Genetic parameters can reflect some regulars of quantitative traits in the population, and the estimation of genetic parameters is very important for cow breeding. The conformation traits of dairy cows are closely related to their milk production and reproduction, which are among the most important traits in dairy breeding. At the same time, milk production of dairy cows has always been the focus of attention at the cattle farm. Studies have shown that there are genetic correlations between conformation traits and milk production traits. For example, taller cows produce more milk. The aim of this study was to investigate the genetic correlations between conformation traits and milk production traits in Chinese Holstein dairy cows. In this study, we estimated the heritability of 23 conformation traits and five milk production traits and the genetic correlation between them using DMU software combined with single-trait and multiple-trait animal models. The heritability of conformation traits was medium in Holstein cows in Ningxia, China. In addition, there were positive genetic correlations between most conformation traits and milk production traits.The objective of this study was to explore the genetic parameters of conformation traits and milk production traits in Chinese Holstein cattle and to provide a reference for dairy cattle breeding. We collected the phenotypic data of 23 conformation traits and five milk production traits of Chinese Holsteins and used animal models to estimate the genetic parameters of conformation traits and milk production traits. The estimated heritability of conformation traits ranged from 0.11 (angularity) to 0.37 (heel depth) and the genetic correlation between conformation traits ranged from −0.73  to 0.76 (chest width and loin strength). The heritability of milk production traits ranged from 0.23 (somatic cell score) to 0.50 (305-d milk yield). The estimated values of genetic correlation between conformation traits and milk production traits ranged from −0.56 (heel depth and 305-d milk yield) to 0.57 (udder texture and milk fat percentage). There was a positive genetic correlation between most conformation traits and milk fat percentage, but a weak negative genetic correlation with milk yield. Strengthening the moderately and highly heritable milk production and conformation traits, especially the selection of rear udder traits and body shape total score, will be beneficial in improving the performance of dairy cows. Conformation traits are closely related to production, reproduction, mastitis resistance, and herd life of dairy cows ,2,3,4,5.The data of this study were collected at the Ningxia Farming Cultivate Helanshan Dairy Co., Ltd. We collected data on the conformation traits and milk yield of 64,972 Chinese Holstein dairy cows from 74 farms in the Ningxia Hui Autonomous Region from 2007 to 2020. The five milk production traits studied were milk yield, milk fat percentage, milk protein percentage, somatic cell score, and 305-d milk yield. Conformation traits included 23 linear traits: stature, height at the front end, trunk size, chest width, body depth, loin strength, pin setting, pin width, foot angle, heel depth, bone quality, set of rear legs, rear leg-rear view, udder depth, udder texture, median suspensory, fore attachment, fore teat placement, fore teat length, rear attachment height, rear attachment width, rear teat placement, and angularity. Cows’ conformation traits were recorded using a 9-point scale to evaluate linear character, including frame capacity, rump, feet and legs, mammary system, and dairy character. The assessors determined the conformation trait scores 30~180 days after the first calving. SAS V9.0 was used to trace the pedigrees to three generations and Excel 2019 was used to organize the pedigrees. A total of 2974 bull and 81,256 cow pedigrees were traced. Trait names, abbreviations, and definitions are presented in https://www.holstein.org.cn/newsIndex.jsp (accessed on 15 July 2022)).Values for the conformation and milk production traits were determined by a professional assessor. DHI data is determined in strict accordance with national standards. The measurement standards of conformation traits and milk production traits were based on the Code of Practice of Type Classification in Chinese Holstein and Technical of Chinese Holstein Cattle Performance Test in Dairy Data Center of China  All herds measuring fewer than 100 cattle were eliminated; (2) The offspring of bulls with fewer than 50 daughters were deleted; (3) Deleted offspring of bulls with fewer than 7 daughter distribution farms. A total of 11,019 conformation observations and 17,462 DHI observations were selected for heritability analysis. Subsequently, 7923 first parity cows that met the criteria were selected as the analysis objects for further statistical analysis of the correlation between conformation traits and milk production traits. After quality control, the cattle from 18 farms in the Ningxia Hui Autonomous