The identified significantly linked SNP markers, associated putative genes, and developed GP models could potentially facilitate breeding programs targeting to improve the overall genetic gain of wheat breeding for grain yield and biofortification of micronutrients via marker-assisted (MAS) and genomic selection (GS) methods.Globally, over 2 billion people suffer from malnutrition due to inadequate intake of micronutrients. Genomic-assisted breeding is identified as a valuable method to facilitate developing new improved plant varieties targeting grain yield and micronutrient-related traits. In this study, a genome-wide association study (GWAS) and single- and multi-trait-based genomic prediction (GP) analysis was conducted using a set of 252 elite wheat genotypes from the International Center for Agricultural Research in Dry Areas (ICARDA). The objective was to identify linked SNP markers, putative candidate genes and to evaluate the genomic estimated breeding values (GEBVs) of grain yield and micronutrient-related traits.. For this purpose, a field trial was conducted at a drought-prone station, Merchouch, Morocco for 2 consecutive years (2018 and 2019) followed by GWAS and genomic prediction analysis with 10,173 quality SNP markers. The studied genotypes exhibited a significant genotypic variation in grain yield and micronutrient-related traits. The GWAS analysis identified highly significantly associated markers and linked putative genes on chromosomes 1B and 2B for zinc (Zn) and iron (Fe) contents, respectively. The genomic predictive ability of selenium (Se) and Fe traits with the multi-trait-based GP GBLUP model was 0.161 and 0.259 improving by 6.62 and 4.44%, respectivelyThe online version contains supplementary material available at 10.1007/s00438-023-02074-6. Wheat is the world's leading food crop and the cradle of human civilization with an average annual production of 770 million tons in about 220 million hectares FAOSTAT . It has n = 4x = 28, AABB) with the diploid wild relative, Aegilops tauschii  followed by chromosome doubling of the corresponding diploids  to evaluate the available genotypic variation in micronutrient content and grain yield; (II) to identify significantly linked MTAs with targeted traits via GWAS and localize putative candidate genes; and (IV) to estimate the GEBVs of target traits through single-trait- and multi-trait-based genomic prediction analysis. For this purpose, the panel was evaluated for 2 consecutive years  under field condition and genotyped with 15 K SNP array.2 plot size following an alpha lattice design with two replications. Planting was done on the first week of December during the two trial seasons similarly at a seeding rate of 100 kg per hectare. Merchouch station is characterized with a cambisol soil and was received an annual precipitation of about 320 mm over the two cropping seasons. Merchouch station has a moderate humidity with annual temperature ranging between 10 and 40 °C.The current study used a panel comprising 252 elite wheat genotypes developed from ICARDA (Supplementary Table 2O2). Then samples were filtered followed by diluting with hydrochloric acid (HCl) after the digestion process was completed and the solution became colorless. The determination of micronutrient content was performed using a simultaneous multi-element inductively coupled plasma emission spectrometer  at the Cereal and Legume Quality Laboratory, ICARDA, Morocco. Grain yield was measured in kg/plot after threshing the whole plot.The currently used panel was phenotyped for four traits including grain yield (GY), iron (Fe), zinc (Zn), and selenium (Se). The micronutrient content of 252 elite wheat samples was evaluated following the standard protocol explained by Pequerul et al. . BrieflyRcmdrMisc package in R  software  is the expected value of r2 under drift-recombination equilibrium; c is the recombination fraction between sites and n is the sample size.The pairwise SNP markers linkage disequilibrium (LD) was calculated as CMplot package in the R environment using −log10(p) > 3.0 as exploratory threshold (LiLin-Yin https://plants.ensembl.org/Triticum_aestivum/Info/Index) was explored to identify putative functional genes-associated SNP markers identified as MTAs with studied traits following the variant effect predictor method, and the UniProt (https://www.uniprot.org) database was used to determine associated protein functions.Genome-wide association analysis of the current panel from the absolute mean values of grain yield and micronutrient traits was performed using the mixed linear model (Q + K) in TASSEL including the kinship matrix (K) and principal components (Q) to control possible false-positive results raised from kinship and population structure, respectively. Manhattan plots were generated to display significant markers across chromosomes using u is the vector of additive effects derived from SNP markers following the multivariate normal μ∼MN , where G is the marker-based genomic relationship matrix calculated using the method developed by VanRaden , where I is the identity matrix and σ2e is the residual variance.Both the single- and multi-trait genomic prediction analysis was conducted using the genomic best unbiased prediction (GBLUP) model. The single-trait GBLUP-based genomic prediction analysis was done following the mixed linear model formula:VanRaden  as \docuThe micronutrient-related traits genomic prediction analysis was further analyzed with multi-trait based genomic prediction models following the mixed model formula:Yij i = 1, 2, 3, and j = 1, …, n is a matrix of the response of the ith trait related to micronutrients in the jth wheat genotype, β1- β3 is a matrix from intercepts of each of the three micronutrient-related traits, Z1- Z3 the matrices of random SNP markers effect, u1—u3 is a matrix of vector derived from markers effect, and ε1 – ε3 is a matrix of the error terms from each traits.where A fivefold cross-validation analysis was employed to evaluate the single- and multi-trait genomic prediction models where 80% of the panel was used to develop the prediction model. The developed model was used to estimate the genomic breeding values (GEBVs) of the remaining 20% individuals. The prediction accuracy was derived from the correlation between the GEBVs with recorded phenotypic values of yield and micronutrient-related traits. The cross-validation analysis was done for 100 iterations.All the single- and multi-trait genomic prediction analysis was done in BGLR software package (Pérez and Los The highly performed elite genotypes for grain yield can be found in Supplementary Table 2) 0.23 and 0.26, respectively  with phenotypic scorings clustered the 252 genotypes into 4 distinct groups Fig. . Furtherr2 value of 0.21 followed by the sub-genomes B and A with 0.20 and 0.19, respectively. The genome-wide linkage disequilibrium revealed the LD decay started at r2 value of 0.46. The half-LD decay (r2 = 0.23) intersected with the LD decay trend at 8.77 Mbp  linked to the 4 tested traits (Table IAAV2271 located on chromosome 4B was particularly highly significantly (P < 0.000001) linked with grain yield with −log10P value of 6.06 followed by the markers BobWhite_c4502_252 and RAC875_c46194_201 which are both located on chromosome 3B with −log10P value of 4.8 and 3.6, respectively. Other six MTAs were identified for Fe located on chromosomes 3A, 4A, 7A, 2B, and 1D  was identified significantly . Environment, management, and the interaction between genotypes with these factors further affect the performance of wheat genotypes for these traits. The current study assessed the genotypic variation of 252 elite bread wheat genotypes for yield and micronutrients under rainfed conditions followed by GWAS and genomic prediction analysis. A significant variation was observed among evaluated genotypes for grain yield and micronutrient-related traits agreeing with several previous studies Below is the link to the electronic supplementary material.


