The improvement of crop yield is a major breeding target and there is a long history of research that has focussed on unravelling the mechanisms and processes that contribute to yield. Quantitative prediction of the interplay between morphological traits, and the effects of these trait-trait relationships on seed production remains, however, a challenge. Consequently, the extent to which crop varieties optimise their morphology for a given environment is largely unknown. This work presents a new combination of existing methodologies by framing crop breeding as an optimisation problem and evaluates the extent to which existing varieties exhibit optimal morphologies under the test conditions. In this proof-of-concept study using spring and winter oilseed rape plants grown under greenhouse conditions, we employ causal inference to model the hierarchically structured effects of 27 morphological yield traits on each other. We perform Bayesian optimisation of seed yield, to identify and quantify the morphologies of ideotype plants, which are expected to be higher yielding than the varieties in the studied panels. Under the tested growth conditions, we find that existing spring varieties occupy the optimal regions of trait-space, but that potentially high yielding strategies are unexplored in extant winter varieties. The same approach can be used to evaluate trait (morphology) space for any environment. Crop plants are often studied from the perspective of how to increase yield. Yield is known to be a complex trait, depending on multiple genes, plant architecture and plant-environment interactions. Total seed yield can be decomposed into multiple simpler yield components or yield traits . Based oCrop improvement through genetic selection corresponds to the choice of traits with the highest yield. This process can be viewed as a search for the point on the manifold which results in the highest yield potential. To achieve this, trait space is sampled by iterating the two steps of 1) making crosses between chosen parent varieties, and then 2) selecting within the progeny based on desired attributes for further evaluation . These sCausal relationships due to physiological linkage between traits within an individual plant have been experimentally demonstrated through perturbation experiments –6, 16–18Complex, non-linear trait-trait interactions mean that i) a particular trait value may be beneficial for yield in combination with one set of traits, but detrimental given another , ii) thaWhen a theoretical ideotype is identified in terms of a single desirable trait, then the heritability of the target trait must be confirmed to ensure it can be produced through selective breeding. When an ideotype is defined in terms of multiple traits, the independent heritability of these traits may be important to ensure that the ideotype can be produced through selective breeding. Sources of variation in a trait can be partitioned into genetic, environmental and stochastic variability . HoweverGenome Wide Association Studies (GWAS) are used to identify candidate alleles with genetic variation underlying yield traits. Trait-trait relationships also have complicating effects on these studies, and it has previously been shown that crop models can be used to constrain genomic prediction models to improve their performance , 30. MedHere, we explore the consequences of causal trait-trait relationships for crop improvement by modelling their effect on predictions for yield improvement, trait heritability and genetic association studies. Oilseed rape is an excellent system for studying trait-trait relationships due to its complicated and plastic developmental morphology –3, 35, aB. napus diversity set consisted of 94 varieties [The studied arieties , 41. Thearieties . For GWAPlants were pot-grown as described in  arrangedTransformations were applied to each trait to make them more normally distributed, To learn the structure of the relationships between the measured traits, Sequential Path Analysis models  were sepThe inferred trait relationship structures for spring and winter oilseed rape , with its parent nodes the corresponding independent variables (X). Their relationship was modelled using Gaussian Process (GP) regression,N denotes a normal distribution over f with mean μ and covariance K. The covariance matrix K was defined using the Automatic Relevance Determination kernel,q is the number of parent trait nodes (independent variables), and δij is Kronecker’s delta (1 if i = j and 0 otherwise). GP hyperparameters α, ρ, σ were estimated through regularised maximum marginal likelihood, with priors α ∼ N, ρk ∼ IG, σi ∼ N. To estimate posterior probabilities of dependent variables, No-U-Turn sampling was carried out, implemented in Stan [Each trait (node in the DAG) was considered as a dependent variable  sampled. These sampled predictions were then used as inputs to the next level of GP models, (with response and explanatory variables as defined by the DAG structure), to estimate the values of the modified-trait’s grandchildren and so on, respecting the dependencies identified in the DAG, until all descendants of the modified node were predicted. This sampling approach was used to allow uncertainty in predictions to correctly propagate through the DAG structure. Traits which were not descendants of the modified trait were held to their observed To identify optimal crop ideotypes for maximum seed yield and propose other promising regions of trait space for exploration, we used a Bayesian Optimisation framework. As the “surrogate model” we used sub-models of the GP models for predicted seed yield described above, such that seed yield was predicted only from its parent traits shown in .f(x) is the predicted value of seed yield at x, and x+ is the observed point in trait-space with the maximum seed yield. Expected Improvement thus allows us to balance exploitation of regions with high predicted seed yield, with exploration of regions with high posterior uncertainty. Proposed trait values were constrained to be between the maximum and minimum observed values. Because generating crosses is costly, but parallelizable, it is desirable to be able to propose multiple optimal points for exploration given the current observed data. In order to obtain an approximately optimal design for q proposed sampling points, we used the “Constant Liar” heuristic strategy proposed in [The relationship between seed yield and these parent traits was modelled by GP regression as before. As the “acquisition function” for maximisation we used Expected Improvement (EI)  defined posed in . BrieflyRelationships between the traits exist in the observed data, and so it is unlikely that all points of trait-space can be occupied by a real plant. To incorporate correlation between the parent traits as constraints in the optimisation of proposed points, we followed the approach proposed in . An indePublicly available RNA-seq derived SNPs , were fiSTRUCTURE , and Str-6, which results from a false discovery rate of 0.1, correcting for multiple testing following [The significance of associations between SNPs and traits was based on the threshold p < 5.8x10ollowing  with a pollowing .See Supplemental Methods  for detahttps://github.com/AlexCalderwood/identification-of-ideotypes, together with exemplar input and output data files.All code used in data processing, modelling and analysis is made available at Oilseed rape exhibits complex morphology with many traits which might be expected to affect seed yield. We measured 27 traits in spring and winter oilseed rape to examine the relationships between them and with seed yield. We modelled spring and winter oilseed rape varieties separately, as studies conducted in spring type and winter type oilseed rape have reached different conclusions regarding which features are key for crop yield , 37, 42,From the correlation matrix it can be seen that many traits are statistically dependent , yet theWe used the inferred graph structures to model the effect that changing yield traits can be expected to have on seed yield in spring and winter oilseed rape, accounting for non-linear, and interactive effects see . Fig 4 sSurprisingly, we see little clear evidence that individually modifying many previously identified yield traits will affect seed yield. The traits which are not predicted to affect Seed Yield can be divided into several classes based on whether they correlate with seed yield , and wheThe traits whose alteration are most confidently predicted to modify seed yield are the number of secondary inflorescences (NumberSecondInfl), pod length (PodLength), total seed number (SeedNumber), seeds per pod (SNPP), thousand grain weight (TGW) and seed area . InteresHaving identified the traits expected to affect seed yield when varied individually, we were interested to identify crop ideotypes in terms of multiple traits simultaneously accounting for interactions between them. To identify ideotype plants for our experimental conditions, we performed Bayesian optimisation over trait values to maximise the expected improvement (EI) in seed yield see . The EI If the empirically observed relationships between the yield traits are ignored, then unsurprisingly, optimisation finds many ideotype plants that are expected to have higher seed yield than the experimentally observed plants in both the spring and winter oilseed rape panels . PredomiConversely, among winter oilseed rape, only one observed plant yields higher than the mean expected seed yield of the best hypothetical point. This suggests that the optimal yield trait region under these growth conditions may not have been fully explored by varieties in the winter oilseed rape panel. In winter oilseed rape increased seed number is expected to lead to increased seed yield, and the best ideotypes produce the most total seeds (seedNumber) of small size (TGW). Ideotype production on the main inflorescence is suggested to be similar to that observed among the experimentally high yielding plants. The best ideotypes have approximately 90 pods on the main inflorescence and produce 0.75 to 0.9g of seed per 10 pods on the main stem (seedWeight M). Whilst the identified ideotypes respect the trade-off between seed number and TGW, the most promising ideotype plants occupy the upper edge of the cloud of experimentally observed plants in both these metrics. This means that they are expected to be able to produce heavier seeds than observed plants which produce the same number of seeds, and more seeds than plants which produce the same TGW. They have high values for traits associated with increased photosynthetic capacity relative to the winter oilseed rape average. Among the three highest yielding ideotype plants, the number of secondary inflorescences is between 10 and 12, (compared to an average of 8 among winter oilseed rape), and more than 850 pods on secondary branches. This overall strategy is qualitatively similar to that of the single best performing observed plant.When choosing traits for genetic selection, it is important to ascertain that genetically controlled variation in the trait exists within the available breeding material. However, we find that many of the measured traits can be well predicted from observations of their parent traits . This imWhen traits can influence each other through some physiological link, a gene may have an indirect causal link to the (child) trait of interest, mediated by its direct effect on intermediate (parent) traits . This isUnder simulation, we find that when variation in the child trait is predominantly caused by variation in the parent trait . Highly connected networks were inferred for both spring and winter oilseed rape, supporting the view many yield traits affect each other.We inferred the trait network structures that are most parsimoniously able to explain the empirically observed trait-trait relationships, consistent with prior beliefs about trait relationships. Sequential path analysis may be viewed as an intermediate between statistical yield modelling (in which a trait’s relationship with yield is inferred from the data only) and mechanistic crop models , pod length (PodLength), total seed number (SeedNumber), seeds per pod (SNPP), seed area (SeedArea) and thousand grain weight (TGW).Identification of the individual traits whose modification is expected to alter seed yield ignores potential interactions between traits in the function mapping traits to yield. We therefore identified the points in