation against random optimisation.We ran two batches of optimisation. For the first batch, with each scenario and each optimisation method, we repeated 16 optimisation runs with However, we conducted these optimisations independently. Thus, there was no direct association between each run of Bayesian optimisation and random optimisation (for each scenario). To compare the runs of Bayesian optimisation and random optimisation, such an association is required. To be as representative as possible in our comparisons, for each method and run, we calculated the cumulative maxima of the objective function values over the optimisation iterations. Subsequently, we calculated the average cumulative maxima for each method. Finally, to represent one method, we chose the run that was closest (in terms of mean square error) to the average cumulative maxima of the method.itern = 15 with q = 2 parallel sampling points at each iteration. After each run, we simulated a evaluationn = 1 breeding scheme using the parameters returned by the optimisation run.For the second batch, in a more pragmatic manner, we tried to mimic what a breeder who would like to use Bayesian optimisation can do with relatively limited computer power and time available for optimisation. We set the number of iterations to We planned to run these optimisations 1024 times for each scenario and method. Over the 1024 × 8 = 8129 Bayesian optimisations planned, the library used for Bayesian optimisation encountered an unexpected error in 16 cases. Because the number of failed runs was low (approximately 0.2%) and identifying and fixing this bug would require a huge effort, we decided not to consider those runs. This explains why the total number for each optimisation is not exactly 1024 for some scenarios (1017 in the worst case for scenario g.). Moreover, in practical cases, the probability that a user encounters such a bug is quite low; however, in such cases, the user can re-run the optimisation with a different random seed to solve the problem.2H = 0.3 and the other with 2H = 0.7. Because one optimisation required q cores, 64/q optimisations were run in parallel.The optimisations were conducted simultaneously on two computers, each with 256GB of RAM and an AMD Ryzen Threadripper 3990X @2.9GHz 64 cores CPU. One computer performed the optimisations of all scenarios with 33.13.1.1Here, we detail the behaviour of the optimisation using examples.itern = 50 iterations of q = 8 parallel-sampled points at each iteration for all scenarios. The figures we present associate the representative runs of Bayesian optimisation with random optimisation, as defined in the Materials and Methods section (see. 2.4.3).The results presented in this section are obtained from the batch in which the optimisations were repeated 16 times with 3.1.2i, initi, repB, ppheno as variables, whereas the points sampled by the Bayesian optimisation algorithm were active individuals , and the points sampled by random optimisation were supplementary individuals .It is difficult to visualise the parts of the research space that have been explored by optimisation algorithms, because it has four dimensions. To ease this visualisation, we performed principal components analysis (PCA) on the data generated using the optimisations presented in 3.2itern = 50 iterations of q = 8 parallel sampled points. .In this section, we compare for each scenario, the optimised breeding schemes returned by one run of Bayesian optimisation and one run of random optimisation . Thus, we present the “evaluation results” (see. 4.2.1) returned by the optimisations with We present in 2), whereas the highest performance was for scenario f. where it was better 86.43% of the time (885 comparisons out of 322).Bayesian optimisation yielded the lowest performance for scenario e. where it was better 52.2% of the time .Next, we compared for each scenario, all optimised breeding schemes returned by all the 1017 to 1024 runs of Bayesian and random optimisations of As we compared the optimised breeding schemes from different runs of optimisation, the variance of these data was derived from both the stochastic nature of the optimisation algorithms and the stochastic nature of the simulated breeding campaigns. This reflects what a breeder would face by performing one optimisation and then one breeding scheme.2 = 1034289 to ∼ 10242 = 1048576 comparisons , Bayesian optimisation showed the lowest performance for scenario a. where it was better in 57% of the cases and was the best for scenario f. where it was better in 75% of the cases. For half of the scenarios, the Bayesian optimisation results were better in more than 65.6% of the cases.3.3itern = 15 iterations with q = 2 parallel sampled points at each iteration for each scenario. These marginal distributions do not have simple shapes and differ from scenario to scenario. For example, the distributions of the parameter Initi are bi-modal and relatively flat for some scenarios, and the distributions of the parameter i have different expected values.Visually compare the joint distributions of the optimised parameters is not possible as it would require four dimensions. Therefore, a PCA was performed on the optimised parameters of the above results. Like the empirical marginal distributions, the distribution shapes were found to differ for the two scenarios, and to be quite complex. This may suggest a complex covariance structure between the optimised parameter which may also depend on the constraints.4In this study, an optimisation method applied to a simulated breeding scheme was described and tested. Even if, by its design, Bayesian optimisation cannot ensure finding the global maximum, it can rapidly find breeding scheme parameters that yield good results regarding the entire parameter space. Further, Bayesian optimisation outperformed a naive optimisation method.Moreover, the empirical distribution of the optimal parameters found using Bayesian optimisation differed according to the constraints applied, even if the constraints and breeding parametrisation used were relatively simple. Additionally, the shape of those distributions may suggest that some breeding parameters may be more or less important depending on the breeding constraints, and that several local optima may be present which makes this optimisation difficult. This supports the work of  and highAlphaSimR (BreedingSchemeLanguage (BreedSimulatR (The method is also quite flexible, and if a breeding simulation algorithm can provide an objective function, Bayesian optimisation algorithms can theoretically be applied. Currently, there are several computing libraries and software that can ease the creation of breeding simulations, including lphaSimR , BreedinLanguage , and BreSimulatR . TherefoThe optimisation algorithm itself can be adapted for practical use. In this test, because of the nature of the analysis, we used a fixed number of iterations as the stopping criterion for optimisation. However, other criteria, such as ending the optimisation after a specific time or when a set of parameters returns a value above a specific target can also be used.The proposed approach still has some limitations. First, the breeding scheme parametrisation and constraints presented in this paper are simple, and we used only four numeric parameters for parametrisation of the breeding scheme. Moreover, only one selection criterion and one mating method were used, which were not part of the scheme parameters. Further, most generations had the same selection intensity, and there were no constraints on the number of phenotyping or genotyping at each generation. Further investigations should thus be conducted with more complex breeding scheme parametrisation to obtain insights into the robustness of this method. Such studies will require the use of categorical parameters, which could be more difficult to include in a Bayesian optimisation framework, because they usually assume a continuous change in the value of an objective function  Zhang e. MoreoveSecond, optimisation uses breeding simulations rather than actual breeding campaigns. Therefore, it is not possible to guarantee that the results are valid for actual breeding. These results depends on how well the implemented simulation reflects reality.The main sources of divergence can be derived from the simulation algorithm itself and the information used.For example, during our breeding simulation, even if the ßimulated breeder” does not know the true marker effects, all the QTNs related to the phenotypic trait of interest are available in the genetic data used to build the genomic prediction models, which is usually not the case in actual breeding campaign. Additionally, the genetic architecture and probability distributions implemented in the simulation are not the same as those in reality. All the parameters of the simulation like the “real marker effects” and “real linkage map position” may not be known by the user. One could still use values based on estimates and assumptions derived from empirical information, but those estimations can be noisy and/or biased and their effects on the accuracy of thesimulation are not yet known.5This study is one of the first to apply Bayesian optimisation to the design of breeding schemes while considering constraints. The presented approach has some limitations and should be considered as a first proof of concept that demonstrates the potentialof Bayesian optimisation when applied to breeding schemes. It also presents the integration of breeding constraints into the breeding scheme design in aims of optimisation which can provide a basis for further research. Bayesian optimisation applied to simulated breeding campaigns may provide a new tool to breeders that, rather than providing strict guidelines, may give insight to design new breeding schemes. For example, a practical application could be to estimate some marker effects for a particular phenotypic trait, phase the genotypes of a population, write the simulation function of a potential breeding scheme under some constraints, and finally optimize this function. The results returned by the optimizer could be an interesting line of thought when compared with the intuition of the breeder or their expectations. This may provide information which can assist the breeder in making decisions on designing breeding programs.Since the code we wrote for this article is freely available online , and we Further studies, focusing on the sensibility of the presented approach regarding the errors brought by the assumptions of the simulations, would be necessary to improve the confidence on its usage in a more practical manner.The data analyzed in this study is subject to the following licenses/restrictions: This data-set can potentially be shared on request to the authors. Requests to access these datasets should be directed to Hiroyoshi IWATA, hiroiwata@g.ecc.u-tokyo.ac.jp.JD developed the simulation algorithm, wrote the optimisation code, created the figures, and drafted the manuscript. HI conceived and designed the study, provided administrative support, and supervised the study. Both authors read and approved the final manuscript.