These selected founders strategically represented the four most prevalent botanical races  to capture a significant source of genetic variation to study the quantitative traits in grain sorghum [Sorghum bicolor (L.) Moench]. MBL was phenotyped at two field locations for seven yield-influencing traits: panicle type (PT), days to anthesis (DTA), plant height (PH), grain yield (GY), 1000-grain weight (TGW), tiller number per meter (TN) and yield per panicle (YPP). High phenotypic variation was observed for all the quantitative traits, with broad-sense heritabilities ranging from 0.34 (TN) to 0.84 (PH). The entire population was genotyped using Diversity Arrays Technology (DArTseq), and 8,800 single nucleotide polymorphisms (SNPs) were generated. A set of polymorphic, quality-filtered markers  and phenotypic data were used for genome-wide association studies (GWAS). We identified 52 marker-trait associations (MTAs) for the seven traits using BLUPs generated from replicated plots in two locations. We also identified desirable allelic combinations based on the plant height loci , which influences yield related traits. Additionally, two novel MTAs were identified each on Chr1 and Chr7 for yield traits independent of dwarfing genes. We further performed a multi-variate adaptive shrinkage analysis and 15 MTAs with pleiotropic effect were identified. The five best performing MBL progenies were selected carrying desirable allelic combinations. Since the MBL population was designed to capture significant diversity for maintainer line (B-line) accessions, these progenies can serve as valuable resources to develop superior sorghum hybrids after validation of their general combining abilities via crossing with elite pollinators. Further, newly identified desirable allelic combinations can be used to enrich the maintainer germplasm lines through marker-assisted backcross breeding.Multi-parent populations contain valuable genetic material for dissecting complex, quantitative traits and provide a unique opportunity to capture multi-allelic variation compared to the biparental populations. A multi-parent advanced generation inter-cross (MAGIC) B-line (MBL) population composed of 708 F Sorghum was domesticated in Africa circa 3000 B.C.E., most likely in the Sahel area, where it is one of the most important cereal crops due to its drought tolerance . Seconda1 hybrid seed production is dependent on the cytoplasmic male sterility (CMS) system for cross-fertilization. In the CMS system, three distinct line types  are required, and crossing is performed using specific parental pairs  to produce a hybrid seed   . The A/B and 9E) . The B-l and 9E) . To date and 9E) , which i2s, doubled haploid (DH), and backcrosses. Among these populations, RILs and DHs have the distinct advantage that they are “immortal” and can be used in multiple experiments. Conversely, RILs and DHs that are derived from biparental crosses exhibit relatively low genetic recombination and diversity with a high probability for parents to carry the same alleles at a locus. To overcome these limitations, multi-parent genetic populations like MAGIC and nested association mapping (NAM) have been established in various crops.In the last two decades, the accessibility of relatively inexpensive genotyping costs makes it more effective to study complex traits through association studies . DevelopArabidopsis  between plant height and yield component traits. In this study, we selected best-performing lines based on the new allelic combinations as the unique genetic resources for future breeding efforts to facilitate the pyramiding of desirable alleles using marker-assisted selection. These selected B-lines can be used directly in a hybrid breeding program after validation of general combining abilities.In the present study, a recently developed MBL population was leveraged to mine the B-line  parent gene pool for novel and favorable genetic variation . A set oMBL population was developed from crosses between four founder lines SC630 (PI533937), SC605 (PI534096), BTx642 (PI656029), and BTxARG-1 (PI561072). These founders were selected to capture genetic diversity across multiple botanical races  as well as broad phenotypic variation across various qualitative, agronomic and yield related traits . In addi6 RILs and the four founder lines were phenotyped for PT, DTA, PH, GY, TGW, TN, and YPP. The first and last plants in each row were not phenotyped since they served to eliminate confounding results caused by border effects. Panicle type (PT) was visually assessed  at physiological maturity at a single field location  and used in the analysis as numerical values in 1–4 scales, . DTA was measured as days after planting to when 50% of the plants in the plot were at mid-bloom. PH was measured at physiological maturity in centimeters from the ground to the apex of the primary panicle. The tiller number per plot (TN) was estimated by counting the total number of plants from a representative 1-m row. All panicles were harvested from this 1-m section and subsequently threshed to process grain yield (GY) and individual yield component traits. Harvested panicles were dried for 3–4 days in an electric dryer to a constant moisture content (∼12% moisture) and threshed individually with a BT-14E belt thresher . GY was estimated as a total grain weight of 1-m harvested plot, which was also used for estimating TGW by counting 1,000 grains of every individual using Model U electric seed counters . To estimate the grain yield per panicle (YPP), GY was divided by number of panicles. All seven traits were divided into three major categories: 1) panicle morphology (PT), 2) agronomic (DTA and PH) and 3) yield related traits .As previously mentioned, a set of 708 F6) RILs along with their four founders were grown at two field locations including Simpson Research Farm, Pendleton, SC near to Clemson University (designated as CU) and Pee Dee Research and Education Center, Florence, SC (designated as FL) during summer 2021. The first location was planted on 2 June 2021 at Simpson Research Farm  in Pendleton, SC included 701 RILs, while the second location was planted on 7 June 2021 at the Pee Dee Research and Education Center  in Florence, SC, which included 708 RILs. An alpha lattice field design was used across locations to evaluate the MBL RILs and founder lines, with two replications in each location and four incomplete blocks per replicate. Each incomplete block contained 182 RILs, four founders, and a F1 hybrid check. Each genotype was grown as a single-row plot at 3 m in length and a row spacing of 0.76 m. A plant density of ∼130,000 plants ha−1 was calculated based on a plant establishment of 75% using a seeding rate (MBL (708 Fing rate .S-metolachlor. A post-emergent application of atrazine was administered approximately 40 days after planting. Sugarcane aphids (Melanaphis sacchari) were controlled with one or more applications of flupyradifurone (Sivanto Prime), and chlorantraniliprole (Prevathon) was administered in a single application to prevent corn earworm (Helicoverpa zea) and fall armyworm (Spodoptera frugiperda) infestation. The field trials were irrigated as required to prevent puzzling effects on genotypic performance due to maturity and varying degrees of drought tolerance.Before planting the field trials, the seeds were treated with a blend of fluxofenim , clothianidin , mefenoxam , and fludioxonil . Pre-plant N-P-K was applied at a variable rate based on point soil samples and worked into the soil using conventional tillage. To prevent the germination of weeds, fields were sprayed just after planting with a pre-emergent herbicide containing atrazine and https://www.diversityarrays.com/); for Diversity Arrays Technology sequencing at low density (DArTseqLD). For DArTseqLD analysis, DNA wasdouble digested using PstI and MseI . Resulting FASTQ data were processed using fastp (BWA (GATK)   best praA (GATK) . In brieA (GATK)  with higA (GATK) . The recA (GATK) . SNPs wemetan package and the corr_plot and plot.corr_coef functions were used for the visualization of correlation matrices for each trait in R software  were estimated using the lme4 package in R  GWAS program  to use as covariates and control for false positives during analysis . FarmCPUr2) between markers located on each chromosome was measured to estimate the LD relationship between loci. The r2 was plotted on the y-axis and physical distances (Mb) on the x-axis.The linkage disequilibrium (LD) decay was estimated using PopLDdecay  program Dw1, Dw2, and Dw3). For performing this analysis, we used two haplotypes  based on the closest associated SNP with each gene  and performed GWAS analysis.GWAS were also performed for yield and yield component traits  within a subset of RILs with genetic backgrounds fixed for the three dwarfing genes  using the R package e traits . The estshr in R . The LFSshr in R  due to ishr in R . A contrR  did not show any relationship with DTA, TGW, and TN, while it showed poor positive correlations with PH (r = 23), GY (r = 0.12), and YPP (r = 0.11) . DTA wasi.e., linkage blocks) across the 10 chromosomes of sorghum, except chromosome 8  for seven targeted traits. These 52 MTAs were distributed on 32 genomic regions (mosome 8 . Of thesDw2 (Sobic.006G067700).The FarmCPU model identified three significant MTAs for PT, which were located on Chr1 (∼24 Mb), Chr2 (∼58 Mb), and Chr6 (∼43 Mb). No significant association was identified by MLM. Of these three associations, a sole SNP  overlapped with PH and GY. This genomic region at ∼43 Mb was within 0.50 Mb of the dwarfing gene In total, 12 MTAs were significantly associated with DTA, which were located on seven different chromosomes . Among these, nine MTAs were uniquely detected by FarmCPU, whereas two MTAs were detected using MLM, and a single SNP was commonly detected by both models. Two significant MTAs between DTA and PH overlapped, which were located on Chr3  and on Chr5  .Dw1 (Sobic.009G229800). Another MTA was detected on Chr6  in the same LD block being ∼0.43 Mb to Dw2 (Sobic.006G067700). For PH, a significant MTA was identified on Chr7 , in proximity (0.21 Mb) to Dw3 (Sobic.007G163800). In addition to these PH loci, seven additional MTAs that passed the significant threshold were identified including three on Chr3, two on Chr2, and one each on Chr4 and Chr10. Interestingly, a common MTA was detected on Chr3  around ∼62 Mb for both the agronomic (DTA and PH) traits  were identified for PH, which were located on seven different chromosomes . A single MTA carrying 10 significant SNPs was associated with Chr9, which spanned a ∼1.3 Mb region (∼57 Mb) within a LD block. Out of 10 significant SNPs, one SNP was positioned on Chr9  near ) traits .Dw1, Dw2, and Dw3). Of these five SNPs, three were independently detected by FarmCPU and a sole SNP  was detected by MLM, whereas only one SNP  detected by both models (FarmCPU and MLM) within the same LD block. All four MTAs showed shared associations with all traits  except DTA.Altogether, GWAS identified 27 significant MTAs for grain yield and its components  including several shared associations among the traits . Four MTFive MTAs (eight SNPs) were identified for TGW in total, which were located on five different chromosomes . Of these five associations, one MTA (four SNPs) was significantly associated with a locus on Chr9 at ∼57 Mb that spanned a genomic region (0.75 Mb). This genomic region showed significant association with multiple traits including TGW . In total, GWAS identified 12 SNPs (10 MTAs) significantly associated with TN, which were located on seven different chromosomes . All the significant associations were detected by FarmCPU. Three genomic regions, each located on Chr5 (∼25 Mb), Chr7 (∼59 Mb), and Chr9 (∼57 Mb) showed common associations with multiple traits .Application of FarmCPU to YPP resulted in the identification of eight MTAs, which were located on six chromosomes . Of these eight MTAs, a locus on Chr2 (∼59 Mb) showed a significant association with PH. Similarly, another significant MTA on Chr7 (∼57 Mb) showed association with multiple traits including YPP, PH, and GY .mashr program. A total of 15 significant MTAs (29 SNPs) were detected with pleiotropic effects on multiple traits (All the significant SNPs identified through GWAS following two models (FarmCPU and MLM) were used to assess pleiotropic effects for all the seven traits using the e traits . These Me traits . OverallDw1, Dw2, and Dw3) that were segregating in the MBL population. Using this approach, two additional MTAs were identified on two different chromosomes (Chr1 and Chr7) that were all independent from Dw1, Dw2, and Dw3  identified in the four founders of this population  parents . As a result, the high phenotypic variance was observed for all the phenotypic traits  in the MBL population. In this population, our major focus was to identify desirable allelic combinations and