The meat production of beef cattle is a major economic factor that is closely linked to body conformation traits. Qinchuan cattle are a local breed in China with a unique flavor. However, meat production needs improvement compared to commercial breeds. In this study, we selected 254 individuals, measured 14 phenotypic data points, and used genome-wide association analysis to identify loci and genes closely related to these traits. In total, we identified 250 suggestive loci and 37 candidate genes. These genetic markers will aid in the exploration of traits for Qinchuan cattle and provide a reference for breeding.p < 3.54 × 10−6) were screened and 37 candidate genes were annotated. Furthermore, we performed a linkage disequilibrium analysis of SNP loci and considered published studies, identifying the eight genes  most likely to be involved in growth traits. This study provides new insights into the regulatory mechanisms of bovine body size development, which can be very useful in the development of management and breeding strategies.A genome-wide association study (GWAS) is an effective tool for identifying the dominant genes of complex economic traits in livestock by statistical analysis of genotype data and measured phenotype data. In this study, we rigorously measured 14 body conformation traits in 254 Qinchuan cattle, comprising body weight (BW), body height (BOH), back height (BAH), buttock height (BUH), chest depth (CD), chest width (CW), hip cross height (HCH), body length (BL), hip width (HW), rump length (RL), pin bone width (PBW), chest girth (CG), abdomen circumference (AG), and calf circumference (CC). After quality control, 281,889 SNPs were generated for GWAS with different traits. A total of 250 suggestive SNPs ( Cattle are widely farmed around the world for meat, milk, skin, and leather. For beef cattle, meat yield and quality are the main concerns ,2. SeverCattle body conformation traits are comprehensively and delicately regulated by multiple genes, and it is a considerable inconvenience to find the main effector genes; however, a genome-wide association study (GWAS) can be used for the genetic analysis of economic traits and breeding. A GWAS is a statistical analysis method that associates genome-wide genetic variations with phenotypic traits and selects genetic variations that are significantly related to phenotypic traits . By inteHowever, comprehensive research on the body size traits of indigenous Chinese cattle is still lacking. Qinchuan cattle have a significantly larger body size index after many years of breeding compared to non-core groups , and theEthics approval for all animal experiments was granted by the Institutional Animal Care and Use Committee of Northwest A&F University following the recommendations of the Regulations for the Administration of Affairs Concerning Experimental Animals of China.We collected blood samples and measured each body trait from 254 individuals, including the new strain of Qinchuan cattle , unselected Qinchuan cattle , and Zaosheng cattle , from the National Beef Cattle Improvement Center’s experiment farm , the Genetic Resource Conservation of Qinchuan Cattle , and Longshang Tianyuan Agriculture and Animal Husbandry Co., Ltd. , respectively. These experimental animals were raised in the same environment. When selecting samples, close relationships between individuals were avoided. The cattle were placed in a hold frame or head lock and sterilized with an alcohol cotton ball. Then, blood samples were taken from the jugular vein, collected by negative pressure into EDTA-containing 5-milliliter gatherers, gently mixed in reverse several times, and, in the case of ice packs, transferred for long-term storage at −20 °C for DNA extraction.The body conformation traits were assessed as follows. BW: weight on the same weighbridge  before morning feeding; BOH: vertical height from the apex of the bun nail to the ground, measured by a measuring stick; BAH: vertical height of the last section of the thoracic spine to the ground, measured by means of a measuring stick; BUH: vertical height of the outer edge of the hip to the ground, measured by means of a measuring stick; CD: distance between the upper and lower chest of the back of the scapula, measured with a measuring stick; CW: distance between the shoulder blades measured by a measuring stick; HCH: the height of the ground vertically between the corners of the waist, measured by means of a measuring stick; BL: shoulder-to-hip distance, measured with a measuring stick; HW: the distance between the outer edges of the two waist corners, measured by a measuring stick; RL: the distance between the front edge of the waist corner and the back edge of the sciatica node, measured with a measuring stick; PBW: the distance between pin bones, measured by means of a measuring stick; CG: chest circumference measured by a tape measure along the back corner of the shoulder blade, the tightness of which was determined by sliding up and down the index and middle fingers; AG: a vertical line at the back edge of the waist was made and measured with a tape measure; CC: horizontal circumference was measured by a tape measure at 1/3 of the thinnest part of the left forearm tub.In total, 14 phenotypic data points were statistically analyzed for number, mean, standard deviation, minimum, maximum, and coefficient of variation using Excel. If the absolute value of the difference between the measured value and the mean value was greater than 3 times the standard deviation, the measured value was considered an outlier and this data point was deleted. The frequency distribution histogram of each phenotype was drawn and tested for normality by R-4.2.1 software.Overall, genomes from 127 QNC underwent mutation detection using a 600 K high-density chip designed by our lab, and genomes from 88 QNC, 20 QCC, and 19 ZSC underwent 10× genome resequencing by Xinjiang Compass Agritechnology Co., Ltd. . DNA extraction, detection, fragment purification, library construction, and whole genome sequencing using the BGI T7/G2000 technique with paired end reads of 150 bp from the samples were completed. Before subsequent analysis, we filtered the original data according to the following conditions: remove reads with adapters; when the N content in the sequencing read exceeds 10% of the base number of the read, remove the paired reads; when the number of low-quality (Q <= 5) bases contained exceeds 50% of the number of bases in the read, remove the paired reads.Quality control of the clean data obtained from the company was performed using FastQC software version 0.12.0 with default parameters , and theWe used Plink v1.9 software  to conve−6 . LDBlockShow [We carried out the GWAS with a mixed linear model using the genome-wide efficient mixed-model association software package v0.94.1 (GEMMA) . At the lockShow  was usedhttps://www.ncbi.nlm.nih.gov/pubmed (accessed on 1 October 2023).In order to further analyze the function of SNPs, we used a perl script and identified significant SNPs in 100 kb upstream and downstream as candidate regions for screening genes based on the Ensembl database. Then, in order to determine the function of annotated genes, PubMed was used to search for relevant published papers with A total of 254 cattle were measured for body conformation traits, and the descriptive statistics of the phenotypes are presented in Samples underwent a principal components analysis (PCA) using GCTA software v1.94.1, and the top two components were visualized by R language, indicating no population stratification and no need for correction B. Kinship < 3.54 × 10−6) were screened for genome-wide association analysis of 14 body conformation traits in 254 cattle and annotated to 37 candidate genes  (UBE2E2)  J. Regard (FRMD6) K. For CG(UBE2E2) L. Eighte(UBE2E2) M,N. OverUsing the GWAS for each trait, we found that 27 SNPs closely associated with BOH were concentrated on chromosome 21 , 15 SNPs strongly associated with CD were concentrated on chromosome 24 , 10 SNPs strongly associated with CW were concentrated on chromosome 3 , and 14 SNPs strongly associated with HCH were concentrated on chromosome 8 . The physical locations of these SNPs are very close together, and in order to explore the interlocking relationships between them, a linkage disequilibrium analysis was performed. The distribution of haplotypes is shown in In general, animal weight and growth rates are considered profit drivers of animal production systems . QuantitBiological breeding can effectively increase the amount of meat production and improve the quality of meat per beef cattle, on the one hand saving feedstock and controlling breeding cost, and, on the other hand, improving the economic value of high-quality meat. However, the quantitative traits of beef cattle are co-regulated by multiple genes, which are harder to find. GWAS provides a good example of screening for dominant genes associated with quantitative traits in livestock and poultry, such as pigs ,25, chicADAMTS17 is a pathogenic gene for Weill–Marchesani syndrome (WMS) and Weill–Marchesani-like syndrome, often characterized by lens heterotopia and small stature [ADAMTS17 is implicated in skeletal formation through modulation of the BMP-Smad1/5/8 pathway, possibly through inhibition of fibrillin-2 incorporation into microfibrils [ADAMTS17 hints at adaptive evolution [ALDH1A3 is a key enzyme involved in the production of retinoic acid required for head formation [ALDH1A2 as a probable primordial gene that originated in invertebrate genomes and underwent a sequential gene duplication event to generate two additional genes, ALDH1A1 and ALDH1A3, in the majority of vertebrate genomes [CHSY1, and targeting human chondrocytes with CRISPR/Cas9 technology demonstrated that this locus modulates CHSY1 expression suggesting it as a key gene for human growth [CHSY1 being identified as a candidate gene for bone weight [GABRA5 plays a role in reproductive processes, giving it potential for use in marker-assisted selection programs in Markhoz goats [GABRG3 was related to teat number [IGF1R gene in 1716 sheep from six breeds in New Zealand revealed an association between lifespan and variation in IGF1R [MAGEL2 is a patrilineal regulatory gene that may play an important role in muscle maturation, according to a study of piglets [MEF2A has been shown in several studies to play a significant role in myocardial muscle [MEF2A regulates the MEG3-DIO3 miRNA mega cluster-targeted PP2A signaling in bovine skeletal myoblast differentiation [CNTNAP5 was speculated to be a candidate gene associated with growth and carcass traits including body weight and body fat deposition in four Beninese indigenous cattle breeds [CTNNA3 was associated with development and growth by iHS and ROH detection based on genome-wide SNP markers in Youzhou dark goats [We conducted a literature search of 37 selected candidate genes and identified several important genes that may be associated with quantitative traits in beef cattle. It has been firmly established that  stature , and hasofibrils . Two daivolution . Likewisormation . Phyloge genomes . GWASs hn growth . Genotype weight . Furtheroz goats . In Luzht number . Genotypin IGF1R . In addi piglets , and it  piglets . MEF2A hl muscle , skeletal muscle , and smol muscle . MEF2A rntiation  and sheentiation . In a GWe breeds , as welle breeds . CTNNA3 rk goats  and in Hrk goats , as wellrk goats . A largeA haplotype is a combination of alleles at multiple loci that are shared across the same chromosome. The distribution of haplotype blocks throughout the genome can reflect the genetic structure and variation of a population . The hapADAMTS17, ALDH1A3, CHSY1, MAGEL2, MEF2A, SYNM, CNTNAP5, and CTNNA3 as promising candidates affecting growth traits in beef cattle.Therefore, based on functional studies of these candidate genes, we identify ADAMTS17, ALDH1A3, CHSY1, MAGEL2, MEF2A, SYNM, CNTNAP5, and CTNNA3) can be considered potential candidates for body conformation traits, which could serve as important marker information for genome selection and contribute to the improvement of cattle breeding programs.This study shows that a GWAS can be a powerful tool for identifying novel loci and unraveling the genetic basis of complex economic traits. Furthermore, the important candidate genes and molecular markers of body conformation traits were selected from the genome to analyze the potential genetic mechanism of dominant traits and provide a scientific basis for breeding improvement of beef cattle. According to published studies, eight genes (