Genome‐wide association studies have provided many genetic markers that can be used as instrumental variables to adjust for confounding in epidemiological studies. Recently, the principle has been applied to other forms of bias in observational studies, especially collider bias that arises when conditioning or stratifying on a variable that is associated with the outcome of interest. An important case is in studies of disease progression and survival. Here, we clarify the links between the genetic instrumental variable methods proposed for this problem and the established methods of Mendelian randomisation developed to account for confounding. We highlight the critical importance of weak instrument bias in this context and describe a corrected weighted least‐squares procedure as a simple approach to reduce this bias. We illustrate the range of available methods on two data examples. The first, waist–hip ratio adjusted for body‐mass index, entails statistical adjustment for a quantitative trait. The second, smoking cessation, is a stratified analysis conditional on having initiated smoking. In both cases, we find little effect of collider bias on the primary association results, but this may propagate into more substantial effects on further analyses such as polygenic risk scoring and Mendelian randomisation. When seeking genetic effects through pathways other than through X, the natural direct effect is of interest, whereas effects estimated within strata of X are controlled direct effects. In a study of disease progression within cases, the controlled direct effect is defined for each individual in the population, as the effect on progression if (possibly counter to fact) the individual was to experience the disease.Figure G and X in their effect on Y, the controlled direct and natural direct effects are equivalent , but any suitable variable could be used as an IV, and indeed a nongenetic variable could potentially be analysed as G in Figure For simplicity, we assume that the gene d et al. . To fix 2.1X in Figure As part of their mtCOJO, Zhu et al.  developeSimplified to scalar variables with zero mean, mtCOJO assumes linear models relating the variables in Figure βGY=βGYT−βGXβXY, where βGYT is the total effect of G on Y, the terms in Z and U being absorbed into the residual error owing to the collapsibility of the linear model. Subtracted from the total effect is βGXβXY, which is the indirect effect of G on Y via X. This requires an estimate of the causal effect βXY, which is obtained by MR of X on Y, using Z as the IV. In principle, any MR method could be used to estimate βXY, so many SNPs may be used for Z and need not be valid IVs as long as the assumptions for the particular method are met.The direct effect of interest is G and Z on both X and Y. This approach is useful when those marginal effects are available, as is often the case with summary GWAS data. Where only the collider‐biased conditional effects are provided, mtCOJO cannot be used to adjust them toward unbiased effects. In particular, this approach cannot be used to correct within‐stratum effects, as in case‐only studies of prognosis.The direct effect is obtained from the marginal effects of 2.2G on X:βGYC is the conditional effect of G on Y given X. By conditioning on X, we open pathways between G and U  and instrument effects on the covariate.Dudbridge et al.  developeU Figure , so thatb by a genome‐wide regression of βGYC on βGX, including the intercept. This is logically equivalent to MR‐Egger  the effects on covariate βGX. As discussed below, this assumption applies to the signed effect sizes, which reflect the allelic coding. We will therefore use the acronym InCLUDE (instrument coefficient linearly uncorrelated with direct effect), as opposed to the instrument strength of InSIDE, which does not recognise the dependence on allele coding.Dudbridge et al. did not assume that any SNPs are valid IVs, and estimated b, so that its sampling variance has little effect on the power of testing βGY. They showed that under a positive correlation between direct genetic effects on incidence and progression, the increase in Type‐1 error remains less than for an unadjusted analysis, with a similar level of power.The InCLUDE assumption may be disputed. GWAS have been performed predominantly on the risk of disease, with the premise that associated SNPs indicate targets for treatment, thus assuming some genetic correlation between incidence and progression. The motivation of Dudbridge et al. was to use as many SNPs as possible in estimating the slope 2.3b is then used to adjust the conditional effects as above.In contrast, Mahmoud et al.  set out Slope‐hunter. SNPs that affect X are first identified by p  value thresholding. Model‐based clustering is then used to fit a bivariate normal mixture model to the conditional effects on outcome βGYC and the effects on the covariate βGX. One component of this model assumes a proportional relationship between βGYC and βGX and is assumed to identify the valid IVs.Identification of the valid IVs follows an algorithmic approach called βGYCβGX comes from the valid IVs. This resembles the zero modal pleiotropy assumption of the mode‐based estimator in MR ≠0. Slope‐hunter has the same assumption as mode‐based MR and is similar to MR‐Mix , MR‐Mix fits a univariate model to the residual βGYC−bβGX. Median‐based estimators +(βˆGYiC−βGYiC) includes both the direct effect and sampling error, and E(εGYi)=0. By setting b as a constant, we assume that the confounding is the same for all SNPs, which is approximately true when SNPs have small effects. Note that the sampling variances of βˆGYiC and βˆGXi may differ across SNPs. To allow for variable precision in the βˆGYiC, we can estimate b by weighted least squares, specificallywi is the weight of SNP i, typically the inverse sampling variance of βˆGYiC normalised so ∑wi=1. Note that the residual variance in Equation (var(βGY) and so the inverse sampling variance weighting is not the most efficient estimator of b.Replacing parameters by their estimates, rewrite Equation  as(3)βˆGXi, write bˆ in terms of the true but unknown βGXiεGXi is the sampling error in βˆGXi, assumed to have zero mean. Approximate ∑wiεGXi2 in the denominator by ∑wiσGXi2, where σGXi is the (estimated) standard error of βˆGXi. Then, compared to the estimate based on the true βGXi, the numerator is the same, but the denominator is increased by ∑wi∑wiσGXi2. We, therefore, subtract that term from the observed denominator and obtain the weak‐instrument corrected slope asTo deal with the imprecision in E(βGY)=0, hence α=0. Then,A special case is a zero‐intercept model assuming var(βGY), there is residual heteroscedasticity in Equation . By accounting for that variance in the estimation of b, MR‐RAPS might be more efficient than CWLS.We call this approach corrected weighted least squares (CWLS), and will compare it to MR using the robust adjusted profile score (MR‐RAPS) =0 and the InCLUDE assumption holds. This is the assumption made by MR‐RAPS, and as a simple alternative, we also suggest the CWLS estimator . The underlying intuition is an assumption that all individuals have a latent  outcome Y whether or not X=1, which is modelled by linear regression with a normally distributed error. Conditional on X=1, the error becomes truncated normal with its mean estimated from the first‐stage model and then included in the second‐stage model.In the second step, the fitted probability is included, after transformation, as a covariate in the regressionG and Z. Moreover, the probit model is rarely used in the analysis of single‐SNP data. For these reasons, the Heckman selection model appears problematical if only summary‐level GWAS data are to hand  and writeHowever, when SNP effects on G now has the same form as the conditional effect in Equation  and well‐imputed (r2≥0.98) SNPs identified in our previous study  estimator from standard MR analysis, which can be regarded as CWLS without correction for weak instrument bias. We further applied mtCOJO using the total effects on WHR for comparison with index effect regression.To formally quantify the degree of collider bias, we applied IV methods to 143,000 pruned , using the full set and also with p  value thresholds of 5×10−8, 10−5, 0.001 and 0.05. We again compared CWLS, MR‐RAPS and Slope‐hunter and additionally applied MR‐Mix. MR‐RAPS was performed within the TwoSampleMR R package (version 0.5.6) and MR‐Mix using the MR‐Mix package (version 0.1.0). For MR analyses, SNPs were pruned using default parameters in the TwoSampleMR package . To compare results with those from nonoverlapping samples, we repeated these analyses using only the UK Biobank subjects for smoking initiation  and non‐UK Biobank subjects for smoking cessation .To adjust these associations for collider bias we used summary statistics for smoking initiation from the same study in 632,802 individuals excluding 23andMe. We estimated the regression slope 33.1Properties of the methods have been extensively explored in previous studies , where the alleles are coded to have positive estimated effects on Table Y and nominally significant associations after adjustment for collider bias. For each method, we estimated the probability that the adjusted effect changes sign to either the correct direction or the incorrect direction. The results in Table Finally, we considered how often the direction of effect was changed by adjustment for collider bias. Because estimates close to zero could change sign stochastically, we considered just the SNPs with true effects on the outcome 3.2p≤1) may reflect instability in the clustering algorithm in the presence of many null SNPs for both WHR and BMI. This would support the default procedure of initially thresholding SNPs, but at the cost of increased standard error.Table βGYC, and then with mtCOJO, adjusting the total effects βGYT, which were also available as summary statistics. The adjusted effects from the two methods were very similar for all SNPs, with a correlation of 0.988. The correlation increased to 0.995 among the 681 SNPs associated with BMI at p<10−6. These