A method to improve meat production at guinea pig farms is by applying crossbreeding schemes. Our study evaluates the application of a two-way crossbreeding scheme using four genetic lines  to estimate the heterosis for productive traits. Positive heterosis effects in both types of crosses were found only for birth weight: 3.7% for paternal crosses and 12.7% for maternal crosses. However, the heterosis was not observed for weight at 10 days of age, weaning, or at 60 days of age (slaughter age). Based on this, applying a two-way crossbreeding scheme of paternal and maternal guinea pig lines for meat production would not be recommended for our population. On the other hand, assessing the three- or four-way crossbreeding schemes could be interesting.This study aimed to estimate the heterosis for productive traits in a two-way crossbreeding scheme. Four guinea pig lines were originally selected for the following traits: line P1 for the growth rate, P2 for the partial feed conversion rate, M1 for the growth rate of the litter at 10 days of age, and M2 for the litter size at birth. The comparison included 176 purebreds  and 150 crosses ; body weights at birth, 10 days, weaning and 60 days of age were analyzed. A linear fixed-effect model was used, and heterosis was estimated as the difference between the average performance of the crossbred and pure-line animals. The pure line comparisons showed that P2 was lower than P1 for weight at 10 days and weaning weight, while all other comparisons between the paternal and maternal pure lines were not significant. The results indicated significant positive heterosis effects for both types of crosses, but only for birth weight: 3.7% for paternal crosses and 12.7% for maternal crosses. The heterosis estimates were mostly positive but not significant for all other traits. A reason for the low levels of heterosis could be that the lines are not very genetically differentiated. These results suggest that applying a two-way crossbreeding scheme within paternal and maternal guinea pig lines for meat production is not recommended due to the absence of heterosis for growth traits. In Peru, guinea pig production is part of the Andean people’s life and culture. However, the high quality and nutritional value of guinea pig meat have transformed guinea pig production from a subsistence activity into a profitable business . Guinea During the last few decades, the guinea pig population has nearly tripled  [Nowadays, in Peru, there are different breeds and genetic lines used for meat production, such as the Peru, Inti, Andina, and Kuri lines, developed by the INIA  ,8,9. TheAll of these breeds enable farmers to increase their incomes. However, each breed or genetic line is managed under different production systems; for example, guinea pigs developed by the INIA or UNALM are reared under a traditional system, while the animals developed by IVITA are reared under a crossbreeding system . The traNowadays, the meat production of guinea pigs in Peru is carried out using breeds or genetic lines with improved growth traits. Breeds such as Peru, Inti, and Kuri are widely promoted in Peru because they reach the typical slaughter weight of 1000 g at around eight weeks of age. Moreover, their reproductive and fitness traits, such as the average litter size and high viability up to weaning, are highly remarkable in these breeds . HoweverCrossbreeding schemes for livestock are used for meat production. Two- and three-way crossbreeding schemes are used in beef, pork, rabbit, chicken, and broiler production systems . These cCrossbred females (F1) are important, as they have larger litter sizes, better milk production, and better maternal abilities ,19. FurtTherefore, the Instituto Veterinario de Investigaciones Tropicales y de Altura (IVITA) proposed a four-line crossbreeding scheme for the meat production of guinea pigs. Between 2007 and 2009, IVITA established two paternal (P1 and P2) and two maternal (M1 and M2) genetic lines. The P1 and P2 lines were selected for their high growth rate and low partial feed conversion rate . The four genetic lines were phenotypically similar to the main commercial breeds  with regard to body shape, type of hair, and ear size ,21. The Therefore, this study aimed to estimate heterosis for growth traits in a two-way crossbreeding scheme for producing paternal and maternal types of F1 animals.This research was carried out at the Instituto Veterinario de Investigaciones Tropicales y de Altura (IVITA) research center, located 3320 m above sea level in the El Mantaro district, Huancayo Province, Peru. This research center developed four guinea pig lines: two paternal (P1 and P2) and two maternal (M1 and M2). Line P1 is based on the genetic line “Peru”, and line M2 is based on the genetic line “Andina”, purchased from a commercial farm in Lima, Peru. Both are two well-established breeds that were developed by the INIA ,8. TheseEight different crosses were performed, and four traits were recorded. The crosses were performed between September 2021 and September 2022. In total, 176 pure mates  and 150 crosses  were included in the analysis. M1M2 has a lower number of records due to its high mortality at birth. The statistical analyses were performed separately for each group of crosses . The effect of the type of cross was estimated through two different statistical linear models. Thus, for birth weight, weight at 10 days of age, and weaning weight, the following statistical model was used:For weight at 60 days of age, the statistic model used was:y = observations Type cross = type of mate or cross performed Sex = two levels Period = related to the season of the year, and was created using the birthday of animals Litter size at birth = number of pups born in total Litter size at weaning = number of animals at weaningBody condition of the mothers = 5 levels, where 1 is extremely thin, and 5 is extremely fat [mely fat Heterosis was estimated as the difference between the average of crossbred and pure mates:The effect of the crossbred type was estimated through an analysis of variance, and the significance of the heterosis was estimated using orthogonal contrast, coding as −0.5 for pure lines and 0.5 for crossbred lines, according to the type of cross aforementioned in p-value > 0.05).The results for P2 were lower than those of P1 for weight at 10 days of age and weaning weight. However, P2P1 reached higher means for birth weight than P2. However, all paternal crosses were similar for weight at 60 days of age . However, the results for M1M2 were higher than those for the other types of crosses.On the other hand, the performance for all the maternal crosses was similar for weight at 10 days of age, at weaning, and at 60 days of age . Moreover, both maternal lines, as well as the crosses, were similar weights at 10 days of age. The means estimated for weaning weight followed the trend of weight at 10 days of age in the paternal and maternal lines. Furthermore, some of the experimental groups  reached higher values for weaning weight than the range between 248 and 291 g reported in the literature [The weight at 10 days of age is a trait that has scarcely been reported. Our results show that P1 reached higher values for this trait than P2 , daily weight gain (−3.0 and 3.9%), and the feed conversion rate (7.4 and 7.8%) in each cross, the Peru × Andina and Peru × Inti, respectively . In a CoIn a crossbreeding experiment of Peruvian and Bolivian guinea pigs estimating components of heterosis, based on a complex, multi-generation crossbreeding experiment, no dominant components of heterosis were found for growth traits or for reproductive traits. The results were conflicting, depending on the model of heterosis applied . Our stuOur results could also be explained by the close relationship between the pure lines involved. Very small Fst values (0.0013) were reported between the Peru and Andina breeds , which wThe results of this study suggest low levels of heterosis for growth traits in first-generation crosses (F1) of guinea pigs. Yet, it would be interesting to evaluate a three-way crossbreeding scheme using the F1 maternal crossbreed as a dam line and P1 or P2 line as a terminal sire. Thus, to take advantage of the combined desirable traits of these three different guinea pig lines, heterosis for growth, litter size, and other economically important traits should also be estimated for these types of crosses.