genetic loci associated with yield influencing traits that can be used for sorghum yield improvement in the future.Sorghum is predominantly grown as a hybrid crop in the United States. Currently, cytoplasmic male sterility (CMS) is a popular method of hybrid seed production. The CMS system requires three lines  for cross-fertilization . As indi parents . This poInterestingly, we observed several colocalized or overlapped significant MTAs between four yield influencing traits , which was reflected by the relationship among these traits . The ovei.e., morphology) is an important trait that facilitated racial classification in sorghum. Panicle morphology also influences traits related to crop adaptation such as grain maturity, grain yield, and grain size (Dw2), which has been mapped earlier in sorghum  on Chr9 (Dw2 (Sobic.006G067700) on Chr6 (Dw3 (Sobic.007G163800) on Chr7 (Dw2), Chr7 (Dw3), and Chr9 (Dw1). In addition, three common genetic loci associated with PH on the three chromosomes , have been mapped earlier by As we know, three major dwarfing genes have been previously reported in sorghum on different chromosomes, such as  on Chr9 , Dw2 , BTxARG1 (Dw2), and SC605 (Dw3). As a result, all eight gene combinations were present across MBL progeny, which led to significant height variation. Because plant height confounds grain yield and related traits, these dwarfing genes tended to dominate explained phenotypic variance and thus masked the effects of other genetic variants segregating in the population. To identify novel alleles associated with yield related traits, GWAS were performed using a subset of RILs that contained matching plant height gene combinations. Using this approach, seven significant genomic loci were identified on five different chromosomes  for yield influencing traits. These genomic regions overlapped with multiple yield component traits . Of these seven, the two significant loci (Chr1 and Chr7) identified appear to be novel allelic combinations. This MBL population was designed to identify and recombine useful genetic variation present in diverse maintainer (B-line) germplasm. Desirable allelic combinations were indeed elucidated based on phenotypic comparisons of all the MBL lines for yield related traits. Considering plant height allelic combinations, five best-performing MBL progeny lines were selected , Dw2 (Chr6), and Dw3 (Chr7). Additionally, two novel MTAs each on Chr1, and Chr7, were identified for grain yield and yield component traits when dwarfing genes  were fixed in a subset of MBL progeny. Favorable alleles at these loci can be leveraged for continued improvement of sorghum seed parent productivity and performance. Direct selection on these pleiotropic loci, in the absence of linkage between favorable and deleterious alleles, can be used to simultaneously improve correlated traits. Finally, at least five best-performing RILs were selected for validation of grain yield and yield related traits associated with dwarfing genes allelic combinations segregating in the MBL population. The favorable progeny lines can serve as valuable germplasm in sorghum hybrid breeding programs.The MBL is a structured multi-parent population that encompasses a rich source of genetic variation for the seed parent gene pool . Seed parent genetic diversity within the A/B/R CMS system is limited due to the majority of sorghum genotypes being partial or full fertility restorers based on one or more nuclear restoration genes. Here, allelic variants segregating among four diverse founder lines were mined for association with complex, quantitative traits using GWAS. Specific traits included PT, agronomic (DTA and PH), grain yield (GY), and yield components . GWAS identified 52 MTAs located on all the chromosomes of sorghum except Chr8, representing both novel and previously identified genetic loci for the above traits in narrow genetic regions. In addition, 15 significant MTAs were identified with pleiotropic effects involving grain yield and yield influencing traits. Desirable allelic combinations were identified based on plant height haplotypes associated with plant height genes:


Bos taurus): during these contests, pairs of cows compete in ritualized interactions to assess social dominance. The outcomes of 37,996 dominance interactions performed by 8789 cows over 20 years were combined with individual data for fertility, mammary health, milk yield and morphology and analysed using bivariate animal models including indirect genetic effects.Within the same species, individuals show marked variation in their social dominance. Studies on a handful of populations have indicated heritable genetic variation for this trait, which is determined by both the genetic background of the individual (direct genetic effect) and of its opponent (indirect genetic effect). However, the evolutionary consequences of selection for this trait are largely speculative, as it is not a usual target of selection in livestock populations. Moreover, studying social dominance presents the challenge of working with a phenotype with a mean value that cannot change in the population, as for every winner of an agonistic interaction there will necessarily be a loser. Thus, to investigate what could be the evolutionary response to selection for social dominance, it is necessary to focus on traits that might be correlated with it. This study investigated the genetic correlations of social dominance, both direct and indirect, with several morphology and fitness traits. We used a dataset of agonistic contests involving cattle (We found that winning agonistic interactions has a positive genetic correlation with more developed frontal muscle mass, lower fertility, and poorer udder health. We also discovered that the trends of changes in the estimated breeding values of social dominance, udder health and more developed muscle mass were consistent with selection for social dominance in the population.We present evidence that social dominance is genetically correlated with fitness traits, as well as empirical evidence of the possible evolutionary trade-offs between these traits. We show that it is feasible to estimate genetic correlations involving dyadic social traits.The online version contains supplementary material available at 10.1186/s12711-023-00845-8. An individual’s phenotype is determined not only by its own genetic background, but also by the genetic backgrounds of its interacting counterparts , 2. IndiThe evolutionary trajectory of social dominance, or ‘a dyadic social relationship that emerges from sequences of agonistic interactions, where one individual exhibits subordination’ as defined by Strauss et al. , is espeWhenever IGE are considered in studies of dyadic contests with a win/lose outcome as a dominance trait, the total genetic variance is the sum of the direct genetic effect, the indirect genetic effect, and twice their covariance. In 1/0 contests, since for each direct winner there is necessarily an indirect loser and vice versa, the direct and indirect genetic variances and their covariance are all equal in magnitude, but the covariance has a negative sign. This postulates that the net genetic variance associated with social dominance, i.e., the genetic variance available for selection of the trait, is zero. The few empirical studies that have focused on the IGE underlying social dominance , 12 haveBos taurus), which is correlated with traits linked with morphological and physiological masculinization, but it is also negatively correlated with fertility, milk production and longevity ), 0.060 , and 0.224 , respectively. The estimates of heritability for the morphological traits ranged from low for thinness , to moderate for traits linked with udder size and morphology  and the variance attributed to the genetic makeup of the opponent (indirect) individuals. Since the phenotype associated with each fight is the same for both the focal and opponent individuals, variances are expected to be equal, and their covariance to have the same magnitude but with opposite signs. On the one hand, certain software or packages  can add this constraint to the animal model, which ensures that the correlation between direct and indirect components is − 1 and the total heritability for social dominance is 0. On the other hand, an unconstrained model can test the size and completeness of the dataset, since the values that it estimates freely approximate these features. Our unconstrained single-trait model obtained a correlation of − 0.989 between direct and indirect components of social dominance and a total heritability of 0.003. These very close approximations to − 1 and 0 were also obtained in all subsequent bivariate analysis  and life history and fitness traits has been discussed for several years. Our work uncovers evidence of a genetic correlation between a behavioural trait  and key life-history, fitness, and morphological traits. We show that dominance interactions, such as those that are ubiquitous within animal groups, are genetically correlated, via both direct and indirect components, to other traits. Artificial selection for social dominance in our population plays a significant role in the evolution of these traits, by reducing the cows’ reproductive efficiency and increasing morphological masculinization. We argue that social dominance in livestock can be a useful model to investigate the genetics of social behaviour and other social traits.Additional file 1. In-depth description of the single-trait models for milk yield, somatic cell score, fertility and morphological traits [l traits , 104.Additional file 2: Table S1. Descriptive statistics of the traits, including the extreme phenotypic values, the phenotypic mean and its standard deviation (SD). Table S2. Variance components of the traits expressed as mean ± standard error (SE) of posterior density intervals of Gibbs sampling estimates, obtained by running single-trait analysis. Table S3. Genetic (σa1a2) and permanent environmental covariances (σpe1pe2), obtained by running bivariate analyses for each pair of traits including social dominance vs. the other traits. Table S4. Heritability estimates for the traits of interest obtained with bivariate analyses. Posterior means for heritability estimates with their respective standard error (SE) and the 95% high posterior density confidence interval (CI) of Gibbs sampling estimates obtained for the traits of interest by running bivariate analyses. Table S5. Regression coefficients (slopes) of the trait variations over time calculated using the average estimated breeding values (EBV) for the newborns of target years (genetic trends) and % of replicates with slope > 0.Additional file 3: Figure S1. Distribution of the regression coefficients (slopes) for the EBV variations in the direct component of social dominance over time calculated using MCMC replicates of average cohort breeding values. Figure S2. Distribution of the regression coefficients (slopes) for the EBV variations in the indirect component of social dominance over time calculated using MCMC replicates of average cohort breeding values. Figure S3. Distribution of the regression coefficients (slopes) for the EBV variations in milk yield over time calculated using MCMC replicates of average cohort breeding values. Figure S4. Distribution of the regression coefficients (slopes) for the EBV variations in somatic cell score over time calculated using MCMC replicates of average cohort breeding values. Figure S5. Distribution of the regression coefficients (slopes) for the EBV variations in fertility over time calculated using MCMC replicates of average cohort breeding values. Figure S6. Distribution of the regression coefficients (slopes) for the EBV variations in fore udder attach over time calculated using MCMC replicates of average cohort breeding values. Figure S7. Distribution of the regression coefficients (slopes) for the EBV variations in udder overall over time calculated using MCMC replicates of average cohort breeding values. Figure S8. Distribution of the regression coefficients (slopes) for the EBV variations in rear udder attach over time calculated using MCMC replicates of average cohort breeding values. Figure S9. Distribution of the regression coefficients (slopes) for the EBV variations in udder width over time calculated using MCMC replicates of average cohort breeding values. Figure S10. Distribution of the regression coefficients (slopes) for the EBV variations in thinness over time calculated using MCMC replicates of average cohort breeding values. Figure S11. Distribution of the regression coefficients (slopes) for the EBV variations in front muscularity over time calculated using MCMC replicates of average cohort breeding values. Figure S12. Distribution of the regression coefficients (slopes) for the EBV variations in thorax depth over time calculated using MCMC replicates of average cohort breeding values.