results give empirical support for the analytic equivalence shown in the Methods section.We also calculated adjusted effects for all 143,000 pruned SNPs first using instrument effect regression, adjusting the conditional effects 3.3p  value thresholds. Again, standard errors increase with stricter p  value thresholding, although weak instrument correction has less effect. Adjusted effect sizes for the 24 associated SNPs are given in Supporting Information: Table p value thresholds less than 1) results consistent with, although less precise than, those of Table Table 4There are strong parallels between the corrections for collider bias described here and the mature field of two‐sample MR. Both approaches aim to estimate a linear relationship between SNP effects on two traits, under IV assumptions on the SNPs. Whereas the same mathematical procedures can be applied to both problems, there are some key differences in context. In the collider bias setting the interest is on the direct effects of SNPs on the outcome trait, with these direct effects expected to exist and being the target of inference. The linear relationship itself is not of primary interest, and we believe that precision is more important than bias in estimating that relationship, to retain the power to detect the direct effects. This contrasts with MR in which unbiased inference of a causal effect is the priority.For this reason, we advocate methods designed for a larger number of IVs. Here, we have used Slope‐hunter, MR‐RAPS and MR‐Mix, but many other methods are now available  or conditional effects βGYC (for instrument effect regression), along with the effects on exposure βGX.The link with MR is explicit in the mtCOJO approach, which subtracts an estimate of the indirect effect from the total association of an SNP with the outcome. In our example of WHR adjusted for BMI, we have shown that this gives equivalent results to instrument effect regression, and both approaches are appropriate for conditioning on a continuous trait, depending on which summary statistics are available. Indeed, both approaches estimate the direct effect of βGX effects, the net bias in a GWAS can be close to zero and it is difficult to characterise a typical bias on individual SNPs. For this reason, we have focussed on error rates as a proxy for estimation bias. In our earlier work, we showed that Type‐1 errors at the nominal level correspond to low absolute bias on parameter estimates (Dudbridge et al., We have presented results for Type‐1 error, power and false discovery rate, reflecting the emphasis on testing over estimation in GWAS. Parameter estimates are nevertheless important for subsequent analyses, including polygenic scoring and MR and are the direct subject of collider bias. Because the bias varies with the In our data examples, the primary results were barely changed by adjustment for collider bias. This is reassuring but should not be taken to mean that adjustment is unnecessary. Estimating the adjustment, even if small, increases the standard error of the estimates and this may play into subsequent analyses.In the situations we have considered, SNP associations on the conditioning trait are readily available, as are the associations with the outcome. However, there are other scenarios in which collider biases are not so amenable to the present methods. In case/control GWAS using prevalent cases, the analysis is conditional on individuals surviving until study recruitment and genotyping, with collider bias arising if there are common determinants of short‐term survival and longer‐term disease. This may be a particular issue for acute events such as myocardial infarction (Hu et al., Representativeness is a common concern in epidemiological studies, but determinants of study participation may give rise to collider bias (Yaghootkar et al., Currently, therefore, the approaches we have described have limitations. Nevertheless, when there may be unknown confounding between a conditioning trait and an outcome of interest, they do offer a useful addition to the existing toolkit of sensitivity analysis, inverse probability weighting and other model‐based corrections. Indeed genetic IV methods may be applied in conjunction with those approaches to provide improved assessment of and adjustment for collider bias. We expect further developments of genetic IV approaches to address collider bias in applications such as secondary analyses of case/control studies and MR of factors affecting disease progression.Supporting information.Click here for additional data file.Supporting information.Click here for additional data file.Supporting information.Click here for additional data file.


The aim of this study was to identify genomic regions and genes associated with the fiber diameter (FD), clean fleece weight (CFW), live weight (LW), body condition score (BCS), pregnancy rate (PR) and lambing potential (LP) of Uruguayan Merino sheep. Phenotypic records of approximately 2000 mixed-age ewes were obtained from a Merino nucleus flock. Genome-wide association studies were performed utilizing single-step Bayesian analysis. For wool traits, a total of 35 genomic windows surpassed the significance threshold (PVE ≥ 0.25%). The proportion of the total additive genetic variance explained by those windows was 4.85 and 9.06% for FD and CFW, respectively. There were 42 windows significantly associated with LWM, which collectively explained 43.2% of the additive genetic variance. For BCS, 22 relevant windows accounted for more than 40% of the additive genetic variance, whereas for the reproduction traits, 53 genomic windows  reached the suggestive threshold of 0.25% of the PVE. Within the top 10 windows for each trait, we identified several genes showing potential associations with the wool , live weight , body condition score  or reproduction traits  of the mixed-age ewes. The genetic improvement of livestock has traditionally been based on phenotypic and pedigree information. Advances in molecular DNA technologies offer the opportunity to increase the rate of genetic gain using genetic markers  ,2. For sIn sheep, GWAS analyses have been performed for economically relevant traits such as the fiber diameter (FD), clean fleece weight (CFW), live weight (LW) and reproduction ,9,10. ThIn Uruguay, few GWAS of livestock species have been undertaken or published ,19,20. TAll animal work was approved by the INIA Animal Ethics Committee (INIA_2018.2).The data were derived from a Uruguayan Merino nucleus flock involved in a genetic program, as described by Ramos et al. ,22. The ® Genomic Profiler™ Ovine 50K panel. Quality control was performed to remove SNPs with a minor allele frequency (MAF) lower than 1% and call rate below 85%, as well as animals with a call rate lower than 90%. After applying the quality control measures, 1133 animals and 40,036 SNPs were retained and utilized in the analyses.Genomic DNA extraction from the blood samples was performed as described by Carracelas et al. . The anihttps://reworkhow.github.io/JWAS.jl/latest/theory/theory accessed on 1 July 2022.Genome-Wide Association Studies were performed utilizing single-step Bayesian regression analyses implemented in the JWAS package . The sofy = vector of phenotypes for the genotyped individuals,β = vector of fixed effects,u = vector of random animal genetic effects not explained by the markers,pe = vector of random permanent environmental effects accounting for the covariance between observations of the same individual,α = vector of marker effects,e = vector of random residual effects,X, Z and W = incidence matrices relating records to fixed, animal and permanent environmental effects,M = genotype covariate .A Bayes C linear mixed model that included the genotyped and non-genotyped ewes was constructed. The model equation for the single-step Bayesian GWAS was as follows for the genotyped animals:y = vector of phenotypes for the non-genotyped individuals,M = genotype covariate matrix for the non-genotyped individuals imputed from the genotyped relatives,n = incidence matrix corresponding to the imputation residual,Zϵ = vector of imputation residuals accounting for errors in the genotype imputation.The model equation for the non-genotyped individuals can be written as:All the other terms are as described in Equation (1).A Markov chain Monte Carlo (MCMC) method was utilized to obtain samples from the posterior distributions of all unknown parameters, including the marker effects. A total of 70000 iterations were run after a burn-in of 5000 cycles and a sampling interval each 10 interactions. The probability that the markers would have null effect was set to 99% (parameter π = 0.99), that is, 1% of the 40036 SNPs (approximately 400 SNPs) was assumed to contribute to the genetic variance.http://www.ensembl.org/Biomart accessed on 10 August 2022)  ∗ 100). The 1 Mb windows that explained at least 0.25% of the genetic variance, which was 5 times the expected proportion of variance (0.05 × 5 = 0.25%), were considered to be the most important regions associated with the trait ,25. The st 2022) . The biost 2022) . Gene oner 2022) . PathwayA summary of the phenotypic records of the traits analyzed is shown in The GWAS results are presented as the proportion of additive genetic variation explained by the windows of 20 consecutive SNPs, as reported by other authors . ManhattFor the wool traits, a total of 35 windows (13 for FD and 22 for CFW) surpassed the significance threshold , biological processes (BP) and/or cellular components (CC). In total, 20 GO terms were enriched . More information from the GO analysis is available in the A gene ontology (GO) enrichment analysis of the genes within the top 10 windows for each trait was performed. GO terms with a The present study reports on chromosome segments associated with economically relevant traits of Uruguayan Merino sheep. The genomic regions of interest on chromosomes 1, 3, 4, 5, 6, 8, 9, 11, 12, 13, 16, 19 and 22 identified in this study contain known candidate genes related to the wool traits, live weight, body condition score and reproduction traits of sheep and other species (see the following paragraphs). This section focused on some of the genes located within the top 10 windows for each trait that explained the largest proportion of the additive genetic variance.Wool follicle regulation involves several genes, including IGF and TGFB . The canThe identification of genes associated with the live weight is of particular interest for sheep breeding programs ,39. SomeIn sheep, the body condition score is an indicator of the available body reserves (fat and muscle) . SeveralThe present study identified two common regions associated with pregnancy and the lambing potential, suggesting that the same genes may play a role in the regulation of these reproduction traits. These regions are located on chromosomes 4 and 16 and contain genes that have previously been associated with several reproduction traits. For example, ADCY1 has been related to pubertal initiation in sheep , fertiliThis work was the first to perform a single-step Genome-Wide Association Study of the production and reproduction traits of mixed-age ewes in Uruguay. As mentioned above, several of the candidate genes detected were also reported in other studies, which provides confidence in our results. A limitation of this study was the small sample size, which affected the power of detection. Future analyses based on larger populations would improve the identification of candidate genes for traits of interest in sheep. Our results will contribute to the Uruguayan Merino genetic evaluation, as some of identified genes are good targets for selection. In addition, the genomic regions identified here should be utilized as targets in further studies.This study performed a single-step GWAS of six traits in a Uruguayan Merino sheep population. A total of 13, 22, 42, 22, 24 and 29 genomic regions were significantly associated with the fiber diameter, clean fleece weight, live weight at mating, body condition score at mating, pregnancy rate and lambing potential, respectively. We detected several genes, some of which were novel, showing potential associations with the wool , live weight , body condition score  or reproduction traits  of mixed-age ewes. These results require validation using a larger dataset before their implementation in genomic selection among Uruguayan Merino sheep. Overall, our findings will be useful for further genomic studies and genetic improvement programs in Uruguay.


The emergence of genome-wide association studies (GWAS) has led to the creation of large repositories of human genetic variation, creating enormous opportunities for genetic research and worldwide collaboration. Methods that are based on GWAS summary statistics seek to leverage such records, overcoming barriers that often exist in individual-level data access while also offering significant computational savings. Such summary-statistics-based applications include GWAS meta-analysis, with and without sample overlap, and case-case GWAS. We compare performance of leading methods for summary-statistics-based genomic analysis and also introduce a novel framework that can unify usual summary-statistics-based implementations via the reconstruction of allelic and genotypic frequencies and counts (ReACt). First, we evaluate ASSET, METAL, and ReACt using both synthetic and real data for GWAS meta-analysis (with and without sample overlap) and find that, while all three methods are comparable in terms of power and error control, ReACt and METAL are faster than ASSET by a factor of at least hundred. We then proceed to evaluate performance of ReACt vs an existing method for case-case GWAS and show comparable performance, with ReACt requiring minimal underlying assumptions and being more user-friendly. Finally, ReACt allows us to evaluate, for the first time, an implementation for calculating polygenic risk score (PRS) for groups of cases and controls based on summary statistics. Our work demonstrates the power of GWAS summary-statistics-based methodologies and the proposed novel method provides a unifying framework and allows further extension of possibilities for researchers seeking to understand the genetics of complex disease. The availability of such alternative sources of information has spurred intense interest into the development of methodologies seeking to leverage such records effectively in order to retrieve as much information as possible. Besides overcoming barriers in individual-level data access, summary-statistics-based methods also offer advantages in computational costs, which do not scale as a function of the number of individuals in the study2.Genome-wide association studies (GWAS) have emerged as a powerful tool, leading to the identification of thousands of common genetic variants that underlie human complex disorders and traits. They also led to the creation of large repositories of human genetic variation creating enormous opportunities for further analysis. However, sharing and transferring of individual-level genotype data is often restricted due to privacy concerns as well as logistical issues. On the other hand, GWAS summary statistics, typically including information such as odds ratio (OR)/effect size (beta), standard error (SE), 4; GWAS meta-analysis and fine mapping9; allele frequency and association statistic imputation11; heritability and genetic correlation estimation15; case-case GWAS16; and polygenic prediction19. Many of these methods have to incorporate additional information from publicly available sources, such as linkage disequilibrium (LD) statistics from a reference population20. Most of the existing methodologies analyzing GWAS summary statistics use the summary statistics  from the input “as is”, often via relatively complicated estimation and modeling.Summary statistics methodologies have been developed to allow a wide array of statistical analyses, including effect size distribution estimationwithout accessing individual level genotype data. Our framework leverages a very straightforward observation: summary statistics information can be expressed as a function of case/control allele frequencies for each SNP. This allows us to recover case/control allele frequencies from summary statistics by solving a non-linear system of equations. Additionally, if one assumes that the SNPs satisfy Hardy-Weinberg Equilibrium (HWE) , the allele frequencies can be used to infer genotype counts. These simple observations allow us to use information from case-control GWAS summary statistics to develop a simple, user-friendly alternative to summary-statistics-based methods for fixed effect meta-analysis and cc-GWAS. Furthermore, we are able to compute group-wise polygenic risk score (PRS) from summary statistics of both a base and a target population. We note that even though there have been summary statistics based methods estimating the variance explained by SNPs using results from existing PRS associations22, to the best of our knowledge, no existing method could return reliable estimates of PRS without any access to individual-level data in the validation cohort prior to our work.The objective of our work is three-fold. First, we seek to unify GWAS summary-statistics-based analyses  under a common framework, as well as determine whether such frameworks can lead to novel analyses using only summary statistics. Second, we seek to compare existing summary-statistics-based analyses to each other and also our newly introduced method. Third, we present a novel approach to compute statistics that can be used to evaluate the performance of polygenic risk scores Here, we describe the mathematical foundations of our framework and its applications to fixed effect meta-analysis, cc-GWAS, and group-wise PRS estimation. We compare the performance of existing methods as well as our novel method using both simulated and real data. Our methods are implemented in the software package Reconstructing Allelic Count (ReACt).(i) the computation of the joint effect of a SNP in a meta-analysis involving multiple studies; (ii) to obtain the mean polygenic risk score of cases and controls in a population; and (iii) to investigate the genetic differences between traits using a case-case GWAS. All of these can be done using only summary statistics, which circumvents the hassle of individual level data sharing and, as an added bonus, considerably reduces the necessary computational time. We start by introducing some notation that will be useful in this section. Let a and u represent effective and non-effective allele counts respectively; let superscripts OR, SE, and N be the odds ratio, standard error (of log(OR), as presented in most of the GWAS summary statistics), and sample sizes obtained from the summary statistics. Thus, for SNP i, i; similarly, i; i in case-control GWAS summary statistics can be expressed as follows:i for effective and non-effective alleles in cases and controls, by solving for the four unknowns OR and SE statistics for each SNP, or perhaps other statistics that can be used to compute OR and SE; it may not be applicable to GWAS reporting other types of summary statistics. Furthermore, these frequencies will be different from those observed from individual level data due to model covariates; the recovered frequencies correspond to the allele counts after corrections have been applied. See Section Our framework is motivated by the fact that the summary test statistics from publicly available GWAS can be expressed as a function of allele counts of the effect and the non-effective allele in cases and controls; as a result, the allele counts can be exactly recovered by solving a system of non-linear equations. Interestingly, this rather straight-forward observation has not received much attention in prior work. Additionally, assuming that SNPs included in GWAS studies are in Hardy-Weinberg Equilibrium (HWE), we can also reconstruct the structure of the genotype vectors for publicly available GWAS studies from just summary statistics. We can leverage this information in multiple applications, including: complete logistic regression on each SNP. Thus, we can essentially proceed with the analysis in exactly the same way as