Region were selected. The cattle were from 18 farms in the Ningxia Hui Autonomous Region. The farms were of different sizes and had different feeding and management conditions. Because of this, we included farm as a fixed effect in the statistical model.Data were analyzed with a linear animal model using the Average Information Restricted Maximum Likelihood (AI-REML) algorithm in the DMU package . The DMUAnalysis model of conformation traits:ijkly was the observed value of conformation traits; ijmly was the observed value of milk production traits; μ was the population mean of all observed values; ifarm was the fixed effect of the i-th farm; jtys was the fixed effect of the j-th year-season of measurement; ki is the fixed effect of the k-th assessor; mbys was the fixed effect of the k-th year-season of calving; lid was the individual additive genetic effect vector; and ijkle and ijmle were random residual effect vectors.Analysis model of milk production traits:Heritability calculation formula:Genetic correlation calculation formula:h2 = heritability, AV = additive genetic variance, and EV = environmental variance. Ar is the genetic correlation of traits, Cov is the additive genetic covariance of trait 1 and trait 2, and aσ12 and aσ22 represent the additive genetic variances of trait 1 and trait 2. Pr is the phenotypic correlation between traits, Cov is the phenotypic covariance of trait 1 and trait 2, and pσ12, pσ22 is the phenotypic variances of each trait.Phenotypic correlation calculation formula:The means of 23 linear scores for conformation traits ranged from 4.97 (UD) to 6.85 (ST). The standard deviation ranged from 0.72 (BD) to 1.38 (UD). The optimal gap from the ideal score ranged from −0.21 (FTP) to 3.85 (RAH). The coefficients of variation were all less than 30% .Descriptive statistics for milk production traits in Chinese Holsteins are shown in h2 < 0.5) were moderately heritable traits.Heritabilities of conformation traits ranged from 0.11 (ANG) to 0.37 (HD) . ST, CW,h2 ≥ 0.5), all the other milk production traits were moderately heritable (0.2 < h2 < 0.5).Heritabilities of milk production traits ranged from 0.23 (SCS) to 0.50 (305MY) . Except The genetic correlations between conformation traits in Chinese Holstein cattle ranged from −0.75 (UD and ANG) to 0.95 (ST and TS). ST and TS (0.95), HFE and TS (0.54), HFE and PW (0.93), HFE and FTL (0.69), TS and PW (0.65), TS and FTL (0.72), TS and ANG (0.69), CW and LS (0.76), CW and PW (0.60), LS and PS (0.62), LS and UT (0.88), UT and FTL (0.64), UT and ANG (0.88), MSL and FTP (0.67), and MSL and RTP (0.69) showed high correlations (shown in bold). Phenotypic correlations between conformation traits ranged from −0.52 (UD and ANG) to 0.68 (ST and TS) . DetaileThe genetic correlations among frame capacity traits ranged from 0.15 to 0.76. The genetic correlation between BD and CW was low (0.15), but the genetic correlation between CW and LS was large and positive (0.76), indicating a moderately positive genetic correlation. The phenotypic correlations among frame capacity traits ranged from 0.13 (BD and LS) to 0.24 , and all the phenotypic correlations were moderate to weak in magnitude. Both genetic and phenotypic correlations were positive between PS and PW, but the phenotypic correlation was small (0.08).The genetic correlations between feet and leg traits ranged from −0.73 to 0.58, in which there were negative genetic correlations between BQ and HD (−0.56), BQ and SORL (−0.73), FAN and SORL (−0.22), HD and RLRV (−0.21), and SORL and RLRV (−0.41). The other traits showed positive correlations. In addition, phenotypic correlations between feet and leg traits ranged from −0.16 to 0.20, with negative correlations including BQ and FAN (−0.02), BQ and SORL (−0.12), FAN and SORL (-0.09), and SORL and RLRV (−0.16). Phenotypic correlations among other traits were positive but weak.The genetic correlations between mammary system traits ranged from −0.57 to 0.69. The genetic correlations between MSL and FTL (−0.09), FTP and FTL (−0.57), FTL and RAH (−0.03), UD and RAW (−0.24), RAH and RAW (−0.22), and FTL and RTP (−0.12) were negative. The genetic correlations were positive for the remaining trait combinations. The phenotypic correlations among the selected traits ranged from −0.13 to 0.26, with weak positive correlations among the traits except between UD and RAW (−0.13).The genetic correlations between MY, FP, PP, SCS, 305MY, and conformation traits ranged from −0.31 (PS and MY) to 0.34 (ST and MY), −0.33 (PS and FP) to 0.57 (UT and FP), −0.32 (PS and PP) to 0.38 (UT and PP), −0.43 (HD and SCS) to 0.34 (BD and SCS), and −0.56 (FAN and 305MY) to 0.32 (BD and 305MY) . The pheh2 < 0.5). Heritabilities of the other traits were moderate, except for 305MY (0.50), which may be mainly due to conformational traits. The moderate heritabilities may be due to the fact that conformation traits can be directly observed at an early age and are relatively simple to record [Heritabilities of conformation traits ranged from 0.11 (ANG) to 0.37 (HD). The results were similar to those of Kadarmideen et al.  (0.08~0.o record . In addiThe range of phenotypic correlations between conformation traits was −0.52 (UD and ANG) to 0.68 (ST and TS) and the range of genetic correlations were −0.75 (UD and ANG) to 0.95 (ST and TS). The phenotypic correlation of PS and PW (0.08) was close to that of Oliveira et al.  (0.05).There were weak phenotypic correlations among all the feet and leg traits and the genetic correlation of FAN and SORL (−0.22) was close to that of Roveglia et al.  (−0.27).Of the mammary system traits, the genetic correlation between FTP and RTP (0.49) was lower than that reported by Oliveira et al.  in CanadST, BD, and BQ in Chinese Holsteins had strongly positive genetic correlations with production traits. This indicated that cows with larger TS, deeper BD, and better bone quality had more milk, their milk contained more fat and protein, and it had a better somatic cell score, which finding is similar to the results of Bohlouli et al. . It furtThere were moderately strong genetic correlations of SCS with BD, PS, and HD , indicating that frame capacity, rump, and feet and leg traits also affected somatic cell score and therefore affected the occurrence of mastitis. There was little correlation between HFE and SCS (−0.02), which was similar to a previous report . Too lonAlthough conformation traits do not produce direct economic benefits, some conformation traits can affect the milk production performance of dairy cows through direct or indirect effects and can be used to identify and breed high quality, high yield, healthy, and long-lived dairy herds, which would improve the economic standing of dairy farms. Therefore, strengthening the selection of moderately to highly heritable conformation traits and milk production traits is beneficial to the production performance of dairy cows.Among the conformation traits of Chinese Holstein cattle in Ningxia, LS (0.32), BQ (0.37), HD (0.37), and RTP (0.33) were medium heritability traits. Among the milk producing traits, 305MY (0.50) was highly heritable. These moderately heritable traits can be improved through systematic breeding and selection. Compared with other studies, the genetic correlation of the conformation traits of feet and leg traits has some deviation. The correlations between mammary system traits and milk production traits are very important. In this study, UD showed moderately negative genetic correlations with milk production traits, while both FTP (0.07) and RAW (−0.43) showed low genetic correlations with 305MY. There were also negative genetic correlations of UD (−0.17) and FTL (−0.02) with SCS.


Association analysis has been proven as a powerful tool for the genetic dissection of complex traits. This study was conducted to identify association of recovery, persistence, and summer dormancy with sequence related amplified polymorphism (SRAP) markers in 36 smooth bromegrass genotypes under two moisture conditions and find stable associations. In this study, a diverse panel of polycross-derived progenies of smooth bromegrass was phenotyped under normal and water deficit regimes for three consecutive years. Under water deficit, dry matter yield of cut 1 was approximately reduced by 36, 39, and 37% during 2013, 2014, and 2015, respectively, compared with the normal regime. For dry matter yield of cut 2, these reductions were approximately 38, 60, and 56% in the same three consecutive years relative to normal regime. Moreover, water deficit decreased the RY and PER of the genotypes by 35 and 28%, respectively. Thirty primer combinations were screened by polymerase chain reaction (PCR). From these, 541 polymorphic bands were developed and subjected to association analysis using the mixed linear model (MLM). Population structure analysis identified five main subpopulations possessing significant genetic differences. Association analysis identified 69 and 46 marker-trait associations under normal and water deficit regimes, respectively. Some of these markers were associated with more than one trait; which can be attributed to pleiotropic effects or tightly linked genes affecting several traits. In normal and water-deficit regimes, these markers could potentially be incorporated into marker-assisted selection and targeted trait introgression for the improvement of drought tolerance of smooth bromegrass. Under the changing climatic context, drought is becoming the most significant and acute problem affecting crop growth, survival, and persistence in many regions of the world, particularly in arid and semi-arid regions , 2. DeveSuccessful adoption of perennial forage species depends on the persistence of the species, which is mainly defined as ‘plant survival through repeated summer droughts and its ability to produce continuously high forage yields across several growing seasons’ , 6. In oBesides, plant survival and recovery have been associated with summer dormancy, dehydration tolerance in surviving