Genetic correlations between complex traits suggest that pleiotropic variants contribute to trait variation. Genome-wide association studies (GWAS) aim to uncover the genetic underpinnings of traits. Multivariate association testing and the meta-analysis of summary statistics from single-trait GWAS enable detecting variants associated with multiple phenotypes. In this study, we used array-derived genotypes and phenotypes for 24 reproduction, production, and conformation traits to explore differences between the two methods and used imputed sequence variant genotypes to fine-map six quantitative trait loci (QTL).We considered genotypes at 44,733 SNPs for 5,753 pigs from the Swiss Large White breed that had deregressed breeding values for 24 traits. Single-trait association analyses revealed eleven QTL that affected 15 traits. Multi-trait association testing and the meta-analysis of the single-trait GWAS revealed between 3 and 6 QTL, respectively, in three groups of traits. The multi-trait methods revealed three loci that were not detected in the single-trait GWAS. Four QTL that were identified in the single-trait GWAS, remained undetected in the multi-trait analyses. To pinpoint candidate causal variants for the QTL, we imputed the array-derived genotypes to the sequence level using a sequenced reference panel consisting of 421 pigs. This approach provided genotypes at 16 million imputed sequence variants with a mean accuracy of imputation of 0.94. The fine-mapping of six QTL with imputed sequence variant genotypes revealed four previously proposed causal mutations among the top variants.Our findings in a medium-size cohort of pigs suggest that multivariate association testing and the meta-analysis of summary statistics from single-trait GWAS provide very similar results. Although multi-trait association methods provide a useful overview of pleiotropic loci segregating in mapping populations, the investigation of single-trait association studies is still advised, as multi-trait methods may miss QTL that are uncovered in single-trait GWAS.The online version contains supplementary material available at 10.1186/s12864-023-09295-4. Genome-wide association studies (GWAS) combine genotype and phenotype information to identify trait-associated variants. Genotypes at polymorphic loci are tested for association with phenotypes to determine their impact on traits of interest. Multi-trait GWAS can increase the statistical power over single-trait GWAS because they exploit cross-phenotype associations at pleiotropic loci –3.Several methods have been developed to detect pleiotropic variants. These methods can be divided into two groups based on their underlying statistical framework , 5. FirsThe power to detect trait-associated variants increases as the marker density increases –17. Low-MC4R, BMP2, IGF2, and CCND2 [Only few genome-wide association studies have been conducted in the Swiss Large White (SLW) population. Becker et al.  performend CCND2 –35.In this paper, we compare single-trait, multivariate and meta-GWAS in 5,753 genotyped pigs from a Swiss breed to investigate the genetic architecture of 24 traits. We inferred sequence variant genotypes from a sequenced reference panel to identify candidate causal variants for six pleiotropic QTL.A detailed description of the 24 traits considered in our study including their grouping into four categories  is shown in Table P < 1.24 × 10−6). Fifteen out of 24 traits had at least one significantly associated variant (Table Mixed model-based single-trait genome-wide association studies (stGWAS) between 40,382 SNPs and deregressed estimated breeding values (drEBV) for 24 traits in 5,753 Swiss Large White (SLW) pigs revealed 237 significantly associated variants , and between lifetime daily weight gain (LDWG) and a variant on SSC1 . 57 SNPs were significantly associated with more than one trait  suggesting that pleiotropic effects are present and detectable in our dataset.The 237 associations were detected at 99 unique SNPs located at 11 QTL on SSC1, 5, 11, 15, 17 and 18. The two strongest associations were detected between the number of teats (NT) and a variant on SSC7  for traits within the four trait categories. For unbiased investigation of differences between both methods, we considered between 1,074 and 2,689 individuals with complete phenotypic records for all traits within a trait category  association testing and performed a multi-trait meta-analyses of the single-trait GWAS , but slightly lower  in the metaGWAS1 than the mtGWAS , and one QTL on SSC10 that was associated with MT (P = 5.60 × 10−7) and MES (P = 8.44 × 10−7). From the seven QTL detected by both multi-trait and single-trait methods, six were associated with more than one trait in stGWAS. For the six pleiotropic QTL, at least one single trait analyses revealed more associated variants, and smaller P-value of the top SNP, than the mtGWAS or metaGWAS1  increased by 10, 14 and 48 to 28, 60 and 55 in the conformation, production, and all groups, respectively  between true and imputed allele dosages, cross-validated proportion of correctly imputed genotypes (concordance ratio – CR) and model-based accuracies from Beagle5.2 (Beagle DR2), respectively. Although the model-based estimate from Beagle was highly correlated with the Pearson R2 (0.81), the Beagle DR2 values were consistently higher.Whole-genome sequence-variant genotypes from 421 pigs were used to impute the medium-density genotypes of 5,753 pigs to the whole-genome sequence level. The principal components analysis  > 8.5). The QTL was associated with ADFI, DWG, and LDWG, and explained 5.16, 3.45, and 4.49% of the trait variance, respectively. The lead SNP at this QTL was an imputed intergenic sequence variant (rs692827816) located at 159,637,589 bp (P = 5.26 × 10–13). rs692827816 was in high LD (R2 > 0.90) with 935 variants that had similar P values. One of the variants in high LD was a missense variant (rs81219178) at 160,773,437 bp within the MC4R gene, which has previously been proposed as candidate causative variant for growth and fatness traits [A QTL on SSC1 was between 157.73 and 162.74 Mb and encompassed 34,312 imputed sequence variants including 1,157 that were significantly associated in the sequence-based metaGWAS (-logs traits . rs81219P = 2.85 × 10–36), approximately 13 kb downstream from ASS1 and 53 kb upstream from FUBP3. The lead variant was in high LD (R2 > 0.90) with 747 variants. The MAF of the lead variant was 0.14 and its genotypes were imputed from the reference panel with high accuracy (DR2 = 0.98).Another QTL on SSC1 was between 270.33 and 272.32 Mb and encompassed 21,683 imputed sequence variants including 3,765 that were significant in the sequence-based metaGWAS  which was in high LD (R2 > 0.90) with seven other variants. rs346219461 was 7.60 kb downstream the fibroblast growth factor 6 (FGF6) encoding gene and it had MAF of 0.41, and it was imputed from the reference panel with high accuracy (DR2 = 0.99). The rs346219461 was in LD (R2 = 0.82) with a non-coding variant  in the third intron of CCND2, that was previously proposed as putative causal variant for backfat thickness [A QTL on SSC5 was located between 65.75 and 66.21 Mb and encompassed 5,065 imputed sequence variants, from which eight were significantly associated with BFT  located 12.7 kb downstream VRTN. The rs333375257 had MAF of 0.34 and was imputed from the reference panel with high accuracy (DR2 = 0.99). The rs333375257 was in high LD (R2 > 0.90) with 424 sequence variants. A previously described candidate causal variant  for the number of thoracic vertebrae [R2 > 0.99) with the rs333375257. In addition, 334 significant variants in the ATP binding cassette subfamily D member 4 (ABCD4) gene were detected. This gene was proposed to impact NT in a Duroc population [P = 1.49 × 10–42) with CL, NT and NUT, and in almost complete LD (R2 = 0.98) with the lead SNP.A pleiotropic QTL on SSC7 was between 93.12 and 99.26 Mb and it encompassed 31,013 imputed sequence variants including 2,341 that were significant in the sequence-based metaGWAS. The QTL was associated with BFT from the production group (0.37% variance explained), and CL, NT, and NUT from the conformation group, where it explained 6.11, 5.00, and 3.32% of the trait variance, respectively. The lead SNP was an imputed variant  in an intergenic region 106 kb upstream the BMP2 gene. The variant had MAF of 0.2 and it was imputed from the reference panel with high accuracy (DR2 = 0.97). The lead SNP was in high LD (R2 > 0.90) with two other variants. One of them was a previously proposed candidate causative variant for carcass length [P = 8 × 10–82) in an intergenic region upstream of the BMP2 gene, 17 kb away from the lead SNP.A pleiotropic QTL on SSC17 encompassed 82,132 imputed sequence variants including 2,112 that were significantly associated residing between 10.80 and 20.93 Mb. The QTL was associated with DWG, LDWG, and BFT from the production group , and with BL, BFL, and XOH from the conformation group . The strongest association was from an imputed sequence variant  in complete LD, which were imputed sequence variants located at 10,678,235 bp (rs338817164) and 10,678,293 bp (rs334203353). Both variants had MAF of 0.32. The imputation accuracy was 0.80. They were in high LD (R2 > 0.90) with other 11 intergenic variants.A QTL on Single- and multi-trait genome-wide association studies involving array-derived and imputed sequence variant genotypes from 5,753 SLW pigs enabled us to investigate the genetic architecture of 24 complex traits from three trait groups. The response variables for the association tests were deregressed breeding values because the genotyped pigs had progeny-derived phenotypes. Progeny-derived phenotypes have been frequently used to perform association studies in animals that lack own performance records for the traits of interest. To avoid false-positive associations arising from the accumulation of family information in the progeny-derived phenotypes , we usedP-values, and perhaps because of higher penalty for multiple testing their effects were too small to be detected in our medium-sized cohort. The phenomenon that associations detected by stGWAS might disappear in multi-trait analyses has been reported earlier [The single-trait association analyses revealed 26 trait × QTL associations at eleven QTL of which seven were associated with at least two traits. Exploiting genetic correlations among the traits in a multi-trait framework revealed association for six out of the seven pleiotropic QTL detected in the stGWAS. Despite considering up to 10 phenotypes in the mtGWAS and up to 24 phenotypes in the metaGWAS, the multivariate methods applied in our study revealed three QTL, that were not detected by the single-trait association studies. The multi-trait methods did not reveal association at four QTL that were revealed by stGWAS. These QTL had low  earlier , 40.1 detected largely the same associated SNPs for almost all trait-groups. However, the metaGWAS1 revealed more associated SNPs, and lower P-values for the lead SNPs. Combining all 24 traits in the mtGWAS revealed five associated SNPs, and the metaGWAS1 conducted with the same individuals detected association of seven