standard GWAS  as well as ReACt on synthetic data under various conditions. The simulation was carried out using the Balding-Nichols model26, assuming a minor allele frequency of 0.3. For each setting, we predefined the risk for effective alleles of the causal SNPs by setting 27. Since the latest stable release of METAL does not include an implementation for sample overlap correction, we used the GitHub version of METAL from28. ReACt allows the user to provide the overlapping sample sizes as an input parameter (ReACt(Exact) in Figs. Z-scores in input GWAS summary statistics from28 (ReACt(Est.) in Figs. First, we tested the performance of two leading methods used for fixed-effect meta-analysis , could also be used to implement cc-GWAS under the same umbrella. We proceed here to describe this implementation and comparison of the two methods.Case-case GWAS (cc-GWAS) based on summary statistics has only very recently been described by Peyrot et al.Similar to our proposed approach for meta-analysis of multiple GWAS datasets using summary statistics, we can also carry out cc-GWAS using regression by simply swapping the labels of the phenotypes. Perhaps the biggest challenge in cc-GWAS is the separation of the differential genetic effects from between-study stratification. To circumvent this issue, we leverage the difference of SNP effects in control groups to estimate the extent of stratification  trait differential SNPs (ii) null SNPs; and (iii) stress SNPs . We compared our statistics for SNPs that were also analyzed in16 and results for this comparison are shown in Table  16 only used summary statistics as input, and could not apply the individual level quality control steps of34. As a result, both methods identified additional significant loci showing divergent genetic effects between BD and SCZ compared to34, mainly due to a much larger effective sample size. Results for all genome-wide significant risk loci are shown in Table Next, we evaluated the performance of our method on real GWAS summary statistics and compared our method with the recently released method oft-test in order to get a p-value comparing the difference of PRS between cases and controls.We realized that our new method opens up a new opportunity for summary-statistics-based analysis which was not possible before: even though we still cannot compute individual level PRS without access to raw genotypes, we observe that, under the additive model, the mean and standard deviation of PRS for a population are just functions of SNP allele frequencies in the target group . After generating GWAS summary statistics for synthetic base and target datasets, we compared the estimated group means and standard deviations using our method (which operates on summary statistics) with the real group means and standard deviations of PRS computed from the individual level genotypes using PRSice236 as the base study and assessing its predicting power on 18,368 independent depressive episode cases and 312,849 ancestry-matched controls in UK biobank. We did not choose the latest MDD GWAS to be a base study because the latest one has included samples from UK biobank. To run ReACt, we generated GWAS summary statistics for the target dataset as described. We compared the estimated PRS statistics using our methods with the real PRS statistics computed using PRSice2. The results are shown in Table t-test p-values. For this experiment, we took into account potential sample overlap between all pairs of base and target studies; see Section 7.We further tested our method on real GWAS data, using GWAS summary statistics for MDD37. Moreover, as our allelic frequency reconstruction framework does not make any assumptions, our approach is unlikely to introduce additional bias into the results. However, just like any other summary-statistics-based method, it is still possible that the performance of ReACt might get affected by preexisting ascertainment bias from the input GWAS.Extracting as much information as possible from easily accessible GWAS summary statistics can help accelerate research that aims to elucidate the genetic background of complex disease, allowing fast sharing of results and datasets while alleviating privacy concerns. In prior work, GWAS meta-analyses and cc-GWAS were treated as separate tasks with different theoretical foundations. In our work, we compare and evaluate leading methods and present a novel framework that unifies analyses under the same methodological umbrella, while expanding capabilities of summary-statistics-based analysis even further allowing, for the first time, group PRS estimation. Our methods do not affect the differential privacy established by sharing GWAS summary statisticsknown sample overlap, whereas METAL’s development version is able to correct for unknown sample overlap. Our work here presents, for the first time, a thorough evaluation of correction for known and unknown sample overlap; our sample overlap correction is theoretically founded and more intuitive compared to previous methods27.In terms of GWAS meta-analysis, we found that all three methods we tested are comparable in terms of power and type I error rates. However, both METAL and ReACt greatly outperform ASSET in terms of running time. The reconstruction of the allelic counts for each SNP in ReACt allows us to run a full logistic regression model instead of doing the conventional inverse-variance weighted fixed-effect meta-analysis, under the assumption of HWE. Our results on real GWAS data showed that just the standard HWE filtering threshold of 16. In our work, we achieve this objective in a straightforward manner: we directly compare the reconstructed allele frequencies of each SNP in two groups of cases, without the need to estimate heritabilities or prevalence of disorders as in16. The fact that case-case GWAS using ReACt analyzes each SNP independently also allows the user to run the analysis even when only a subset of the GWAS results are made available, which is a common scenario in practice and could not be addressed by prior work. Further, we do not need any extra assumptions on the distribution of SNP effects.We further propose a novel perspective on case-case association studies (cc-GWAS), allowing analysis without the need for complicated assumptions or side information apart from sample sizes. To the best of our knowledge, the only prior work on summary statistics based case-case GWAS is16. As also pointed out by16, we do not expect the existence of stress SNPs to be particularly common in practice.ReACt showed good control of type I errors in null SNPs (type II SNPs) given sufficiently large control sample sizes for both input studies. In practice, our experiments demonstrated that we can get accurate results with 2000 controls from each input GWAS, which is a reasonable sample size in modern GWAS. It also shows slightly higher, but under-controlled, type I errors in the stress test SNPs (type III SNPs), which is also observed by the method of16 is that we do not filter for SNPs showing association due to differential tagging effects. While analyzing such SNPs, our method behaves more like a direct case-case GWAS using individual level data.A notable difference between our method and the work oft-test can be carried out instead place of logistic regression, which is commonly used for predictability evaluation when the individual level PRS are available. It is worth noting that, for case-control studies, t-tests and logistic regression are testing the same hypothesis: whether scores generated from the SNP effect of a base study can differentiate individuals in the target study, or, equivalently, whether the base study can predict the case/control status of samples in the target study. We applied our method to summary statistics of eight psychiatric disorders from PGC for predicting group PRS and found the results in general concordance with the genetic correlation obtained by the work of Lee et al.7.Our framework also introduces a novel perspective on case-control PRS. Conventionally, PRS for a target study is only accessible from individual level genotype data. However, we notice that the group means and standard errors of PRS can in fact be estimated using only summary statistics of both the base and target studies. With such statistics available, a p-value based clumping and thresholding (PC+T) approach. However, the methodology underlying the ReACt group PRS can be easily adapted to any other PRS computation model, e.g.,38 .In our work, PRS evaluations use the 22 to quantify the predisposition to a particular disorder that is explained by a certain SNP set. Finally, a notable feature of ReACt is that, theoretically, it can handle known and unknown sample overlap between base and target populations; to the best of our knowledge, this is done for the first time for PRS computations. Sample overlap has long been known as a problem in PRS and our approach provides a good starting point for future work. We do note that, recently, a different method has also been proposed to specifically correct the inflation due to known sample overlap between base and target studies in PRS evaluations with individual level data44. An interesting future research direction would be to combine the results of44 with summary statistic methods such as ReACt.It is worth noting that given SNP effect sizes and weights as input, ReACt does not require the base summary statistics to be from a case-control GWAS because, in PRS computations, the base summary statistics provide the predictor weights and we do not need to convert them back into allele frequencies. This fact makes ReACt easily applicable on any of the aforementioned SNP re-weighting schemes. To date, most PRS improvements target the selection and prioritization of SNPs or the adjusting of the weights to build a better prediction model using the base study. Our work contributes from a different perspective: it allows the user to evaluate the performance of models without access to individual level genotype data. Moreover, results from group PRS using our approach can be further connected with19 and the predictive power