tissues, extensive root system, and the ability of roots to extract water at low soil water potentials , 16. SumDrought tolerance is a complex quantitative trait involving diverse and multiple molecular and physiological mechanisms, signal transduction, and metabolic pathways . TherefoAs an essential tool for accelerating the rate of genetic gain, MAS’s first step is dissecting marker-trait associations (MTAs) . Genome-Lolium perenne) :PE:PE7]:Young leaf tissues of smooth bromegrass plants were used to extract genomic DNA using the modified method described by Murray and Thompson . The qua2b) was estimated for normal and water deficit regimes on a phenotypic mean basis averaged over replications as follows  was calculated as:The Kolmogorov–Smirnov and Bartlett’s tests were used to test the normal distribution of the data and homogeneity of variances, respectively. Subsequently, a combined analysis of variance, proposed by Steel and Torrie , was con follows  52]:h:h2b) wpic mean .Structure analysis and stratification of the studied population into subpopulations with different genetic structures were done based on SRAP marker data in STRUCTURE version 2.3.4 . This anP-values for marker–trait associations using TASSEL version 4.2.1 [Association analysis was run by MLM  to calcuon 4.2.1 . The pheon 4.2.1 . A correon 4.2.1 .As stated earlier, before analysis of variance, normality test was performed for all of the studied traits using the Kolmogorov–Smirnov test and its results verified the normality of all data . The BarMean comparisons of all measured traits for the two moisture regimes are given in In the fourth year (2016), after withholding irrigation in both moisture regimes and re-watering, no mortality of smooth bromegrass genotypes was observed and all recovered. Mean comparisons of recovery-related traits indicated that genotypes 9 and 28 had higher values of recovery vigor in normal regime, and genotypes 7 and 32 had the higher values under water deficit. The lowest recovery vigor was observed in genotypes 3 and 21 under normal regime and in genotype 15 under water deficit . MoreoveGenotypes of this germplasm showed incomplete summer dormancy; therefore, the SDI was calculated as the ratio of a genotype’s summer yield to the same genotype’s spring yield. Using the SDI based on summer forage production under normal regime, genotypes 4 and 17 were scored the least dormant. While genotypes 8, 14, and 19 showed the higher summer dormancy .In the present study, the genetic coefficient of variation (GCV) was from 2.86% for SDI-Y1 to 68.15% for PER in normal regime and from 22.26% for DMY2-Y3 to 55.10% for RY under water deficit regime. Except for DMY2-Y1 and PER, the values of genetic variation under water deficit were higher than the ones for normal regime .Broad-sense heritability estimates for combined data ranged from 50.93% for RY to 76.33% for PER. Moreover, the heritability estimates were calculated for each moisture regime separately and are given in K were used to calculate the number of subpopulations (K). The optimum number of sub-populations (K) was determined based on 30 SRAP primer combinations using the largest value of Delta K in the STRUCTURE 2.3.4 software. The maximum value of DK was obtained at K = 5 suggesting that there were five subpopulations in the smooth bromegrass panel , 69 SRAP markers showed significant associations with means of the studied traits at the 0.01 probability level (R2) range was from 2.72% to 11.38% (R2) range was variable from 1.77% to 16.03%  between years. Results revealed five and three stable MTAs between years, under normal and water deficit regimes, respectively . For exaThe quantitative inheritance of drought tolerance and interaction between gene expression and environment has challenged the understanding of the genetic basis of drought tolerance-related traits in plants . In the Poa pratensis L.), Saeidnia et al. [Bromus inermis), and Majidi et al. [Dactylis glomerata) reported similar results. Wide genetic variation observed for all studied traits revealed the potential for genetic gain from selection in this germplasm. Moreover, higher estimates of GCV for most of the evaluated traits under water deficit than non-stress indicate that water deficit has increased genetic variation. Therefore, selection under this condition would be more effective. The findings in this context are contradictory. For example, some researchers believe that genetic gain through selection is higher in normal regime than water-limited conditions [All traits were significantly affected by water deficit, more likely due to decreased water potential of the soil and decline in net assimilation and photosynthesis of leaves . Shariata et al.  in smooti et al.  in orchanditions , 61. At nditions . Moderatnditions .Festuca arundinacea Schreb.) reported that, after withholding irrigation, some of the genotypes died under both well-watered and deficit irrigation regimes. The superiority of smooth bromegrass in this respect