SNPs. Multivariate linear mixed models may suffer from over-parametrisation and loss of power when more than ten traits are considered [N = 1,074 pigs with non-missing records for all 24 traits).The mtGWAS and metaGWASnsidered . The low2 increased to 55 (tenfold higher than before). According to Bolormaa et al. [The metaGWAS approach enabled us to establish a larger sample size by considering summary statistics from stGWAS that were conducted with a various number of individuals . In this setting, the number of associated SNPs detected by the metaGWASa et al. , in situa et al. –44, but Our comparisons between GWAS approaches considered microarray-derived SNP genotypes. The imputation of array-derived genotypes up to the sequence level provides more statistical power to identify associated loci because causal variants are in the data and directly tested for association with traits of interest , 26. TheThe imputed sequence variants were used in metaGWAS to fine-map QTL and prioritise candidate causal variants. The top associated variants in four QTL were variants that had been previously identified as candidate causal variants –38, indiP = 1 × 10–62) and daily weight gain (P = 6 × 10–13). These two traits are barely correlated with each other (r = 0.02—0.05). Another pleiotropic QTL on SSC1 at ~ 270 Mb was associated with six traits ADFI, DWG, LDWG, LMC, BFT, and IMF, that were moderately to highly correlated (mean r ± SD = 0.37 ± 0.27). This chromosomal region harbours QTL for backfat thickness and feed efficiency-related traits in other pig populations [ASS1 gene. Expression of ASS1 has been associated with digestive tract development, cell adhesion, response to lipopolysaccharide, and arginine and proline metabolism in pigs [ASS1 as a positional and functional candidate gene for a pleiotropic QTL at ~ 270 Mb. Further functional annotations of the trait-associated variants in the non-coding regions might help elucidating the genetic mechanism underpinning this pleiotropic QTL.Our meta-analyses approach using imputed sequence variants revealed six QTL of which five were associated with multiple traits. Several porcine pleiotropic loci are underpinned by heterozygous loss-of-function alleles that may have fatal consequences in the homozygous state –51. Pleiulations , 53. How in pigs , 55. ConMulti-trait associations analyses provide strength of evidence for the presence or absence of a QTL segregating in populations. Here, we compared the multivariate linear model with meta-analyses of single-trait summary statistics using real data. Both approaches performed similarly in correlated groups of traits with complete datasets. The ability of meta-analyses to include different sets of individuals and unrestricted number of traits promoted the detection power. Thus, we recommend using the meta-analyses for getting overview of pleiotropic QTL in cohorts with more than 10 traits. For analyses of reduced and correlated groups of traits, the choice of the method seems to provide indifferent results. Yet, we stress the importance of the single-trait analyses for accurate interpretation of the pleiotropic effects and for the assignment of the affected traits.The reference-guided imputation to whole-genome sequence level assigned genotypes to 22 million variants with high accuracy. Putative causal variants found in literature were among the top variants in the fine-mapped QTL. Our analyses provide overview of the QTL affecting economically important traits in Swiss Large White and might serve as catalogue for future research examining the causal variants for complex traits.2drEBV) and weights (wdrEBV) for 24 traits were provided by the Swiss breeding company SUISAG for 5,753 pigs of the SLW breed. Breeding values were estimated using BLUP multiple trait animal models neglecting genomic information and subsequently deregressed according to Garrick et al. [2drEBV > 0.3 and were within five standard deviations from the mean values. We considered only traits for which at least 2,000 genotyped animals had records. The final number of animals with phenotypes was between 2,025 for gait (GAIT) and 5,419 for number of teats (NT). Up to 37% of the pigs had missing records for at least one trait. The traits were assembled in four trait groups  with their corresponding degrees of determination  or 61,565 (v. 2) SNPs; 13,342 pigs were genotyped using customized 60 K Bead Chips comprising either 62,549 (v. 1) or 77,119 (v. 2) SNPs; and 546 pigs had genotypes at 68,528 SNPs obtained with the GeneSeek Genomic Profiler (GGP) Porcine 80 K array.P < 0.00001), had SNP- and individual-level genotyping rates above 80%, and minor allele frequency (MAF) greater than 0.5%. Finally, sporadically missing genotypes for the resulting 44,733 variants were imputed for 14,292 animals using Beagle  to merg.For the array-based GWAS and for the comparison of the multi-trait GWAS methods, we considered 40,382 SNPs that had MAF greater than 5% in 5,753 animals of the SLW breed.drEBV). The genomic relationship matrix was built for 14,292 individuals with 44,733 SNPs, but only up to 5,753 SLW animals were used to estimate genomic heritability.We used the Fisher-scoring algorithm implemented in the GREML module of GCTA  to estiWhole-genome sequence (WGS) data were available for 421 SLW pigs that had been sequenced at an average read depth of 7.1x, ranging between 2.35 × and 37.5x. This panel also included 32 key ancestors of the genotyped SLW pigs that explained a large fraction of the genetic diversity of the current breeding population .Raw sequence data were trimmed and pruned for low-quality bases and reads with default parameter settings of the fastp software  and subThe filtered WGS dataset containing 421 pigs and 22,018,148 variants  with MAF greater than 0.01 was used for the imputation of sequence variant genotypes into the array dataset. The reference panel was pre-phased using SHAPEIT4  using t2) between imputed and true genotypes.The accuracy of imputation was assessed empirically by five-fold cross-validation in the 421 animals as follows: 40 animals which were sequenced at high coverage (> 10x), were used as target panel. The remaining 381 animals served as the reference panel. The SNP density in the target panel was reduced to 44,733 SNP chip genotypes and subsequently imputed to the sequence level based on 381 reference animals as described above. The imputed and actual genotypes of the target samples were compared to derive concordance ratio  and squared correlation  analysis. First, a genomic relationship matrix (GRM) was built among 14,629 pigs that had 16,387,582  imputed biallelic SNPs with MAF > 5%  using GCTA . Then, tP < 10–8), resulting in a total of 16,051,635 biallelic variants  which were used for association analyses and the fine-mapping of QTL in 5,753 SLW pigs.Post-imputation quality control excluded SNPs with MAF < 5%, model-based accuracy of imputation (Beagle DR2) < 0.6, and deviations from Hardy–Weinberg proportions ; The linear mixed model fitted to the data was in the following form: P-value of each SNP was estimated by the score test implemented in GEMMA (-lmm 3).The centred GRM was calculated with GEMMA (–nk 1) using either array-based or imputed sequence variant genotypes. The The stGWAS was run with either all available individuals or considering only preselected individuals that had non-missing phenotypes within a group of traits. The markers were separately filtered for MAF > 5%. Thus, in the latter run, the stGWAS for the reproduction traits included 41,242 variants typed in 2,553 samples; the stGWAS for the production traits included 40,557 variants typed in 2,689 samples; the stGWAS for the conformation traits included 41,168 variants typed in 1,927 samples, and the stGWAS for the 24 traits together included 41,152 variants typed in 1,074 samples with non-missing records.d (number of traits) columns, and d. d × d symmetric matrices of genetic and environmental variance components, respectively. Because multivariate association testing as implemented in GEMMA requires phenotype data for all individuals and traits, we considered only 2,553, 1,927, 2,689 and 1,074 individuals, respectively, for the reproduction (4 traits), conformation (10 traits), production (10 traits), and all-trait (24 traits) mtGWAS. The GRM, used during the mtGWAS, was the one from the stGWAS.Multi-trait association tests (mtGWAS) were conducted using a multivariate mixed model-based approach implemented in the GEMMA software . The mu2 statistic was subsequently calculated as t is a j × d matrix of signed t-values at the jth marker across d traits, and V−1 is the inversed d × d variance–covariance matrix.A multi-trait meta-analysis (metaGWAS) was conducted with the summary statistics from stGWAS as suggested by Bolormaa et al. . BrieflyP-values for the j markers were calculated with pchisq function with d-1 degrees of freedom, as implemented in R. We carried out the meta-analyses with the 24 traits classified into the same four trait categories as in mtGWAS . First, to enable unbiased comparison between the metaGWAS and the mtGWAS results, we considered summary statistics obtained from stGWAS based on individuals with complete records within the respective trait-group. Second, to increase the power of the association tests through maximizing the volume of entering information, hence exploiting the benefit of the metaGWAS approach, we used the stGWAS summary statistics based on all available individuals for the trait (the first run). For clarity we denote the first and second meta-analyses as metaGWAS1 and metaGWAS2, respectively. The latter approach was repeated for the fine-mapping of the QTL with imputed sequence variant genotypes.−6 and 3.11 × 10−9 for array and imputed sequence variant genotypes, respectively) to consider multiple testing. Genomic inflation factors were calculated to compare the distributions of the expected and observed test statistics.We used a 5% Bonferroni-corrected significance threshold , A is the number of significant variants and T is the total number of variants tested.The statistical power was assessed using false discovery rate (FDR). Following Bolormaa et al. , the FDRP-value within a QTL was defined as lead variant. Linkage disequilibrium (LD) between the lead variant and all other variants was calculated with the PLINK (v. 1.9) –r2 command. Variants within QTL were annotated with Ensembl’s Variant Effect Predictor  tool usor  and meta-analyses GWAS (bottom) based on array genotypes. Suggestive line is at 5.9.Additional file 5. Results of multi-trait GWAS based on array genotypes. Table contains numbered QTL regions, chromosome, start and stop positions of the QTL, lowest P-value, top SNP with MAF, number of significantly associated variants within the QTL and the trait-group.Additional file 6. Results of meta-analyses GWAS based on array genotypes and complete dataset. Table contains numbered QTL regions, chromosome, start and stop positions of the QTL, lowest P-value, top SNP with MAF, number of significantly associated variants within the QTL and the trait-group.Additional file 7. Comparison of pleiotropic QTL across methods based on array genotypes. Summary table of P-values and number of significantly associated variants within the 6 pleiotropic resulting from single-trait GWAS, meta-analyses GWAS with complete datasets and multi-trait GWAS.Additional file 8. Results of meta-analyses GWAS based on array genotypes and dataset including missing phenotypic records. Table contains numbered QTL regions, chromosome, start and stop positions of the QTL, lowest P-value, top SNP with MAF, number of significantly associated variants within the QTL and the trait-group.Additional file 9. Number of significantly associated pleiotropic variants in the meta-analyses GWAS within groups using all possible samples and stGWAS of individual traits. The groups are denoted as $ (production