the multidimensional trait space which maximise the expected improvement in seed yield over the best observed yield in each of the spring and winter oilseed rape panels. The identified points define “ideotypes”—hypothetical plants which are expected to produce high seed yield based on a model of the way that yield traits interact to produce seed yield.In spring oilseed rape, we found that when observed trait trade-offs were imposed as constraints, many of the observed plants were higher yielding than the best ideotypes identified (which are required to be unlike observed plants). This implies that the high yielding regions of trait space are already well explored by the existing varieties, and it is not expected that other plants can be defined in terms of these experimentally observed traits, which are likely to yield better under these growing conditions. Therefore, our modelling suggests that under these conditions, spring oilseed rape plants should ideally produce 35 to 60 pods on the main branch, approximately 150 to 250 pods should be produced on secondary branches, seed number should be > 4000, seed weight per 10 pods on the main branch should be > 0.6g, and TGW should be approximately 3g to 5g. In winter oilseed rape, ideotypes were identified which are expected on average to perform better than all but the single best observed plant. This suggests that regions of the trait space exist, which are likely to yield better than the existing varieties when grown under these conditions. This difference between spring and winter oilseed rape may reflect greenhouse conditions being more like the conditions that spring oilseed rape has been selected for. The best identified winter oilseed rape ideotype has 90 pods on the main branch, and approximately 900 pods across 11 secondary branches. It produces just under 6000 seeds in total, with a relatively small TGW of 4g. ovule area, and seed weight per 10 pods on the main branch is approximately average for winter varieties. Interestingly, many optimal trait values are intermediate rather than extremely high or low. This is consistent with the largely non-linear, saturation type relationships individual traits exhibit with yield. These results suggest that the linear models widely used in yield-trait association modelling may be misleading in recommending selection either for or against a particular trait to improve seed yield. Instead, for the majority of traits an optimal region exists, with suboptimal regions both above and below it.In identifying ideotypes through optimisation, we have constrained correlation relationships among yield traits to be the same as those observed in the experimental plants. This assumes that the observed relationships have unavoidable, physiological causes (such as competition between traits for seed filling capacity) rather than correlated underlying genetic causes which could potentially be broken through selection. We have shown that if the observed correlations between traits can be broken, then large increases in potential seed yield may be possible. However, it is doubtful that the key relationships can be relaxed, as it is generally considered that in oilseed rape seed number and seed size directly compete for seed filling resources .The importance of different traits in determining yield varies with environment . Here, tYield is a complex trait and its decomposition into contributing yield traits leads to non-linear relationships between these yield traits. Ideally, we would be able to infer associations between genetic variants and seed yield directly from available data. However, we have shown that causal associations between traits mean that genes which affect each trait directly also affect its descendent traits indirectly. This results in accumulating genetic complexity as the trait hierarchy is descended, and consequently relatively large sample sizes are required to provide the statistical power to identify genetic variants which affect yield as a relatively large numbers of genes with relatively small effects can be expected compared to upstream traits. By factorising the effect of the genome on yield into its effects on each upstream yield trait, the problem is simplified, and association can be detected through less statistically powerful experiments assuming known trait-trait relationships. As we have seen, many other traits in oilseed rape besides yield are also “complex”  and can also benefit from a decomposition approach. By testing for association to the residuals of a trait-trait model, rather than to observed traits directly, the effects of trait-trait relationships are controlled for, and the undesirable accumulation of genetic complexity can be avoided. Additionally, by using the modelled trait-trait relationships to identify mediated pleiotropy, genes directly regulating the trait of interest can be identified, rather than its parent traits.The potential benefit of correcting for causal trait-trait relationships depends on the relative contribution of direct, physiologically caused trait relationships over classical pleiotropic gene action, in which a gene independently influences two traits. Whether a link is directly causal (rather than due to a confounding latent variable) cannot be distinguished without either perturbation experiments or making the uncounfoundedness assumption—that all relevant variables are observed and included in the model. The requisite perturbation experiments generally have not been performed within the context of yield trait association studies , 62–70. Overall, we have shown the importance of accounting for trait-trait relationships in many facets of applied plant science, from identifying crop ideotypes, to estimating which traits can be selected for, to identifying gene-trait association. Taking these relationships into account is becoming increasingly tractable due to the introduction of phenomics platforms, allowing the simultaneous measurement of numerous traits. The analysis presented here exemplifies a novel methodological framework to make best use of this emerging experimental design.S1 FigPrincipal component analysis was carried out using the 108,653 SNPs detected by Havlickova et al., 2017, to check that the identified SNPs reflect the expected population structure.(TIF)Click here for additional data file.S2 FigQQ plots are shown for models associating SNPs to the trait directly (raw), and to the residuals of models in which the trait is predicted from its parent traits.GLM-Q, GLM-PCA5d, GLM-PCA10d are general linear models (GLM), which use the SNP data, as well as either the population structure matrix (Q), first five (PCA5d), or first ten principle components (PCA10d) of the SNP matrix. MLM-K, MLM-Q+K, MLM-PCA5d+K are mixed linear models (MLM) which use the SNP data, as well as the Q-matrix, PCA components, or kinship matrix (K). A Bayesian-information and Linkage-disequilibrium Iteratively Nested Keyway (BLINK) model was fit using the GAPIT package implementation. Based on these plots, GLM-PCA5d was used for NumberPods M, and GLM-PCA10d was used for OilContent.(TIF)Click here for additional data file.S3 Figa) silique beak length, b) pod length on other yield traits and seed yield in Winter oilseed rape. Changing beak length is not expected to much affect yield. Modifying pod length is expected to strongly affect both beak length and seed yield. Consequently beak length and seed yield are correlated. c) a trade-off exists between seed number and thousand grain weight (TGW) in both Spring and Winter oilseed rape.Predicted trait values (y-axis) as traits a singly varied (x-axis). Median prediction shown with a heavy line, confidence quantiles are shown with 10% intervals. The predicted effect of modifying (TIF)Click here for additional data file.S4 Figa) Spring and b) Winter oilseed rape panels. Coloured points show mean predicted seed yield for identified hypothetical ideotype plants with the indicated yield trait values. Their order reflects their probability of having greater seed yield than the best experimentally observed plant, with 1 having the best chance, and 10 the worst of the calculated ideotypes. These points are identified through Bayesian optimisation as the unobserved points with the best chance of exceeding the maximum observed seed yield in each panel. Thin black line shows the maximum observed seed yield produced by any of the plants in the experimental panels, thick black line shows mean observed seed yield. Black dashes show yield trait values for experimentally observed plants. If expected improvement in seed yield is maximised without the constraint of respecting observed correlations between yield traits, then more optimal regions of trait space exist than the regions occupied by observed plants for both spring and winter oilseed rape. As might be expected, this is largely via breaking the negative trade-off between seed size, and seed number in both Spring and Winter oilseed rape.Facets show traits which affect seed yield directly in the (TIF)Click here for additional data file.S5 Figa) Spring oilseed rape, b) Winter oilseed rape. Variance in the trait explained by the parents is given above each plot. It can be seen that variation in many traits can be well explained by their parent traits, indicating that this variation may not be due to direct genetic or stochastic variation in the trait itself, but instead due to variation in its parent traits.Plots show the mean predicted trait values vs observed trait values for each plant in (TIF)Click here for additional data file.S6 Figa) Model from which simulated data were generated for each plant independently (j). Parent-trait (p) is the weighted sum of “parent SNPs”  which affect it directly. Child-trait (c) is the weighted sum of “child SNPs” , as well as the parent trait. So, it is affected directly by the child SNPs, and indirectly by the parent SNPs. Parent SNPs will therefore exhibit mediated pleiotropy. Noise  was added to both traits. “Non-causal SNPs”  do not affect either trait. All SNPs were independently sampled from a Bernoulli(0.5) distribution. b) The number of “parent SNPs”, “child SNPs” and “non-causal SNPs” statistically associated with variation in the child-trait, using either observations of the child trait directly (red), or correcting for trait-trait relationships, by using the residuals of a model in which child-trait was predicted by parent-trait (blue). (See methods section for details of associated SNP inference). In all tested cases, , the power to detect child SNPs is greater when the effect of p on c is controlled for. The greater the value of γ, the bigger the difference. When γ is large relative to σ, parent SNPs are identified rather than child SNPs due their indirect effect . When γ is similar to σ, association using the direct observations of c was less able to detect any directly or indirectly causal SNPs . Neither method was more associated with spurious identification of non-causal SNPs.(TIF)Click here for additional data file.S7 FigThe measures of yield shown are “SeedYield” (weight of seed produced per plant), or “SeedYield (kg / hectare)”. Lu et al, Jeromela et al, Aytac & Kinaci 2003 & 2004 use Winter oilseed rape panels. Chen et al use Spring and Winter oilseed rape, the remainder use Spring oilseed rape panels. Years indicate repeated trials in the same study, with the exception of Khan 2000, Khan 2006 which are separate studies. Sabaghnia et al., and Ivanovska et al., alter environmental conditions within a study, either experimentally or through trial location. Referenced studies are –68, 70.(TIF)Click here for additional data file.S1 TableList of macrotrait (n = 5) and microtrait (n = 3) names and abbreviations measured in the diversity set population.(DOCX)Click here for additional data file.S2 TableList of transformations applied to normalise trait distributions. Id = identical (no transformation applied).(DOCX)Click here for additional data file.S3 TableStatistical significance of genotype effect on measured traits estimated by one way ANOVA. Reported p-values were adjusted for multiple hypothesis testing by Benjamini-Hochberg method. Broad sense heritability Click here for additional data file.S1 FileThe ASSYST code, genotype names, crop type description and the 4 oilseed rape groups are presented.(XLSX)Click here for additional data file.S2 File(XLSX)Click here for additional data file.S3 FileHeritability estimation and causal SNP identification.(DOCX)Click here for additional data file.