To identify whether there exists a genetic correlation and causal relationship between 25(OH)D and autism spectrum disorder (ASD).Based on large-scale genome-wide association studies, a series of genetic approaches were adopted to obtain summary statistics. Using linkage disequilibrium score regression, we assessed the shared polygenic structure between traits and performed pleiotropic analysis under composite null hypothesis (PLACO) to identify pleiotropic loci between complex traits. A bidirectional Mendelian randomization (MR) analysis was applied to investigate whether there is a causal relationship between 25(OH)D and ASD.rg = − 0.227, P < 0.05), and PLACO analysis identified 20 independent pleiotropic loci matched to 24 pleiotropic genes, of which the function reveals an underlying mechanism on 25(OH)D and ASD. In Mendelian randomization analysis, the inverse variance-weighted (IVW) method with OR = 0.941  and p < 0.474 did not show a causal relationship between 25(OH)D and ASD, while, in the reverse Mendelian randomization analysis, IVW method showed OR = 1.042 , indicating no causal relationship either.The linkage disequilibrium score regression (LDSC) showed a negative genetic correlation between 25(OH)D and ASD (This study provides evidence for a shared genetic overlap between 25(OH)D and ASD. Bidirectional MR analysis also did not show a definite causal relationship between 25(OH)D and ASD.The online version contains supplementary material available at 10.1186/s12263-023-00727-0. Autism spectrum disorder (ASD), a group of neurodevelopmental disorders that begin in infancy, is characterized by a lack of social communication and interaction, stereotypical patterns of behavior, and narrow interests. It is accompanied by depression, irritability, and intellectual impairment  and is oVitamin D is a neurosteroid hormone essential for neural development . As the In conclusion, controversy with regard to the true link between 25(OH)D and ASD has been unsettled.Based on this, we systematically evaluated the complex relationship between 25(OH)D and ASD from the perspective of shared genetics, pleiotropy, and causality. Traditional observational epidemiological studies encounter many challenges in discovering disease etiology and causal inference, including reverse causal associations, potential confounders, minimally effective exposure factors, and multiple testing, and randomized controlled trials based on chronological order may be unethical. Since the association between genes and underlying outcomes is not confounded by common factors such as postnatal environment and socioeconomic status, and the causal time sequence is reasonable, genes acting as instrumental variables in disease association studies  play an important role in causal inference. Serum 25-hydroxyvitamin D (25(OH)D) reflects the combined effects of supplementation and skin production, so the level of serum 25 (OH)D is considered to be the standard measure of vitamin D status in subjects . ConsideIn this paper, the genetic correlation between 25(OH)D and ASD was assessed at first; then, pleiotropic genes associated with both phenotypes were identified using PLACO, an effective method. Finally, the biological function of the localized pleiotropic genes was explored using enrichment analysis , and bidhttps://cnsgenomics.com/content/data; the ASD data download link is https://pgc.unc.edu/, data updated to June 2022. Ethical approval and informed consent were achieved for all original studies.To reduce population stratification bias, all samples used in this study were obtained from European populations. 25(OH)D was taken from a genome-wide association study of 25(OH)D in 417580 Europeans , and ASDLDSC was first used to assess shared polygenic structure between traits, where linkage disequilibrium scores (LDS) could be calculated from European samples in the 1000 Genomes Project and then used as a reference panel. For single-nucleotide polymorphism (SNP), strict quality control was implemented, by removing all non-biallelic allele SNPs, SNPs with strand-ambiguous alleles , SNPs with MAF < 1%, SNPs without rs numbering, duplicate rs IDs, and SNPs that were not presented or whose alleles did not match phase 3 of the 1000 Genomes Project.Z score’s square was calculated for each variant, and SNPs with very high Z2 (> 80) values were removed. In addition, given the potential correlation between the two diseases, the correlation matrix of Z was estimated, and its matrix was included in the performed analysis. Finally, the hypothesis of no pleiotropy was tested using the level-α cross-over-unit test (IUT) method, and the final pleiotropy p-value was determined. Functional mapping and annotation of genome-wide association studies (FUMA) were used to determine the biological function of pleiotropic [r2 threshold defining independent significant SNPs was set to 0.2, and the maximum distance between LD blocks merged into a locus was set to 500 kb. The identified locus was then mapped to nearby genes, and a series of pathway enrichment analyses were used to determine the function of the mapped genes based on the Molecular Signature Database (MSigDB) [As a new way to study pleiotropic loci between complex traits, PLACO uses aggregated levels of genotype-phenotype association statistics . During iotropic . The r2 (MSigDB) .The bidirectional MR analysis was adopted to assess the causal relationship between 25(OH)D and ASD. Notably, the MR analysis should be performed under three basic assumptions: (1) genetic variation is strongly associated with exposure, (2) genetic variation is independent of any potential confounding factors, and (3) genetic variation is not associated with outcomes except through the stated exposure pathway. This study was reported in accordance with the latest guidelines in Strengthening the Reporting of Observational Studies in Epidemiology Using Mendelian Random Assignment (STROBE- MR).F statistic < 10 were excluded to avoid weak instrumental bias. The instrumental variable had a PVE of 5.82% and a minimum F value of 10.1. A total of 123 SNPs were screened for 25(OH)D by taking P < 5 × 10−8 and linkage disequilibrium r2 < 0.1 as screening parameters. Detailed information on IVs is provided in the supplementary material , and this suggests a potential shared mechanism between 25(OH)D and ASD.In this study, we observed a negative correlation between genetically determined 25(OH)D and ASD risk P < 5 × 10−6), which were mapped to 24 pleiotropic genes, as indicated in Fig. The pleiotropy analysis identified 20 independent pleiotropic loci P < 5 × 0−6, whicLater, we further performed enrichment analysis for pleiotropic genes and found that they were enriched through three pathways, i.e., regulation of secretion by cell, cellular response to hormone stimulus, and intracellular protein-containing complex. In the process, no significant enrichment of pleiotropic genes was observed in different tissues, but differential expression of GC gene appeared in the liver, pancreas, and stomach. In addition, two of the top 3 enriched tissues were both brain tissues. See supplementary material for details  and did not show a causal relationship between 25(OH)D and ASD.P < 5 × 10−8 and chain imbalance r2 < 0.1 as screening parameters, and a total of 1 SNP was screened. Only the IVW method and a fixed model were respectively conducted and taken, yielding an OR = 1.042 . No statistical significance was found either.A reverse MR analysis was also performed simultaneously in this study, again taking In the Mendelian randomization analysis on VD–ASD correlation, the accuracy of IVW was subsequently verified using the weighted median and weighted-mode methods in order to ensure that the results were convincing. See Fig. Both MR-PRESSO’s global test and MR-Egger’s intercept term were used to identify variants with disproportionate effects and directional pleiotropy. Besides, scatter plots and funnel plots were adopted to assess the robustness of the association. The scatter plot indicated that there were no multi-effect outliers that affected causal inference; the funnel plot showed that the causal estimates were evenly distributed on both sides of the effect estimates and the results were robust. Moreover, LOOCV results demonstrated that, after excluding one instrumental variable at a time for analysis, the frequency distribution plot found no SNPs interfere with causal inference. See Fig. P < 5 × 10−6), which are matched to 24 pleiotropic genes. This is followed by further enrichment analysis. Finally, we used bidirectional MR to analyze whether there was a causal relationship between 25(OH)D and ASD, and the result was negative.This study is the first to combine three methods, LDSC, pleiotropy analysis, and bidirectional MR to comprehensively assess the relationship between 25(OH)D and ASD. Based on large-scale GWAS summary statistics, we first used LDSC to find evidence of negative shared genetic overlap between 25(OH)D and ASD. Then, we used PLACO to identify 20 independent pleiotropic loci D deficiency. It has also been suggested that 25(OH)D deficiency leads to increased expression of CaV1.2 and CaV1.3 channels [2+ levels and affects neurodevelopmental processes [Several possible mechanisms may explain the relationship between 25(OH)D and ASD. Vitamin D deficiency in the developing brain affects many important processes, for example, the synthesis of neurotransmitters including NO, cellular differentiation, Caactivity . In the activity . Furtherelopment . One of elopment . These schannels , which irocesses .The mechanism of 25(OH)D production is linked to other factors that further influence ASD development, so it is important to investigate the direct causal relationship between 25(OH)D and ASD. As a result, after identifying the definitive evidence on a link between 25(OH)D and ASD, we attempted to further investigate it in depth.In previous studies, partial cause-to-effect studies with high-level evidence-based medical evidence concluded that the causal relationship between 25(OH)D and ASD is unclear. Kerley found in a double-blind, randomized, and placebo controlled trial developed in 2017 that vitamin D supplementation had no effect on the primary outcome with limited and inconsistent effects in children with ASD , whereafThis study also has some limitations. On one hand, the data applied in this study were only available from the GWAS summary data as individual data were not available. This limits stratification and thus the assessment of the non-linear relationship between 25(OH)D and ASD. On the other hand, the data included in the study were all European sources, and the findings were somewhat limited. And the GWAS for 25OH-vitamin D comes from a different population as the GWAS on ASD, which is from a unique and homogenous Danish population.The present study provides evidence for a shared genetic overlap between 25(OH)D and ASD. However, bidirectional MR analysis do not show a definite causal relationship between 25(OH)D and ASD, so the results need to be treated with caution. Larger, multicentered, and double-blinded RCTs are needed to validate our results.Additional file 1: Supplementary Material, Table 1. DetailedAQ information on IVs.Additional file 2: Supplementary Material, Figures 1–3. Enrichment analysis for pleiotropic genes.