group) and ¥ (conformation group). The single traits are: BFL - Bent to pre-bent curve of forelegs; NUT - Number of underdeveloped teats; IMF - Intramuscular fat content in MAS; BFT - Back fat thickness; NT - Number of teats (both sides); LMC - Lean meat content; CL - Carcass length; LDWG - Lifetime daily weight gain; DWG - Daily weight gain on test; ADFI - Average daily feed intake.Additional file 10. PCA plot of reference panel samples (red and blue) and the target samples with array-genotypes (black +). First and second principal components captured 7.55 and 0.83% of the genomic variation.Additional file 11. Accuracy of imputation to whole-genome sequence versus minor allele frequency. Dosage R2 values from Beagle, and empirical measures of concordance and accuracy (R2 ) derived in 5 cross-validation sets, were assessed in 421 animals based on 44,733 downsampled chip genotypes.Additional file 12. Results of single-trait GWAS based on imputed sequence variants. Table contains numbered QTL regions, chromosome, start and stop positions of the QTL, lowest P-value, top SNP with MAF, number of significantly associated variants within the QTL and the single-trait abbreviation.Additional file 13. Manhattan plot of multi-trait meta-analyses GWAS of all 24 traits after fitting the 6 QTL as covariates, which resulted in loss of the peaks. Suggestive line is at 8.5.Additional file 14. Results of meta-analyses GWAS based on imputed sequence variants and dataset including missing phenotypic records. Table contains numbered QTL regions, chromosome, start and stop positions of the QTL, lowest P-value, top SNP with MAF, number of significantly associated variants within the QTL and the trait-group.Additional file 15. List of genes ID and genes symbols within the 6 pleiotropic QTL.


N < 1000) or when there is considerable assortative mating. Within families, ORIV is the best choice since the PGI-RC correction factor is generally not available. We verify the empirical validity of the simulations by predicting educational attainment and height in a sample of siblings from the UK Biobank. We show that applying ORIV between families increases the standardized effect of the PGI by 12% (height) and by 22%  compared to a meta-analysis-based PGI, yet estimates remain slightly below the PGI-RC estimates. Furthermore, within-family ORIV regression provides the tightest lower bound for the direct genetic effect, increasing the lower bound for the standardized direct genetic effect on educational attainment from 0.14 to 0.18 (+29%), and for height from 0.54 to 0.61 (+13%) compared to a meta-analysis-based PGI.Measurement error in polygenic indices (PGIs) attenuates the estimation of their effects in regression models. We analyze and compare two approaches addressing this attenuation bias: Obviously Related Instrumental Variables (ORIV) and the PGI Repository Correction (PGI-RC). Through simulations, we show that the PGI-RC performs slightly better than ORIV, unless the prediction sample is very small ( Measurement error in polygenic indices attenuates their power to predict complex traits. Here, the authors compare two approaches addressing this attenuation bias and provide guidance on which approach to apply in various scenarios. A natural consequence of this has been the widespread adoption of so-called polygenic indices (PGIs), weighted sums aggregating the small effects of numerous genetic variants (single-nucleotide polymorphisms (SNPs)), which enable out-of-sample genetic prediction of complex traits5. It is common practice to meta-analyze GWAS summary statistics from as many samples as possible to foster the identification of genome-wide significant SNPs1. Through the law of large numbers, this strategy has also proven to be very effective in reducing measurement error in the PGI and thus to boost the power and accuracy for analyses involving PGIs4. PGIs are now able to explain a non-negligible proportion of the variance in health and behavioral traits6.Genome-wide association studies (GWASs) have firmly established that, with few exceptions, most human  traits are highly “polygenic”—that is, influenced by many individual genetic variants, each with a very small effect size6) because, amongst other reasons, the GWAS underlying the construction of the PGI is based on a finite sample8. The noise in the GWAS coefficients translates into noisy measures of the PGI and leads to what is typically known as “attenuation bias”  in the coefficient of a PGI in a regression. As a result, the predictive power of today’s PGIs is still substantially smaller than the SNP-based heritability estimates, which constitute an upper bound for the predictive power of PGIs9. For example, the current predictive power, or variance explained in the phenotype (R2), of the educational attainment (EA) PGI is about 12–16%10, whereas the SNP-based heritability is estimated to be in the range 22–28%13. Importantly, the returns to increasing GWAS sample size in terms of gaining predictive power of the PGI are rapidly decreasing15. For example, for a trait with a SNP-heritability of 25%, it takes a sample of ~1 million to construct a PGI explaining 20%, but it would take a 7-fold increase in discovery sample size to achieve an R2 of 24% . Hence, reducing measurement error in the conventional way of meta-analyzing ever-larger discovery samples has rapidly diminishing payoffs.Empirically constructed PGIs are nonetheless a noisy proxy for the true latent PGI  studies that estimate direct genetic effects using within-family estimation to understand the mechanisms through which molecular differences translate into education differences18; (iii) studies that investigate gene-environment interplay in EA20, (iv) studies that investigate intergenerational persistence of education23; and (v) studies including a PGI as a control variable to reduce omitted variable bias or improve precision24. While there have been important advances in our understanding of how functional priors and optimizing construction methods can enhance the predictive power of a given PGI27, most existing applications still employ a  PGI without applying a correction for measurement error.There is a burgeoning body of papers exploiting PGIs in medicine, biology, and the social sciences. In many applications, the goal is to estimate the regression coefficient of the true latent PGI. Taking the EA PGI as an example, a non-exhaustive list of applications includes (i) studies estimating the pathways through which the EA PGI influences lifetime outcomes including speech and reading skills28 or IV estimation29 to deal with measurement error in the PGI. First, DiPrete et al.29 suggested an instrumental variables (IV) approach to reduce measurement error in the PGI as a by-product of their Genetic Instrumental Variable (GIV) method. The intuition of the IV approach is simple: when we split the GWAS discovery sample into two, we can obtain two PGIs that both proxy the same underlying “true” PGI. For example, when splitting the UK Biobank (UKB) at random into two discovery samples, both resulting PGIs approximate the same true latent PGI. Hence, theoretically, their correlation should be 1. However, in practice, their correlation will be smaller than 1 since the GWAS sample sizes used to construct these PGIs are finite and therefore each PGI will be subject to measurement error. In case (i) of polygenicity, (ii) the sources of measurement error are independent, and (iii) the relative variance of measurement error in the PGI is the same across the two discovery samples, then the correlation between the two PGIs reveals the degree of measurement error. The IV approach in turn uses this information to correct  the observed association between the PGI and the outcome. Second, Becker et al.6 developed an approach to disattenuate estimated effects of the PGI on a trait, on the basis of external information of the trait’s SNP-based heritability . Intuitively, in this approach the SNP-based heritability is estimated in a first step, after which the coefficient of the PGI is re-scaled to match this SNP-based heritability in a second step. Since this approach was proposed alongside the introduction of the PGI repository project6, we will refer to this approach as the PGI repository correction (PGI-RC).Recent studies have laid out the advantages of a measurement error correction and either suggested OLS estimates that have undergone some reasonable correction for attenuationR2) for which ever-increasing GWAS discovery samples remain important. We are also not primarily interested in estimating the direct  effect of a PGI. In between-family designs, PGIs typically capture not only effects of inherited variation (direct effects), but also effects of population stratification, demography, and relatives 30. In the simulations, we compare the performance of a meta-analysis based PGI (benchmark) to (1) results obtained using an IV approach29, which relies on two PGIs constructed from two non-overlapping GWASs, and (2) the PGI repository correction6. For the IV approach, we avoid the arbitrary choice of selecting one PGI as the independent variable and the other as IV by using the recently developed Obviously-Related Instrumental Variables (ORIV) technique31. In our comparison of the benchmark, ORIV, and PGI-RC we consider various degrees of (i) genetic nurture , (ii) assortative mating, and (iii) genetic correlation across discovery and prediction samples. In turn, in the empirical application, we compare the benchmark, ORIV, and PGI-RC using data on height and educational attainment from the sibling sample of the UK Biobank32. We conclude that ORIV is preferred over the PGI-RC if the prediction sample is very small (N < 1000), when there is assortative mating, and in within-family designs. For small discovery samples, or when there is imperfect genetic correlation between the discovery and prediction sample, PGI-RC is the preferred choice.In this study, we use simulations and empirical analyses to compare the IV and PGI-RC approach to reduce attenuation bias in analyses involving PGIs. Our goal is to estimate a coefficient that is free from attenuation bias due to measurement error in a PGI. As such, we are not primarily interested in boosting the out-of-sample predictive power of a given PGI . This tool allows users to efficiently simulate multi-generational genotype and phenotype data under genetic nurture (GN) effects and assortative mating (AM). The GNAMES tool itself, a description of its technical details, and a tutorial, are freely available on the following GitHub repository: In the simulations, we partition simulated genotype and phenotype data into three sets: two sets with equal sample size to perform non-overlapping GWASs (discovery) and one set to construct PGIs (prediction). For both sets of GWAS results, we construct a separate PGI. In addition, we also construct a PGI based on the meta-analysis of the two GWASs, in line with the common practice of using the largest possible discovery sample to construct a PGI. Since the SNP effect sizes are simulated to be independent, the PGI simply equals the sum of the SNPs, weighted by the respective GWAS coefficients.30 for a recent exception). The outcome of each simulation is a data file with the individual’s ID, the father and mother’s ID, a simulated outcome, two PGIs constructed from the two non-overlapping GWAS discovery samples, and the meta-analysis PGI. Further details of the simulation design are provided in Supplementary Methods In our setup, every family has two children. Importantly, in the GWASs, data for only one sibling per family are used, and so the resulting PGIs are based on between-family GWAS results, as is common in the literature . For example, a recent study on EA33 employs prediction samples of similar sizes (the Dunedin Study (N = 810), the Environmental Risk Longitudinal Twin (E-Risk) Study (N = 1860), AddHealth (N = 5526), the Wisconsin Longitudinal Study  and the Health and Retirement Study ).The prediction sample—To create realistic variation in prediction sample sizes we vary 8, the predictive power of a PGI mainly depends on the variance explained by each SNP, and the ratio of the GWAS discovery sample to the number of SNPs. In particular, under the assumption that all SNPs explain an equal proportion of the SNP-based heritability, De Vlaming et al. (Eq. 2)34 approximate the predicted R2 of a given PGI asM is the number of SNPs and NGWAS denotes the GWAS discovery sample size. To maintain a manageable simulation space, we hold the number of SNPs fixed at M = 5000, and set NGWAS in the range . These values for the GWAS discovery samples then respectively generate an expected R2 of 2.3% , 4.2% , 7.1%, 11.1%  and 15.4% . To verify that our results are not sensitive to downsizing both M and NGWAS for computational reasons, we also analyze an additional setting with M = 100,000 and NGWAS = 100,000. The results remain very similar, see Supplementary Results The size of the GWAS discovery sample—As described in refs.  al. Eq. 34 approx37. If the direct genetic effects and genetic nurture components are independent, the heritability is given by: h2 denotes the phenotypic variance accounted for by direct genetic effects, and n2 denotes the phenotypic variance accounted for by genetic nurture. In scenarios without genetic nurture, we fix the SNP-based heritability at h2 = 0.2 and n2 = 0.1 such that the genetic nurture is half the size of the direct genetic effect. This parametrization is again loosely following empirical evidence for EA, where the indirect genetic effect represents roughly half of the additive SNP factor18.The presence of genetic nurture—We hold 40.The degree of assortative mating—We vary assortative mating on the outcome variable  between 0 and 1 in increments of 0.25. Plausible levels of assortative mating on education vary between 0.1 and 0.6The genetic correlation—We vary the genetic correlation between the GWAS discovery samples and the prediction sample between 0.25 and 1 in increments of 0.25, where the two GWAS discovery samples have a perfect genetic correlation. An imperfect genetic correlation may for example arise if the discovery GWAS is performed in a UK sample and the prediction is performed in a non-UK sample. Additionally, we vary the genetic correlation between discovery sample 1 and discovery sample 2 from 0.25 to 1 in increments of 0.25, where we maintain a perfect genetic correlation between discovery sample 2 and the prediction sample.The simulations are calibrated based on educational attainment (EA), but we show that our conclusions hold for traits with a different level of heritability in Supplementary Results 41. In these GREML analyses, individuals with a genetic relatedness larger than 0.05 are excluded.For each scenario, we perform 100 simulation replications (“runs”), and we average the results over these runs. In all scenarios, we estimate (i) a linear regression of the outcome on the meta-analysis PGI; (ii) ORIV, on basis of a two-stage least squares (2SLS) regression where we simultaneously use both independent PGIs as instrumental variables for each other; and (iii) the PGI-RC where we scale the coefficient obtained under (i) by the estimated SNP-based heritability. The SNP-based heritability is estimated in the prediction sample using MGREML (Multivariate Genome-based Restricted Maximum Likelihood)6). This may lead to anti-conservative standard errors, and in the univariate case the true standard error is equal to the standard error of the square root of the estimated SNP-heritability. We will therefore present both the default PGI-RC (“PGI-RC (Default)”) application, as well as the PGI-RC that incorporates the uncertainty in the scaling factor (“PGI-RC (GREML unc.)”). Details of the estimation procedures can be found in Methods.The default PGI-RC estimator ignores the estimation error in the estimation of the scaling factor . Since we simulated a scenario without genetic nurture and assortative mating, the between-family , the PGI-RC slightly underestimates the true coefficient. When using a small prediction sample, the uncertainty in the PGI-RC (GREML unc.) is considerably larger than suggested in the default application PGI-RC (default). With a relatively large GWAS sample, the RMSE for ORIV tends to be slightly smaller than for the PGI-RC  to 15.4% (EA4). Whereas a meta-analysis PGI clearly benefits from an increased GWAS sample size, for a relatively large prediction sample size of N = 16,000 both ORIV and the PGI-RC are approximately unbiased irrespective of the GWAS sample size. The PGI-RC shows a narrower confidence interval compared with ORIV, even when taking into account uncertainty in the GREML estimates, with the difference becoming smaller for larger GWAS discovery samples  for different prediction sample sizes, and a relatively large GWAS sample size, calibrated to be resulting in an R-squared of 15.4% (roughly EA4), again for different prediction sample sizes. As expected, when the GWAS sample size increases, the coefficient of a meta-analysis PGI comes closer to the true value (0.5), but even for a relatively large GWAS sample size it is still considerably biased. As before, an increasing prediction sample size does not decrease the bias in the coefficient of a meta-analysis PGI, but only shrinks the confidence interval.In Table F-statistic43. As we derive in the “Methods” section, in this context, the first-stage F-statistic is given by (see Eq. (N. The correlation between the two PGIs is determined directly by the R-squared of the independent PGIs (see Eqs. (N = 1000) or the prediction sample size increases , ORIV quickly converges to the true point estimate.ORIV is somewhat biased when both the GWAS sample and the prediction sample are relatively small. This is not surprising, since it is well known that the bias of IV is inversely proportional to the first-stage (see Eq.  for a desee Eqs.  and 17)F-statistIn this baseline scenario, the point estimates of the PGI-RC are largely independent of the GWAS sample size, but are very sensitive to a small prediction sample. The picture that emerges from Table 29. The most important difference is that with genetic nurture, the between- and within-family results are starting to diverge. In within-family designs, ORIV continues to outperform a meta-analysis PGI, yet slightly underestimates the true coefficient because of the noise introduced in the PGI since the GWAS did not control for genetic nurture (see Methods for a more extensive discussion).When genetic nurture is present, the results Fig.  and 2. T45.Figure 45. Similar to our baseline simulation, in Supplementary Results The simulations are designed such that the true coefficient does not change with increasing levels of assortative mating (AM). This is achieved by standardizing the true PGI to mean zero and unit variance in each generation of the forward simulation, and then assigning effect entclass1pt{minimaIn our simulations, we draw independent effect sizes for the direct genetic effects and the genetic nurture components in the founding population, such that the direct genetic and genetic nurture components are independent. However, when modeling genetic nurture and assortative mating simultaneously, over generations, a correlation emerges between the direct genetic effects and the genetic nurture effects  prediction samples, and its confidence intervals are wider than those of ORIV when incorporating the uncertainty of the SNP-based heritability estimates. In those cases, as well as in within-family settings where the PGI-RC correction is generally not available, ORIV seems the preferred alternative.The simulations suggest that it is virtually always beneficial to apply a measurement error correction compared to just using a meta-analysis based PGI. We find that both the PGI-RC as well as ORIV outperform the benchmark in all scenarios. Comparing among them, the PGI-RC is particularly valuable in between-family analyses when the GWAS discovery sample is very small  or when there exists an imperfect genetic correlation between the discovery and prediction sample . In the presence of assortative mating, however, the PGI-RC is both more biased and less precise than ORIV. Moreover, the PGI-RC tends to be imprecise in case of small . In the prediction sample, we first residualized the outcomes EA and height for sex, year of birth, month of birth, sex interacted with year of birth, and the first 40 principal components of the genetic relationship matrix. For both EA and height, we consider three PGIs: (i) a PGI based on the UKB discovery sample that excludes siblings and their relatives; (ii) a PGI based on the 23andMe, Inc. sample (EA) or the GIANT consortium ; and (iii) a PGI based on a meta-analysis of (i) and (ii). In addition, we construct two additional PGIs on the basis of randomly splitting the UKB discovery sample into two equal halves. All PGIs are constructed with the LDpred software47 using as parametrization a default prior value of 1. We standardize the PGIs to have mean 0 and standard deviation 1 in the analysis sample. The standardization of the PGI has the advantage that the square of its estimated coefficient in a univariate regression is equal to the R-squared (see “Methods”). Additionally, by standardizing a given PGI we can interpret the resulting coefficient as a one standard deviation increase in the true latent PGI . More details on the variables and their construction can be found in Supplementary Methods In this section, we use OLS , ORIV, and the PGI-RC to predict EA and height in a subsample of European ancestry siblings in the UK Biobank N = 35,82. In thN = 18,989 for EA, N = 18,913 for height). To obtain the LDSC estimate, the GWAS summary statistics were computed using FastGWA48. In the GREML analysis49, the analysis sample was slightly lower  because we excluded closely related individuals using the default relatedness cut-off of 0.025. These SNP-based heritabilities are a useful benchmark, as they constitute an upper bound on the R2 we can achieve in our sample using a PGI9. The SNP-based heritabilities additionally are a crucial input for disattenuating the OLS estimator in the PGI-RC6. In fact, in the univariate case, the SNP-based heritability obtained using GREML is actually the PGI-RC estimate.Using LDSC and GREML, we estimate the SNP-based heritability of EA in the prediction sample to be 0.162 (s.e. 0.027) and 0.155 (s.e. 0.019), respectively, which is somewhat lower than estimates from other samples in the literature. For height, the SNP-based heritability is estimated to be 0.497 (s.e. 0.039) using LDSC, and 0.530 (s.e. 0.020) using GREML. These results were obtained using one randomly selected sibling per family in the sibling subsample . At the same time, when accounting for the uncertainty in the GREML estimate of the heritability entclass1pt{minima51. It bears repeating that in this context the PGI-RC cannot be applied, and so the relevant comparison here is between regular OLS and ORIV. The standardized effect estimates are substantially smaller within-families than they are between-families. This finding reflects an upward bias in the between-family estimates as a result of population phenomena, most notably genetic nurture52. More specifically, in line with the literature54, our within-family ORIV estimates are around 45% smaller than the between-family ORIV estimates. As in the between-family analyses, applying ORIV within-families increases the coefficient of the PGI compared to a meta-analysis or using standalone individual PGIs. ORIV estimates the standardized direct genetic effect to be 0.184 (using PGI from two samples—UKB and 23andMe) and 0.170 (using split-sample UKB PGIs). This estimate may still be prone to attenuation bias