of output PRS will be subject to the power of the base study21 and the p-value threshold chosen by the user.Our framework is robust against within-study stratification effects. However, users should keep in mind that general rules of thumb for conventional PRS also apply to our method. For instance, the SNPs used for PRS computations are expected to be independent to a certain extent (clump/prune/LASSO shrink the summary statistics)OR and SE for each SNP. There exist summary statistics-based methods transforming GWAS results obtained from linear mixed model association to odds ratio45, and it will be interesting to further explore how such methods could interface with our approach. Also, our meta-analysis module only investigated the most straight forward application of ReACt in a fixed-effect model. It would be interesting to explore methodologies that carry out random-effect meta-analyses using the same framework. Another interesting topic for future work would be to incorporate information beyond GWAS summary statistics. For example, one could consider incorporating external information such as LD structure using LD reference maps, or eQTL and SNP to gene annotations. Such information could be used to improve the accuracy of sample overlap estimation and to extend the group-PRS applications. Furthermore, although outside the scope of our analysis here, one could investigate expanding towards methods that perform haplotype (instead of genotype) reconstruction. Overall, our work here highlights the power of summary-statistics-based methodology and opens up additional avenues for research.Our work opens many future research directions. First, the reconstruction scheme that our framework is built upon is based on input summary statistics that are generated using a logistic regression or a i to denote SNP number: given, say, M SNPs, i will range between one and M. Similarly, we will reserve the subscript L studies from which summary statistics will be meta-analyzed, L. We assume that all L studies released summary statistics on a common set of M SNPs. For simplicity, we will first describe our methods for the case Prior to introducing our methods, we discuss notational conventions. We will reserve the subscript a to represent counts of the effective allele and the variable u to represent counts of the non-effective allele. We also reserve the variable N to represent counts for the number of cases or controls. Given the above conventions, we now present the following table of allele counts  for SNP i (We will use the three-letter shorthand cse for cases and the three-letter shorthand cnt for controls. We reserve the variable L) Table .Table 8Ti . This approach has been reported with stable performance in both balanced and unbalanced studies, as well as with rare SNPs50.Note that a drawback for logistic regression is that it can produce anti-conservative results under imbalance, which in our case, includes unbalanced sample sizes in cases and controls, as well as unbalanced sample sizes among input studies. We apply Firth bias-corrected logistic regression testp-value for the logistic regression formulation of Eq.  of 0.3 for each SNPs and a relative risk r (r = 1.15/1.2/1.3) for the effective alleles of the causal SNPs in each population. The simulation was carried out under a range of Synthetic data. We used the Balding-Nichols model29. The simulation was carried out using quality controlled genotypes  SNPs with non-zero effect in only one of the studies and zero effect in the other; (ii) SNPs with zero effect in both input studies; and (iii) SNPs with the same non-zero effect size (predefined r) in both input studies. All of the three types of SNPs would suffer from population stratification at a predefined value of For the cc-GWAS, inspired by30. Only independent European ancestry samples identified through PCA and IBD check are included for the analysis. We applied basic quality control filters on those samples, which were removing SNPs and samples with a missing rate exceeding 53.Individual level genotype data. We tested the performance of our fixed-effect meta-analysis method and group PRS method on the depressive episode trait in UK biobank dataset47, correcting for the top ten principal components (PCs) in the case of admixed datasets. For real individual level genotype data, we divided the samples randomly into two equal sized subsets and ran a GWAS on each subset separately to obtain summary statistics for each subset. We performed ten such random iterations in our experimental evaluations. For the fixed-effect meta-analysis, on top of two independent subsets, we also introduced 100/500 sample overlap for synthetic data under the Balding-Nichols model; 5000/10,000 sample overlap for synthetic data under the gcta model; and 500/1000 sample overlap for the real GWAS data on depression.Generating summary statistics. For synthetic data and individual level genotypes, summary statistics were generated using PLink36 as the base study. Most recent large-scale GWAS often include UK biobank as part of the samples. We chose to use an earlier GWAS published before the release of UK biobank data in order to minimize sample overlap between the base and target populations as much as possible. The summary statistics contains in total 1,235,109 SNPs on genome build hg18. After liftover54 to hg19, 1,234,855 remained for the analysis.Publicly available summary statistics. As part of the performance evaluation for our group PRS method, we used summary statistics from an MDD GWAS published in 2013For group PRS and cc-GWAS, we demonstrated the applicability of our methods using publicly available summary statistics. We chose the summary statistics of eight neuropsychiatric disorders made available by the Psychiatric Genomics Consortium (PGC), since the underlying relationships between this set of disorders has been relatively well-studied. Information on the eight summary statistics can be found in Table Fixed-effect meta-analysis. For synthetic experiments using the Balding-Nichols model where all SNPs were simulated independently, results after performing the meta-analysis were compared with the predefined causal variants. Under each experimental condition, we reported the average true positive rate , as well as the false positive rate  out of ten independent iterations.p-value For experiments under the gcta simulator it was unreasonable to report power and type I error rates by comparing with the predefined causal SNPs, since the SNPs were not independent in the input genotypes. Therefore, for this experiment, the performance of ReACt and the other tools was evaluated by comparing results to the outcome of a GWAS on all 50,000 cases and 250,000 controls, where 1,886 SNPs were identified as genome-wide significant  showing, on average, how many times a SNP reported as a “true signal” in the overall GWAS got picked up by each meta-analysis method (true positive) as well as how many extra SNPs each method identified . The performance on real genotype data was also evaluated under 0/100/500 sample overlap. Sample size for each subset under different conditions was 482 cases, 993 controls with no sample overlap; 532 cases, 1043 controls with 100 cases and 100 controls overlap; and 732 cases, 1243 controls with 500 cases and 500 controls overlap.24 and ASSET25, which are both widely used tools for fixed-effect meta-analysis. Note that the latest stable release of METAL does not have the sample overlap correction functionality implemented. Therefore, for performance comparison, we used the development version available on GitHub28.We compared the performance of ReACt in terms of accuracy as well as running time with METALith SNP in the first study and ith SNP in the other study. Since all three types of SNPs suffered from population stratification, we evaluated the performance of ReACt under a challenging scenario. Besides simulation, experiments using summary statistics for schizophrenia (SCZ)33 and bipolar disorder (BIP)32 were also carried out. These two disorders were chosen due to the existence of case-case association study using the individual level genotypes34. We tested ReACt using the summary statistics and compared the results with the existing case-case association study between SCZ and BIP to see whether it could detect possible genetic differences between the two disorders. Since no individual level quality control could be carried out, we expected our results to correspond to a case-case GWAS including 36,989 cases from SCZ and 20,352 cases from all three sub-types of BIP . SNPs on the X-chromosome were excluded from this analysis. Further, to make our protocol comparable to the ones used in31, we also removed variants on the MHC region . From a theoretical perspective, our approach analyzes each SNP independently. Therefore, removing MHC is not mandatory to run ccGWAS using ReACt, unless the study design requires to do so. As a result, a total of 9,018,199 SNPs shared between both summary statistics were used for the analysis. The results were compared in detail with the results reported by the cc-GWAS in16.cc-GWAS. Out of the three types of SNPs generated for the cc-GWAS evaluation see “” sectionp-values strictly less than 36 as the base study .Group PRS. In order to show that our method outputs reliable estimates of the group-wise statistics for PRS without accessing individual level genotypes, we compared the output of our method to the true group mean and standard deviation computed from the individual level PRS on synthetic data, as described in “t-test p-values for the difference between case/control group PRS means. Prior to the group PRS computation, each base summary statistics was clumped using PLink47 using parameters –clump-p1 1 –clump-kb 250 –clump-r2 0.1, with the European samples from 1000 Genome Project as a reference. All PRS values were estimated using independent SNPs with p-values strictly less than Finally we applied ReACt to summary statistics of eight neuropsychiatric disorders (OCD, TS, ED, ADHD, ASD, BIP, SCZ and MDD, see “Supplementary Information 1.Supplementary Information 2.