FROH > 1 Mb) and excess of SNP homozygosity (FSNP). The effect of inbreeding was estimated by regressing phenotypes of semen traits on inbreeding coefficients. Associated variants with inbreeding depression were also detected by regressing phenotypes on ROH state of the variants.The reduction in phenotypic performance of a population due to mating between close relatives is called inbreeding depression. The genetic background of inbreeding depression for semen traits is poorly understood. Thus, the objectives were to estimate the effect of inbreeding and to identify genomic regions underlying inbreeding depression of semen traits including ejaculate volume (EV), sperm concentration (SC), and sperm motility (SM). The dataset comprised ~ 330 K semen records from ~ 1.5 K Holstein bulls genotyped with 50 K single nucleotide polymorphism (SNP) BeadChip. Genomic inbreeding coefficients were estimated using runs of homozygosity . A 1% increase in FROH reduced SM and SC by 0.28% and 0.42% of the population mean, respectively. By splitting FROH into different lengths, we found significant reduction in SC and SM due to longer ROH, which is indicative of more recent inbreeding. A genome-wide association study revealed two signals positioned on BTA 8 associated with inbreeding depression of SC . Three candidate genes of GALNTL6, HMGB2, and ADAM29, located in these regions, have established and conserved connections with reproduction and/or male fertility. Moreover, six genomic regions on BTA 3, 9, 21 and 28 were associated with SM . These genomic regions contained genes including PRMT6, SCAPER, EDC3, and LIN28B with established connections to spermatogenesis or fertility.Significant inbreeding depression was observed for SC and SM (Inbreeding depression adversely affects SC and SM, with evidence that longer ROH, or more recent inbreeding, being especially detrimental. There are genomic regions associated with semen traits that seems to be especially sensitive to homozygosity, and evidence to support some from other studies. Breeding companies may wish to consider avoiding homozygosity in these regions for potential artificial insemination sires.The online version contains supplementary material available at 10.1186/s12864-023-09298-1. F) based on genomic data and provides them with some benefits. Genomic inbreeding does not depend on the quality and completeness of the pedigree [A reduction in the mean performance of a quantitative trait owing to mating between relatives is known as inbreeding depression. The genetic background of inbreeding depression is increased homozygosity because of descending two copy of the same allele from a common ancestor . Inbreedpedigree , and is pedigree . Genomicpedigree . Genomicpedigree , 7.The approximate age of inbreeding can be estimated by the ancestor . Indeed,ancestor . Consequmentclasspt{minimaancestor .Another relative merit of ROH-based inbreeding is that it enables researchers to identify genes and genomic regions underlying inbreeding depression. However, only a few researchers have used ROH to do so. Pryce et al.  used a sRecent studies have shown that inbreeding influences many quantitative traits in livestock including fertility . FertilBased on this background information, the objectives of the current study were: (i) to estimate genomic inbreeding coefficients and inbreeding depression; (ii) to explore inbreeding depression resulted from old and recent inbreeding; and (iii) to identify genetic variants contributing to inbreeding depression, for semen quality traits in Holstein bulls.8 spermatozoa per ml [8 mL for SC, and 330,199 records for SM ranged between 0.1 and 0.98, remained for further analyses. Across the three semen quality traits, the minimum and maximum number of records per bull were 6 and 1,415, respectively.Phenotypic records of three semen quality traits including EV, SC, and SM belonging to 1,508 Holstein bulls genotyped using the Illumina Bovine SNP50 BeadChip were used in this study. This dataset was published in a study by Yin et al. . Phenotya per ml . SM was a per ml . Phenotyhttps://genome.ucsc.edu/cgi-bin/hgLiftOver) and the default options. In this process, 231 SNPs were removed, which were duplicated or with unknown positions in ARS-UCD1.2 reference genome; therefore, 52,655 SNPs remained for the following analyses.A total of 52,886 SNPs across 29 autosomes were available in the initial dataset. There were no missing genotypes in the dataset as they were already imputed . GenotypF were used.All SNPs, even those with very low minor allele frequency (MAF) were used to calculate inbreeding as suggested by Meyermans et al. . Two estFSNP was defined as the excess of homozygosity [FSNP can take positive or negative values when zygosity , which iThis inbreeding coefficient represents the proportion of the genome that is covered with ROH segments, and is calculated as follows :\documenLINK 1.9  softwarementclasspt{minimaF−2ROH1, F−4ROH2, F−8ROH4, FROH8−16, and FROH>16 respectively). Inbreeding coefficients relevant to different length classes were calculated by summing up ROH segments of that class for each animal divided by To study the contribution of length of ROH on inbreeding depression, we classified ROH into five groups: (i) 1 to 2 Mb, (ii) 2 to 4 Mb, (iii) 4 to 8 Mb, (iv) 8 to 16 Mb, and (v) > 16 Mb  to fit the inbreeding depression models. We adapted the model of Yin et al. [FROH or FSNP) in the model to study the effect of inbreeding depression on semen traits:The SNPs with MAF < 0.01 and Hardy–Weinberg equilibrium chi-square test LINK 1.9 . We utilLINK 1.9 , as we an et al.  and inclF−2ROH1, F−4ROH2, F−8ROH4, FROH8−16, and FROH>16 were fitted simultaneously in the model, as follows:This model is similar to the Model (1) except FROH) depression for these traits in Model (1). To map genomic regions underlying inbreeding depression, each of the autosomal SNPs  were fitted one at a time and sequentially in the model (3):F: (i) the regression coefficient SC and SM were used in the association analyses since we found significant inbreeding , FROH1−2, FROH2−4, FROH4−8, FROH8−16, and FROH>16 in the model (2), and We used AIREMLF90 1.148  to estimted in R  to test p value significance threshold. Assuming p value threshold, and N is the proportion of significant SNPs. Different p value thresholds were tested to find the p value with the minimum FDR for SC and SM.Model (3) was run for each individual SNP at a time because the aim was to map genomic regions associated with inbreeding depression. To consider multiple testing problem, a method proposed by Bolormaa et al.  was usedThe descriptive statistics and estimated heritability and repeatability are shown in Table FSNP, FROH, FROH1−2, FROH2−4, FROH4−8, FROH8−16, and FROH>16, and Fig. FROH was 0.09 ± 0.02. In a previous study by Bjelland et al. [FROH of 0.04 ± 0.02 was reported for US Holstein cows, which was less than the amount we obtained for Holstein bulls of China. The average ± SD of FSNP -0.006 ± 0.03 in our study was less than FROH  increases. Therefore, the difference in the correlation reported by previous studies in pig and sheep breeds [Figure d et al.  FROH of  in pigs  and shee in pigs , respectr et al.  suggestep breeds , 33 compFROH>16 Mb) had the highest correlation with FSNP (0.75) and FROH (0.81), while more distant inbreeding  had the lowest correlation (~ 0.09) with the same measures . However, both FROH and FSNP had significant effects on SM (p < 0.01). A 1% increase in FROH, reduces SM by 0.28% of the mean population. Inbreeding depression for SM has been documented in livestock species including cattle [In this study, we hypothesized that inbreeding would negatively affect semen quality traits. Inbreeding depression estimates of the semen traits are shown in Table g cattle , sheep [g cattle , as wellg cattle , 36, endg cattle  and zebrg cattle . Dorado g cattle  studyingg cattle , 40. In g cattle  reportedg cattle . TherefoFROH on SC was significantly different from zero (p < 0.01), and a 1% increase in FROH decreased SC about 0.42% of the population mean , significant effect of inbreeding depression on testicular SC was reported [Our result show that the effect of reported . To the reported . A similreported . TherefoFROH>16 Mb was significant on SM (p < 0.01) and the effect of FROH8-16 Mb was significant on SC (p < 0.05). This level of inbreeding traces back to common ancestors approximately 3 and 6 to 3 generations ago, respectively. Our findings regarding the importance of more recent inbreeding on semen traits were in line with previous studies on semen and other traits. Antonios et al. [We also hypothesized that recent inbreeding would have a more detrimental effect on SM and SC. All the ROH length classes were simultaneously fitted in the model to account for the correlations between classes . No signs et al.  reporteds et al.  studied s et al. . Studyins et al.  observeds et al.  showed as et al.  reporteds et al.  and our 10p values for the association between SNP in ROH state and inbreeding depression for semen quality traits across chromosomes are shown in Fig. To link inbreeding depression of semen traits to genomic regions, a statistical test was performed for every autosomal SNP, considering the potential additive effect and ROH state of the SNP in the model. The distributions of -logp value thresholds. As shown in Fig. p < 0.00001; FDR < 2%) with the SC trait  is a member of the HMGB protein family, which includes the ubiquitous HMGB1 and the embryo-specific HMGB3. It has been reported in mice that adult males lacking