Xinggao sheep are a breed of Chinese domestic sheep that are adapted to the extremely cold climatic features of the Hinggan League in China. The economically vital reproductive trait of ewes  and productive traits of lambs  are expressed in females and later in life after most of the selection decisions have been made. This study estimated the genetic parameters for four traits to explore the genetic mechanisms underlying the variation, and we performed genome-wide association study (GWAS) tests on a small sample size to identify novel marker trait associations (MTAs) associated with prolificacy and growth. We detected two suggestive significant single-nucleotide polymorphisms (SNPs) associated with LS and eight significant SNPs for BWT, WWT, and ADG. These candidate loci and genes also provide valuable information for further fine-mapping of QTLs and improvement of reproductive and productive traits in sheep. Sheep domestication began approximately 9000 years ago in southwestern Asia, in today’s Iran and Turkey . China hFecB gene was the first to be identified as a major gene that has been shown to affect fertility [BMP15 [GDF9 [B4GALNT2 [Improved reproductive and growth performance of ewes and lambs have been associated with greater profitability and life cycle efficiency. Individual birth weight, body growth, and reproductive traits are vital indicators in litter-bearing animals and increase productive efficiency ,11. Geneertility ; this way [BMP15 , GDF9 [215 [GDF9 , and B4GB4GALNT2  genes. GB4GALNT2 . There iOur study utilized Xinggao sheep, with a record of all three consecutive births (from 2018 to 2020) at the Inner Mongolia DuMei pasture Bio Technology Co., Ltd. Jalaid Banner, Hinggan League, China . Xinggao sheep are characterized by a high reproduction rate, rapid growth and development, high resistance to cold, suitability for captive housing, and good nutritional benefits and are a new breed of mutton sheep suitable for confined feeding on a largescale in northeastern regions of Inner Mongolia. In 2018, 618 ewes were mated to 5 sires to produce 1186 lambs; in 2019, 380 ewes mated to 6 sires gave birth to 867 lambs; and 404 ewes mated with 5 sires in 2020 to produce 930 lambs. Male and female lambs are produced annually at a ratio of 1:1. To remove outliers during data statistics, the following methods were used: (1) the first and third quartiles for each trait were calculated, (2) the interquartile range was assessed, (3) the upper and lower bounds of the data range were returned, and (4) data below the lower limit or above the upper limit were outliers. After the removal of outliers and missing data, the pedigree contained a total of2289lambs born to 11 sires and 946 dams. Data on the reproductive traits of ewe on litter size (LS) came from the spring ), summer ), and autumn ) of these years. Lamb’s three growth performances were recorded: birth weight (BWT), weaning weight (WWT), and average daily gain (ADG). Among them, WWT was used to linearly adjust for weaning age at 50 days and to record lamb wean days (WD). The weaning weight formula corrected to 50 days was: Phenotypes records for Xinggao ewes and lambs were gathered, and there were 2289 records of the reproductive trait litter size (LS) in 946 ewes and three growth traits for lambs: birth weight (BWT), weaning weight (WWT), and average daily gain (ADG), which had 2281, 2284 and 2284 phenotypic records, respectively , which wY is the LS observation; b is fixed effects; a is the vector of solutions for the coefficients of direct animal (additive) genetic random effects; pe is the vector of solution for permanent environmental effects; s is the random effect of the service sire in the year of calving; e is the vector of residual effects; X, Z, W, and T are the correspondent incidence matrices of the fixed effects and additive genetic, permanent environmental random effects due to repeated records per ewe and random effect of the service sire, respectively.Genetic parameters are principal components in any breeding program for selecting animals based on their phenotypic performance. The multivariate mixed models (DMU) program was used to evaluate the genetic parameters of re(production) and estimated breeding values (EBV)using the restricted maximum-likelihood algorithm ,25. To dQ is the observed BWT, WWT and ADG, b is the fixed effects; a is the random direct effects; pe is permanent environmental random effects; m is the maternal genetic effects; e is the random residual. E, F, H and K are incidence matrices of the fixed effects, additive genetic random effects, permanent environmental effects, and maternal genetic random effects, respectively.Three traits were analyzed for lamb performance: BWT, WWT, and ADG, with the permanent environment, animal genetics, maternal genetics, and residuals as random effects. Fixed effects included the following variables: lambing year ; lambing season ; and gender . The module was:h2 is heritability; r is repeatability; σ2a is direct additive genetic variance; σ2pe is permanent environment variance related to repeats; σ2s is service sire in the year of calving random variance; and σ2e is residual variance.Heritability of the LS trait was estimated as the ratio of the additive genetic variance to total phenotypic variance; repeatability as the ratio of the sum of the additive genetic variance and permanent environmental variance to phenotypic variance .h2=σa2σh2 is heritability; σ2a is direct additive genetic variance; σ2pe is permanent environment variance related to repeats; σ2m is maternal genetic variance; and σ2e is residual variance. EBV for 4 traits and the top 500 breeding values were drawn in a Venn diagram for 4 traits [BWT, WWT, and ADG were used to estimate the variance component for heritability and genetic correlation.4 traits .®, Wilmington, DE, USA), and integrity was visualized on a 1.5%agarose gel. The 95 sheep were genotyped using a medium-density GGP Ovine 50K array  in the NEOGEN Genomics. In the future, we will follow up on this batch of sheep and perform more comprehensive related tests to verify.We first chose 49 ewes and 46 lambs for genotype testing in order to perform an in-depth study on Xinggao sheep. At LS 1 to 4, 49 ewes were selected for blood collection. Blood tests taken from 46 healthy and active lambs revealed that the ranges for birth weight, weaning weight, and average daily growth were 2.3 kg to 6.3 kg, 10.2 kg to 21.51 kg, and 0.14 kg to 0.37 kg, respectively. DNA was extracted from 200μL of blood using the Quick-DNA-Miniprep kit  according to the manufacturer’s protocol. The DNA obtained was purified using the DNA Clean & Concentrator-5 kit (Zymo Research). The DNA concentration was measured with a Nanodrop Lite (Thermo Scientificp < 1 × 10−6) were excluded [A total of 51,867 SNPs were obtained from genotyping data. Quality control (QC) of genotyping data was performed using the PLINK v1.9 software  to filteexcluded . Finally−5)) were used for calculation, the chromosome-wide significance level was used as the criterion to call significant association, and a suggestive association corresponds to a p < 10−4 [p_value = gwas$p, (b) z = qnorm, (c) λ= round/0.454, 3).The TASSEL 5.2.81 software was used to analyze LS, BWT, WWT, and ADG traits as well as high-quality SNPs for genome-wide association study (GWAS)  and to ip < 10−4 . Associap < 10−4 . The calp < 10−4  is: (a) https://www.ncbi.nlm.nih.gov/genome/gdv/?org=ovis-aries, accessed on 20 October 2022).Potential candidate genes were located within 50kb upstream and downstream from the detected significant SNPs using Oar_v4.0 sheep reference genome from the NCBI’s Genome  of LS, BWT, and ADG have been basically floated within 3 years, and WWT has gradually increased, with EBV reaching18.23 ± 0.28 in 2020. Male lambs were found to have a higher EBV than female lambs .The top 500 EBVs for each trait were sorted to select the most favorable lambs for breeding , includip < 10−4). The Q-Q plot showed the total distribution of the observed p (−log10 p) of 45,147 SNPs versus the expected values, showing that some deviated from the expected with an inflation factor (λ) of 1.002. Of the lambs’3growth traits, a total of 8 SNPs on 5 chromosomes crossed the chromosome-wide significance thresholds (p < 2.99 × 10−5), and 24 were suggestively significant SNPs (p < 10−4)  traits in ewes and lambs was performed. A total of 45,147 SNPs on 27 sheep autosomes were selected for association analyses. These SNPs passed the quality control criteria. The genome-wide scan revealed a significant association between SNPs and BWT, WWT, and ADG . For the < 10−4) . Out of  < 10−4) B, WWT (F < 10−4) C, and AD < 10−4) D, respec2P) in litter size was low due to additive inheritance (0.06 to 0.08) and the influence of the permanent environment (0.05 to 0.07) in ewes [Reproduction is a key factor in breeding sheep and herd expansion. Fecundity is a complicated trait reflecting differences in rates of ovulation/fertilization, the uterine environment, and embryo/fetus survival. Heritability mainly reflects the magnitude of the additive effects of gene and is a comprehensive reflection of traits, populations, and environments . Therefo in ewes . Masood  in ewes . For the in ewes ,38 and i in ewes . In Suff in ewes . In Egyp in ewes . For Tar in ewes . Throughp < 0.01) across all traits [p < 0.05) effect on litter size and weight at weaning [We selected three growth traits for lamb performance, namely BWT, WWT, and ADG, which were analyzed in animal models with multi-traits considering the permanent environment, animal genetics, and maternal genetics. In this study, the heritabilities of BWT, WWT, and ADG were 0.22, 0.28, and 0.29, respectively. There is a certain relationship between the traits shown by organisms. When a trait is selected, it also indirectly produces selection effects on other