may be related to greater rhizomes in length and volume, which would serve as soluble carbohydrate storage organs during stress [In the present study, mortality of smooth bromegrass plants was not observed due to withholding irrigation. Meanwhile, Abdollahi et al.  in orchag stress . Moreoveg stress . It seemg stress .In this study, summer forage yields were significantly lower than the spring ones under normal and water-deficit regimes. Therefore, genotypes of this germplasm showed low to moderate summer dormancy. Norton et al.  and SaeiAssociation analysis of evaluated traits under normal and water deficit regimes revealed that MTAs were mostly different in the two moisture regimes. Moreover, results showed that more genes were involved in controlling traits under normal regime than in water deficit. The percentage of variation explained by each identified association was low. This may be explained by the role of many minor genes controlling the trait, outcrossing nature of smooth bromegrass, markers exhibiting minor quantitative effects, rare alleles, and complex allelic interactions , 69. SimBased on the results, some markers showed significant associations with more than one trait which would be attributed to the pleiotropic effects or several tightly linked genes affecting these traits , 70. TheImproving the persistence, recovery, and survivability of perennial forage grasses is one of the main objectives of grass breeders in areas with prolonged periods of drought , 5. ThisIn this study, most of the MTAs were different for the two moisture regimes and the three experimental years, indicating the considerable role of environmental effects in these associations . These fIn conclusion, the advantage of the association analysis technique as a powerful tool to identify and detect genes and markers linked to complex traits of agricultural and economic importance was demonstrated. Satisfactory levels of polymorphism and genetic diversity were observed for the studied traits in the polycrossed population. Five subpopulations were identified in the smooth bromegrass panel, and 69 and 46 significant MTAs were detected under normal and water deficit regimes, respectively. Environmental specificity of MTAs indicated that genotype × environment interactions affect association analysis; nevertheless, eight markers showed stable association with different traits across moisture regimes; and five and three markers showed significantly stable expression across two or three years of study, under normal and water deficit regimes, respectively. However, most of the MTAs were different for the two moisture regimes and the three experimental years, indicating the considerable role of environmental effects in these associations. This suggests that different genes contribute to the same trait in different environments and years or the same genes may change the expression level between different environments. Based on the results, some markers showed significant associations with more than one trait demonstrating reliability of these markers for marker-assisted selection (MAS) of correlated traits concurrently. The molecular markers identified in the present study are suggested as valuable genomic resources in the future breeding programs of smooth bromegrass. However, these SRAP–trait associations must be validated in independent germplasm for use in marker-assisted selection programs.S1 Table(DOC)Click here for additional data file.S2 Table(DOC)Click here for additional data file.S3 TableThe p-value larger than 0.05 indicates that the data follow normal distribution.(DOC)Click here for additional data file.S4 Table(DOC)Click here for additional data file.S5 Table(DOC)Click here for additional data file.S1 Raw data(XLSX)Click here for additional data file.S2 Raw data(XLSX)Click here for additional data file.


H), which is a linear combination of unobservable individual traits’ breeding values, weighted by the trait economic values; thus, the target of LSI is not a parameter but rather the unobserved random H values. The LSI can be single‐stage or multi‐stage, where the latter are methods for selecting one or more individual traits available at different times or stages of development in both plants and animals. Likewise, LSIs can be either constrained or unconstrained. A constrained LSI imposes predetermined genetic gain on expected genetic gain per trait and includes the unconstrained LSI as particular cases. The main LSI parameters are the selection response, the expected genetic gain per trait, and its correlation with H. When the population mean is zero, the selection response and expected genetic gain per trait are, respectively, the conditional mean of H and the genotypic values, given the LSI values. The application of LSI theory is rapidly diversifying; however, because LSIs are based on the best linear predictor and on the canonical correlation theory, the LSI theory can be explained in a simple form. We provided a review of the statistical theory