HMGB2 (Hmgb2−/−) have reduced fertility, that correlates with Sertoli and germ cell degeneration in seminiferous tubules and immotile spermatozoa [ADAM29 is highly expressed in the testis and involved in human spermatogenesis [ADAM29, GLRA3 and HPGD were reported to be associated with early pregnancy in Nelore heifers [Identification of genomic regions contributing to inbreeding depression of female fertility and semen quality traits has already been reported in cattle , 15. Yety cattle . Parker y cattle  did a germatozoa . The proogenesis . Gene On heifers . The funp < 0.0001; FDR < 8%). There were 74 genes within the 500 Kb upstream and downstream of the significant SNPs such as PRMT6, SCAPER, EDC3 and LIN28B  on BTA 3 at the region 36.73–36.94 is highly expressed in testis and influences cell migration and apoptosis of germ cells (http://www.genecards.org/) [PRMT6 include methyltransferase activity and protein methyltransferase activity, and the diseases associated with PRMT6 include male infertility in humans [PRMT6 was previously reported to be associated with sperm concentration in Holstein bulls (P = 6.5E-6) [SCAPER (S Phase Cyclin A-Associated Protein in The Endoplasmic Reticulum) gene in this region is highly expressed in testis and involved in spermatogenesis  (http://www.genecards.org/). SCAPER, originally identified as a cell cycle regulator, was also suggested to be a ciliary protein. It has been reported to be associated with sterility in male and reduced fertility in female mice [SCAPER is a crucial component in both the male and female reproductive systems [SCAPER expression in spermatogonia are azoospermic due to early defects in spermatogenesis [We also identified inbreeding depression signals with significant adverse effects on SM trait Fig.  Table S2ds.org/) . Gene Onn humans –54. PRMT 6.5E-6) . In the ale mice . The sam systems . In humaogenesis .EDEC3 is overexpressed in testis and may play a role in spermiogenesis and oogenesis (https://www.genecards.org/). LIN28B is another gene that is overexpressed in testis (https://www.genecards.org/) and is reported to be associated with globozoospermia, which is a rare but severe cause of male infertility in mammals [CLDN20, AFGL1, TIAM2, RCN2, ODF3L1, and ARID3B are other genes that overexpressed in testis (https://www.genecards.org/), suggesting a potential role for these genes in relation to the SM trait. However, FDR for SM trait was calculated 7.5%, so these candidate genes need to be validated by using a different dataset to confirm their role in bull fertility.There were also other genes associated with inbreeding depression of SC in our study which could be associated with male fertility in bulls.  mammals , 57. MorWe aimed to address: (i) which semen traits are more susceptible to inbreeding depression (ii) Whether the detected inbreeding depression is due to recent or ancient inbreeding (iii) Where the associated genes with these traits are in the genome. In summary, we found sperm concentration and sperm motility were unfavourably affected by inbreeding. The effect of recent inbreeding on semen traits was more harmful, probably because of genetic purging effect reducing its impact in older inbreeding. Our results provide novel insights into where genes causing inbreeding depression of semen traits are in the genome. Breeding companies may wish to consider avoiding homozygosity in these regions for potential artificial insemination sires. Validation of these regions and additional research into candidate genes located in these regions using larger sample sizes and denser markers may shed light on the molecular mechanisms and the causal variants underlying inbreeding depression of semen traits in dairy cattle.Additional file 1:Table S1. Genome-wide association with inbreeding depression of sperm concentration.Additional file 2:Table S2. Genome-wide association with inbreeding depression of sperm motility.Additional file 3:Table S3. Genes associated with inbreeding depression of sperm motility trait.


Genomic evaluations in pigs could benefit from using multi-line data along with whole-genome sequencing (WGS) if the data are large enough to represent the variability across populations. The objective of this study was to investigate strategies to combine large-scale data from different terminal pig lines in a multi-line genomic evaluation (MLE) through single-step GBLUP (ssGBLUP) models while including variants preselected from whole-genome sequence (WGS) data. We investigated single-line and multi-line evaluations for five traits recorded in three terminal lines. The number of sequenced animals in each line ranged from 731 to 1,865, with 60k to 104k imputed to WGS. Unknown parent groups (UPG) and metafounders (MF) were explored to account for genetic differences among the lines and improve the compatibility between pedigree and genomic relationships in the MLE. Sequence variants were preselected based on multi-line genome-wide association studies (GWAS) or linkage disequilibrium (LD) pruning. These preselected variant sets were used for ssGBLUP predictions without and with weights from BayesR, and the performances were compared to that of a commercial porcine single-nucleotide polymorphisms (SNP) chip. Using UPG and MF in MLE showed small to no gain in prediction accuracy (up to 0.02), depending on the lines and traits, compared to the single-line genomic evaluation (SLE). Likewise, adding selected variants from the GWAS to the commercial SNP chip resulted in a maximum increase of 0.02 in the prediction accuracy, only for average daily feed intake in the most numerous lines. In addition, no benefits were observed when using preselected sequence variants in multi-line genomic predictions. Weights from BayesR did not help improve the performance of ssGBLUP. This study revealed limited benefits of using preselected whole-genome sequence variants for multi-line genomic predictions, even when tens of thousands of animals had imputed sequence data. Correctly accounting for line differences with UPG or MF in MLE is essential to obtain predictions similar to SLE; however, the only observed benefit of an MLE is to have comparable predictions across lines. Further investigation into the amount of data and novel methods to preselect whole-genome causative variants in combined populations would be of significant interest. Genomic evaluations have been successfully implemented in pig breeding programs to increase the accuracy of predicting genomic EBV (GEBV) and better identify the best animals to be parents of the next generation. However, most evaluations are single-line, or at most, include phenotypes from crossbred progeny to help better evaluate purebred parents. Combining multiple lines could be a possible strategy to allow the comparison of animals across lines and the identification of the best gene combinations . Additio(H) ((A) and genomic (G) relationship matrices (A and G are missing pedigrees and heterogeneous base populations (G would correspond to the ones from the base population in the pedigree (Single-step genomic BLUP (ssGBLUP) has been commonly used for genomic evaluation in pigs . The fun(H) . For thimatrices . Two sigulations . In theopedigree ; howeverpedigree . The incpedigree . Legarrapedigree  and singpedigree  evaluatiNe) than dairy and beef cattle, resulting in smaller numbers of independent chromosome segments (Me) (Another factor affecting the performance of genomic predictions in MLE is the inconsistent linkage disequilibrium (LD) structure between single-nucleotide polymorphisms (SNP) and quantitative trait loci (QTL) across lines. Pig populations have a smaller effective size ( (Me) . Therefo (Me)  showed tIn the current study, we aimed to 1) investigate strategies to combine different lines in a multi-line evaluation through the use of unknown parent groups or metafounders and 2) evaluate the impact of using jointly preselected SNP from WGS in multi-line evaluations under ssGBLUP without and with weights from BayesR .All datasets were provided by Pig Improvement Company . We investigated the average daily feed intake (ADFI), average daily gain (ADG), backfat thickness (BF), ADG recorded in crossbred animals (ADGX), and BF recorded in crossbred animals (BFX) in three terminal pig lines named TL1, TL2, and TL3. These three terminal lines were chosen from the seven lines analyzed in a previous study by The WGS data used in this study were generated by Before the SNP preselection, all animals with (imputed) WGS data were split into two non-overlapping datasets: training and test by line. Test sets were generated by extracting genotyped individuals in the litters from the last generation of the pedigree to resemble the selection candidate evaluation performed by pig breeding companies. Only litters with a minimum of five full-sibs were considered as a way to balance the representation of different families in the test sets, and all remaining individuals were considered training sets. The training sets were additionally defined as those individuals with a pedigree relationship coefficient with any individual in the test set that was lower than 0.5 to reduce the relationship between the training and test sets. Therefore, we can avoid the initially high accuracy generated by the close relationship between the two sets. For multi-line scenarios, the training set was formed by merging the training sets that had been defined for each line. Training datasets were included in the discovery sets for multi-line GWAS in this study. Although using the same sets for training and GWAS