Wolbachia are endosymbiotic alpha-proteobacteria infecting a wide range of arthropods and nematode hosts with diverse interactions, from reproductive parasites to obligate mutualists. Their taxonomy is defined by lineages called supergroups , while their evolutionary history is complex, with multiple horizontal transfers and secondary losses. One of the least recently derived, supergroup E, infects springtails (Collembola), widely distributed hexapods, with sexual and/or parthenogenetic populations depending on species. To better characterize the diversity of Wolbachia infecting springtails, the presence of Wolbachia was screened in 58 species. Eleven (20%) species were found to be positive, with three Wolbachia genotypes identified for the first time in supergroup A. The novel genotypes infect springtails ecologically and biologically different from those infected by supergroup E. To root the Wolbachia phylogeny, rather than distant other Rickettsiales, supergroup L infecting plant-parasitic nematodes was used here. We hypothesize that the ancestor of Wolbachia was consumed by soil-dwelling nematodes, and was transferred horizontally via plants into aphids, which then infected edaphic arthropods (e.g. springtails and oribatid mites) before expanding into most clades of terrestrial arthropods and filarial nematodes. However, studies on Wolbachia–host coevolution suggest that horizontal transfer events from bacteria to a new host have occurred during the evolution of the species [Wolbachia supports the existence of horizontal transmissions on an evolutionary scale [Wolbachia gene fragments ranging from about 200 bases to more than 3 kb [Wolbachia genome from Brugia malayi was transferred to the genome of its filarial host [The alpha-proteobacterium thropods  and nemathropods –6). Wolb species . Moreovery scale ,6,9,10. han 3 kb –13. In pial host . These tial host ,15.Wolbachia symbionts of arthropods and nematodes is the nature of the symbiosis. Wolbachia is mainly localized in the reproductive tissues of arthropods and it is responsible for the induction of a number of reproductive alterations including feminization, parthenogenesis, male-killing and cytoplasmic incompatibility (CI). Wolbachia induces changes in the reproduction of its host to promote its own transmission [Wolbachia in arthropods are not limited to parasitism and the bacteria can act as obligate mutualists. In the bedbug Cimex lectularius, Wolbachia supplements the blood diet by provision of the B vitamins that are deficient, and the nature of the association is described as nutritional mutualism [Wolbachia and filarial nematodes, as worms lacking their symbionts cease to produce viable embryos and exhibit reduced longevity [Wolbachia is essential for enabling embryonic development and supporting adult survival, consistent with their location in the female germline and somatic hypodermal cords. However, most of the time the nature of the association is often undefined and in particular it is not established for plant nematodes [One difference between smission –18. Howeutualism ,20. Antiongevity ,22. Wolbematodes  and poorematodes –27.Wolbachia have been divided into distinct lineages known as supergroups [Wolbachia supergroups have been described and labelled from A to U [Based on phylogenetic studies, ergroups ,28,29. Tm A to U ,30–32. Am A to U ,34. HoweWolbachia is still a subject of scientific debate. One issue is that phylogenetic analysis methods are prone to long-branch attraction artefacts and therefore the origin of infection remains unresolved. The most recent comprehensive phylogeny suggests that L (in plant nematodes), M (aphids) and E (Collembola and mites) are sister clades to all the other Wolbachia supergroups [Wolbachia up to now and 13 are infected [Wolbachia of supergroup E have only been detected in parthenogenetic populations of Collembola [2 in forest soils [www.collembola.org). Springtails also participated in the emergence of life from an aquatic environment to a terrestrial environment by colonizing soils [Despite the significant number of phylogenetic and taxonomic studies, the evolutionary root of ergroups . Regardiinfected –39. Notallembola ,36,38,39st soils . Collembst soils . These sst soils   . PCR products were purified and sequenced by Eurofins Genomics using the Sanger method. Chromatograms were analysed and edited using CodonCode Aligner. Supergroups of Wolbachia were identified as described in previous studies [The presence of  studies . A total. 2.3Wolbachia sequences from all valid supergroups were extracted from the GenBank database for the six genetic markers . These sequences were then reviewed to delete contaminated or misassigned sequences. Taxa with only one genetic marker available were not included in the dataset. The dataset of GenBank sequences consisteds of 100 taxa . The selected outgroups are other Rickettsiales belonging to the genera Anaplasma spp.  and Erlichia spp. (E. chaffeensis and E. ruminantium) or the Wolbachia genotypes of the supergroup L infecting Pratylenchus penetrans and Radopholus similis.To analyse the newly obtained sequences, a dataset of GenBank sequences was built. Sequences were aligned, for each locus independently, using MAFFT v7.505  with def. 3. 3.1Wolbachia was detected with the amplification of at least one genetic marker in 11 species. Among them, six are new genotypes of Wolbachia  [Out of the 58 studied species of springtails, the presence of olbachia . Thus, a/6 (66%) ; 3/9 (33/6 (66%) ; and 4/1/6 (66%) . The dif/6 (66%) ). Hence,Wolbachia infecting Collembola are distributed in two separate clades: supergroup A and supergroup E (figures Wolbachia of supergroup E infecting the springtails are more diverse than previously described [Wolbachia genotypes infecting the springtail Mesaphorura yosii (wMyos) and the mite Opiella nova (wOnov). Previously, wOnov was not associated with a supergroup, and it was positioned at the root of the filarial and arthropod supergroup dichotomy [wMyos, it was already inferred as the sister group of the remainder of supergroup E [wOnov and wMyos do not have every genetic marker available, sharing only the 16S rDNA and fbpA genes. Thus, the E1 clade lacks robustness. The Wolbachia genotypes of E2 form a strongly supported clade. Interestingly, the springtails of Neelipleona order are strictly infected by the E2 genotypes, although only two genera are represented in this study . With two springtail orders and two from Acari infected, the E3 clade is the most diverse of the three E subclades.The genotypes of  figures  and 2. Tescribed ,38,39. Wichotomy . Regardic–e) [Wolbachia of supergroup E, the genotype infecting Folsomia candida (wFol), has unique genomic characteristics compared to other genomes of Wolbachia. While the depletion of Wolbachia in haplo-diploid arthropod hosts leads to the development of males, in F. candida the presence of antibiotics strongly limits the production of eggs and once the treatment ends, the egg production recovers to its normal level, suggesting wFol is able to enter a reversible persister state [Wolbachia studied infecting arthropods [wFol genome encodes for an Abi (abortive infection) Type IV TA system and has only been found in this genome [Wolbachia has yet to be deciphered. One hypothesis would be the involvement in the reversible persister state of wFol.The springtails infected by the genotypes of the supergroup E are euedaphic: they live in the soil, are small and have lost their pigmentation c–e [40,4c–e) . Moreovec–e) . One of er state –27,51. Bthropods . The expthropods . These Ts genome . HoweverWolbachia infecting springtails were classified in supergroup A. Among them, the presence of a genotype infecting Sphaeridia pumilis is also the first observed case of a symphypleonan infected by Wolbachia, indicating that all four springtail orders  are infected by this bacterium. All the springtail species infected by Wolbachia from supergroup A live at the surface either in semi-aquatic environments (Anurida maritima and Podura aquatica) [S. pumilis) [Three quatica) ,55 or inpumilis) . They hapumilis) . More imWolbachia of supergroup E or supergroup A seem strongly correlated with their ecology.In springtails, infections either by . 3.2Wolbachia phylogeny. First, the usual Rickettsiales outgroups , Ehrlichia and Anaplasma, were used . Second, the phylogeny was inferred with the supergroup L as the outgroup . However, when the topology is rooted and supergroup L is present in the phylogeny, in 14 out of 18 phylogenies (82%), supergroup L is consistently positioned as the sister group of the remaining Wolbachia .Usually, an outgroup should be selected from outside the studied phylum, but the other Rickettsiales are too phylogenetically distant. Inferring lytomies ,48,58–74Wolbachia infects plant-parasitic nematodes (PPN) and up to now three genotypes have been described. The first two genotypes are known to infect nematode hosts: Radopholus similis (wRad) [Pratylenchus penetrans (wPpe) [Wolbachia-positive samples originating from the same soil of a Texan farm (USA). The contigs obtained from the sequencing of the six Wolbachia-positive samples were highly similar between themselves. Thus, to increase the coverage of the assembly, the authors decided to pool the 192 contigs and named the final assembly wTex [wTex, two genera might be associated this pool: Helicotylenchus spp. and Rotylenchus spp. The genomes of wRad, wPpe and wTex share multiple characteristics [cifA and cifB genes (which are linked to cytoplasmic incompatibility i), WO phage, and homologues of biotin synthesis genes. These absent features make these genotypes closer to the Wolbachia infecting filariae  than those infecting arthropods (e.g. supergroups A and B). However, the core genes of wRad, wPpe and wTex share the most similarities with the core genes of other Wolbachia infecting plant-feeding arthropods such as Bemisia spp., Bryobia spp. or Cinara spp. [Wolbachia lineage [Anaplasma spp. and Ehrlichia spp. are bacteria infecting hematophagous ticks, the most anciently derived Wolbachia supergroups  infect hosts dwelling in the soil and/or arthropod plant pests, which are ecologically closer to PPN. The supergroup organization in the phylogeny with supergroup L as outgroup , the cladogenesis events are distally aggregated while in the phylogeny rooted in L, the cladogenesis events are more evenly distributed   and Prats (wPpe) . The thibly wTex . Regardieristics ,23, mosttributed . The lat. 3.3Wolbachia is influenced by two opposite processes: (i) bacteria coevolve with their host or shift to new hosts and (ii) the host becomes extinct, or symbiont loss occurs [Wolbachia, with host shift events between supergroups whose existence has in most cases been lost. Though ‘paleosymbiosis’ may be inferred with the detection of nuclear Wolbachia transfers in Wolbachia-free host genome [Wolbachia shifted hosts in order to attain the current supergroup distribution  a physical transfer of the endosymbiont to a new species must occur, through a predator–prey interaction, or a host–parasite interaction, or by sharing plant and other food sources; (ii) the bacteria must be able to develop inside the new host; (iii) a maternal transmission of Wolbachia must be possible; and (iv) Wolbachia must successfully spread in the new host population.The evolutionary history of s occurs . The latt genome . With thribution . Sanaei   old [Buchnera–aphid coevolution is sufficiently ancient for the Buchnera genome to have become highly reduced (from 600 kb to 400 kb) and