O. rufipogon accessions using a set of 15,083 high-quality single nucleotide polymorphic markers.The phenotypic data analysis indicated large continuous variation for all the traits under study, with a significant negative correlation observed between grain parameters and agronomic parameters like plant height, culm thickness. The presence of 74.28% admixtures in the panel as revealed by investigating population structure indicated the panel to be very poorly genetically differentiated, with rapid LD decay. The genome-wide association analyses revealed a total of 47 strong MTAs with 19 SNPs located in/close to previously reported QTL/genic regions providing a positive analytic proof for our studies. The allelic differences of significant MTAs were found to be statistically significant at 34 genomic regions. A total of 51 O. rufipogon accessions harboured combination of superior alleles and thus serve as potential candidates for accelerating rice breeding programs. The present study identified 27 novel SNPs to be significantly associated with different traits. Allelic differences between cultivated and wild rice at significant MTAs determined superior alleles to be absent at 12 positions implying substantial scope of improvement by their targeted introgression into cultivars. Introgression of novel significant genomic regions into breeder’s pool would broaden the genetic base of cultivated rice, thus making the crop more resilient.Being one of the most important staple dietary constituents globally, genetic enhancement of cultivated rice for yield, agronomically important traits is of substantial importance. Even though the climatic factors and crop management practices impact complex traits like yield immensely, the contribution of variation by underlying genetic factors surpasses them all. Previous studies have highlighted the importance of utilizing exotic germplasm, landraces in enhancing the diversity of gene pool, leading to better selections and thus superior cultivars. Thus, to fully exploit the potential of progenitor of Asian cultivated rice for productivity related traits, genome wide association study (GWAS) for seven agronomically important traits was conducted on a panel of 346 The online version contains supplementary material available at 10.1186/s12284-022-00582-4. To feed nearly 10 billion people by 2050, agricultural production must be increased by 60% from 2005 base year  and 22 wild species  representing the AA, BB, CC, BBCC, CCDD, EE, FF, GG, KKLL, and HHJJ genome types , spikelet number per panicle , yield  and four QTLs for thousand grain weight  in BC2F1 population derived from O. rufipogon IC22015. Septiningsih et al. , spikelet number per panicle , yield  and 1 QTL, gpl1.1, for grain number. Likely, Xiao et al.  for grain number in BC2 population derived from IRGC 105,491. Fu et al.  at p-value ≤ 1e-4 , pentatricopeptide, OsSCP46—Putative Serine Carboxypeptidase homologue, DEAD-box ATP-dependent RNA helicase. Also, some other loci encoding F-box family protein, OsWAK33—OsWAK receptor-like protein OsWAK-RLP, polygalacturonase inhibitor 3 precursor, OsSCP44—Putative Serine Carboxypeptidase homologue, tetraspanin family protein, harpin-induced protein 1 domain containing protein (DS), Arabidopsis-LEA (LEA) hydroxyproline-rich glycoprotein family/other ortho NHL25, leucine-rich repeat family protein, cytochrome P450, dynamin-2B, OsGrx_S17-glutaredoxin subgroup II, lysine ketoglutarate reductase trans-splicing related 1, hydrolase, alpha/beta fold family domain containing protein, pentatricopeptide repeat-containing protein , pyruvate kinase, hydrolase, alpha/beta fold family domain containing protein, acyl-desaturase, chloroplast precursor were present in the LD region of strongly associated SNPs.For GW, the most significant MTA, S8_24,621,885, on chromosome 8 altered the trait over the mean value by 3.92% and was localized in LOC_Os08g38960 encoding conserved expressed protein. The SNP was also associated with PB and PH at lesser significant levels. Among all the significant MTAs obtained for GW, S4_12374542, on chromosome 4, had the highest effect, which altered the trait by a value of 4.88% over the mean value. SNP S5_28157471, altered GW by 2.44% over the mean, was also associated with HGW and PH at p-value of 0.005, altering it by 2.61% over the mean value. SNP S2_2875772, strongly associated with HGW was also associated with GL and PH, was in LOC_Os02g05830, encoding ribulose bisphosphate carboxylase small chain, chloroplast precursor with highest FPKM values reported in embryo. Another SNP, S2_3873759, present on chromosome 2, was also associated with GL. Three more SNPs, S4_26914103, S4_31316844, S4_35115087, altered the trait by ~ 6%. SNP S4_26914103 were part of LOC_Os04g45480, encoding heat shock protein with highest reported FPKM values in seed. SNP S5_24316574 and S11_19062952, strongly associated with HGW, were localized in LOC_Os05g41530 and LOC_Os11g32260, encoding ZOS5-11-C2H2 zinc finger protein and lysosomal alpha-mannosidase precursor, respectively.The most significant SNP for HGW, S4_4499266, on chromosome 4 also had the highest effect. Two loci, LOC_Os04g08390 and LOC_Os04g08410, encoding Leucine Rich Repeat family protein, and ELM2 domain containing protein were present within 10 kb region of the SNP. The latter locus had its highest FPKM expression value reported in anthers. Also, this SNP was associated with GL at p-values have been presented in Table The allelic effects of significant MTAs for each trait were evaluated using Kruskal–Wallis test and their chi-square values along with O. rufipogon accessions possessing superior allelic combinations for CT, PB, PL, HGW, GL and GW, at significant genomic regions were found to be 5, 13, 1, 3, 34 and 1, respectively  threshold, lead to loss of true rare SNPs. Therefore, ideally an association mapping (AM) panel with minimum genetic structure or accounted genetic structure in models is employed. Due to absence of strong differentiation in the current AM panel, no structure co-variates were used in the GWAS model in the current study. Generally, a test is statistically significant if the p-value is smaller than the pre-defined alpha value. Since GWAS is based on hundreds to thousands of multiple comparisons/testings, the average probability of false positives increases dramatically. For choosing an appropriate significance threshold that distinguishes false positives from false negatives, many corrections have been demonstrated. Out of them, Bonferroni correction is too stringent, gives a very conservative p-value threshold, resulting in a huge loss of power and leads to loss of true positives and underrate the goal of genome-wide studies. In our case, LD based determination of significance threshold was also too stringent. Therefore, another threshold was defined based on minimum Bayes Factor using the formula mBF =  − e*P*ln(P) as documented by Wakefield  to affect rice panicle branching, explaining association with PB. Most of these MTAs also showed association with PH and PB at p-value ≤ 0.05, evident from positive correlation of CT with PB and PH. For PL, 60% of the significant SNP associations obtained in the present study localized in previously identified QTLs/genes pl2a, qPL-3–2, qPL-6, AQCU085 and AQCU149 as reported by Zhuang et al. , whose overexpression in roots led to increased grain yield under drought stress , AQEO021 , culm thickness (CT), panicle length (PL), number of primary branches per panicle (PB), grain length (GL), grain width (GW) and hundred grain weight (HGW). Data for HGW was recorded in all the three years, while all the other traits were recorded in two years, 2014 and 2015. Briefly, PH and PL was recorded from two different plants and four panicles per accession. The culm thickness was measured from four and six plants respectively with a Vernier caliper. The number of primary branches were counted manually from four panicles. Grains were dehulled and grain parameters; GL and GW, were recorded for 10 grains/accession with grain analyzer. Grain weight was recorded for hundred grains with an electronic weighing balance.2) on a line mean basis was calculated.The phenotypic data was statistically analyzed in R version 3.4. Distribution of averaged phenotypic data was checked by plotting histogram using hist function and by Shapiro–Wilk test. Statistical analysis of phenotypic data was done in R using lme4 package . Restriction enzyme ApeKI was used to generate GBS library.Large scale DNA was isolated from each accession from 10-day old leaves using Cetyltrimethyl ammonium bromide (CTAB) method  and AMOVA were calculated by stamppFst function of StAMPP package  and StrAuto program was used to investigate population structure of 346 Endelman . PCA wasEndelman , based oEndelman . The lenEndelman  based onEndelman . The outEndelman  in R. Oty Wright  and upday Wright . The numGenome-wide association study (GWAS) was carried out for seven traits, namely, PH, CT, PB, PL, GL, GW, HGW in R using GAPIT 3 (Wang et al. Additional file 1: The detailed information of Oryza rufipogon accessions investigated in current study along with their countries of origin.Additional file 2: Genotypic data of studied 346 accessions.Additional file 3: Map positions of genotypic dataset.