can reduce accuracy and increase the bias of genomic predictions , Ros-Frep-value (not necessarily below the significance threshold) in each consecutive non-overlapping 55-kb window along the genome, based on multi-line GWAS analyses, so that a similar number of variants than in Chip was retained. TopSign only included significant variants (p ≤ 10−6), where the significance threshold is based on a significance level of 0.05, accounting for multiple testing through the Bonferroni correction, assuming that markers from Chip were independent, from the multi-line GWAS in a way that when a 55-kb window contained more than one significant variant, only that with the lowest p-value was selected. ChipPlusSign combined TopSign and Chip because sometimes the number of significant variants was too small, and empirical results have shown that augmenting marker arrays with WGS variants can be a successful strategy to improve prediction accuracy  to (3) were trait-specific, and the preselected variants for the traits included in each of the ADFI and GROWTH models were combined for our analyses. Thus, in the ADFI model, preselected variants for ADFI and ADG were combined and used for the genomic predictions. Likewise, preselected variants from ADG and BF were merged and used for genomic predictions in the GROWTH model; variants were not selected for ADGX and BFX because crossbred animals were not sequenced. For a fair comparison to the commercial SNP chip data , 206,634 animals were extracted from each preselected variant set. Afterward, quality control was performed with the same aforementioned criteria, and those individuals with parent–progeny conflicts were also removed. Five different preselected SNP panels were created from WGS for genomic predictions. Two of them were as described in accuracy . The LDTy is the vector of phenotypes; b is a vector of fixed effects; c is a vector of random litter effects; u is a vector of random additive genetic effects; and e is a vector of residual effects. Matrix X is an incidence matrix relating phenotypes in vector y to the fixed effects , and carcass weight as a covariate only for BF and BFX, respectively) in vector b; matrix W is an incidence matrix for random litter effects in vector c; and matrix Z is an incidence matrix for random additive genetic effects in vector u.To compare the performance of genomic prediction using single-line and multi-line evaluation, Chip data were tested for SLE. Multi-trait linear mixed models were used to perform genomic predictions with two and four traits for the ADFI and GROWTH models, respectively. Only a four-trait GROWTH model  of TL1, TL2, and TL3 is described here:Q is an incidence matrix relating animals in vector u to the UPG in vector g. All other matrices and vectors were already described.Multi-trait linear mixed models were used to carry out genomic predictions with two and four traits for the ADFI and GROWTH models, respectively. Two UPG methods or MF were used to model the heterogeneous base across the lines. Herein, only the four-trait GROWTH model  is described:G was created with method 1 of I in ssGBLUP and I in WssGBLUP); and G was blended with 5% of The genomic predictions were performed with ssGBLUP without and with weights from BayesR (WssGBLUP) using the BLUPF90 family of programs . In ssGB follows :H−1=A−1A, A22, and G  . H−1 wapproach  through approach .For UPG and MF models, five groups of base animals were defined based on the lines of origin. The first three were assigned to TL1, TL2, and TL3; one was assigned to another terminal line (TL4), and the other represented a crossbred line (CL). TL4 is another important terminal line, and it contributed to generating the commercial crossbred data that were used in this study, so we defined it as one group of base animals. Due to an issue of estimating G, the algorithm for proven and young (APY) (G constructed with Chip data (To efficiently compute ng (APY)  was applhip data . TherefoD was equal to the number of SNP. More details about using BayesR weights in WssGBLUP are described in In WssGBLUP, BayesR  was usedr. On the other hand, the whole data, with no phenotype truncation, will be represented by the subscript w. Under the LR method, the accuracy of GEBV was calculated as 1) from 1, where b1 was obtained from the regression of The LR validation method  was usedPrincipal component analysis (PCA) was performed to investigate the population structure of the three terminal lines . Chip daThe relationships within and between MF  of GEBV whole on GEBV reduced when using SLE and MLE with UPG1, UPG2, and MF for five traits of each line using Chip data are given in 1 greater than 1 indicate deflation of reduced GEBV, and those smaller than 1 indicate inflation. In general, negligible deflation (1.01) and slight inflation (0.92) of GEBV were observed . A similar pattern was identified for ADFI in TL2, which showed the greatest increase of GEBV with SLE (0.93). For the other four traits, no considerable differences were found between the models (0.96–1.01). TL3 reported the most consistent results in each trait. For ADFI, all four models showed b1 values equal to 0.96.The regression coefficients  than Chip for some traits . Likewise, Top40k (0.96–1.00), TopSign (0.99–1.01), and ChipPlusSign (0.95–1.00) displayed similar results to Chip (0.97–1.00) in TL2, but a large increase of GEBV with LDTags (0.78–0.90) and AllComb (0.88–0.96). Similar patterns were observed in TL3. Remarkably, TopSign showed less increase of GEBV than Chip for all five traits, although the differences are small (0.02). The results of dispersion for EBV with traditional BLUP using SLE and MLE are given in To compare the same scenarios with preselected genotype panels from WGS as was performed for prediction accuracy and bias, only the results with UPG1 are given in Top40k and ChipPlusSign were used for WssGBLUP with BayesR weighting because these two panels showed the best performance among the preselected genotype panels. Additionally, BayesR weights were considered in ssGBLUP to make the results of this study more comparable to those of In this study, we investigated the impact of using UPG and MF when combining different pig populations in multi-line evaluations. Additionally, we explored the potential benefits of using preselected SNP from WGS in these joint evaluations when SNP received equal or different weights in ssGBLUP. The novel aspect of this study is the amount of sequence data used for multi-line genomic predictions, that is, over 200k pigs. This study provided an insight into how accounting for different genetic bases in three large pig populations could affect the performance of joint genomic evaluations. It also proved that forecasting regarding the usefulness of sequence data for cross-breed predictions does not hold , at leasCombining populations with different genetic backgrounds in genomic evaluations has been actively investigated in cattle , where tIn the MF theory, a matrix of relationships within and across metafounders  in multi-line evaluations could help adjust the genetic base for each line . The major difference between UPG1 and UPG2 was that groups were assigned to ach line . Howeverach line . Among tach line . More beIn addition to the size of the reference population and the relationships between the reference and test animals , anotherNe, such as pigs and chickens with small Me and a strong LD extent, which makes the identification of causative variants more difficult  only in TL3. On average, ChipPlusSign showed the greatest prediction accuracy among all preselected scenarios but was still smaller than Chip, although the difference was slight. For the multi-line GWAS, we used seven lines  as described in ifficult . Severalifficult ; howeverNe. They showed that identifying significant quantitative trait nucleotides (QTN) is more difficult in populations with smaller than larger Ne because of the strong LD extent across the genome in the former. Accordingly, improvements in the accuracy of genomic predictions using those selected QTN combined with a 50-k SNP chip in the population with smaller Ne were very limited (∼1.98%) compared to the population that had larger Ne (∼9.01%). Therefore, although multi-line GWAS could benefit across lines, the benefits would be still limited when it comes to genomic prediction.Adding significant SNP to the regular chip data or using only them could potentially improve prediction accuracies only if those are real causative variants with known effects, positions, and variances explained . To accuIn the current study, we used WssGBLUP with weights computed from BayesR. We did not use the standard weights proposed by This is the first study in large-scale pig lines using MLE with WGS selected variants using the WssGBLUP approach. In practice, the benefits of using WssGBLUP seemed very limited, especially with large datasets and many genotyped animals . Our MLEThis study revealed limited benefits of using preselected whole-genome sequence variants for multi-line genomic predictions, even when hundreds of thousands of animals had imputed sequence data. Correctly accounting for line differences with unknown parent groups or metafounders in multi-line evaluations is essential to obtain predictions similar to single-line evaluations; however, the only observed benefit of a multi-line evaluation is to have comparable predictions across lines. Further investigation into the amount of data and novel methods to preselect whole-genome causative variants in combined populations would be of significant interest.