since the PGIs are based on a GWAS that did not consider indirect genetic effects from relatives 56, but this estimate does represent a tighter lower bound on the direct genetic effect.The results in the bottom panel of Table R-squared of 34%. It is also noteworthy that for height the between- and within-family results do not differ as much as they do for EA. Again, this is in line with the literature18 which generally finds genetic nurture to be more important for behavioral outcomes such as EA than for anthropometric outcomes like height.Table 49, these empirical findings build confidence that our conclusions from the simulations apply more broadly.Despite the differences in heritability and in the role of genetic nurture, we reach similar conclusions for height as for EA in the comparison of OLS , ORIV, and the PGI-RC. The two-sample ORIV estimation is 25% (between-family) and even 30% (within-family) higher when using ORIV compared to a meta-analysis PGI. The confidence interval for the PGI-RC incorporating uncertainty in the GREML estimate is again larger than the confidence interval of the ORIV estimate, yet the point estimate of the PGI-RC is significantly higher than the one for ORIV. In the within-family analysis, ORIV delivers the tightest lower bound on the direct genetic effect for height, which is estimated to be above 0.6. With height being a typical trait to test new quantitative genetics methodologies57, it is not entirely clear whether the estimated SNP-based heritability of 15.5% (and corresponding standardized effect of ~0.4) is the correct target, or that it is overestimating the true effect. Similarly, estimates of assortative mating for height are in the order of 0.2358, again leaving potential for bias in the GREML estimates that underlie the PGI-RC. For this reason, in Supplementary Results 59. In line with the absence of assortative mating, the between-family and within-family results are highly concordant in case of DBP. Similar to the results for EA and height, ORIV increases the estimated coefficient by 27% compared to an OLS regression on basis of the meta-analyzed PGI. Again similar to EA and height, the confidence interval around the PGI-RC estimates is wider, but ORIV tends to produce slightly lower point estimates than the PGI-RC. In sum, it seems that our empirical findings are not driven by assortative mating in EA or height.As shown in the simulations, depending on the degree of assortative mating, the SNP-based heritability may be overestimated, and hence the PGI-RC could be overestimating the true effect size more severely than ORIV. With EA being a trait that exhibits a considerable degree of assortative matingThe increasing availability of genetic data over the last decade has stimulated genetic discovery in GWAS studies and has led to increases in the predictive power of polygenic indices (PGIs). Phenotypes such as educational attainment (EA) and height are currently at a critical turning point at which boosting the GWAS sample size further will only increase the predictive power of the PGIs at a marginal and diminishing rate. As a result, for the foreseeable future, regressions involving PGIs will be subject to an attenuation bias resulting from measurement error in the estimated GWAS coefficients. In this study, we compared two approaches that attempt to overcome attenuation bias in the estimated regression coefficients of a PGI: Obviously-Related Instrumental Variables (ORIV) and the PGI Repository Correction (PGI-RC).Extensive simulations show that in between-family analyses, the comparison of ORIV versus the PGI-RC is subject to a bias-variance trade-off that will differ across applications. The PGI-RC performs especially well (in terms of Root Mean Squared Error) compared with ORIV when there exists imperfect genetic correlation between the discovery and prediction sample, or when the GWAS discovery sample is relatively small. These conclusions hold even when incorporating the additional uncertainty induced by first estimating the SNP-based heritability. However, when a sizable discovery sample is available that has a near-perfect genetic correlation with the prediction sample, ORIV tends to perform better than the PGI-RC. Moreover, when there exists considerable assortative mating on the basis of the outcome variable, ORIV also tends to perform significantly better compared with the PGI-RC. In within-family analyses, ORIV is the most convenient way of estimating direct genetic effects. The simulations suggest that applying ORIV within-families provides consistent estimates in the absence of genetic nurture and assortative mating, and tightens the lower bound on the direct genetic effects in the presence of genetic nurture and assortative mating.We empirically tested these predictions using UK Biobank data on educational attainment (EA) and height and largely confirm the simulation results. Both ORIV and the PGI-RC outperform a meta-analysis based PGI in terms of bias and RMSE. Among them, in our application, ORIV tends to underestimate the SNP-heritability somewhat , but tends to have smaller standard errors than the PGI-RC, especially when the PGI-RC incorporates the uncertainty in the GREML estimates of the SNP-based heritability. On the basis of within-family analyses, in our application, ORIV estimated the standardized direct genetic effect to be around 0.18 for EA and 0.6 for height, respectively a 30% (EA) and 14% (height) increase compared with a meta-analysis PGI. Similar findings for diastolic blood pressure provide reassurance that assortative mating on EA and height was not driving these empirical findings.https://github.com/geighei/ORIVfor implementations in STATA and R). Moreover, ORIV does not require external information on the SNP-based heritability. This feature makes ORIV particularly attractive for within-family studies aiming to estimate direct genetic effects, because SNP-based heritability estimates do not solely capture direct genetic effects, but also incorporate indirect genetic effects from relatives, assortative mating, and population stratification60. Whereas Young et al.37 and Eilertsen et al.61 developed approaches for separating direct and indirect genetic effects, both methods require genetic data from unrelated individuals and both of their parents, samples of which are currently very rare. Hence, ORIV is more flexible since it does not require external information on the “true” level of disattenuation, which is hard to obtain in within-family settings or for traits that are subject to considerable assortative mating.In sum, the application of the PGI-RC or ORIV in empirical applications requires a careful assessment of the setting in which the correction is applied. From a practical point of view, it is a particular advantage of ORIV that it can be easily implemented in all standard statistical software packages  and Structural Equation Modeling (SEM65). Since IV can be seen as a special case of GMM and SEM models66, the differences between the approaches are typically negligible in linear models. The distributional assumptions are somewhat stronger in SEM compared with IV, and including family fixed effects tends to be cumbersome69. A possible advantage of SEM is however its flexibility in allowing the factor loadings of the two individual PGIs to be different. This could be especially relevant when the sample sizes and/or genetic correlation with the prediction sample differ substantially across two GWAS discovery samples. An extensive comparison of ORIV versus (genomic) SEM or GMM is beyond the scope of this paper, but we anticipate that differences will typically be small unless the precision of the two independent PGIs differs substantially.There are alternative approaches to deal with measurement error than those analyzed in this study. Simulation-extrapolation (SIMEX)30, but their sample sizes are still comparatively small. The results of the present study suggest that the application of ORIV could help to reduce attenuation bias in regression coefficients for PGIs constructed using the results of within-family GWASs.Whereas we have shown that both ORIV as well as the PGI-RC are superior to a regular meta-analysis PGI in terms of overcoming attentuation bias, this does not mean that further collection of additional genotyped samples is useless. In contrast, larger sample sizes are essential in identifying specific genetic variants that affect the phenotype of interest, allowing one to investigate the biological mechanisms driving these effects. Further, whereas the PGI-RC and ORIV are useful “scaling tools” to estimate the regression coefficient of the latent “true” PGI, they do not boost the predictive power of a PGI in out-of-sample applications. Finally, applying measurement error corrections using ORIV or the PGI-RC are not a substitute for within-family GWASs. The collection of family samples is the only way to explicitly control for the indirect genetic effects from relatives that plague the interpretation of the effects of PGIs in between-family studies. The collection of genetic data of family samples is on the riseY  is influenced by many genetic variants:J is the number of genetic variants  included, SNPj represents the number of effect alleles an individual possesses at locus j, and j. The true data generating process would also include the effects of maternal and paternal SNPs, because only conditional on parental genotypes the variation in SNPs is random and hence exogenous. We discuss the use of family data briefly below, but ignore the effects of parental genotype as well as environmental factors in the following discussion for simplicity.Consider a simple linear fixed effects model in which a dependent variable Y is assumed to be standardized with mean zero and standard deviation 1 (σY = 1). The true latent polygenic index PGI* is then defined as:*, then running the OLS regressionβ measures what happens to the outcome Y when the true latent PGI* increases with 1 unit in the analysis sample. Since a 1 unit increase in the PGI* is not straightforward to interpret, researchers are typically more interested in R-squared of this univariate regression. That is, in a univariate regression, the R-squared measures the squared correlation between the outcome and the independent variable, and it can be compared to the upper bound represented by the SNP-based heritability.The dependent variable * because it is measured with error:ν is classical in the sense that it is uncorrelated to the error term in Eq. . Therefore, Eq.  of Becker et al.72 that when at least two independent measures of the same construct (independent variable) are available, it is possible to retrieve a consistent effect of this construct on an outcome through IV estimation.An alternative way of addressing measurement error is instrumental variables (IV regression). It has long been recognized in the econometrics literature1 = PGI* + ν1 and PGI2 = PGI* + ν2, with Cov = 0 and 2 as an instrumental variable (IV) for PGI1, then the IV estimator is the ratio of the reduced form (regression of Y on PGI2) and the first stage (regression on PGI1 on PGI2):st1, and PGIst2, by the same scaling factor:1,+ and PGI2,+, then the resulting estimator is given by29. In Supplementary Results In terms of formulas, if we have two measures for the true PGI*, PGI73. LDSC can also be used to estimate the cross-trait LDSC intercept. With a trend towards ever-larger GWAS meta-analyses and reuse of genetic data, it is important to verify that GWAS summary statistics used to create the two independent PGIs are not computed from partially overlapping samples . Using two PGIs based on such related GWAS summary statistics would be in violation of the ORIV assumptions. The cross-trait LDSC intercept can be used to test for sample overlap, although the precise threshold depends on the GWAS samples sizes and the heritability of the