of the LSI from phenotypic to genomic selection showing their relationships, advantages, and limitations, which should allow breeders to use the LSI theory confidently in breeding programs.A linear selection index (LSI) can be a linear combination of phenotypic values, marker scores, and genomic estimated breeding values (GEBVs); phenotypic values and marker scores; or phenotypic values and GEBVs jointly. The main objective of the LSI is to predict the net genetic merit ( Linear selection indices (LSIs) are useful in plant and animal breeding.The main LSI objective is to predict the net genetic merit of individuals.We did a complete review of the statistical theory of LSIs.We described the bases of genomic LSIs. BLPbest linear predictorBLUPbest linear unbiased predictorBPbest predictorCCAcanonical correlation analysisCDMLPSIconstrained decorrelated multi‐stage linear phenotypic selection indexCLGSIconstrained linear genomic selection indexCLPSIconstrained linear phenotypic selection indexCOMLPSIconstrained optimum multi‐stage linear phenotipic selection indexDMLPSIdecorrelated multi‐stage linear phenotypic selection indexESIMeigen selection index methodGEBVgenomic estimated breeding valueGESIMgenomic eigen selection index methodLGSIlinear genomic selection indexLMSIlinear molecular selection indexLPSIlinear phenotypic selection indexLSIlinear selection indexMESIMmolecular eigen selection index methodMSPEminimum mean square prediction errorMVNmultivariate normalOMLPSIoptimum multi‐stage linear phenotipic selection indexPPG‐ESIMpredetermined proportional gain eigen selection index methodQTLquantitative trait lociREMLrestricted maximum likelihoodRESIMrestricted eigen selection index methodRLrate of layRLPSIrestricted linear phenotypic selection indexSMage at sexual maturity.1H). The LPSI is a linear combination of observable phenotypic records, and H is an unobservable linear combination of trait breeding values (g) weighted by the trait economic values (W).Smith  publisheg  is composed entirely of additive effects of genes and thus is the plant or animal breeding value, and H is the total individual genotypic value  and H have a joint multivariate normal distribution. By those assumptions, the joint distribution of H and LPSI is bivariate normal, whereas the conditional distribution of H given LPSI  is univariate normal. Under those assumptions, the regression of H on y is linear  is the conditional expectation of g given the linear selection index (LSI) values  are useful in plant and animal breeding.The main LSI objective is to predict the net genetic merit of individuals.We did a complete review of the statistical theory of the LSI.We described the bases of genomic LSI.The LPSI theory has been extended, and now a LSI can be a linear combination of marker scores  or the first eigen vector of a linear transformation of the matrix of multi‐trait heritabilities (in the RESIM and PPG‐ESIM case) in the prediction. Cerón‐Rojas et al.  are particular cases of the best linear predictor (BLP) theory, which is based on the minimum mean square prediction error (MSPE), because this approach allows us to obtain the unique best MSPE predictor  is a particular realization of X. Vectors are denoted with bold lowercase letters  and matrices with bold uppercase letters (X). Sometimes it is necessary to denote vectors with bold capital letters; for example, E denotes the vector of expected genetic gain per trait.We use uppercase letters to denote random variables and lowercase letters to denote their particular realizations. For example, X[E(X)] is scalar, whereas the conditional expectation of X given Y = y [E(X|Y = y)] is a function of y. Thus, before we observe Y, the value of E(X|Y = y) is unknown, so it is a random variable that we denote as E(X|Y).The expectation of the random variable 3Pearson  and Pear3.1X and Z are design matrices and that θ and g are vectors of fixed and random  effects, respectively, associated to the trait of n individual candidates (y) to selection, and assume that y and g have multivariate normal (MVN) distribution, also denoted as “∼”, i.e., y ∼ MVN and g ∼ MVN. Consider the following linear mixed model:y ∼ MVN is a n × 1 vector of observations for one trait, Xθ and V = σg2ZAZ′ + Inσε2 are the unconditional expectation and variance of y, respectively, and var(g) = Aσg2 is the unconditional variance of g ∼MVN ; var(y|g) = Inσε2 is the conditional variance of the vector of residuals ε ∼MVN , 0 denotes the null expectations of g and ε, A is the numerator relationship matrix −1X′V−1y is called the BLUP of g and is one of the main tools to make genotypic selection in plant , as shown in Equations W = X ⊗ Z, and ⊗ denotes the Kronecker product  = 1nμ, where 1n is a vector of n ones, μ is the population mean, V = Xvar(γ)X′ + σg2ZAZ′ + Wvar(π)W′ + Inσε2, and var(γ) and var(π) are the variance of γ and π, respectively. Note that in this last case, nongenetic effects (included in γ), such as location and year effects, are treated as random