traits. Generally, genetic correlation and phenotypic correlation are used to measure the degree of correlation traits. The genetic correlation of BWT was 0.34 with WWT and 0.12 with ADG. The genetic correlation was 0.97 between WWT and ADG, indicating that WWT may be able to effectively increase ADG. The phenotypic correlation between the WWT and ADG traits was 0.97, and the lowest correlation was 0.13 between BWT and ADG. The results indicated that there were different degrees of correlation between the growth traits of Xinggao sheep. Others have reported in sheep heritabilities of the birth weight and body weight of lambs at 60, 90, and 120 days of age ranged from 0.015 to 0.19 . Hanfordl traits , and a s weaning . The gen weaning . The higPHLPP2 gene is a type of serine/threonine phosphatase that has been identified in cells and is involved in the regulation of AGC kinase and plays an important role in cell signal transduction and cell function regulation [TAT (tyrosine amino transferase) gene is assigned to chromosome 14 and spans 9995 bp composed of 12 exons and is expressed in the liver, kidney, brain, oviduct, and ovary. TAT is an estrogen-dependent gene that can be used in hormone therapy in the monitoring and treatment of ovarian carcinomas [Reproductive traits are complex traits that are regulated by multiple factors. The heritability of LS in Xinggao ewes, assessed by genetic parameters, was higher than in previous studies. Genome-wide association studies help to find genetic variants linked to one or more interesting phenotypes . Thus, igulation . The TATrcinomas ,55 and srcinomas . MRVI1, LYVE1, RNF141, DOK7, RGS12;DLG2, AGRN, ISG15, PERM1, PLEKHN1, KLHL17, NOC2L,SAMD11, and NLRP9) (DOK7 gene was underlined as a member of the kinase (DOK) protein family, which involves in intracellular signal transduction pathways downstream of receptor tyrosine kinases (RTKs) [DOK7 inhibits the proliferation, migration, and invasion of breast cancer cells through the PI3K/PTEN/AKT pathway [RGS12 is a regulator of G protein signaling (RGS proteins), and GTPase-accelerating proteins (GAPs) in cells attenuate G protein-dependent signals that are received by cells from the external environment [RGS12 expression is temporally and spatially regulated in developing mouse embryos and is significantly expressed in somites and developing skeletal muscles and also expressed in primary myoblasts [DLG2 (Disks Large Homolog 2) as a tumor suppressor candidate, of which its structural mutation is associated with osteosarcoma, and DLG2 deletion accelerates the growth of bone tumors in canine and human cell lines [DLG2 was pointed towards the hippo signaling pathway; this pathway is evolutionarily conserved and it controls organ size by regulating cell proliferation, apoptosis, and stem cell self-renewal [NLRP9 (NLR family pyrin domain containing 9) is one of the NLRP gene family genes with replication and functional diversity in the mammalian reproductive system, especially in reproductive functions such as the specific amplification of NLRP9 in mice [NLRP genes, 10 NLRPs  were detected in oocytes and/or early embryos [For growth traits, we found two SNPs , three SNPs , and three SNPs  associated with BWT, WWT, and ADG, respectively, and mapped near or within regions of genes (d NLRP9)  that wers (RTKs) . DOK7 inyoblasts . DLG2  traits in Xinggao sheep and attempted to highlight possible disciplines that are expected to benefit local sheep breeding. This was the first study on genetic parameters in Xinggao sheep for reproduction and growth. Estimates of heritability were low to moderate. Genetic correlations between traits were positive and generally high, and genetic and phenotypic correlations between these traits are favorable. The estimated genetics parameters indicate that the LS and weight in Xinggao sheep could be genetically improved. Therefore, weight traits or related traits can be used to select growth-promoting breeds. Additionally, our results suggest a differential genetic regulation mechanism for a novel set of genes involved in local sheep reproductive and growing functions. These novel MTAs have important applications in future molecular breeding and in providing important insights into the regulation of breeding in sheep and other mammals.


In a genome-wide association study, analyzing multiple correlated traits simultaneously is potentially superior to analyzing the traits one by one. Standard methods for multivariate genome-wide association study operate marker-by-marker and are computationally intensive.MendelIHT.jl. In simulation studies with up to 100 quantitative traits, iterative hard thresholding exhibits similar true positive rates, smaller false positive rates, and faster execution times than GEMMA’s linear mixed models and mv-PLINK’s canonical correlation analysis. On UK Biobank data with 470 228 variants, MendelIHT completed a three-trait joint analysis (We present a sparsity constrained regression algorithm for multivariate genome-wide association study based on iterative hard thresholding and implement it in a convenient Julia package https://github.com/OpenMendel/MendelIHT.jl.Software, documentation, and scripts to reproduce our results are available from  Current statistical methods for genome-wide association studies (GWAS) can be broadly categorized as single variant or multi-variant in their genomic predictors. Multi-variant sparse models ignore polygenic background and assume that only a small number of single-nucleotide polymorphisms (SNPs) are truly causal for a given trait. Model fitting is typically accomplished via regression with penalties, such as the least absolute shrinkage and selection operator (LASSO) , minimaxAlthough there is no consensus on the best modeling framework for single-trait GWAS, there is considerable support for analyzing multiple correlated traits jointly rather than separately . When prExisting methods for multivariate-trait GWAS build on the polygenic model or treat SNPs one by one. For instance, GEMMA  implemenMendelIHT.jl, part of the larger OpenMendel statistical genetics ecosystem  that can represent four sample genotypes simultaneously in a single 8-bit integer. Extracting sample genotypes can be achieved via bitshift and bitwise and operations. Genotypes are stored in little-endian fashion, with 0, 1, 2, and missing genotypes mapped to the bit patterns 00, 10, 11, and 01, respectively. For instance, if a locus has four sample genotypes 1, 0, 2, and missing, then the corresponding UInt8 integer is 01110010 in binary representation. Finally, because the genotype matrix is memory-mapped, opening a genotype file and accessing data are fast even for very large files.genotype  stores ege Julia  supportsSingle instruction, multiple data (SIMD)-vectorized and tiled linear algebra. In IHT, the most computationally intensive operations are the matrix-vector and matrix-matrix multiplications required in computing gradients. To accelerate these operations, we employ SIMD vectorization and tiling. On machines with SIMD support, such as Advanced Vector Extensions, our linear algebra routine on compressed genotypes is usually twice as fast as Basic Linear Algebra Subroutines (BLAS) 2 (https://github.com/OpenMendel/SnpArrays.jl/blob/master/docs/SnpLinAlg.ipynb.(BLAS) 2  calls wifor loops needed for computing products LoopVectorization.jl package, which performs automatic vectorization on machines with SIMD support. Furthermore, this routine can be parallelized because individual blocks can be multiplied and added independently. Because multi-threading in Julia is composable, these parallel operations can be safely nested inside other multi-threading Julia functions, such as IHT’s cross-validation routine.Computation of the matrix product r traits, traits are simulated according to the matrix normal distribution  . The oriribution  asusingation.jl . Here, X studies . To captby GEMMA . We let Finally, using PLINK , we genemIHT) to running multiple separate univariate IHT (uIHT) analyses  cross-validation. Following FlashPCA2  by calling the relevant OpenMendel packages. If binary PLINK files are used, MendelIHT.jl avoids decompressing genotypes to full numeric matrices.This article presents mIHT for analyzing multiple correlated traits. In simulation studies, mIHT exhibits similar true positive rates, significantly lower FP rates, and better overall speed than LMMs and CCA. Computational time for mIHT increases roughly linearly with the number of traits. Since IHT is a multivariate regression method, the estimated effect size for each SNP is explicitly conditioned on other SNPs and non-genetic predictors. Analyzing three correlated UK Biobank traits with MendelIHT.jl’s superior speed is partly algorithmic and partly due to software/hardware optimization. Internally, each iteration of mIHT requires a small r is the number of traits. Each iteration also requires a dense matrix–matrix multiplication for computing gradients. For MendelIHT.jl is capable of utilizing hundreds of compute cores on a single machine.P-values and ignores hidden and explicit relatedness. IHT can exploit principal components to adjust for ancestry, but PCA alone is insufficient to account for small-scale family structure (IHT’s statistical and computational advantages come with limitations. For instance, it does not deliver tructure . To overk. Although our vectorized linear algebra routine partially overcomes many of the computational barriers, we feel that further gains are possible through GPU computing (We will continue to explore improvements to IHT. Extension to non-Gaussian traits is hindered by the lack of flexible multivariate distributions with non-Gaussian margins. Cross-validation remains computationally intensive in tuning the sparsity level omputing , 2022b. omputing , especiabtad193_Supplementary_DataClick here for additional data file.