to have lost key metabolic functions, such as the biosynthesis of peptidoglycan or nucleotides [Buchnera presents a mutualistic association with their aphid host by providing essential amino acids, the role of Wolbachia in aphid remains a matter of discussion [Wolbachia in populations of aphids Pentalonia nigronervosa might be because Wolbachia and Buchnera symbionts complement each other in several important pathways to provide essential amino acids and vitamins to the aphid host [Wolbachia being a nutritional co-obligate in this aphid [Close to supergroup L is supergroup M, which until now has only been detected in sap-sucking aphids. Cases of horizontal transfer to an aleyrodid host, the whitefly cumented : Wolbachle event . Contrar(Ma) old ). The Buleotides . If it hscussion ,83. A stis aphid . Thus, tWolbachia can be either scattered in different tissue or confined in bacteriomes in the abdomen [et al. [Wolbachia-free mites became infested by the bacteria by consuming fly corpses, indicates arthropods may become infected by Wolbachia via detritivory.Honeydew is excreted by aphids and can fall on soil, promoting the growth of microbial biomass and attracting fungivorous and bacterivorous springtails . The ori abdomen ,86. Give abdomen , the hor [et al.  marked i [et al. , where WWolbachia infecting termites and pseudoscorpions, may be explained by an ancient springtail relationship with social insects. In an approximately 16 Ma old amber fossil, springtails were observed grasping a winged termite and an ant, apparently in a phoretic association [Wolbachia between springtails and termites. Moreover, pseudoscorpions are also phoretic with other insects [Wolbachia when springtails and pseudoscorpions were in phoresy with the same termite host. In supergroups A and B numerous horizontal transfers [Wolbachia infecting sexual springtails in supergroup A underlines the complexity of the evolutionary history of these bacteria. Their origin of infection may be linked to their ecological niche, but the nature of their interaction has yet to be deciphered.The evolutionary history of the supergroups positioned as sister clades of the supergroup E is complex. The origin of infection of supergroup H , composeociation . This cl insects , so therransfers ,93 and sransfers  occurredransfers : predato. 3.4Wolbachia phylogeny. Close to half of the cladograms were inferred with one gene . Therefore, the record of the highest number of genotypes (n = 236) is held by a phylogeny inferred with this single gene [Published topologies including the supergroups E, M or L were analysed to better understand how to calibrate the one gene , mostly gle gene . Howevergle gene ,74,95,96Wolbachia of the supergroups A, B and M infecting Chinese aphids [Although 16S rDNA is the most sequenced gene, the phylogenies with the highest number of taxa were inferred with two genes . This die aphids . Thus, tgatB, coxA, hcpA, ftsZ and fbpA) [Wolbachia, and they were designed solely on Wolbachia genotypes infecting arthropods. Thus, different authors calculated the recombination rates of the MLST genes to check whether their use was pertinent to discriminate the different supergroups of Wolbachia [To ensure a reliable and robust estimation of the relationships and especially the relevance of the clades/supergroups, the concatenation of multiple genetic markers with various evolutionary rates is needed ,98. The nd fbpA)  are widend fbpA) , not forolbachia ,97,98, aWolbachia phylogenomics involves numerous drawbacks: (i) the poor terminal sampling with the lack of data for some supergroups  and (ii) lower supergroup intra-diversity than in the phylogenies with multiple genes . These problems are linked to the cost and the technical difficulties of sequencing Wolbachia's genome. Indeed, currently, the bacteria cannot be cultivated in an axenic culture system [Wolbachia has a small genome, from 550 kb [If one objective of a study is to explore the intra-supergroup relationships where recent infection occurred, such as in supergroups A, B or F, then complementing the MLST genes with other orthologous genetic markers having a higher evolutionary rate would be needed . This hae system , thus mam 550 kb , the prem 550 kb . These c. 4Wolbachia genotypes infecting springtails is not straightforward. The nature of the infection depends on the host's biology and ecology: the infection of parthenogenetic soil-dwelling springtails is high and belongs to supergroup E; sexually reproducing springtails living in wet environment are infected by another group of Wolbachia (A). The Wolbachia of supergroup E are known to be involved in the manipulation of their host reproduction. The nature of the interaction between Wolbachia of supergroup A and their springtails hosts will have to be deciphered. Increasing the number of genotypes in sexual springtails would give a better picture of the infection in these arthropods.Altogether the evolutionary history of


The MendelianRandomization package is a software package written for the R software environment that implements methods for Mendelian randomization based on summarized data. In this manuscript, we describe functions that have been added or edited in the package since version 0.5.0, when we last described the package and its contents. The main additions to the package since that time are: 1) new robust methods for performing Mendelian randomization, particularly in the cases of bias from weak instruments and/or winner’s curse, and pleiotropic variants, 2) methods for performing Mendelian randomization with correlated variants using dimension reduction to summarize large numbers of highly correlated variants into a limited set of principal components, 3) functions for calculating first-stage F statistics, representing instrument strength, in both univariable and multivariable contexts, and with uncorrelated and correlated genetic variants. We also discuss some pragmatic issues relating to the use of correlated variants in Mendelian randomization. Mendelian randomization uses genetic variants as instrumental variables to assess evidence for the causal effect of an exposure on an outcome.Assessing causality in relationships between risk factors and outcomes is tricky and subject to many potential pitfalls. Mendelian randomization is an epidemiological technique that assesses such questions using genetic variants. Formally, an instrumental variable should be associated with the exposure (relevance), not associated with the outcome via any confounding pathway (exchangeability), and does not affect the outcome directly, only potentially indirectly via an effect of the exposure (exclusion restriction). Generally speaking, genetic variants are plausible candidate instrumental variables, as they tend to have specific effects on particular biological mechanisms, they are uncorrelated with genetic variants that influence other biological mechanisms (due to Mendel’s laws of inheritance), and they are fixed at conception .An instrumental variable is a variable that behaves analogously to randomization in a randomized trial. It divides the population into groups with different distributions of the exposure, but the groups are otherwise indistinguishable  that implements methods for Mendelian randomization based on summarized data. By summarized data, we mean genetic associations with traits (beta-coefficients and standard errors) taken from regression analyses of a trait on a genetic variant. Such associations are estimated in genome-wide association studies. They have been publicly reported for millions of variants by many large studies and consortia, and can be accessed through several different public portals.MendelianRandomization is a software package written for the R software environment, and updates up to version 0.5.0 have been discussed. In this paper, we present updates to the package since version 0.5.0 up to the current version 0.9.0. A complete list of functions in the package is given in. We also encourage users to consult the documentation provided with the package, which describes all the options available for each method in greater detail. In this paper, we aim to provide an overview of recent updates to the package.This package has previously been introducedConstrained maximum likelihood methods. Themr_cML andmr_mvcML functions perform the constrained maximum likelihood method, for the univariable Mendelian randomization (mr_cML) and multivariable Mendelian randomization (mr_mvcML) settings. These methods are robust to the violation of any of the three instrumental variable assumptions under mild assumptions. In a maximum likelihood framework, these methods constrain the number of invalid instruments with horizontal pleiotropy. The number of invalid instruments is asymptotically consistently selected by the Bayesian information criterion. To further account for the selection uncertainty with a finite sample, a data perturbation approach is employed.mr_cML function takes anMRInput object as input, created using themr_input command. The syntax is:Themr_cML,n=17723)ldlc andldlcse are genetic associations with low-density lipoprotein (LDL) cholesterol and their standard errors for 28 uncorrelated genetic variants previously reported as associated with at least one of LDL-cholesterol, high-density lipoprotein (HDL) cholesterol, or triglycerides by Waterworthet al., andchdlodds andchdloddsse are genetic associations with coronary heart disease risk for the same variants. These data variables are provided with the package.n is the sample size for the genetic associations. In practicen could be the sample size either for the exposure or the outcome, the smaller value is recommended to get appropriately-sized confidence intervals; see reference for detailed discussions.wheremr_mvcML function takes anMRMVInput object as input, created using themr_mvinput command. The syntax is:Themr_mvcML,   bxse = cbind,   by = chdlodds, byse = chdloddsse), n = 17723)hdlc andhdlcse are genetic associations with HDL cholesterol and their standard errors,trig andtrigse are genetic associations with triglycerides and their standard errors for the same 28 variants, andn is the sample size for the genetic associations with the exposures . Again, these data variables are provided with the package.whereDP = TRUE, which performs data perturbation and generally results in wider confidence intervals, but is recommended to provide appropriately-sized confidence intervals;MA = TRUE (formr_cML), which performs model averaging, which again is recommended to provide appropriately-sized confidence intervals; andrho_mat (formr_mvcML), which specifies the exposure correlation matrix between the summarized data for the exposures and outcome. If this is unspecified, then it is taken as the identity matrix, implying that the summarized data for the exposures and outcome are independent . Other options are provided to change the settings of the optimization functions used in the methods.The main options for these methods areDebiased inverse-variance weighted method. Themr_divw function performs the debiased inverse-variance weighted method. This method is an extension of the inverse-variance weighted (IVW) method that is more robust to weak instruments, with better bias and coverage properties.mr_divw function takes anMRInput object as input, created using themr_input command. The syntax is:Themr_divw)over.dispersion = TRUE, which allows for overdispersion in the variant-specific estimates (similar to a random-effects model for the IVW method), anddiagnostics = FALSE, which