effects, whereas in Equation θ. For additional details, see Xu  = R ⊗ In, where R=σε12σε12σε21σε12 is a residual covariance matrix. The unconditional covariance matrix of [y′1y′2] is V=vary1y2=Z∗[C⊗A]Z′∗+R⊗In, where Z∗=Z100Z2, and In is as defined in Equation Equation g1 can be partitioned into additive genetic (a1) effects , dominant genetic (δ1) effects, and epistasis (ι1) effects, such that g1 = a1 + δ1 + ι1. Epistasis refers to the interaction among additive and dominance genetic effects, for instance, additive by additive, dominance by dominance, additive by dominance, additive by additive by dominance, etc.; that is, ι1 = aa1 + dd1 + ad1 + …For one trait, g1 is var(g1)=Aσa2+Θσd2+(A∘A)σaa2+(A∘Θ)σad2+(Θ∘Θ)σdd2+ …When we separate non‐additive genetic effects from additive effects, we remove the confounding factors that would otherwise bias the results of the analysis  is the only component that can be selected and is thus the main component of interest in breeding programs  and that the level of inbreeding among individuals is null; then A = In, and the matrix V can be written asWe have indicated that in the LSI context the assumption that only additive effects are transmitted from generation to generation is acceptable. For V is associated with the BLP (LPSI) theory. Hereafter we denote it as P, which is the usual notation in the LPSI context.As we shall see later, in this last case matrix A. However, matrix A is used for the estimation of C th selection cycle are not correlated. (i) The correlation between the LPSI and the net genetic merit should be at its maximum in each selection cycle. These conditions are valid for all unconstrained or constrained LPSIs in the single‐stage selection context. Cerón‐Rojas and Crossa  The phenotypic value is additively composed of two main parts: the genotypic value and the environmental contribution. (b) The genotypic value is the individual breeding value; thus, it is composed only of additive genes effects. (c) The net genetic merit of an individual is the genotypic economic value  individual net genetic merit isw′ = [w1w2 … tw] is a 1 × t (t = number of traits) vector of known economic weights. The variance of qH is denoted as σH2 = w′Cw. Smith , a predictor of Equation h^′=[H^1H^2…H^n] is a vector of estimated net genetic merit values for n indviduals, and jw is the thj element of the vector of economic weights w  differs from the expected value of qH  element of vector μ is equivalent to Xθ (Equation X = 1 (a vector of 1s) and θ = μj. We have assumed that qe has a normal distribution, null expectation, and variance σe2 and that the covariance between any pairs of qe and re (q ≠ r) is zero. As we will see later, β′(yq − μ) = qI is the LPSI. It is possible to show that the random error (qe) and the LPSI are uncorrelated  and dependent (qH) variables are random and have a joint multivariable normal distribution  = Hm, E(Y) = μ, and that var(H), var(Y) = P, and the covariance between H and Y [cov = w′C] exist and are finite, then the BLP of H (or the LPSI) is0 = Hm − β′μ, w′C and μ are as defined above, P−1 is the inverse of the matrix P, and β = P−1Cw. Cochran  isE(H|Y=y)PSI) isE(H|Y=y)PSI) isE(H|Y=y)PSI) isE(H|Y=y)g and Y have joint multivariate normal distribution with mean E(g) = 0 and E(Y) = μ, and covariance matrix varYg=PCCC, where P and C are as defined above. Moreover, the distribution of LPSI and H = w′g values is a bivariate normal distribution, with mean E(H) = Hm and E(I) = Im and covariance matrix varHI=w′Cww′Cβw′Cββ′Pβ=σH2σHIσHIσI2, where β, w′, σH2, and σI2 were defined earlier and σHI = w′Cβ is the covariance between H and LPSI.By Equations 0 = Hm − β′μ is a fixed contant, the maximized LPSI expected genetic gain per trait (Equation H and LPSI (Equation E(H) = 0 and E(Y) = 0. Likewise, when we select parents for the next generation, they will be the same in both cases, that is, when E(H) = 0 and E(Y) = 0 and when E(H) = Hm and E(Y) = μ. Thus, we think that there are not loss of generality or loss of accuracy to describe the LSI theory if we assume that E(H) = 0 and E(Y) = 0. In practice, E(Y) = 0 only if y is centered with respect to μ.As we will see later, because we can assume that βEquation , the maxEquation , and theEquation , are not4.6β = P−1Cw, then σHI = w′Cβ and σI2 = β′Pβ are the same, from where the variance of the LPSI prediction error isBecause e2 = σH2(1 − ρHI2) are the same, where ρHI2 = (σHI2)/(σH2σI2) and σHI2 = (w′Cβ)2 are, respectively, the square of correlation and the square of covariance between H and LPSI. We defined the other parameters earlier.Equation H is dependent on y /σI, where E(I) = Im and σ1 is the standard deviation of I. Assume that all I values higher than I*value corresponding to any proportion of selected individuals (p) will be selected; then the variables T* = (I* − Im)/σI and T are related to selection intensity (k) asz(τ∗)=exp{−0.5τ∗2}2π is the height of the ordinate of the normal curve at the lowest value of τ* retained, and p is the proportion