Although genome-wide association studies (GWAS) have identified tens of thousands of genetic loci, the genetic architecture is still not fully understood for many complex traits. Most GWAS and sequencing association studies have focused on single nucleotide polymorphisms or copy number variations, including common and rare genetic variants. However, phased haplotype information is often ignored in GWAS or variant set tests for rare variants. Here we leverage the identity-by-descent (IBD) segments inferred from a random projection-based IBD detection algorithm in the mapping of genetic associations with complex traits, to develop a computationally efficient statistical test for IBD mapping in biobank-scale cohorts. We used sparse linear algebra and random matrix algorithms to speed up the computation, and a genome-wide IBD mapping scan of more than 400,000 samples finished within a few hours. Simulation studies showed that our new method had well-controlled type I error rates under the null hypothesis of no genetic association in large biobank-scale cohorts, and outperformed traditional GWAS single-variant tests when the causal variants were untyped and rare, or in the presence of haplotype effects. We also applied our method to IBD mapping of six anthropometric traits using the UK Biobank data and identified a total of 3,442 associations, 2,131 (62%) of which remained significant after conditioning on suggestive tag variants in the ± 3 centimorgan flanking regions from GWAS. Identity-by-descent (IBD) segments shared from a common ancestor can be used in association mapping to identify genomic regions associated with complex traits such as height, weight and body fat distribution. In this work, we developed FiMAP, an efficient IBD mapping test that can be applied to hundreds of thousands of individuals from biobank-scale cohorts, by leveraging IBD segments from recent efficient IBD detection software programs. Built upon the classical variance component model which has its deep root in linkage analysis, FiMAP utilizes sparse linear algebra and random matrix algorithms to speed up the computation in large samples. We demonstrated accuracy, type I error control and statistical power of the FiMAP algorithm, and illustrated how IBD mapping using FiMAP could identify associations complementary to genome-wide association study findings. With the availability of an abundance of genetic data from large biobank-scale cohorts, FiMAP provides a unique angle to leverage such rich resources and better understand the genetic architecture of complex traits. Identity-by-descent (IBD) segments between two individuals are inherited from their common ancestor, without recombination . They haIBD mapping is the study of association between the sharing of IBD segments and phenotypic similarities. Early IBD mapping studies were mostly in linkage analysis for family studies –15. IBD Although IBD mapping is not yet a widely-applied method, it holds promise to uncover untyped rare variant and haplotypic effects, that are still escaping the search of the “missing heritability”. While most common variants are tagged by genotypes from arrays and can be well-imputed, rare variants are still not well covered. IBD mapping can indirectly test the effects of rare variants co-segregated with the IBD segments and recover the association signal. Further, the combination of multiple variants on an IBD segment, each individually weakly associated with the phenotype and thus difficult to be identified by single variant association tests, can be captured by IBD mapping. As the phase information is often ignored in traditional GWAS approaches and variant set tests for rare genetic variants –31, IBD O(N) complexity, where N is the sample size, and have made IBD segment calling computationally feasible in biobank-scale cohorts with hundreds of thousands to millions of individuals. However, even with the advancements of abovementioned technologies, existing IBD mapping methods are still not scalable to large sample sizes due to lack of efficient statistical tests for IBD mapping.While promising, IBD mapping were not popular due to several technical challenges. First, IBD mapping requires accurate phasing of haplotypes. This difficulty is mostly addressed by current phasing software –34. SecoO(N2) computational time complexity or higher would quickly become infeasible, we have developed FiMAP, a fast IBD mapping test for biobank-scale cohorts. FiMAP leverages IBD segments identified by IBD callers such as RaPID [Fig 1). The genotype similarity tensor is a collection of N×N pairwise local IBD matrices that change across the whole genome. Each local IBD matrix is a sparse matrix indicating the proportion of IBD sharing between any two individuals in the given genomic region, with most elements being 0. The average of local IBD matrices across the whole genome is the global IBD matrix representing the average proportion of IBD sharing, which can be used as a kinship coefficient matrix to account for genetic relatedness in the study samples. FiMAP leverages a random matrix-based algorithm to speed up the computation, so that a genome-wide IBD mapping analysis for hundreds of thousands of study samples can be finished within a few hours. We also demonstrate the application of FiMAP to the IBD mapping analysis of six anthropometric traits in the UK Biobank.To address the main computational challenge for biobank-scale cohorts that any IBD mapping methods with as RaPID  and hap-as RaPID , and conYi = Xiβ+bi+δil+εi for individual i, where Yi is the phenotype, Xi are c fixed-effect covariates with effect sizes β, bi is the polygenic random effect accounting for global relatedness, δil is the random effect of local relatedness at genetic locus l, we stack N individuals to get length N vectors Y, b, δl, ε, and an N×c matrix X, from Yi, bi, δil, εi, Xi, respectively. The vector N×N global IBD (or kinship) matrix Φ denoting the average copies of the genome shared IBD between any two individuals, N×N local IBD matrix Ψl, defined as the copies of the genome shared IBD at this genetic locus l, and ε are independent and identically distributed errors that follow a normal distribution with mean 0 and variance l, the random effects δl are equal to 0 and this is equivalent to testing variance component hypotheses Variance component models have long been used in linkage analysis to identify genetic loci linked with quantitative traits –15. Consδl = 0) in the matrix-vector form Y = Xβ+b+ε. Let N residual vector, where both jζjχj1,2 [χj1,2 are independent chi-square distributions with 1 degree of freedom, and ζj’s are the eigenvalues of N×N projection matrix [Let jζjχ1,j2 ,42,43, wn matrix –47.Ψl for genetic locus l is often sparse, which could invalidate asymptotic inference on the quadratic form P, where ξj’s are the eigenvalues of Although under regularity conditions, le sizes –50. DespΨl are N×N matrices (and O(N2) memory footprint and O(N3) computational time complexity, which becomes infeasible for hundreds of thousands to millions of individuals. To solve this computational challenge, we use an N×B random matrix R to approximate the eigenvalues of B×B matrix Φ = LLT, then simulate length N random vectors r1 and r2 from a standard normal distribution to compute B times to get an N×B random matrix R. In reality, the global IBD matrix Φ often has non-zero but small off-diagonal elements and we can set them to zero to ensure that it is block-diagonal (with a bounded largest block size) and sparse, then both L and r3 and r4 do not require any matrix-vector multiplications involving full dense N×N matrices. If both the largest block size and B are O(1), then the overall computational complexity of FiMAP is O(N). Moreover, for the finite-sample adjustment, we can pre-compute B×B matrix As both analysis ,52, herianalysis  and geneanalysis  estimatianalysis ,55, and analysis . SpecifiΨl with 2 identical copies of the genome shared IBD at this genetic locus l for each individual with themselves, we use L equally spaced genomic regions across the whole genome, the global IBD matrix Ψl’s. In our implementation, we pre-compute an offset l, we then compute the finite-sample p-value as l with an observed local IBD matrix B×B matrix In reality, we observe the start and end positions of each IBD segment between two individuals. Therefore, instead of a local IBD matrix Φ for 487,252 individuals. For individuals with no inbreeding, the diagonal elements of Φ is equal to 2, which is 4 times the theoretical kinship coefficient, indicating the total copies of the genome shared IBD by each individual with themselves. The off-diagonal elements of Φ are the average copies of the genome shared IBD by each pair of individuals. To ensure the sparsity of the global IBD matrix Φ, we set off-diagonal elements less than 0.088 to 0, which is equivalent to including fourth-degree relatives or closer in the kinship matrix (kinship coefficient ≥ 0.022). We first compared the performance of asymptotic and finite-sample variance component tests, using either the dense N×N projection matrix B = 100, 1,000 or 10,000. Given the O(N3) complexity and O(N2) memory footprint to directly use the dense N×N projection matrix N = 10,000 samples and fitted a null linear mixed model for waist circumference adjusting for age, age2, sex, their interactions, and top 10 ancestral principal components (PCs), and used the global IBD matrix to model the covariance structure of the random intercept. We then simulated random N×N symmetric local IBD matrices with 10,000, 100,000 and 1 million non-zero off-diagonal elements from a uniform distribution between 0 and 2. These matrices were not associated with waist circumference. We evaluated the p value calibration of asymptotic and finite-sample tests, as well as the accuracy of approximation in the FiMAP algorithm.We used RaPID and hap-IBD IBD segment calls from the UK Biobank array-typed genotype data to simulate phenotype data, and evaluated type I error rates and power of FiMAP in identifying genetic associations. This genotype dataset consists of 658,720 variants on 22 autosomes. Specifically, RaPID and hap-IBD IBD segments were called using the phased haplotypes released by the UK Biobank team , on all S1 and  show simulation results using a random subset of N = 10,000 samples from the UK Biobank, under the null hypothesis of no genetic association with waist circumference. Finite-sample p values from both the FiMAP algorithm and directly using dense S1A, S1B and S1C Fig). However, asymptotic p values were not well calibrated if the number of random vectors B used in the FiMAP algorithm was small, or the number of non-zero off-diagonal elements in the simulated local IBD matrices was small . Of note, with 10,000 non-zero off-diagonal elements in the simulated local IBD matrices, even the asymptotic p values from directly using dense S2A, S2B and S2C Fig show that finite-sample FiMAP p values were highly concordant with finite-sample variance component test p values using dense S2D Fig), but much closer as local IBD matrices became denser . Compared to asymptotic variance component test p values using dense B and/or denser local IBD matrices . It is also worth noting that the asymptotic p values in S2D, S2E and S2F Fig were the same as the dense S2G, S2H and S2I Fig, respectively, and they only performed well with denser local IBD matrices . Based on these results, we only presented finite-sample FiMAP p values with B = 100 in subsequent simulations and real data analysis.