Zhou et al. (2022) (PMID: 35766061) has further extended it under the longitudinal response by proposing a quadratic inference function-based penalization method in gene–environment interaction studies. This study introduces “springer,” an R package implementing the bi-level variable selection within the QIF framework developed in Zhou et al. (2022). In addition, R package “springer” has also implemented the generalized estimating equation-based sparse group penalization method. Alternative methods focusing only on the group level or individual level have also been provided by the package. In this study, we have systematically introduced the longitudinal penalization methods implemented in the “springer” package. We demonstrate the usage of the core and supporting functions, which is followed by the numerical examples and discussions. R package “springer” is available at  In gene–environment interaction studies, a central task is to detect important G×E interactions that are beyond main G and E effects. Although the main environmental factors are usually preselected and of low dimensionality, in the presence of a large number of G factors, conducting G×E analysis can be performed in the variable selection framework. Recently, springer.springer, which has implemented not only the proposed and alternative regularized QIF methods from via bypassing the nuisance parameters, leading to consistent and optimal estimation of regression coefficients even when the working correlation is misspecified  is repeatedly measured over ti time points, which naturally results in ti repeated measurements that are correlated for the same subject and are assumed to be independent with the measurements taken from other subjects. Then, Yij denotes the phenotype measured for the ith subject at time point j (1⩽j⩽ti). p-dimensional vector of genetic factors and the q-dimensional vector of environmental factors, respectively. The bi-level G×E model associates the genetic and environmental main effects and their interactions with the repeatedly measured phenotypic response as follows:αn0 is the intercept, and αnh, γnk, and unhk denote the regression coefficients of environmental and genetic main effects and their interactions, correspondingly. We also define Zijk is a (q + 1)-dimensional vector representing the main and interaction effects with respect to the kth genetic factor. For 1⩽j⩽ti, the random error ϵij has mean zero and a finite variance. For convenience, the random error ϵi is assumed to be multivariate normal as i is the covariance matrix corresponding to the ith subject. From now on, we let ti = t. Combined, we can write ηn is p + pq. Then, model (1) can be equivalently expressed asq + p + pq)-dimensional vectors In a typical longitudinal setting with springer  = δ(μij), where δ(μij) is a known function of the mean μij. Then, the score equation for the longitudinal G×E model is defined asVi is defined as ith subject, the diagonal matrix Ai is defined as Ai = diag{Var(Yi1), …, Var(Yit)}, and the “working” correlation matrix Ri(ν) depends on a finite dimensional parameter vector ν, characterizing the within-subject association. We have The Ri(ν) from the true underlying correlation among intra-subject measurements. ν is consistently estimated, the GEE estimator is consistent even if the correlation structure is not correctly specified. However, there is a cost under such misspecification, that is, the GEE estimator is no longer efficient, and ν cannot be consistently estimated.The term “working” correlation in GEE is adopted to distinguish quadratic inference function overcomes the disadvantage of GEE by avoiding the direct estimation of ν (R(ν) can be calculated by a linear combination of basis matrices within the QIF framework. Specifically, B1 is an identity matrix and B2, … , Bm are symmetric basis matrices with unknown coefficients c1, … cm. The specifications of these basis matrices are dependent on the types of working correlation , for the bi-level G×E model asϕi (βn) is c1, … cm in Eq. Accordingly, for the springer (∞) with the tuning parameter λ and regularization parameter γ  further selects the main and interaction effects that are associated with the phenotype.R package springer  can perfameter γ . The groOur choice of the baseline penalty function is the MCP, and the corresponding first derivative function of MCP is defined as The penalized QIF in (4) is the extension of bi-level variable selection to longitudinal studies, which conducts selections of important groups and individual members within the group simultaneously. It is worth noting that the penalized GEE model proposed by glmnet has included LASSO and its extensions, such as the ridge penalty and elastic net  entries on the main diagonal of the matrix are zero accordingly. With fixed tuning parameters, 1 norm of Optimization of the penalized QIF in (4) demands the Newton–Raphson algorithm that can update ^ngβ^ng,where Pvergence .λ1 and λ2, to determine the amount of shrinkage on the group and individual level, correspondingly. An additional regularization parameter γ further balances the unbiasedness and convexity of MCP. The performance of the proposed regularized QIF is insensitive under different choices of γ  can be searched over the two-dimensional grid through K-fold cross-validation. We first split the dataset into K non-overlapping portions of roughly the same size and held out the kth  fold as the testing dataset. The rest of the data are used as training data to fit a regularized QIF by giving a specific pair of . nk and nk− denote the index sets of subjects as training and testing samples, respectively. We can compute the prediction error on testing data asnk−| is the size of testing data, and K fold for k = 1, 2. ., K, yielding the following cross-validation error:λ1, λ2) can be retrieved across the entire two-dimensional grid. The optimal pair of tunings is corresponding to the smallest CV value. Details of the algorithm are given as follows:λ1, λ2) is provided with an appropriate range.1 The two-dimensional grid of ,2 Under the fixed ((a) g + 1)th iteration, (b) at the ((c) (d) The cross-validation error is calculated using Eq. λ1, λ2) until convergence.3 Step 2 is repeated for each pair of  is found under the smallest cross-validation error. The corresponding 4 The optimal  incorporates two tuning parameters, ces of γ . The besλ1, λ2) can be assessed on the testing data directly. On the contrary, in cross-validation, the prediction error can only be obtained after cycling through all the K folds as shown by Equation The validation approach is a popular alternative of tuning selection to bypass the computational intensity of cross-validation. When the data-generating model is available, the independent testing data with much larger size can be readily generated. Then, the prediction performance of the fitted sparse group PQIF model under .springer  to be used for regularized estimation. One of the three working correlations from AR-1, exchangeable, and independence can be called through the input argument corr. For example, corr = “exchangeable,” corr = “AR-1,” and corr = “independence” denote exchangeable, AR-1, and independent correlation, respectively. In addition to the bi-level structure, this package has also included sparsity structures on the group and individual level, respectively. To use the bi-level PQIF under the exchangeable working correlation proposed by func = ”QIF,” structure = ”bi-level,” and corr = ”exchangeable” at the same time. It is worthwhile noting that the bi-level selection requires two tuning parameters to impose sparsity. When structure = ”group” or structure = ”individual,” only one of the two tuning parameters lam1 and lam2 is needed.The character string argument springer proceed in an iterative manner. The input argument maxits provides the maximum number of iterations determined by the users. We can supply the small positive fraction ϵ that is used to ensure the stability of the algorithm through argument tol.The Newton–Raphson algorithms implemented in the package springer, function cv.springer performs cross-validation based on the regularized coefficients provided by springer. The R code isIn package structure, maxits = 30,tol = 0.001).cv.springer  imposed for variable selection. Both the group- and bi-level penalizations involve the empirical norm k is not unique. For example, Σk can be chosen as an identity matrix, and then 2 norm. While the alternatives might be equally applicable, the default choice of Σk in package Springer is in the form discussed in Package reformat to convert the wide format to long format with dimension 500 by 1. Similarly, the design matrix under sample size 100 and 50 main and interaction effects has a dimensionality of 100 by 50, if they do not vary across time. Then, reformat will return a 500 by 51 wide format matrix including the column of intercept. An “id” column will also be generated by reformat to show the time points corresponding to 500 columns. Moreover, a simulated dataset, dat, is provided to demonstrate the penalized selection in the proposed longitudinal study. We describe more details in the next section.It is assumed that repeated measurements on the response are given in the wide format with the dimension of 100 by 5, where 100 is the sample size and 5 is the number of time points, then we can use function Springer based on simulated datasets. Although model (1) is general in the sense that both the response and predictors are repeatedly measured, it can be reduced to the case where the predictors, consisting of the clinical covariates and environmental and genetic factors, are cross-sectional under the longitudinal response. Model (1) is flexible in which the predictors can have a mixture of cross-sectional and longitudinal measurements. For instance, the repeated measurements are only taken on E factors and not on clinical or G factors.In this section, we demonstrate the fit of bi-level selection using package springer only accounts for such a case. It is worth noting that technically it is not difficult to extend the package to repeatedly measured predictors because the only difference lies in using time-specific measurements rather than repeating the cross-sectional measurements across all the time points in the estimation procedure. We will discuss potential extensions of the package at the end of this section. In the following simulated example, the longitudinal responses are generated together with cross-sectional predictors. The data-generating function is provided as follows:The motivating dataset for the sparse group variable selection developed in function Data <- {matrix ,n,k)y = sig = matrix for (i in 1: p) {for (j in 1: p) { sig  = 0.8^abs (i-j) }}# Generate genetic factorsrep ,sig)g = mvrnorm sig0 = for (i in 1: q) {for (j in 1: q) { sig0  = 0.8^abs (i-j) }}# Generate environmental factorsrep ,sig0)e = mvrnorm E0 = E0 = E0+1cbind e = e.out = ecbind [1]),e)e1 = for (i in 1:p) { e = cbind  }scale(e)x = ll = 0.3ul = 0.5coef = runif mat = x (6*for (u in 1:k){ y  = 0.5 + rowSums (coef*mat) }#Exchangable correlation for repeated measurementsmatrix sig1 = diag (sig1) = 1for (i in 1: k) {for (j in 1: k) { if (j != i){sig1  = 0.8} } }rep ,sig1)error = mvrnorm )dat = return (dat)}n, p, and q represent the sample size, dimension of the genetic factors, and environmental