trait74. In our study, the LDSC cross-trait intercepts do not cross the critical threshold. Moreover, in the UKB split-sample analyses, the GWAS samples do not contain related individuals  may however differ across samples. This would imply that the genetic correlation between the two samples would be lower than 1 for a particular outcome variable. This can be tested, for example with LDSCF-statistic with a factor roughly equal to 1/F43. The F-statistic in a univariate regression is equal to the square of the t-statistic of the first-stage coefficient β and N.Instrumental Variable regression provides consistent estimates, yet it is biased in small samples with the bias in IV regression inversely related to the first-stage (see Eq. ). Equati31. The idea is to use a “stacked” modelN is the number of individuals and ON an N × 1 vector with zero’s. The implementation of ORIV is straightforward: simply create a stacked dataset and run a Two-Stage Least Squares (2SLS) regression while clustering the standard errors at the individual level. In other words, replicate the dataset creating two values for each individual, and then generate two variables  that alternatively take the value of PGI1,+ and PGI2,+. The resulting estimate is the average of the estimates that one would get by instrumenting PGI1,+ by PGI2,+, and vice versa. This procedure makes most efficient use of the information in the two independent PGIs and avoids having to arbitrarily select one PGI as IV for the other. Family fixed effects can also be included in the model, in which case one should include a family-stack fixed effect in order to conduct only within-family comparisons within a stack of the data. Standard errors should then be clustered at both the family as well as the individual level. Supplementary Results https://github.com/geighei/ORIV.The most efficient implementation of the proposed IV estimator is the recently proposed technique “Obviously-Related Instrumental Variables” (ORIV) by ref. 18, the role of parental genotype can be profound. Another way of showing this is by studying the effect of non-transmitted alleles of parents on their children’s outcomes53, to estimate so-called genetic nurture. The true data generating process (DGP) may therefore be:F and M denote father and mother, respectively. When the true DGP is governed by Eq.  is used in its construction, a PGI reflects direct genetic effects as well as indirect genetic effects  arising from parental genotype. When applying these PGIs within-families, some of the differences in the PGI across siblings therefore spuriously reflect the effects of parental genotype, whereas in fact their parental genotype is identical. Hence, genetic nurture can be seen as measurement error in the PGI when applied in within-family analyses, leading to an attenuation bias55.In a between-family design, the bias in the coefficient of the resulting PGI tends to be upward, as the coefficients of the individuals and his/her parents are typically of the same sign75, e.g., arising from siblings. For example, consider a case with two siblings where there is a direct effect γj of one’s sibling’s SNP on the outcome of the other:βj. Again, the only way to overcome this source of bias is to include the parental genotype in the GWAS, since conditional on the parental genotype, the genotypes of siblings are independent.A final source of downward bias could stem from indirect genetic effects, sometimes referred to as social genetic effects76. Therefore, this approach provides a lower bound estimate on the direct genetic effects.In sum, within-family analyses are the gold standard to estimate direct genetic effects, free from bias arising from the omission of parental genotype. However, when using a family fixed effects strategy on the basis of a PGI from a conventional GWAS that did not include parental genotype, the direct genetic effect is biased downward as a result of measurement error, genetic nurture effects and social genetic effectsFurther information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


Selection of desirable genotypes or progenies is perhaps the most important practical method in plant and animal breeding programs. The selection index method is the most useful method to choose superior genotypes based on using simultaneous several traits. The optimum and base selection indices are the two indicators that are most used in plant and animal breeding. In this paper, a simple and practical code was developed for the analysis of optimum, base, and Pesek and Baker selection indices. Four different criteria were used to evaluate the selection index, and the phenotypic and genotypic variance–covariance matrix of traits was obtained based on statistical or genetical design. Moreover, an index that was more efficient on these coefficients was used for the breeding program. The results showed that simultaneous selection for the important traits desired by the breeder through economic values such as heritability, genetic, or phenotypic correlation is the most effective method for selecting the best genotypes. Therefore, the best progeny or genotype can be selected to use in breeding programs. This program provides detailed information on selection indices of segregation and natural populations involving any number of individuals or genotypes. These codes are much easier and simpler than other programs and provide more information than other programs. This code is easy to execute in both R and SAS programs. Selection is often effective in traits with high heritability, compared to those with low heritability. Since one of the important goals in breeding programs is to obtain high-yielding plants and, on the other hand, direct selection for yields and quantitative traits is not very effective because they are controlled by many genes and have low heritability. In most of the correlation studies in plants, it has been determined that yield has a high positive or negative correlation with traits that have high heritability or traits controlled by a small number of genes. Besides, the success of the indirect selection depends mainly on the magnitude and direction (positive or negative association) of the correlation coefficient between the trait of interest and each of the other traits. So, it is better to use indirect selection to improve yields2 or other interested traits because selection based on morphological traits with high measurement accuracy and relatively high heritability may be a quick way to screen plant populations and improve yield and quantitative traits and for this reason, using the index can be effective in improving these traits. Also, indices are one of the best methods for simultaneous breeding of traits in breeding programs3.The selection of desirable genotypes or progenies is perhaps the most important activity in plant and animal breeding programs. Selection efficiency depends largely on the genetic diversity of the population and the heritability of the studied trait4. Since the traits to be considered in selection may not be equally important economically, a type of weighting is required. Unless appropriate weighting is adopted, some traits will receive too much and others too little attention. The amount of weight given to each trait depends on its relative economic value, its heritability, and genetic and phenotypic correlations between different traits4.Selection is made for all traits simultaneously by using a total score or index of the net merit of an individual, constructed by combining scores for component characters. Individuals with the highest score are kept for breeding purposes6. The use of selection indices increases the chances of success of breeding programs because it simultaneously uses different traits to identify the superior genotype. Various information on experimental units is used in selection indices, and the index’s ability is based on a complex economic value of breeder interest traits to increase genetic values6. The economic return of a crop plant is mostly determined by its several trait values. So, plant breeders study simultaneous selection for those numerous traits that maximize a plant’s economic value. Although the number of traits affects the efficiency of a selection index and the less number of traits has a higher efficiency7.Choosing a superior progeny in a plant population can affect other progenies because traits are heavily influenced by the environment and often correlate with each other. Therefore, selection based on only one trait to identify superior genotypes may be slightly effective due to low heritability8. Selection indices have been used in different plants18. In addition, many studies have been used in animals based on selection indices and these indices have been used to improve and increase performance in them21.Using some statistical techniques, we can obtain the necessary information for the indirect selection of traits to improve yields. Among these techniques, we can mention the selection index, including optimum and base selection indices26 such as Genes, MIX, RIndSel, and SelAction. Although these softwares are complete and comprehensive and perform different analysis types and many statistical methods and designs as well as plant breeding method such as diallel analysis, QTL mapping, etc. Besides, these softwares has been widely used by researchers in private and public enterprises, and universities around the world. But some of these softwares that calculate the selection index are complex and do not calculate many of the index evaluation criterias (based on different economic weights) to select the best indices and other programs are not easy and simple in this case.There are various software applications developed to compute selection indices; although, they do not permit the estimation of some parameters27 and base28 selection index methodology, and estimated different criteria for evaluation indices. Simplicity, convenience, and its use in SAS and R softwares are one of the advantages and novelty of this code. Furthermore, the input information of this code can be easily collected by the code written in SAS software 29. In addition, there are various criteria to evaluate these two indices in this code, based on which the best index can be selected.Thus, due to the lack of easy-to-use specialized software for optimum and base selection indices and their application in plant breeding, in this paper, we described a SAS code developed for the analysis of optimum and base selection indices according to the optimum30. This input information for this code is easily and simply estimated through the SAS code29 and saved in Excel format and can be used for this code in both R and SAS programs. Additionally, the variance–covariance matrix (phenotypic or genotypic) can be obtained by other software as well as Excel. Selection indices based on the studied and used traits in the index  were calculated concerning their phenotypic, genetic, and economic values according to the following equation for the optimum selection index:The phenotypic and genotypic variance as well as the covariance between traits were estimated based on the expected value of statistical designs. Then, the broad-sense heritability of traits was calculated based on the formula i is called the index coefficient (the vector of index coefficients) that is assigned to each trait, and Xi is the phenotypic value of each trait as phenotypic trait matrix (n × m). Using the optimum index31, the index coefficients were obtained from the following equation:Here, bIn which, b is the vector of index coefficients, P is the phenotypic variance–covariance matrix (m × m), G is the genetic variance–covariance matrix (m × m), and 28. If the relative economic values of each trait are determinable, but acceptable and valid estimates of the phenotypic and genetic parameters of the traits are not available, in this case, the