Φ to get a submatrix Φ0. We then simulated age from a normal distribution with mean 50 and standard deviation 5, and sex from a Bernoulli distribution with probability 0.5. A continuous phenotype Yi for individual i was simulated as Yi = 0.05agei+0.5sexi+bi+εi, where the vector form b followed a multivariate normal distribution with mean 0 and covariance matrix Φ0, and εi followed a standard normal distribution. Then we tested for the association with 3,403 local IBD matrices after adjusting for age and sex. We simulated 50 phenotype replicates and obtained a total of 170,150 p values under the null hypothesis of no local IBD random effects.For the full UK Biobank samples, we chunked 22 autosomes into 1 cM windows and assembled 3,403 local IBD matrices by computing the proportion of IBD sharing between each pair of individuals in each 1 cM window (the last window on each chromosome is shorter than 1 cM). In type I error simulations, we randomly selected 400,000 individuals and subset the global IBD matrix Fig 2 shows that the number of non-zero off-diagonal elements increased as the IBD length cutoff decreased from 10 cM, 5 cM to 3 cM, for IBD segments called from both RaPID and hap-IBD. For example, when the RaPID IBD length cutoff was 3 cM, the median number of non-zero off-diagonal elements in local IBD matrices was 60.4 × N, with a range of 8.2 × N to 468.0 × N across locations. For hap-IBD IBD segments with length ≥ 3 cM, the median number of non-zero off-diagonal elements in local IBD matrices was 51.6 × N, with a range of 2.4 × N to 276.6 × N across locations. These results suggest that in the UK Biobank data, the largest number of non-zero off-diagonal elements in any local IBD matrix at a given length cutoff  was much less than what we would expect from a dense N×N matrix, regardless of the IBD caller, thus we could consider all these local IBD matrices as sparse and the computational efficiency of FiMAP was guaranteed. On a computing server with dual Intel Xeon E5-2687W v4 CPU , analysis of each simulation replicate with N = 400,000 took about 131 and 115 minutes for RaPID and hap-IBD IBD length cutoff 3 cM, 36 and 33 minutes for 5 cM, and 9.4 and 8.9 minutes for 10 cM, respectively, using 40 threads in parallel.We counted the numbers of non-zero off-diagonal elements in each of the 3,403 local IBD matrices for 1 cM windows, with sample size N = 487,252. The local IBD matrices were constructed from RaPID and hap-IBD IBD segments called with length ≥ 3 cM, 5 cM, 10 cM. Fig 3. FiMAP p values with the finite-sample adjustment were well-calibrated, for IBD segments called from both RaPID and hap-IBD. The median finite-sample FiMAP p value showed genomic inflation factors very close to 1 for IBD segments called with length ≥ 3 cM, 5 cM, 10 cM, suggesting that FiMAP with the finite-sample adjustment is a fast and valid IBD mapping test in large samples, regardless of the IBD caller.Finite-sample FiMAP p values under the null hypothesis of no local IBD random effects were shown in J ultra-rare variants with MAF < 0.0005 in this window were causal variants and simulated a continuous phenotype Yi for individual i as agei, sexi, bi and εi were simulated using the same parameter settings as in the type I error simulations, and the causal variant effect −5, and for GWAS as the proportion of any minimum single-variant p values in each 1 cM testing window being less than the genome-wide significance level of 5 × 10−8, over 1,000 simulation replicates.We also conducted power simulations to benchmark finite-sample FiMAP results with standard GWAS results using the single-variant score test from GMMAT  which fiFig 4A shows that FiMAP had less power than the GWAS single-variant test on imputed variants. Fig 4B shows empirical power to identify the association in the neighboring ± 1 cM windows from the simulated untyped rare causal variants. Compared to Fig 4A, FiMAP power dropped slightly in these neighboring windows, while the GWAS single-variant test power plummeted more dramatically. This is not surprising because the IBD segments used in the FiMAP analysis were all longer than the 1 cM window, while for the GWAS single-variant test, the linkage disequilibrium decayed much faster in the neighboring windows, suggesting that FiMAP had a lower resolution than the GWAS single-variant test in association mapping.Assuming the causal variants in a 1 cM window were ultra-rare (with MAF < 0.0005) and not directly observed, Yi for individual i as agei, sexi, bi and εi were simulated using the same parameter settings as in the type I error simulations, and Hi was the number of causal haplotypes carried by individual i, with possible values 0, 1, 2. The causal haplotype effect was assigned as In addition, we simulated haplotype effects and compared FiMAP with IBD segments called with length ≥ 3 cM, 5 cM, 10 cM with the GWAS single-variant test. Specifically, we estimated the start and end positions for all haplotypes with length ≥ 0.5 cM shared by at least 500 individuals in the UK Biobank using a PBWT-block algorithm . In eachFig 4C). FiMAP with IBD segments called with length ≥ 10 cM suffered from a substantial power loss, likely due to the fact that many shorter IBD segments tagging the causal haplotype in the region were not included. Fig 4D shows empirical power to identify the association in the two closest 1 cM windows that did not overlap with the simulated causal haplotype, and we observed similar patterns as seen in Fig 4B.In the presence of causal haplotype effects, the GWAS single-variant test on imputed variants also had the best power, followed by FiMAP with IBD segments called with length ≥ 3 cM and 5 cM , hip circumference , standing height , sitting height , body mass index , and body weight , from the UK Biobank white British study participants with genetic ethnic group in Caucasians, after excluding individuals with inconsistent gender and biological sex. Each trait was adjusted for age, agevariates  and the −6. In the conditional analysis, for each 1 cM window with a significant finite-sample FiMAP p value < 1.47 × 10−5, we considered all imputed genetic variants with suggestive association evidence from the GWAS single-variant test in the ± 3 cM flanking regions  to account for the fact that the IBD segments we used had a length ≥ 3 cM. Specifically, we clumped these suggestive variants into 100kb windows and selected the most significant variant from each window to form a conditional set. We then adjusted for this set of variants as covariates in the conditional FiMAP analysis.We also analyzed the same anthropometric traits in a GWAS setting using GMMAT  on 22 miΨl. RaPID IBD segment calling in the UK Biobank incurred a one-time cost of 474 CPU hours. As a comparison, the GWAS of 89 million unfiltered imputed variants using GMMAT [Table 2 shows that FiMAP using RaPID IBD segments with length ≥ 3 cM identified 1,587 significant 1 cM windows without any GWAS suggestive evidence as there were no imputed genetic variants with MAF ≥ 0.0001, imputation quality score ≥ 0.3, and GWAS single-variant test p-value < 1×10−6 in these regions. On the other hand, it also shows that 1,855 significant 1 cM windows in FiMAP results had at least one such GWAS tag variant in the same test region. Interestingly, 2,131 out of 3,442 (62%) 1 cM windows remained significant even after a very stringent conditional analysis by adjusting for all tag variants in the conditional set, selected as the most significant variants with GWAS single-variant test p-value < 1×10−6 and physical distance > 100kb in the ± 3 cM flanking regions , suggesting that IBD mapping provided complementary association evidence missed by a traditional GWAS approach. Fig 5 shows that for windows with a significant unconditional FiMAP p-value, conditional p-values were generally larger (less significant) than unconditional p-values as expected, for all 6 anthropometric traits. Of note, we identified a 6 cM region (29–35 cM) on chromosome 21 associated with both standing height and sitting height using RaPID IBD segments with length ≥ 3 cM .For each anthropometric trait from the UK Biobank, the FiMAP analysis using the RaPID IBD length cutoff 3 cM took 90 CPU hours with a maximum memory footprint scaling linearly with the number of nonzero entries in the local IBD matrix ng GMMAT  took an FTO region on chromosome 16, using RaPID IBD segments with length ≥ 3 cM. However, none of the conditional p values remained significant after adjusting for GWAS tag variants in this region, suggesting that FiMAP did not provide additional association evidence beyond GWAS in this case .For four anthropometric traits related to body fat distribution or weight , we found highly significant FiMAP p values near the S4 and  show that FiMAP p values were highly stable with respect to different random number seeds used in IBD segment calling by RaPID (minimum Spearman’s ρ = 0.966) and IBD mapping (minimum Spearman’s ρ = 0.989). These results suggested that our top findings were robust against different random number seeds used in IBD segment calling and/or IBD mapping.As both RaPID and FiMAP rely on random algorithms, we also used a different random number seed in RaPID IBD segment calling, as well as FiMAP IBD mapping, to investigate the numerical stability of our findings. In this work, we have developed FiMAP to conduct fast IBD mapping analysis that scales linearly with the sample size and applied it to the UK Biobank data. Compared to traditional GWAS approaches that utilize only unphased genotypes, IBD mapping with the phase information offers a distinct perspective into the genetic architecture of complex traits by leveraging the genotype similarity in a specific genomic region. The heuristics that if a genomic region is associated with a complex trait, then individuals with greater genotype similarity are also expected to show greater phenotype similarity, have led to the development of linkage analysis methods for family data, including the variance component linkage analysis. However, such methods have not previously been widely applied to biobank-scale population-based cohorts, due to computational challenges.FiMAP is an accurate and computationally efficient IBD mapping method. It leverages a random matrix to approximate the null distribution of the variance component test statistic in quadratic form. We have found in the UK Biobank data analysis that different random number seeds used in either IBD segment calling by RaPID or IBD mapping by FiMAP had minimal impact on the top association findings, as the p values were highly consistent regardless of the different random number seeds. In addition, our simulation results using IBD segments from two different IBD callers RaPID and hap-IBD were also highly concordant given the same length cutoff, suggesting that FiMAP was robust with respect to which IBD caller was used.et al. [Our simulation studies showed that FiMAP is often more powerful using IBD segments with shorter length cutoffs. However, there is also a tradeoff between statistical power and computational efficiency. With an IBD length cutoff of 10 cM, FiMAP was very computationally efficient, but also had the lowest power in our simulation settings. With an IBD length cutoff of 3 cM, FiMAP was often more powerful than using cutoffs of 5 cM and 10 cM, but the run time also dramatically increased. With even shorter IBD length cutoffs , a much larger number of IBD segments are expected to be called , compared to 0.124 × N in a random subset of N = 1,000 individuals. Therefore, we expect FiMAP to have much greater statistical power in biobank-scale cohorts, although the actual power may also depend on other factors such as the genetic architecture of the trait, as well as the degree of relatedness in the samples. On the other hand, when N is much larger than the size of UK Biobank (in a somewhat distant future), the local IBD matrix may not be sparse enough and new method than FiMAP may be needed for efficient IBD mapping.N×N local IBD matrices are large in biobank-scale cohorts, they are often sparse with a limited number of non-zero off-diagonal elements. As a result, the asymptotic p values from FiMAP are extremely conservative in the tail for sparse local IBD matrices. In contrast, our simulation studies under the null hypothesis of no genetic association showed that finite-sample FiMAP p values followed