Migration is a complex phenotypic trait with some species containing migratory and nonmigratory individuals. Such life history variation may be attributed in part to plasticity, epigenetics, or genetics. Although considered semianadromous, recent studies using otolith geochemistry have revealed life history variation within the critically endangered Delta Smelt. Broadly categorizable as migratory or freshwater residents, we examined Restriction site Associated DNA sequencing data to test for a relationship between genetic variation and migratory behaviors. As previously shown, we found no evidence for neutral population genetic structure within Delta Smelt; however, we found significant evidence for associations between genetic variants and life history phenotypes. Furthermore, discriminant analysis of principal components, hierarchical clustering, and machine learning resulted in accurate assignment of fish into the freshwater resident or migratory classes based on their genotypes. These results suggest the presence of adaptive genetic variants relating to life history variation within a panmictic population. Mechanisms that may lead to this observation are genotype dependent habitat choice and spatially variable selection, both of which could operate each generation and are not exclusive. Given that the population of cultured Delta Smelt are being used as a refugial population for conservation, as a supply for wild population supplementation, and currently represent the majority of all living individuals of this species, we recommend that the hatchery management strategy consider the frequencies of life history-associated alleles and how to maintain this important aspect of Delta Smelt biological variation while under captive propagation. Oncorhynchus) is atypical (The causes of migration may be viewed as both proximate and ultimate . UltimatOncorhynchus mykiss) has been demonstrated  has been broadly described as semianadromous , to investigate several important questions regarding the potential genetic underpinning for the MIG behaviors of Delta Smelt and related implications for species management and conservation. Specifically, we examine whether there is evidence for a genetic basis of life-history variation within Delta Smelt, how any such genetic basis might be characterized, and how well the MIG phenotype of an individual can be predicted from its genotype. Results of this work illuminate the control of MIG behaviors in Delta Smelt and how such phenotypic diversity might be incorporated into culture and supplementation strategies.Nu Plasma HR from Nu Instrument Inc.) interfaced with a Nd:YAG 213 nm laser (New Wave Research UP213) (LA–MC–ICP–MS). Strontium isotope ratios (87Sr/86Sr) were obtained from the core to the ventral edge of the otolith, representing the entire lifespan of the fish. Individuals expressing an FWR phenotype exhibit 87Sr/86Sr profiles with values remaining below 0.7075 across the entire profile (entire life span), corresponding with practical salinities <0.5 (i.e. “freshwater”)  with the Burrows–Wheeler Aligner using the MEM algorithm . ResultiWe searched for a signal of neutral genetic structure within sampled Delta Smelt with PCAngsd through principal component (PC) analysis . PCAngsdP(T), that there are 25 genetic variants that may contribute to life history variation within the GWAS, P(T) = 25/number of variants. The significance level can be calculated by the formula: α = Prior Odds * 1/Posterior Odds (e.g. P(T)/(1− P(T)), posterior odds is calculated based on 95% certainty of observing a significant effect that is real (0.95/(1− 0.95)) and an upper bound of power (1) is used.Genetic variants associated with life history variation were identified through a genome-wide association study (GWAS) implemented in ANGSD. We used a generalized linear framework (-doAsso 2) and provided sex as a covariate (-cov). We applied the following options to the program to provide quality control -minMapQ 20, -minQ 20, -SNP_pval 1e6, and -minInd 91. We specified a SAMtools genotyping model (-GL 1) and -minCount 2. To determine a significance threshold, we do not use a Bonferroni correction as sites in the genome are not independent. Instead, we calculated a significance threshold in a Bayesian framework , hierarchical clustering, and machine learning approaches. With these approaches we used called genotypes generated by ANGSD (-doGeno 2) producing a dataset of 0, 1, 2 coded variation. We applied the same quality control thresholds as used previously, with the following changes: an MAF specified (-minMaf 0.05); posterior cutoff specified (-postCutoff 0.9), and a 90% missing individual threshold (-minInd 109). The genotypes were uploaded into R for further analysis .heatmap.2 function of the gplots package. Clustering of samples and loci was done with Ward’s distance, and missing data were not treated. The same set of 200 most-associated SNPs was then analyzed with a k-nearest neighbor (knn) approach for classification after treating missing data with the na.roughfix function of the randomForest package. We divided our dataset into a training dataset of 30 randomly selected FWR and 30 randomly selected MIG fish. With the training dataset, we identified a best k with a repeated k-fold cross validation using the trainControl and train functions of the caret library. A best k was identified as being most accurate after 100 sampling events and 10-folds for odd values of k from 1 to 29. The selected best k was then applied to all samples for the top associated variants and a cross table of accuracy computed with the CrossTable function of the gmodels library.DAPC identifies the largest axis of variation between predefined groups. We conducted DAPC using the adegenet package in R  with MIGAnnotated genes located near SNPs distinguished by GWAS were identified to provide insight into the functional relevance of the polymorphisms to MIG phenotypes. We determined first the extent to which linkage disequilibrium (LD) operates in the Delta Smelt genome given our data, and then obtained annotated features within that window. Descriptions of genes, Gene Ontology (GO) terms, GO names, and GO definitions were obtained with the biomaRt package in R by querying the zebrafish dataset (drerio_gene_ensembl) .SbfI restriction site is 16,578. Genotype likelihoods were identified from 11,638 RAD loci ; while others complete their life cycle entirely within freshwater habitats, foregoing migration (FWR) , howeverH. pretiosus), a widely distributed marine species  also contributed the greatest to the separation of FWR and MIG fish through DAPC . As a whscl6a3), which functions in transmembrane sodium transport  as well as a scl6a3 specifically being identified as a candidate gene from freshwater and marine divergence  is located near a highly associated SNP and is a key regulator of immune response (tfcp2l1). This transcription factor is not well-characterized in fishes, however, the mouse ortholog is known to function in the development of kidneys and the diversification of both intercalated and principal cells regulating acid–base and salt–water homeostasis (The transcription factor nuclear factor of kappa light polypeptide gene enhancer in B-cells 1 (response . MIG fisaamdc) was not assigned GO terms in ENSEMBL, but the UniProt database indicates a function of positive regulation of fat cell differentiation (https://www.uniprot.org/uniprot/Q502H1). Adiposity is previously been linked to MIG behavior in fishes. Examples include vestigial like family member 3 (vgll3) which is an adiposity regulator that exhibits a high degree of control of age-at-maturity in Atlantic Salmon   are found in the key chromosomal inversion underlying divergent MIG phenotypes in Rainbow Trout . Monitoring of Delta Smelt life history adaptive genetic variation would be best served through the identification of genetic or structural variants more closely linked to life history variation through whole-genome sequencing and corroboration across more brood years of Delta Smelt.Though informative, here we only examined a single cohort of Delta Smelt. The importance and prevalence of these associations through time are unknown and genome coverage was not complete. Furthermore, patterns in the relative abundance of each phenotype are likely to covary with changes in environmental conditions that may exert phenotype-specific effects on survival and behavior. For example, variation in freshwater outflow is known to affect dispersal, habitat suitability, and population dynamics of several species . Laborathttps://github.com/MacCampbell/delta-smelt. Sequence data have been deposited with NCBI into the Sequence Read Archive under BioProject PRJNA834474. Otolith microchemistry profiles and assignments are available as part of the Bureau of Reclamation Directed Outflow Technical Report #3 (https://www.usbr.gov/mp/bdo/directed-outflow.html).Sample metadata and scripts for analysis and generation of figures in this manuscript are available at G3 online.jkac133_Supplementary_Document_S1Click here for additional data file.jkac133_Supplementary_Document_S2Click here for additional data file.jkac133_Supplementary_Document_S3Click here for additional data file.jkac133_Supplementary_Document_S4Click here for additional data file.jkac133_Supplementary_Figure_S1Click here for additional data file.jkac133_Supplementary_File_S1Click here for additional data file.jkac133_Supplementary_AppendixClick here for additional data file.