use of the base index is recommended for simultaneous improvement of two or more traits. In this method, an index is calculated for each individual using the phenotypic values observed for that trait and by assigning the economic values associated with each trait as the index coefficients:The second index was the base indexIn which, 32.However, to assign economic weights to the traits is not a trivial task for breeders, because it demands to know many economic variables of the market . To avoid this, an index based on the desired genetic gains for each trait was developed by Pešek and Baker 32. The amount of increase or decrease of the obtained traits can be considered as d. In this study, the standard deviation of the genetic variance of traits is considered as d.where d is the vector of desired genetic gains. The d can be the standard deviation of the genetic variance of the traits or it can be considered as a percentage of the increase or decrease of the traits by the breeder such as study of Pesek and BakerHI) were calculated, which would yield the maximum response if this criterion was maximized. Since in addition to grain yield, simultaneous improvement of the genetic value of several traits was the aim, another comparison criterion, namely, genetic gain of total traits . This code corresponds to the steps necessary to execute the selection indices according to the optimum and base methods31.The SAS code  was written in SAS/IML31. For the analysis to proceed, this code requires an input data file (available at https://www.ebi.ac.uk/biostudies/studies/S-BSST853 as DataFile 3–5) prepared in excel format (CSV). Data can be stored in any format such as xlsx, txt, xls, and others. However, in the proc import section and sub-section (dbms), the format of the data must be specified in the R code, the data format must be specified in the first part of the program and the data introduction. Economic values in the SAS code are entered manually in the code, but in the R code they are stored in an Excel file (available at https://www.ebi.ac.uk/biostudies/studies/S-BSST853 as DataFile 6) and placed in a folder next to the data. The name of the input data file should be changed to the full name (such as DataFile 3-X) in the SAS and R codes. To do these codes, if you do not make any changes to the codes, you must delete the DataFile 3- to 6 in the file names and the file names are changed to X, P, G, and a1.This code is based on the mathematical derivations presented in optimum and base methodshttps://www.ebi.ac.uk/biostudies/studies/S-BSST853 as DataFile 3–5). Genotypic and phenotypic covariance matrices are calculated through the mathematical hope of experimental design and can be calculated by the program written for this purpose in SAS29.On file X, there should be phenotypic measurements of traits, on the file P, there should be a phenotypic covariance matrix of traits, and on file G, there must be a genotypic covariance matrix of traits  according to the user of the data. Data for X, P, and G can be stored separately for each file or can be stored in one file on separate sheets. However, the filename or the special sheet of the file should be specified in the proc import section. Users can create a folder on drive C called “selection index’’ and prepare and store data under the same name without the need to change the path of data in codes. Or the data related to these two codes can be placed in a special folder, and at the beginning of the program, the path of this folder, wherever it is on the computer, must be specified for these two codes.IML section of the SAS code, some information should be provided for the data to be used and should be changed based on user data and the studied trait (Table 2), vector of relative economic values (a1 = {}), and interesting trait number (tr).In the proc it Table . The infFurthermore, for the R code like the SAS code, some parts must be defined before being done. The g is the NT × 1 vector of genotypic variance–covariance of interest trait with other traits (here is the yield which is the seventh trait that is placed in the genotypic or phenotypic matrix) and shown in the genotypic matrix by this G and 7 is the number of this trait. The wg and wp are the genotypic and phenotypic variance values of the interesting trait (here is the yield which is the seventh trait and 7 is the number of this trait) and shown by these G1 and P1, respectively. The results by R code are stored in different sheets of an excel file according to the path to save them and can be accessed. For example, the results when correlations are considered as economic weights are given in Supplementary Material 4 (output_with_EW_corre).An economic value varies and depends on a researcher’s choice; it can also be any value based on heritability, correlation coefficients, etc. In this research, the amount of economic value (a) was considered in three ways. 1: 1 for all traits, 2: Correlation of traits with yield  can also be used), 3: The The phenotypic value matrix is obtained based on trait evaluation. Moreover, the phenotypic and genotypic variance- covariance matrix of traits is obtained based on statistical or genetical designs in plant and animal breeding programs.Data from seven measured traits  on 28 maize inbred lines evaluated in a complete block design with three replications in 2020 at the field were used in this study (DataFile 3). This study complied with relevant institutional, national, and international guidelines and legislation of Iran, and no specific permits were required to collect the plant materials. The phenotypic-genetic covariance matrices of the traits (DataFile 4 and 5) were calculated based on the expected value of statistical designs using the SAS code and saved in Excel format. The PROC IML of SAS was used to estimate the selection index. In the proc iml part, the program needed information including the number of genotypes and traits, genetic variance value of yield, selection intensity, heritability value of yield, NT × 1 vector for relative economic values, and interesting trait number (tr), respectively. The same steps can also be implemented in the R program.In this study, an economic value  were used to estimate the expected genetic advance.The coefficient of the traits in the base index is equal to the economic value of the traits. To evaluate the selection strategies for maximizing the maize grain yield, selection indices were calculated based on optimum, base and Pesek and Baker indices with an equal economic trait value (vector a1), as described by SmithThe Pesek and Baker index should have been calculated only once because d vector (desired genetic gains) has been used as economic weights. But to see the difference in ranking of genotypes, when the economic weights changed, this index was calculated again. According to the results in Table The ranking of genetoypes in Table 6 of Supplementary Material 3 shown that the top five genotypes based on optimum index as well as economic weights as Method 1 were genotypes 2, 4, 19, 22 and 6, respectively. While, the genotypes 1, 4, 19, 22 and 6 were identified as five top genotypes, respectively, in base index with the same conditions. In addition, genotypes 3, 17, 8, 16 and 13, respectively were selected as the five top superior genotypes based on the Pesek and Baker index.Also, for example, the correlation between the indices  and the economic weights as Method 1) shown in the Table 7 of Supplementary Material 3. The results showed that correlation between base-optimum, base-Pesek and Baker, and optimum-Pesek and Baker were 0.99979, 0.31516, and 0.32464, respectively. The correlation of base and optimum indices showed that the rankings of genotypes in these two indices were very similar and more than 99% were the same in ranking. Meanwhile, the correlation of index Pesek and Baker with other two indices (base and optimum) showed that the ranking of genotypes based on this index had a high difference with the other two indices.35. In the base index, the importance of the phenotypic value of each studied trait is directly determined by the factor of economic values, so traits with zero economic value will not be included in the index equation. Additionally, in this index, there is no need to estimate the genetic parameters and the results can be easily obtained and interpreted, so it is preferable to the optimal index.By taking a glance at the optimum and base indices and comparing them with the Pesek and Baker index based on different economic coefficients Table , we can ds Table . Conside8. Although there is a positive relationship between the yield and the number of its components, the existence of negative relationships between some of the components of the yield has led to the fact that selection for all yield components cannot be used as a factor in increasing the yield36. This code has already been used to estimate the optimum and base selection indices at different economic values38 and it has shown its effectiveness.The goal of plant breeding is the genetic modification of a species in the best possible way. The economic value varies depending on its various traits. Therefore, how to apply selection for several traits to achieve the maximum value of the economy has always been of interest to breeders38 examined the base and optimum indices with different economic values  on seven hybrids maize, and finally selected the best index based on the criteria for evaluating the indices. They selected the best genotypes based on the best index. Asghar and Mehdi39 reported that the Smith–Hazel and Brim indices were useful for the improvement of a sweet corn population. However, the Brim index was reported to be more efficient than the Smith–Hazel index in genotype improvement for quality traits in a maize population.Rahimi and Ramezani8. This code has also been developed based on the selection index of optimum and base methods31. This code can assist breeders to choose progeny or genotypes because it is simple and convenient and has criteria for comparing different indices based on different economic values. The economic value can be varied and based on this various selection indices can be obtained. Sometimes, multivariate regression coefficients or trait heritability are considered as an economic value. Therefore, this code can compare different selection indices and is chosen as the best selection index. Thus, the best progeny or genotype can be selected to use in breeding programs.To make choices for yield more reliable, breeders need to identify selection criteria that reduce the phenotypic evaluation of traits and focus more on the effect of several traits on the yield. In general, simultaneous selection for the important traits desired by the breeder through economic values such as heritability and genetic or phenotypic correlation is the most effective method for selecting the best genotypes. In this method, an index is defined and the progeny of the population is selected accordingly as a single traitConsidering that the simultaneous use of traits to improve plants and animals can be more beneficial than breeding plants and animals through single traits. Therefore, this simple and practical code can help breeders by simultaneously breed of traits and the use of selection indices in the breeding of plant and animals programs. Moreover, by using different economic values to calculate different indices, an appropriate index can be used for plant or animal breeding programs by comparing the indices according to different criteria. Furthermore, breeders can select superior genotypes based on the coefficient index of each genotype and use them in breeding programs.Supplementary Information 1.Supplementary Information 2.Supplementary Information 3.Supplementary Information 4.Supplementary Information 5.Supplementary Information 6.Supplementary Tables.Supplementary Information 7.