factors, respectively. The number of repeated measurements is k. Now, we simulate a dataset with 400 subjects, 100 G factors, and 5 E factors. The number of repeated measurements is set to 5. The correlation coefficient ρ of the compound symmetry working correlation assumed for longitudinal measurements is 0.8. In the data-generating function, coef represents the vector of non-zero coefficients, and mat is the part of design matrix corresponding to the main and interaction effects associated with non-zero coefficients. With  = , coef is a vector of length 30, and mat is a 400-by-30 matrix. The R code coef*mat denotes element-wise multiplication by multiplying the non-zero coefficient to the corresponding main or interaction effects. Therefore, rowSums(coef*mat) returns a 400-by-1 vector. The code “0.5 + rowSums(coef*mat)” stand for the combined effects from those important main and interaction effects, and the intercept, with 0.5 being the coefficient multiplied to the intercept. We listed the R codes and output in the following section:In the aforementioned codes, library (MASS)library (glmnet)library (springer)set.seed (123)n.train = n = 400p = 100; k = 5; q = 5dat.train = Datay.train = dat.train$yx.train = dat.train$xe.train = dat.train$eg.train = dat.train$g> dim(y.train)[1] 400 5> dim(x.train)[1] 400 605> dim(e.train)[1] 400 5> dim(g.train)[1] 400 100dat.train$coef saves the non-zero coefficients used in the data-generating model. By setting the seed, we can reproduce the data generated through calling the Data. A total of 100 genetic factors and 5 environmental factors lead to a total of 605 main and interaction effects, excluding the intercept. We first obtain the initial value of the coefficient vector In addition, the R codes cbind ),x.train)x.train1 = data.matrix (x.train1)x.train1 = lasso.cv = cv.glmnet min/2alpha = lasso.cv$lambda.lasso.fit = glmnet as.matrix )[-1])cv.springer to calculate cross-validation errors corresponding to the pair of tuning parameters (lambda1 and lambda2). The number of fold is 5 by setting nfolds to 5 in the following codes. Then, a penalized bi-level QIF model with an independence correlation has been fitted to the simulated data with the optimal tunings. The fitted regression coefficients are saved in fit.beta.With the initial value obtained previously, we call function seq lambda1 = seq lambda2 = tunning = cv.springer lam1 = tunning$lam1lam2 = tunning$lam2> lam1[1] 0.0625> lam2[1] 1> tunning$CV   14.873142 15.37916 16.02844 12.282850 13.23239 13.81465 9.663655 10.62635 11.96531 10.133435 11.00219 12.25365 11.237012 11.79566 13.17813beta = springer fit.beta with the true coefficient vector, which is used to simulate the response variable in Data. Since the codes dat.train$coef only report the true non-zero coefficient, the resulting vector has a length much less than fit.beta, which includes zero coefficient. Therefore, we first retrieve locations of non-zero effects in the coefficient vector used to generate the longitudinal response. In the following codes, tp, tp.main, and tp.interaction represent the locations for all the non-zero effects, that is, the column number of the corresponding effects in the design matrix. Although the coefficients are randomly generated from uniform distributions, the locations of the non-zero effects are fixed. In total, there are 30 non-zero effects, consisting of 5 environmental factors, 7 genetic factors, and 18 gene–environment interactions.To assess the model’s performance, we will compare the fitted coefficient vector ## non-zero effects without interceptc, (q+2), (q+6), (q+4), (2*q+2), (2*q+3), (2*q+7), (2*q+5),tp = q+3), (3*q+4), (3*q+8), (3*q+6), (4*q+4), (4*q+5), (4*q+9), (4*q+7),(3*q+5), (5*q+6), (5*q+10), (5*q+8), (6*q+6), (6*q+7), (6*q+11),(5*q+9), (7*q+7))+1(6*## non-zero main effectsc((q+2), (2*q+3), (3*q+4), (4*q+5), (5*q+6), (6*q+7), (7*q+8))tp.main = ## non-zero interaction effectsc((q+2), (q+6), (q+4), (2*q+3), (2*q+7), (2*q+5),tp.interaction = q+4), (3*q+8), (3*q+6), (4*q+5), (4*q+9), (4*q+7), (5*q+6), (5*q+10),(3*q+8), (6*q+7), (6*q+11), (6*q+9))+1 and 5 environmental factors (q = 5), resulting in a coefficient vector of length 606, including the intercept. To observe the estimation accuracy on a finer scale, we further dissect βn into the component corresponding to tp and calculate the mean square error with respect to the counterpart from fit.beta, denoted as MSE. The mean square error is computed based on the rest of fit.beta, and βn is defined as NMSE. The R codes and output are listed as follows:We run the codes in R console to evaluate the accuracy in parameter estimation. The precision in estimating the regression coefficients has been assessed based on matrix ,1)coeff = rep )coeff.train = coef[-1]coeff.train [tp] = dat.train$mean ((coeff-coeff.train)^2)TMSE = mean ((coeff [tp]-coeff.train [tp])^2)MSE = mean ((coeff [-tp]-coeff.train [-tp])^2)NMSE = > TMSE[1] 0.003455488> MSE[1] 0.06563788> NMSE[1] 0.0002168221dat.train$coef only consists of the non-zero coefficients used to generate longitudinal responses in the data-generating model; therefore, its dimension is not the same as fit.beta as the estimated regression coefficient vector is sparse and includes zero coefficient, thus having a much larger dimension. In regularized variable selection, the non-zero coefficients from fit.beta will not be identical to those in dat.train$coef due to the shrinkage estimation in order to achieve variable selection. The aforementioned output shows the estimation errors in terms of TMSE, MSE, and NMSE, respectively. The NMSE is much smaller than the MSE since it computes the MSE with respect to zero coefficients.The fit.beta and the true coefficient vector used in the data-generating model, we can report the total number of true- and false-positive effects, such as TP and FP. The identification results have also been summarized for the main genetic effects (TP1 and FP1) and G×E interactions (TP2 and FP2). The locations of important effects saved in tp obtained from the chunk of R codes previously also include the environmental main effects that are not subject to selection. When calculating the number of true and false positives in the next section, we only count the effects that are under selection, corresponding to the 7 G factors and 18 G×E interactions. The output is provided in the following section.In addition to evaluating the accuracy in parameter estimation, we also examine the performance in identification in terms of number of true- and false-positive effects. Specifically, by comparing the locations of the non-zero components in abs (coeff) < 0.1] = 0coeff [q)] = 0coeff [1: (1 + which (coeff != 0)ids = length )TP = is.na )]res = ids [length (res)FP = rep )coeff1 = q)] = coeff [1: (1 + q)]coeff1 [1: (1 + q+2):length (coeff)) {for (i in (if ( i%%(q+1)==1) coeff1 [i]= coeff[i]}which (coeff1 != 0)ids1 = length )TP1 = is.na )]res1 = ids1 [length (res1)FP1 = coeff2 = coeffcoeff2 [1: (1 + q)] = 0for (i in (q+2):length (coeff)) {if ( i%%(q+1)==1) coeff2[i] = 0}which (coeff2 != 0)ids2 = length )TP2 = is.na )]res2 = ids2 [length (res2)FP2 = > TP[1] 21> FP[1] 3> TP1[1] 6> FP1[1] 0> TP2[1] 15> FP2[1] 3Results on true and false positives indicate that six out of the seven important main effects have been identified, and 15 out of the 18 interactions used in the data-generating model have been detected. The number of identified false-positive effects is three.springer in the near future.In addition to extensive simulation studies that demonstrate the merit of the proposed sparse group variable selection in longitudinal studies, springer (version 0.1.7) has implemented three working correlation matrices, independence, AR-1, and exchangeable, for individual-, group-, and bi-level variable selection under continuous longitudinal responses in both the GEE and QIF frameworks. The future improvement includes incorporating other working correlations, such as the unstructured working correlation. A question worth exploring is the computational feasibility of unstructured working correlation under QIF as the large number of covariance parameters will potentially lead to much more complicated extended score vectors, incurring prohibitively heavy computational cost for high-dimensional data. We will also consider extensions to discrete responses such as binary, count, and multinomial responses, and longitudinally measured clinical, environmental, and genetic factors, especially after these data are available.The current version of package springer to analyze the high-dimensional longitudinal data from the Childhood Asthma Management Program > [1] 438 12# environmental factors dim(e)> [1] 438 3# genetic factos (SNP)dim(X)> [1] 438 203cv.springer. One can start the process by defining a grid interval for each tuning parameter. We applied the cv.springer function with estimating function type func = ”QIF” and working correlation matrix type corr = ”exchangeable” as follows:Both the environmental and genetic factors are cross-sectional. For example, as shown previously, each of the three E factors is a 438-by-1-column vector, forming a 438-by-3 matrix. We obtained the optimal tuning parameters using function library (springer)> #define input arguments> seq > lambda1 = seq > lambda2 = #run cross-validation> > tunning = cv.springer + #print the results> print (tuning)> $lam1[1] 0.5$lam2[1] 3$CV     0.2827513 0.2838438 0.2846629 0.2855799 0.2865723 0.2858653 0.2867847 0.2877162 0.2885925 0.2894861 0.2884425 0.2897974 0.2906588 0.2916546 0.2925146 0.2919309 0.2927759 0.2936686 0.2945191 0.2954699 0.2948042 0.2954983 0.2962844 0.2971886 0.2979241springer function to the dataset using the optimal tuning parameters as follows:The optimal tuning parameters within the range have been selected as 0.5 and 3 for lambda1 and lambda2, respectively. We have then applied the > #fit the bi-level selection modelbeta = springer springer function returns the estimated coefficients for the intercept, environmental factors, genetic factors, and G×E interactions. We organized the output to show the identified genetic main effects and G×E interactions in The Before the formulation of the bi-level (or sparse group) selection in high-dimensional statistics , the relspringer cannot be applied directly on the ultra-high-dimensional data (g in function springer to one genetic factor at a time, which will return the regression coefficients for all the clinical and environmental factors and main G and G×E interactions with respect to that G factor. The magnitude of the coefficients corresponding to the effects subject to the selection will be used as the measure for ranking and selecting important effects.Package nal data , which inal data . In factnal data . It is mspringer can facilitate further understanding of robustness in bi-level selection models.Robust penalization methods have drawn increasing attention in recent years . In high