provides a quantile—quantile plot of the variant-specific estimates, as a visual inspection for overdispersion and outliers.The main options for this method arePenalized inverse-variance weighted method. Themr_pivw function performs the penalized inverse-variance weighted method. This method is an extension of the IVW method and the debiased inverse-variance weighted method, which handles weak instrument bias by a penalized log-likelihood function, and handles balanced horizontal pleiotropy by accounting for overdispersion in the variant-specific estimates.mr_pivw function takes anMRInput object as input, created using themr_input command. The syntax is:Themr_pivw)lambda = 1, which is the penalty parameter. It plays a role in the bias-variance trade-off of the estimator. It is recommended to chooselambda = 1 to achieve the smallest bias and valid statistical inference;over.dispersion = TRUE, which allows for overdispersion in the variant-specific estimates; anddelta = 0, which is a z-score threshold used for screening out weak instruments.delta should be greater than or equal to zero. Whendelta = 0, all variants provided will be used in the analysis. Whendelta > 0, the optionsel.pval should be specified, which is the p-values of the genetic associations on the exposure. Then, the variants withsel.pval > 2*pnorm will be removed from the analysis. The final option isBoot.Fieller = TRUE, which provides the p-value and the confidence interval of the causal effect calculated by the bootstrapping Fieller method.The main options for this method are:Generalized method of moments method. Themr_mvgmm function performs the generalized method of moments (GMM) method for the multivariable Mendelian randomization setting. The key advantage of the GMM method is that estimates are more robust to weak instrument bias and/or measurement error in the exposures compared with the standard IVW method, which is equivalent to a two-stage least squares approach with individual-level data. Weak instrument bias is particularly important in the multivariable setting, as bias in the multivariable setting can be in any direction, particularly if instrument strength varies between the exposures.mr_mvivw), the multivariable GMM method requires the sample sizes for genetic associations with the exposure, and the sample size for the genetic associations with the outcome. The method offers inferences that are robust to overdispersion in the variant-specific estimates (using the default optionrobust = TRUE).Unlike the multivariable IVW method  should be harmonized with the summary statistics used in the analysis as described in themr_pcgmm section below.If a genetic correlation matrix is not supplied in themr_mvgmm function is:The syntax for themr_mvgmm, bxse = cbind, by = chdlodds, byse = chdloddsse), nx=rep, ny=17723)Principal component generalized method of moments methods. Themr_pcgmm andmr_mvpcgmm functions perform the principal component generalized method of moments (PC-GMM) method, for the univariable Mendelian randomization (mr_pcgmm) and multivariable Mendelian randomization (mr_mvpcgmm) settings. This method is very similar to the GMM method, the key difference being that the GMM method uses individual variants as instruments rather than principal components.. As an alternative to pruning and clumping approaches, which take a large number of variants from a gene region  and select a small number of uncorrelated (or weakly correlated) variants, the principal components approach performs dimension reduction on the full set of variant associations. The aim is to construct a small number of principal components which capture the information in the data, allowing the analysis to be performed using all available data, but avoiding numerical issues that would occur if highly-correlated genetic variants were included in the analysis. Previous investigations have shown that dimension reduction approaches can give additional precision compared with approaches using a small number of selected variants, and are less sensitive to variability arising from the variant selection process. Results from dimension reduction approaches are also less sensitive to misspecification of the variant correlation matrix, compared with other approaches using highly-correlated variants.These methods are designed for use when performing Mendelian randomization using genetic variants from a single gene region, the PC-GMM method has some important advantages: it uses the Continuously Updating Generalized Method of Moments method, which provides estimates that are more robust to weak instruments than some other instrumental variable methods; and it can allow for overdispersion in the variant-specific estimates (using the default optionrobust = TRUE), which is recommended to provide appropriately-sized confidence intervals and valid inferences.Compared with other dimension reduction approaches that have been proposedmr_pcgmm function is:The syntax for the univariablemr_pcgmm,   nx=6351, ny=133010)calcium,calciumse,fastgluc,fastglucse, andcalc.rho are genetic association data on six correlated variants from theCASR gene region provided with the package and their associations with serum calcium levels  and fasting glucose levels (fastgluc and associated standard errorsfastglucse),nx is the sample size for genetic associations with the exposure, andny is the sample size for genetic associations with the outcome. These associations are:where> mr_input    SNP exposure.beta exposure.se outcome.beta outcome.se1 snp_1       0.00625     0.00233      0.02805     0.01222 snp_2       0.00590     0.00338      0.00953     0.01983 snp_3       0.01822     0.00318      0.03646     0.01734 snp_4       0.00598     0.00233      0.01049     0.01195 snp_5       0.00825     0.00229      0.02357     0.01226 snp_6       0.00651     0.00352      0.00204     0.0179mr_mvpcgmm function is:The syntax for the multivariablemr_mvpcgmm,   bxse = cbind,   by = chdlodds, byse = chdloddsse,   correlation = diag(length(ldlc))),   nx=rep, ny=17723)We note that this example does not use genetic variants from a single gene region, and so does not represent a recommended use case for the method. It is provided to demonstrate the code syntax.mr_pcgmm andmr_mvpcgmm functions chooses the number of principal components to explain 99.9% of the variability in a weighted version of the variant correlation matrix. This can be varied, either by setting a different threshold (defaultthres = 0.999 corresponds to 99.9%) or by fixing the number of principal components using ther option . As for themr_mvcML function, themr_mvpcgmm function requires the exposure correlation matrix, set using thecor.x option . If not specified, this is set to the identity matrix, implying that the summarized data for the exposures are estimated in non-overlapping samples. If these associations were estimated in the same dataset and these correlations are not known, a sensitivity analysis may be worthwhile.The default operation of theCorrelated variants. Most methods in the MendelianRandomization package assume that all genetic variants are uncorrelated (seemr_ivw andmr_egger functions (and their multivariable counterparts) can account for correlations between variants, as can themr_mvgmm and related methods (mr_pcgmm andmr_mvpcgmm) introduced above. While using partly correlated variants can provide additional precision compared with using uncorrelated  variants, there are several cautions to the use of correlated variants in Mendelian randomization.ated see. Themr_i.e. the variant having the lowest p-value in its association with the exposure) or with minimally-correlated variants. We would expect the correlated variants to explain slightly more variance in the exposure, and so we would expect the standard error to reduce and confidence intervals to narrow slightly when using correlated variants. However, if the standard error when using correlated variants is much smaller than when using the lead variant only , then inputs should be checked carefully, as the increase in precision may represent a convergence issue. Reporting a sensitivity analysis using fewer variants or principal components is recommended.First, the estimate from the analysis with correlated variants should not be strikingly more precise than the result with the lead variant only (2 matrix) between variants, not the (signed) correlation matrix. Mendelian randomization requires the correlation matrix, which provides correlations between variants.Second, many software tools report the squared-correlation matrix  must be correctly specified. If the correlation matrix is calculated using the same effect and non-effect alleles as the summarized data, then the correlations should have the correct signsflip = ccalc.rho.signed = calc.rho*flip%o%flipflip is +1 for variants for which the alleles are correctly aligned, and -1 for variants for which the alleles are incorrectly aligned, and the elements of flip correspond to the genetic variants in order.whereIn this example, before orientation, the variant correlation matrix is:> calc.rho                        1.000  0.070  0.094 -0.172  0.079 -0.104  0.070  1.000 -0.050  0.081 -0.080 -0.014  0.094 -0.050  1.000 -0.306  0.349 -0.134 -0.172  0.081 -0.306  1.000 -0.129  0.446  0.079 -0.080  0.349 -0.129  1.000 -0.289 -0.104 -0.014 -0.134  0.446 -0.289  1.000After orientation, the variant correlation matrix is:> calc.rho.signed                        1.000  0.070 -0.094 -0.172 -0.079 -0.104  0.070  1.000  0.050  0.081  0.080 -0.014 -0.094  0.050  1.000  0.306  0.349  0.134 -0.172  0.081  0.306  1.000  0.129  0.446 -0.079  0.080  0.349  0.129  1.000  0.289 -0.104 -0.014  0.134  0.446  0.289  1.000flip are -1 , all entries in the third row or column, and the fifth row or column, have flipped in sign . For example, the third entry in the first row was +0.094, but is now -0.094. The element in the third row and fifth column (+0.349) is flipped in sign twice, and so its sign ends up unchanged; similarly for the element in the third row and third column, the fifth row and third column, and so on.As the third and fifth elements ofFourth, the correlation matrix should be obtained from the same population (or same ancestry group) as the original data, to avoid mismatches between the correlation matrix and the summarized data. If the exposure and outcome data are from different ancestry groups, then correlated variant analyses should not be attempted. If the exposure and outcome data are from the same ancestry group, then correlations should be taken from the outcome data by preference, but either (or use of a suitable reference population) is acceptable.mr_pcgmm andmr_mvpcgmm methods. A suggestion when using these methods is to take pairs of variants that are correlated at r2 > 0.95, and remove one of the pair at random until no such pairs remain. This can be implemented using the code below for a correlation matrixrho:Fifth, analysts should consider pruning out extremely highly correlated variants, as these are likely to contribute little to the analysis, even when using theset.seed(496)         # for reproducibilitythres = sqrt(0.95)    # threshold set to r2>0.95omit = NULL           # set up list of variants to be omittedrho.upper = rho       # correlation matrixrho.upper[lower.tri] <- 0                      # only consider upper triangle of correlationsj=1                   # set counter to 1while (max(abs(rho.upper), na.rm=TRUE)> thres) { omit[j] = ifelse==1,    which.max(apply(abs(rho.upper), 1, max, na.rm=TRUE)),    which.max(apply(abs(rho.upper), 