a uniform distribution, after accounting for the variability in estimating the residual variance parameter. Therefore, we recommend the use of finite-sample FiMAP p values for all analyses.Surprisingly, asymptotic p values from FiMAP are not well calibrated even with the sample sizes of the UK Biobank. Although the FiMAP, or IBD mapping in general, has several advantages compared to traditional GWAS approaches on unphased genotypes. It may better tag rare causal variants, especially when they are not directly genotyped, which is often the case for commonly used genotyping arrays. Although it is unlikely that rare causal variants are not genotyped using the whole genome sequencing technology, as the IBD calling algorithm such as RaPID tolerates mismatches, FiMAP is more robust to genotyping or sequencing errors since each individual genetic variant would not dramatically change the IBD segment calling. Moreover, the phasing information is usually ignored in traditional GWAS approaches. Our conditional analysis for UK Biobank anthropometric traits showed that many 1 cM windows identified by FiMAP remained significant after adjusting for all GWAS tag variants in the testing window and flanking regions, suggesting that these associations could not be fully explained by unphased genotypes, and therefore haplotype effects might play an important role in the genetic architecture of complex traits. With recent advances in fully phased human genome assembly from long-read sequencing data ,61, moreFiMAP also has a few limitations. It tests a few thousand 1 cM windows on the human genome, and therefore requires a much less stringent significance level than traditional GWAS approaches which test millions of genetic variants. However, the major limitation of FiMAP is the low resolution of association signals. Follow-up analysis is often needed to identify causal variants or interpret these association findings. On the other hand, we note that IBD mapping has the potential of achieving a higher resolution than admixture mapping as IBD segments tend to be shorter than local ancestry tracts in recently admixed populations. Also, currently FiMAP can only be applied to quantitative traits in a linear mixed model framework, and future work includes extension to binary traits using logistic mixed models, as well as survival traits using Cox mixed models. Nevertheless, FiMAP empowers computationally efficient IBD mapping for complex traits using variance component linkage analysis models with unprecedented sample sizes from biobank-scale cohorts.S1 FigN×N projection matrix B = 100, 1,000 or 10,000 were shown. A random subset of N = 10,000 samples were taken from the UK Biobank, and 3,403 random N×N local IBD matrices with 10,000, 100,000 and 1 million non-zero off-diagonal elements were simulated under the null hypothesis of no association. Finite-sample p values from local IBD matrices with (A) 10,000, (B) 100,000, or (C) 1 million non-zero off-diagonal elements, and asymptotic p values from local IBD matrices with (D) 10,000, (E) 100,000, or (F) 1 million non-zero off-diagonal elements were plotted against expected p values from a uniform distribution.Results from the dense (TIF)Click here for additional data file.S2 FigN×N projection matrix B = 100, 1,000 or 10,000 were compared. A random subset of N = 10,000 samples were taken from the UK Biobank, and 3,403 random N×N local IBD matrices with 10,000, 100,000 and 1 million non-zero off-diagonal elements were simulated under the null hypothesis of no association. Finite-sample FiMAP p values from local IBD matrices with (A) 10,000, (B) 100,000, or (C) 1 million non-zero off-diagonal elements were plotted against finite-sample variance component test p values using the dense N×N projection matrix N×N projection matrix N×N projection matrix N×N projection matrix Results from the dense (TIF)Click here for additional data file.S3 Fig−25 were truncated at 1 × 10−25. For windows with unconditional FiMAP p values < 0.05/3,403 = 1.47 × 10−5, conditional p values after adjusting for all tag variants in each conditional set were shown in blue triangles.(A) Waist circumference; (B) Hip circumference; (C) Body mass index; (D) Body weight. IBD segments called by RaPID with length ≥ 3 cM were used in the FiMAP analysis. GWAS p values were shown in grey and unconditional FiMAP p values were shown in orange. GWAS p values < 1 × 10(TIF)Click here for additional data file.S4 Fig−5. P values < 1 × 10−50 for standing and sitting height were truncated at 1 × 10−50. Spearman’s rank correlation coefficient was calculated for p values from two separate runs.(A) Waist circumference; (B) Hip circumference; (C) Standing height; (D) Sitting height; (E) Body mass index; (F) Body weight. IBD segments called by RaPID with length ≥ 3 cM were used in the FiMAP analysis. Black dashed lines represented the Bonferroni-corrected significance level of 0.05/3,403 = 1.47 × 10(TIF)Click here for additional data file.S5 Fig−5. P values < 1 × 10−50 for standing and sitting height were truncated at 1 × 10−50. Spearman’s rank correlation coefficient was calculated for p values from two separate runs.(A) Waist circumference; (B) Hip circumference; (C) Standing height; (D) Sitting height; (E) Body mass index; (F) Body weight. IBD segments called by RaPID with length ≥ 3 cM were used in the FiMAP analysis. Black dashed lines represented the Bonferroni-corrected significance level of 0.05/3,403 = 1.47 × 10(TIF)Click here for additional data file.S1 TableStart and end positions in centimorgan (cM) are shown, along with the number of non-zero off-diagonal elements in each N × N local IBD matrix and finite-sample p values .(XLSX)Click here for additional data file.S2 TableStart and end positions in centimorgan (cM) are shown, along with the number of non-zero off-diagonal elements in each N × N local IBD matrix and finite-sample p values .(XLSX)Click here for additional data file.S3 TableStart and end positions in centimorgan (cM) are shown, along with the number of non-zero off-diagonal elements in each N × N local IBD matrix and finite-sample p values .(XLSX)Click here for additional data file.S4 TableStart and end positions in centimorgan (cM) are shown, along with the number of non-zero off-diagonal elements in each N × N local IBD matrix and finite-sample p values .(XLSX)Click here for additional data file.S5 TableStart and end positions in centimorgan (cM) are shown, along with the number of non-zero off-diagonal elements in each N × N local IBD matrix and finite-sample p values .(XLSX)Click here for additional data file.S6 TableStart and end positions in centimorgan (cM) are shown, along with the number of non-zero off-diagonal elements in each N × N local IBD matrix and finite-sample p values .(XLSX)Click here for additional data file.


Pinus albicaulis Engelm.) has experienced rapid population declines and is listed as threatened under the Endangered Species Act in the United States. Whitebark pine in the Sierra Nevada of California represents the southernmost end of the species' distribution and, like other portions of its range, faces threats from an introduced pathogen, native bark beetles, and a rapidly warming climate. Beyond these chronic stressors, there is also concern about how this species will respond to acute stressors, such as drought. We present patterns of stem growth from 766 large (average diameter at breast height >25 cm), disease‐free whitebark pine across the Sierra Nevada before and during a recent period of drought. We contextualize growth patterns using population genomic diversity and structure from a subset of 327 trees. Sampled whitebark pine generally had positive to neutral stem growth trends from 1970 to 2011, which was positively correlated with minimum temperature and precipitation. Indices of stem growth during drought years (2012 to 2015) relative to a predrought interval were mostly positive to neutral at our sampled sites. Individual tree growth response phenotypes appeared to be linked to genotypic variation in climate‐associated loci, suggesting that some genotypes can take better advantage of local climatic conditions than others. We speculate that reduced snowpack during the 2012 to 2015 drought years may have lengthened the growing season while retaining sufficient moisture to maintain growth at most study sites. Growth responses may differ under future warming, however, particularly if drought severity increases and modifies interactions with pests and pathogens.Whitebark pine ( Using extensive surveys across the Sierra Nevada, we show whitebark pine had positive to neutral growth prior to drought in our sampled sites. Growth responses measured in terms of basal area increment remained positive to neutral during drought. Genomic analyses found that individual tree growth response phenotypes appear to be linked to genotypic variation in climate‐associated loci, suggesting that growth is not only controlled by site‐level differences, but that some genotypes can take better advantage of local climatic conditions than others. Because of the multiple roles that whitebark pine plays in subalpine ecosystems, it has been described as a keystone and foundation species. Over much of its range whitebark pine is declining due to a suite of stressors including a non‐native, invasive pathogen , outbreaks of native pests , altered fire regimes, and a rapidly changing climate .Whitebark pine  currently appears to be less impacted by biotic and abiotic stressors relative to other portions of its range. A recent synthesis of whitebark pine status from a US national forest inventory database showed slightly over half of the standing whitebark pine ≥12.5 cm DBH (stem diameter at breast height [1.37 m]) were dead; however, within the Sierra Nevada alone only <10% of stems were dead . We expected stem growth responses would provide insight into the important climate drivers over longer time periods, while short‐term patterns in radial growth should provide insight into the short‐term growth responses during meteorological drought. Finally, genetic data describe the degree of local population adaptation to a particular site and may provide a genetic perspective of drought vulnerability.We used growth and genetic data to answer the following questions: (1) how did recent whitebark pine growth respond to variability in interannual climate in the Sierra Nevada? (2) What was the growth response of whitebark pine during the recent meteorological drought years and how did it vary across the Sierra Nevada? and (3) Were growth responses during the drought years linked to climate‐associated genetic variation? By combining measurements of growth and genomic structure over a common set of individual trees and sites, we aim to provide a framework to characterize the environmental and intrinsic genetic drivers of variation in growth. This study potentially provides insight into responses and adaptation of whitebark pine to possible future climatic conditions in its southernmost ecoregion.22.1https://www.fs.fed.us/rm/highelevationwhitepines/About/dist.htm#california, accessed 3/10/2022), growing as vertical stems at low elevations and as a more shrub‐like growth form (krummholz) near treeline , lodgepole pine (P. contorta Dougl.), foxtail pine , western white pine (P. monticola Douglas ex D. Don), mountain hemlock (Tsuga mertensiana (Bong.) Carr.), and red fir (Abies magnifica A. Murray bis var. magnifica).Whitebark pine in the Sierra Nevada of California is found from north of the Lake Tahoe basin to south of Sequoia National Park , which promotes a clumped pattern of individual trees and influences population genetic structure from tree to regional scales. Clark's nutcracker creates caches of 1 to 15 seeds . We determined final sampling sites based on crew safety and the presence of multiple non‐krummholz whitebark pines. We avoided krummholz due to their complex and potentially differing growth responses to environmental conditions  