Selection is one of the essential skills whereby breeders reduce the population size and increase the chance of success. Various selection methods with special applications have been developed. Superior genotypes are assessed according to interesting traits, including univariate, multivariate, phenotypic, genotypic, etc.Mathematical calculation of the traits' importance based on the genetic makeup of investigated population (average degree of dominance/additive involved in the action of genes) and arbitrary genetic parameters is functional. In this paper, a general model for multivariate selection has been presented whereby the selection can be made for (a) more than one interesting trait, (b) the trait(s) with complex inheritance, (c) finding superior genotypes from among a large-scale population, (d) finding superior genotypes in segregating generations and (f) finding tolerant genotypes to stresses. This model is developed based on biometric concepts in four steps. MATLAB script is provided for the model, and users can easily apply that to identify the most suitable genotypes after data collection according to the breeding purposes.The main features of this model are simplicity, precision, repeatability, and speed . All the steps and the analysis of the results are explained step by step in a case study.The online version contains supplementary material available at 10.1186/s13007-023-00992-5. Assessing genetic diversity can help the breeders to obtain important information from the population and to determine the breeding strategy. “Selection” is one of the most important skills in plant breeding which is affected by genetic diversity. In other words, maximizing the response to selection depends on optimizing the use of available genetic diversity . BreedinOne of the reliable parameters to evaluate the variables' effect on each other is to calculate genetic correlation by the statistical designs . This coThe stability of independent variables is an important parameter for evaluation, especially in stress conditions. The lower the effect of stress on a variable, the more stable the variable. The stability of traits and/or genotypes in different environments has been investigated by different researchers , 11. DetIn recent years some scientists used mathematical concepts such as artificial neural network, genetic algorithms, and the Modified Analytical Hierarchy Process (MAHP) to make the best decision , 15, 16.In this paper, a multivariate selection model is presented whereby the traits' importance is calculated mathematically based on the arbitrary genetic parameters, the special purposes of the breeding program, and the particular competitive genotypes. The basis of the model is the characterization of the traits and genotypes before the selection by available parameters. Two new genetic parameters were introduced and used in this paper: the Stability Index (SI) and the Stability Index of Broad-Sense Heritability (SIBH). To better understand, a case study on some tomato families is presented, and the implementation of the model is explained step by step.The proposed model has been developed according to the following well-known biometric concepts;Concept one: It is usually important to improve yield and quality in plant breeding [breeding .Concept two: Traits related to yield and quality are almost polygenic [olygenic .Concept three: Differences in the expression of the different genes (dominant and additive effects) caused complicated inheritance patterns of polygenic traits [c traits , 20.Concept four: Because simple univariate selection has low repeatability and the results cannot be predicted in the next generations, multivariate selection is recommended to improve traits with complicated inheritance patterns [patterns , 21.Concept five: In multivariate selection, traits with higher correlation with interesting traits are more important [mportant .Concept six: Among the traits correlated with interesting traits, one with significant physiological effects is more important in selection [election .This model is done by following steps;Step 1) Determining the inheritance pattern of the interesting traits for breeding, such as fruit yield, quality, ripening time, stress tolerance, etc., by preliminary experiments or literature review.Step 2) Determining all morphological, phenological and physiological traits related to interesting traits (step 1) and determining their inheritance pattern by preliminary experiments or literature review.Step 3) Creating competitive conditions for genotypes and measuring all possible traits (determined in steps 1 and 2) and calculating genetic parameters effective in the selection, such as correlation with interesting traits, heritability, expected genetic advance, and stability in different conditions. Other statistical methods may be needed to calculate the parameters, such as analysis of variance, different types of correlation, distance/similarity matrix, etc.Step 4) Weighting all the measured traits and scoring the competing genotypes by MAHP.A case study on tomato families introduces the following model.In this experiment, 18 physiological, phenological, and morphological traits were measured according to the following descriptions;Number of days to 50% flowering (DTF), number of days to 50% fruit formation (DFF), number of days to harvesting (DTH), plant height (H), cluster number per plant (CN), fruit number per plant (FN), fruit yield per plant (Yld), average of single fruit weight (SFW), average of single fruit volume (SFV), fruit density (FD), fruit water content (FWC), relative chlorophyll content (SPAD), fruit juice acidity (pH), fruit juice electrical conductivity (EC), total dissolved solids (TDS), fruit juice salinity , total soluble solids (TSS), and relative water content (RWC).Plant material, experimental design, and method for measuring traits and genetic parameters are presented in detail  and the Stability Index (SI) were introduced and used as stability parameters to determine the stability of traits in different conditions.2 and BH1 are the estimated The SIBH is calculated by Eq. Trait changes in different conditions appear more in genetic diversity.Genetic diversity plays an important role in trait control, especially in stress conditions.Selection can be effective in improving traits.Thus, the large value of SIBH means a higher trait value in the selection.The large amount of SIBH means;Research on proteome variations has revealed that the impact of the genotype on the proteome variations may be much higher than the stress effect . In suchThe SI is the absolute of the linear regression coefficient of data in the normal condition (as the independent variable) and data in the stress condition (as the dependent variable). The less the amount of SI, the more the stability of the trait. This parameter is just like the simple linear regression of Eberhart and Russell (1966), with the difference that it is calculated for traits.MAHP was used according to  with theStep 1) Aligning traits and families: According to Fig. Step 2) Calculating weights of traits: Forming trait comparison matrices based on each of the calculated genetic parameters and then calculating the eigenvectors of the resulting matrices. In the following, combining the main eigenvectors and obtaining the weights of the traits.Step 3) Calculating weights of families: Forming family comparison matrices based on each trait and then calculating the eigenvectors of the resulting matrices. In the following, combining the main eigenvectors and obtaining the score of families.ij = 1/Aji.). The following points in constructing pairwise comparison matrices must be considered:The construction of pairwise comparison matrices is based on parameters (for weighting traits) or traits (for scoring genotypes). For example, in the case of T traits and P parameters, there are N consistency comparison matrices  The data must be standardized if parameters/traits have different units.i is ith transformed data, Di is ith data, and MD is the maximum value.2) Low amounts for some traits or parameters  are important. Before standardizing data, each must be transformed based on the maximum value (Eq.  Low amou(3) Constructed comparison matrix is a consistency matrix where all its variations are compressed in an Eigenvector –25.(4) The main Eigenvalue (λ) is calculated according to Eq. (5) The main Eigenvector (x) related to the λ is calculated according to Eq. In this paper, calculations of ANOVA, genetic parameters, SI, SIBH, and MAHP were done by MATLAB ver . The totThe proposed model has been implemented step by step in the case study:Step 1) Interesting traits were fruit yield and total soluble solids. Many related studies were previously done to determine their inheritance pattern, such as [ such as . Many otStep 2) Introduced traits in material and methods were investigated in the literature review. In addition, some previous studies were done by [ done by . All of Step 3) Statistical methods were used to calculate genetic parameters after data gathering from competing genotypes. ANOVA, estimating genetic correlation with Yld and TSS, SI, SIBH, EGA, and heatmap clustering were used for this step.Tables Table Step 4) The selection of the remaining traits can be made on the average of the data in two experiments. MAHP has been used to weigh traits, and then scoring families based on them has been done. Figure Genetic diversity in the population helps breeders increase the next generation’s average by selecting superior genotypes. Plant breeding and selection are not possible without genetic diversity. In the following, based on the mentioned concepts, it is explained how the proposed model can be used for selection. In the case study, the measured traits were weighted, the families were scored, and the model’s logic was explained by assessing the statistical and physiological relationships between the traits.Increasing product quality may be as important as increasing product yield in a breeding program, especially in the case of crops and vegetable breeding. Based on the type of yield and quality-related data obtained from counting or measuring, it is obvious the interesting traits are polygenic. Also, the polygenic nature of these traits can be seen in this case study.The dominant or additive effects of genes controlling a trait have special effects on inheritance and make it difficult to predict the selection response. Thus, these effects must be assessed before the selection process. Direct selection of traits controlled by the dominance effect may not show suitable EGA in the next generation, so direct selection cannot be useful for traits with low breeding values . Assessible data , 28.The genetic diversity of the different families was investigated in terms of all measured traits on normal and drought stress conditions. The traits without significant diversity in the competitive families were ignored. Tables Selection of traits with higher values of The more the correlation between a trait with interesting traits, the more it affects them. This effect may be positive or negative. According to Table According to Table Suppose the results of calculating the genetic parameters of traits in different conditions are similar. In that case, selection principles in different conditions can be considered the same, and selection can be made based on the average genotypes in different conditions. If the behavior of the traits in different conditions is completely different, different breeding strategies must be considered in each condition. Tables Additional file 1. Plant material, experimental design, and how to measure traits and genetic parameters (.docx).Additional file 2. Excel of data related to the case study (.docx).Additional file 3. MATLAB script for calculating scores (.M).