2, max, na.rm=TRUE)))                      # find the highest correlation value                      # select either the row or column at random                      # add this to the list of omitted variants rho.upper[omit[j],] <- 0 # set the correlations in this row to zero rho.upper] <- 0 # set the correlations in this column to zero                          # (to avoid selecting the same variant again) j=j+1                # increment the counter  }  # stop when no more pairwise correlations exceed threshold2 < 0.3) or applying another approach for variant selection  is recommended to avoid high levels of variant multicollinearity.For other methods that allow for correlated variants, either pruning at a stricter threshold  function in R applied to the matrix of allele counts. The variant correlation matrix can be obtained from a reference population; for example, correlations for European ancestry individuals from UK Biobank are availablehere, and correlations for other ancestry groups  are available at https://ldlink.nci.nih.gov/?tab=ldmatrix . The exposure correlation matrix can only be estimated from individual-level data, and hence a sensitivity analysis for its value is suggested if it is unknown. However, investigations have indicated that estimates are generally insensitive to the correct specification of the exposure correlation matrix, and so the default value of the identity matrix is likely to be a reasonable choice. A sensitivity analysis varying this matrix is recommended, to assess whether findings change for different values of this matrix.Operation. The R software environment (RRID:SCR_001905) runs on a wide variety of UNIX platforms, Windows, and MacOS, and requires minimal computer resources (256 kilobytes of RAM is recommended). The package requires R version 3.3.0 or higher. As the package now uses C++ code, an up-to-date version of Rtools should be installed to successfully install the package. This current work used R version 4.3.1.mr_ivw andmr_mvivw commands. The first-stage F statistic is a measure of the variance in the exposure explained by the genetic variants in the exposure dataset. It is roughly equal to the R2 statistic (the proportion of variance explained) multiplied by the sample size and divided by the number of instruments. Larger F statistics correspond to stronger instruments. The F statistic cannot be determined exactly based on summary statistics, but the simple formula implemented in the package provides a good approximation. While the mythical threshold of 10 for F statistics is an oversimplification, this value does provide a reasonable yardstick for judging the degree of potential bias due to weak instruments . We note that although weak instrument bias is typically in the direction of the null for two-sample univariable Mendelian randomization, this is not necessarily the case for two-sample multivariable Mendelian randomization.A further update to the MendelianRandomization package is the functionality to report F statistics in themr_ivw andmr_mvivw functions assumes that genetic variants are uncorrelated. Correlations can be specified in themr_input ormr_mvinput command:The default implementation of themr_ivw)mr_ivw function accounts for these correlations in the calculation of the F statistic using a formula that involves the inverse of the Cholesky transform of the correlation matrix.If two uncorrelated variants both explain 3% of the variance in the exposure, then together they explain 6% of the variance in the exposure. If two correlated variants both explain 3% of the variance in the exposure, then the variance explained by both variants could be as low as 3% (if the variants are perfectly correlated) or as high as 6% (if the correlation is negligible), or a value between these two, depending on the magnitude of correlation. This correlation similarly affects estimates of the F statistic. When the variant correlation matrix is specified, the. This is because if the genetic associations with one exposure were strong, but were near-perfectly correlated with the genetic associations with another exposure, then the multivariable model would not be able to differentiate between the effects of the two exposures.For multivariable Mendelian randomization, the key measure of instrument strength is not the univariable F statistic for each exposure separately, but the conditional F statistic, representing the independent proportion of variance explained in each exposure, after accounting for associations with the other exposuresmr_mvivw function calculates estimates of the conditional F statistics based on summarized data. In order to identify and reliably estimate multivariable exposure effects through Mendelian randomization, we require that the genetic predictors of the exposures are not collinear. Hence, conditional F-tests assess whether the genetic predictors of any one exposure can be expressed as a linear combination of genetic predictors of other exposures in the model. Low values of conditional F-test statistics for a given exposure suggest that the exposure may not be identified, and hence estimates may suffer from weak instrument bias. This is generally a bigger concern for two-sample multivariable, rather than two-sample univariable, Mendelian randomization analyses since weak instrument biases in estimation are not necessarily toward the null of no causal effect, and hence inferences can suffer from inflated type I error rates . This calculation of conditional F-statistics involves the sample size for the genetic associations with the exposures, and conditional F statistics are only calculated if these sample sizes are provided. If a single value is provided for the sample size, then it is assumed that this sample size holds for all exposures. For example, the code:Themr_mvivw,   bxse = cbind,   by = chdlodds, byse = chdloddsse), nx = 17723)gives output:Number of Variants : 28 ------------------------------------------------------------------   Exposure Estimate Std Error  95% CI       p-value Cond F-stat exposure_1    1.925     0.439  1.064, 2.786   0.000        20.3 exposure_2   -0.590     0.555 -1.677, 0.498   0.288        12.9 exposure_3    0.723     0.230  0.272, 1.174   0.002        13.3------------------------------------------------------------------By comparison, the code:mr_ivw)gives output including:F statistic = 28.0.We see that the univariable F statistic for LDL-cholesterol is 28.0, but the conditional F statistic is 20.3. Similarly, the univariable F statistic for HDL-cholesterol is 27.6, but the conditional F statistic is 12.9, and for triglycerides, the univariable F statistic is 41.6, but the conditional F statistic is 13.3. The instrument strength for LDL-cholesterol is similar in the multivariable analysis compared to the univariable analyses, but for HDL-cholesterol and triglycerides the instrument strength is weakened considerably.mr_pcgmm andmr_mvpcgmm functions, as these may help the user to judge how varying the number of principal components selected affects instrument strength.F statistics and conditional F statistics are also provided by theIn summary, the MendelianRandomization package has added a number of features since the 0.5.0 version: to implement additional estimation methods, to implement methods for highly correlated variants, and to report F statistics.: while this software simplifies the operational aspects of a Mendelian randomization, the truly difficult parts of an analysis are choosing sensible risk factors and outcomes, selecting genetic variants that are plausible instrumental variables, performing a reasonable range of analyses, and interpreting the results with care and caution. These aspects of an analysis cannot be automated.We conclude by again repeating the warning that we stated at the end of the manuscript accompanying the initial package release No further comments.Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?YesIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:Clinical epidemiology, GastroenterologyI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. No further comments.Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?YesIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:GenomicsI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Thank you Ricardo for the fast and positive response, and for your previous comments on this article, which have helped improve the work. This article describes the updated functions of the MendelianRandomization package, which is widely used to conduct Mendelian randomization analyses. I believe that the updated functions and their detailed descriptions in this article will be immensely beneficial for researchers.At the beginning of the Methods section, the authors introduce ‘three instrumental variable assumptions.’ It would be prudent to both explicate these assumptions in the Introduction section and explain why genetic variants can serve as a proxy for random assignment.This article utilizes data pertaining to LDL cholesterol, HDL cholesterol, triglycerides, and coronary heart disease risk as an illustrative example. It would be beneficial to clearly articulate the research question associated with this example, incorporate a graphical representation to delineate the relationship between genetic variants and these variables, and provide a brief overview of the selection process of these genetic variants within a multivariable Mendelian randomization context.It would be beneficial to include a small portion of the example dataset, comprising the correlation matrix, within the article along with a reference to its availability in the MendelianRandomization package.The authors draw a distinction between the variant correlation matrix and the exposure correlation matrix. A more detailed explanation on how to estimate each of these, as well as guidance on which correlation matrix is pertinent for use with the MendelianRandomization package, would be beneficial.The “Principal Component Generalized Method of Moments” section is extensive and could be made more readable by dividing it into two distinct subsections, each with its own subheading. To enhance the readability and comprehensibility of the article, I propose the following suggestions:Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?YesIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:Clinical epidemiology, GastroenterologyI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Response to reviewers for “MendelianRandomization v0.9.0: updates to an R package for performing Mendelian randomization analyses using summarized data” We would like to express thanks to the reviewers for their time and comments. Replies to points are indicated by angle brackets, and changes to the paper as a result of these comments are clearly indicated. We have numbered the comments for reference as A0, A1, A2, … for the first reviewer; and B0, B1, B2, … for the second reviewer.Reviewer 2: Hajime Yamazaki, Section of Clinical Epidemiology, Department of Community Medicine, Graduate School of Medicine, Kyoto University, Kyoto, Japan B0. This article describes the updated functions of the MendelianRandomization package, which is widely used to conduct Mendelian randomization analyses. I believe that the updated functions and their detailed descriptions in this article will be immensely beneficial for researchers. > We thank the reviewer for their kind comments and positive view of their manuscript. < To enhance the readability and comprehensibility of the article, I propose the following suggestions: B1. At the beginning of the Methods section, the authors introduce ‘three instrumental variable assumptions.’ It would be prudent to both explicate these assumptions in the Introduction section and explain why genetic variants can serve as a proxy for random assignment. > We have added a brief description of the three classical instrumental variable assumptions required to identify a causal effect, as well as a brief motivation as to why genetic variants are often suitable candidate instrumental variables. < B2. This article utilizes data pertaining to LDL cholesterol, HDL cholesterol, triglycerides, and coronary heart disease risk as an illustrative example. It would be beneficial to clearly articulate the research question associated with this example, incorporate a graphical representation to delineate the relationship between genetic variants and these variables, and provide a brief overview of the selection process of these genetic variants within a multivariable Mendelian randomization context. > We have added a brief description of how these genetic variants were chosen. While we appreciate the reviewer’s point, we would not want to give the impression that the data included in the Mendelian randomization package are anything other than illustrative. They are not the ideal data to optimally address any research question. < B3. It would be beneficial to include a small portion of the example dataset, comprising the correlation matrix, within the article along with a reference to its availability in the MendelianRandomization package. > We have added the calcium and fasting glucose data to the manuscript, as well as their correlation matrix . <cor) function in R. Alternatively, some pre-computed correlation matrices are available for download, for example using the LDlink webtool (https://ldlink.nci.nih.gov/?tab=ldmatrix) or the ld_extract function in the TwoSampleMR package. Variant correlation matrices for UK Biobank participants of European ancestries are available athttps://aws.amazon.com/marketplace/pp/prodview-4bhcvjnh4b4cs#resources. > We have listed some of these sources in the manuscript. > Exposure correlation estimates are required by some methods when genetic associations with any two exposures are measured from the same sample. Exposure correlation estimates may be more difficult to obtain than variant correlation estimates, but they could be estimated using individual-level data on the exposures from the same sample or an external sample used to measure genetic variant–exposure associations. In general, we have found estimation and inferences to be quite insensitive to mis-specified exposure correlations, and in practice we would recommend performing a sensitivity analysis that varies the inputted exposure correlations to assess sensitivity of findings to the specification of this matrix. B4. The authors draw a distinction between the variant correlation matrix and the exposure correlation matrix. A more detailed explanation on how to estimate each of these, as well as guidance on which correlation matrix is pertinent for use with the MendelianRandomization package, would be beneficial. > There are various tools to estimate the variant correlation matrix. If you have access to individual-level genetic data , you can calculate the variant correlation matrix using the correlation , users can find functions to perform newer robust methods of MR , consider variant correlation, and obtain F-statistics to estimate the strength of instrument variables. The authors provide good rationale for when and how to implement the newer MR methods. Examples of code are abundant throughout the manuscript, and the package’s functions are explained in detail. The authors set guidelines for implementing genetic correlation in MR and highlight potential caveats of the analysis. Altogether, the manuscript is useful to understand the newer functions available in MendelianRandomization v0.9.0 and has valuable insights into the practical considerations of applying more complex MR methods. Interpretation of newer output (the F-statistic) is included.Minor comments:“where calcium, calciumse, fastgluc, fastglucse, and calc.rho are data on six correlated variants from the CASR gene region provided with the package” – please reword to clarify the meaning of the calcium, calciumse, fastgluc and fastglucse variables.Is “flip = c” length = 6 because “calc.rho” is a matrix of 6 variants? Please show “calc.rho” before and after applying the function “calc.rho*flip%o%flip”. This can help understand the structure and signs of “calc.rho” going into the MR analysis.Regarding the correlation matrix coming from linkage disequilibrium data or exposure correlation: “However, our investigations have indicated that estimates are generally insensitive to the correct specification of this matrix.” – Please expand on this comment. Under the section “Instrument Strength”:“We note that although weak instrument bias is typically in the direction of the null for two-sample univariable Mendelian randomization, this is not necessarily the case for two-sample multivariable Mendelian randomization.” and “This is generally a bigger concern for two-sample multivariable, rather than two-sample univariable, Mendelian randomization analyses since weak instrument biases in estimation are not necessarily toward the null of no causal effect, and hence inferences can suffer from inflated type I error rates.” Please explain why this is the case. Additionally, please comment on how weak instrument bias can affect other types of MR. Under the section of “Principal component generalized method of moments methods”:  Are the conclusions about the tool and its performance adequately supported by the findings presented in the article?YesIs the rationale for developing the new software tool clearly explained?YesIs the description of the software tool technically sound?YesAre sufficient details of the code, methods and analysis (if applicable) provided to allow replication of the software development and its use by others?YesIs sufficient information provided to allow interpretation of the expected output datasets and any results generated using the tool?YesReviewer Expertise:GenomicsI confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. We would like to express thanks to the reviewers for their time and comments. Replies to points are indicated by angle brackets, and changes to the paper as a result of these comments are clearly indicated. We have numbered the comments for reference as A0, A1, A2, … for the first reviewer; and B0, B1, B2, … for the second reviewer.Reviewer 1: Ricardo Costeira, Department of Twin Research and Genetic Epidemiology, King's College London, London, England, UK A0. In this manuscript, the authors describe updates to the MendelianRandomization package implemented in R. In the package’s latest version (v0.9.0), users can find functions to perform newer robust methods of MR , consider variant correlation, and obtain F-statistics to estimate the strength of instrument variables. The authors provide good rationale for when and how to implement the newer MR methods. Examples of code are abundant throughout the manuscript, and the package’s functions are explained in detail. The authors set guidelines for implementing genetic correlation in MR and highlight potential caveats of the analysis. Altogether, the manuscript is useful to understand the newer functions available in MendelianRandomization v0.9.0 and has valuable insights into the practical considerations of applying more complex MR methods. Interpretation of newer output (the F-statistic) is included. > We thank the reviewer for their kind comments and positive view of this manuscript. < Minor comments: Under the section of “Principal component generalized method of moments methods”:  A1. “where calcium, calciumse, fastgluc, fastglucse, and calc.rho are data on six correlated variants from the CASR gene region provided with the package” – please reword to clarify the meaning of the calcium, calciumse, fastgluc and fastglucse variables.calcium represents the genetic associations with serum calcium levels, andfastgluc the genetic associations with fasting glucose levels . < > We have clarified that A2. Is “flip = c” length = 6 because “calc.rho” is a matrix of 6 variants? Please show “calc.rho” before and after applying the function “calc.rho*flip%o%flip”. This can help understand the structure and signs of “calc.rho” going into the MR analysis.flip vector correspond to the six genetic variants in turn. We have added to the manuscript a representation of the correlation matrix before and after orientation. < > We have clarified that the entries in the A3. Regarding the correlation matrix coming from linkage disequilibrium data or exposure correlation: “However, our investigations have indicated that estimates are generally insensitive to the correct specification of this matrix.” – Please expand on this comment. > We have clarified that this sentence refers to the exposure correlation matrix. < Under the section “Instrument Strength”: A4. “We note that although weak instrument bias is typically in the direction of the null for two-sample univariable Mendelian randomization, this is not necessarily the case for two-sample multivariable Mendelian randomization.” and “This is generally a bigger concern for two-sample multivariable, rather than two-sample univariable, Mendelian randomization analyses since weak instrument biases in estimation are not necessarily toward the null of no causal effect, and hence inferences can suffer from inflated type I error rates.” Please explain why this is the case. Additionally, please comment on how weak instrument bias can affect other types of MR. > Weak instrument bias is analogous to classical measurement error bias in a regression model. With a single regressor, error in the regressor leads to underestimation of the regression coefficient; this is known as regression dilution bias. With multiple regressors, bias due to measurement error can be in any direction. For example, suppose we have two regressors that are highly correlated, one of which is a causal risk factor for the outcome, and the other is not. Further, one is measured with substantial error, whereas the other is measured with no error. In a regression model, the coefficient for the regressor measured with error tends towards zero regardless of which is truly the causal risk factor. This could lead the coefficient for the precisely measured exposure being either over- or underestimated, depending on its true value and the true value for the coefficient of the imprecisely measured regressor. In a prediction model, this may be of little consequence, as the predicted values of the outcome may be insensitive to the exposure coefficients – which is large, which is close to zero. However, if the model coefficients have a causal interpretation, then the coefficients matter, as the regressor with the non-zero coefficient is identified as the causal factor. > Generally speaking, bias towards the null in Mendelian randomization is of lesser consequence, as the primary goal of Mendelian randomization is not estimation. Bias towards the null may reduce power to detect a true causal effect, but it will not incorrectly suggest a false causal effect (which is regarded in the Neyman—Pearson hypothesis framework as a more serious error). > As this explanation is fairly involved, we do not think it is appropriate to include in this manuscript. However, given that this journal performs open peer review, it can be found by the interested reader. <