along with one core from a tree with symptoms of white pine blister rust, we had 766 cores used in analysis. At each site, we had an average of 28 individual whitebark pines with usable tree core data (range = 17 to 30 trees). Sampled trees were required to be growing upright, free of significant damage, disease, and beetle activity. Stem diameters for sampled trees ranged from 6.3 to 89.0 cm DBH (average = 25.2 cm), and tree heights ranged from 2.3 to 22.0 m (average height = 7.3 m). There were five trees with missing DBH measurements and two trees with missing height measurements. One sampled tree had symptoms of white pine blister rust and was removed from analysis of growth but was retained for the genetics analyses. If stems were clumped, we cored only one arbitrarily selected stem per clump. Needle samples were collected from cored trees for genetic analysis (NPS Permit #: YOSE‐2019‐SCI‐0061). We were working largely in wilderness areas, and the removal of cone‐bearing branches for herbarium specimens would have been impractical.2.4https://www.sciencebase.gov/catalog/item/5f29c62d82cef313ed9edb39). Data for 26 sites were accessed on 11/12/2020, with data for one site accessed on 9/22/2021. BCM uses gridded instrumental climate products from the Parameter Elevation Regression on Independent Slopes Model  (note that SNOTEL data are captured in the PRISM estimates that are used by the BCM). We considered the period from WY2012 to WY2015 to be “drought years” at our sites, although statewide the drought is described as lasting through 2016  1970 to 2020 , and the Jarbidge Mountains in northeastern Nevada , to act as outgroup sites for genetic analysis. For each tree, five‐needle fascicles were placed into coin envelopes with two 5‐gram silica packets. Upon return from the field site, silica packets were removed, and envelopes were transferred to a 45°C oven to accelerate desiccation and aid in DNA preservation .Genomic DNA was extracted from dried needles using the E‐Z 96® Plant DNA Kit  with Antifoam Y‐30  added to the homogenizer plate and a 60‐min room temperature incubation before elution in 100 μL of water. Reduced representation genomic libraries were prepared using a modified double‐digest restriction‐site associated DNA sequencing (ddRAD) scheme  relative to WY1970 to WY2015. We created average z‐scores for Tmn, PPT, and CWD for correlations with the BAI drought response index. Where the data had a large departure from a normal distribution, we used the Spearman rank correlation (rs).We checked whether the BAI drought response index was sensitive to the years we used to define the drought and predrought periods  was calculated using the populations program in Stacks, while filtering for one biallelic single‐nucleotide polymorphism (SNP) per locus  was calculated on all variable sites within loci using the haplotype‐wise filter, kernel smoothing, and bootstrap resampling flags in the populations program in Stacks. Two genetic clustering analyses were used to explore the pattern of genetic structure among individuals. First, a principal component analysis (PCA) was implemented in adegenet v2.1.1  with increasing geographic distance among Sierra Nevada study sites using a Mantel test for matrix correlation  were used to assess the significance of the full model, each RDA axis, and the marginal effects of each predictor term and r2 was adjusted based on the number of predictor variables. Loci were considered outliers if they were greater than three standard deviations away from the mean RDA axis score. We examined genetic diversity in outlier loci across the range and calculated correlations among individual multilocus genotype RDA loadings and the individual tree BAI drought response indices. These individual‐level measurements were not normally distributed , so we used the Spearman rank correlation to determine relationships.We used BCM 30‐year normals (WY1970 to WY1999) for annual Tmn, Tmx, PPT, PET, AET, and CWD in genetic‐environment multilocus association tests. Although CWD is derived from PET and AET, we had no a priori reason to suspect that any one variable would show stronger associations with genetic data. Climate variables were subject to a forward selection process and variance inflation factor (VIF) cutoff of 10 to remove variables that were highly correlated, resulting in four climate variables to include in the RDA model: Tmn, PPT, AET, and CWD. Because of strong latitudinal gradients in overall genetic differentiation (see results) and temperature and rainfall gradients, we performed a partial RDA with the selected climate variables as predictors and a conditional matrix of plot geographic locations, to control for distance‐based effects. Permutation tests .Across all tree‐ring series, there was high variability within series length, mean series length, first‐order autocorrelation, and mean interseries correlation Table . Mean seβTmn = 0.15 ± 0.02, βPPT = 0.05 ± 0.01). Site‐level average annual sBAI was not related to CWD (p = .42), and models that used CWD had less support relative to models that included independent terms for Tmn and PPT (ΔAICc ≥ 69). While the model with independent terms for Tmn and PPT had the most support by AICc, it explained very little variation in annual site‐level average annual sBAI, with pseudo r2 = .03.From 1970 to 2011, sBAI showed high variance among sites Figure , with gr3.2r = .19, p = .33). There appeared to be a latitudinal gradient in average BAI drought response , with two northern sites having the greatest BAI drought response. Site‐level average BAI drought response was also correlated with average water year PPT and CWD z‐scores during the drought years . The correlation between BAI drought response and Tmn was not statistically significant.Site‐level BAI drought response indices were generally positive Figure , with 223.3We estimated population genomic diversity and structure across our sites. Raw sequence reads totaled 3,560,632,042 with a mean of 3,724,485 ± 1,935,474 reads per individual. Mean read coverage per locus was 10.9× ± 1.6× after individuals with an average read count below 9.0× were dropped. Two sites from Sequoia National Park (sites 18 and 23) and three sites from the eastern slopes of the Sierra Nevada  were dropped from further genomic analysis due to low average read depth, so the final genomic dataset included 327 individuals from 22 Sierra Nevada sites and four outgroup sites. The final filtered genetic marker dataset consisted of 44,287 independent loci with 342,375 variant sites. Reducing to a single SNP per locus resulted in 44,287 independent SNPs, 20% overall missing data, and a minimum minor allele frequency of 0.02. More stringent filtering using a minimum minor allele frequency of 0.05 and/or removing individuals with more than 30% missing data slightly reduced the number of SNPs (<1%) and reduced the number of individuals (20%), but had little to no effect on population structure or genetic diversity estimates Table . TherefoK) ranged from 3 to 5. At K = 3 and K = 4, we found the outgroup sites form a discrete cluster and the Sierra Nevada sites show a distinct geographical pattern from north to south, and K = 5 further separated the Jarbidge Mountains site from the remainder of the outgroup sites . We detected a significant pattern of genetic isolation by geographic distance  indicating a stepping‐stone structure. Expected heterozygosity and nucleotide diversity were correlated with latitude . There were four significant RDA axes (p < .001) and outlier analysis identified 1270 putatively adaptive SNPs that loaded ±3 SD from the mean loading on one or more axes. In total, these outlier SNPs exhibited higher genetic diversity , and significant isolation by distance . Expected heterozygosity and nucleotide diversity of the outlier SNPs were positively correlated with latitude . Based on the strongest correlation between each outlier locus and climate variable, 494 of the identified outlier SNPs were most strongly associated with CWD, 283 with Tmn, 261 with AET, and 232 with PPT.We sought to determine whether climate could help identify patterns of genomic diversity and structure. The full RDA model [genotypes ~ Conditional (UTM northing + UTM easting) + AET + CWD + PPT + Tmn] explained 1.0% of the total observed genetic variation  and RDA4 . RDA3 appeared to differentiate high and low AET, with individuals in sites 3, 6, and 21 exhibiting the highest genotype loading and individuals from sites 14 and 19 lowest  and to better understand patterns of genetic variation associated with this response. Whitebark pine at our sites generally had positive to neutral recent stem growth trends between 1970 and 2011. Stem growth, as measured by sBAI, appeared to be weakly positively correlated with water year annual minimum temperature and precipitation. Counterintuitively, growth indices during the meteorological drought years for BAI were generally positive to neutral. This implies that on average whitebark pine was not strongly moisture limited during the meteorological drought years 2012 to 2015). Individual growth response phenotypes appeared to be linked to genotypic variation in climate‐associated loci, indicating that drought growth response was not only controlled by site‐level environmental differences, but that some genotypes can take better advantage of local climatic conditions than others, as suggested in Millar et al. . Additional analyses using ring‐width data from whitebark pine at our sites suggest that most of our sampled sites were primarily energy limited (rather than water limited), supporting the idea that drought conditions could lead to increased stem growth , so hydraulic limitations found in tall conifers are unlikely. Trees were often in open‐grown conditions, so changes in stand density were unlikely strongly biasing growth responses. One possibility is that larger older trees have established on sites with different conditions due to fine‐scale topographic mediation of snow depth and duration  relative to measurements prior to the drought years. We did not find evidence that the site‐averaged BAI response index was strongly variable across latitude, although the drought was more intense at lower latitudes in the Sierra Nevada  surveyed across the southeastern United States ; data curation (supporting); formal analysis ; funding acquisition ; investigation ; project administration (lead); writing – original draft ; writing – review and editing . Elizabeth R. Milano: Conceptualization ; data curation ; formal analysis ; investigation ; visualization ; writing – original draft ; writing – review and editing . Joan Dudney: Conceptualization ; data curation ; formal analysis (supporting); funding acquisition ; investigation ; methodology ; writing – original draft (supporting); writing – review and editing . Jonathan C. B. Nesmith: Conceptualization ; data curation ; funding acquisition ; investigation ; methodology ; project administration ; resources ; supervision ; writing – review and editing . Amy G. Vandergast: Conceptualization ; data curation ; formal analysis (supporting); funding acquisition ; investigation ; methodology ; project administration ; resources ; supervision ; writing – original draft (supporting); writing – review and editing . Harold S. J. Zald: Data curation ; formal analysis ; investigation ; methodology ; resources ; supervision ; writing – original draft (supporting); writing – review and editing .The authors declare no competing interests with this work.TABLE S1 Comparisons of stringent filtering parameters on dataset characteristics and estimates of population genetic parameters. All filters had minor effects on the number of loci, number of samples, and no effect on estimates of population structure in the Sierra Nevada. Filtering out individuals with <80% data completeness reduced the number of individuals by almost half and was not explored further.Click here for additional data file.