of the population of animal or plant lines that will be selected ] weighted by p and was obtained by Smith (T (weighted by p), over the interval , is equal to 1 + k(τ* − k). Several authors  ask=1pE(T)by Smith . It is p (k) ask=1pE(T)4.8H and g are null, whereas the expectation of I is E(I) = Im; then the conditional expectation of H and g given I is, respectively,H, I) = β′Cw is the covariance between H and I, cov = Cβ is the covariance between g and I, T = (I − Im)/σI, and σI2 = β′Pβ and σ1 are the variance and standard deviations of I, respectively. In Equation H, I)/σI2 is the relationship between E(H|I) and I (or the regression of H on I), and cov/σI2 is the relationship between E(g|I) and I in Equation Suppose that the expectations of tively,E(H|I)=CσITandE(g|I)=C4.9R) and expected genetic gain per trait (E) are, respectively,k was defined in Equation σH=w′Cw is the standard deviation of H, and ρHI = σHI/σHσI is the correlation between H and LPSI. All the other parameters of Equations R is a scalar; in Equation E is a vector t × 1 (t = number of traits) of the expected genetic gain value of each trait. In addition, R is the expectation of H, and E is the expectation of g.By Equations tively,R=kβ′CwσρHIandE=kCβσItively,R=kβ′CwσρHIandE=kCβσItively,R=kβ′CwσρHIandE=kCβσIβ. The result reported by this author was the vector of regression coefficients β = P−1Cw. He indicated that β = P−1Cw should maximize the selection response. Thus, Smith  and correlation are good measures of relationship only for variables with linear trends and are generally unsuitable for non‐normal random variables with a curvilinear relationship 2], is minimized, and a global maximum when the ρHI correlation between I and H is maximized because Equation β = P−1Cw.Cerón‐Rojas et al.  showed t4.12Equations 4.13H and I have a joint bivariate normal distribution; β = P−1Cw; and P, C, and w are known, the statistical LPSI properties are the following:Equations I and the covariance between H and I are the same.The variance of H and I is equal to Equation The maximized correlation between E(I) = E(H)  is maximized  is a linear function of y; (b) is “best” in the sense that it has minimum mean squared error of prediction; and (c) is optimal and unique among the class of linear predictors based on y  = 0 and E(y) = 0 to describe the LSI theory. Thus, to simplify notation, hereafter we will assume that E(H) = 0 and E(y) = 0. These last two assumptions are common in the multi‐stage LPSI context  = 0 only if y is centered with respect to μ = E(y).As we have indicated earlier, there is no loss of generality if we assume that y′ = [Y1Y2 … tY] be a vector of random variables for t traits. We can partition this vector into S  subvectors as y′ = [ω′1ω′2 … ω′S], where ω′i=[Y1Y2…Yti] is a subvector of y at Stage i. Thus, at Stage S, t=∑i=1Sti is the total number of selected traits from vector y and, for each stage, the BLP (or OMLPSI) of H isω′*2 = [ω′1ω′2]. In Equation Let of H isE(H|ω1)=i, let iI = β′iωi be the index, where βi′=[βi1βi2…βiti] is the OMLPSI vector of coefficients, and ωi and it are as defined earlier; then we can construct the OMLPSIs as follows:S − 1, each index is partial, but at Stage S, SI = β′1ω1 + β′2ω2 + … + β′SωS is a whole index. Young  is the ijth  submatrix of P, and let Λ = {cov}′ = [C1C2 … CS] be the covariance matrix between ωi  and g, where cov = Ci is a submatrix of matrix Λ at Stage i. Moreover, let Qii and Λ′i=[C1C2…Ci] be submatrices of P and Λ until Stage i  at Stage i isQii−1 is the inverse of matrix Qii, and w′ = [w1w2 … wt] is as defined earlier.Let 4.17Rmaxi), expected genetic per trait (Emaxi), and correlation (ρmaxi) at Stage i, respectively, areik is the selection intensity. For all stages, the total selection response and expected genetic gain per trait are RT=∑i=1SRmaxi and ET=∑i=1SEmaxi, respectively.By Equation 4.184.18.1ω′(i−1) = [ω′1ω′2 … ω′i−1] and ω′(i) = [ω′1ω′2 … ω′i] (where ω′i=[Y1Y2…Yti] is a subvector of y′ = [Y1Y2 … tY] at Stage i) be subvectors of y until Stages i − 1 and i, and let it−1 and it−1 + it be the size of vectors ω′(i−1) and ω′(i), respectively; then the covariance matrix between ω′(i−1) and ω′(i) (Q(i−1)i) will be of size it−1 × (it−1 + it) and can be written asLet Q(i–1)i is a nonsquare and nonsymmetric phenotypic variance matrix and is useful by making the DMLPSI and constrained decorrelated multi‐stage LPSI (CDMLPSI) values independent among stages. By definition, a covariance matrix is symmetric; however, in this case, matrix Q(i–1)i is of size it−1 × (it−1 + it) because until Stages i − 1 there are it−1 traits, whereas at Stage i there are it−1 + it traits. This means that at Stage i − 1 we have an index with it−1 traits, whereas at Stage i we have an index with it–1 + it; that is, the trait selected until Stage i − 1 and the traits selected until Stage i. By this reasoning, as we indicated earlier, Young  = 0]. By this restriction, iI and iI−1 are not correlated; hence the name “decorrelated multi‐stage linear phenotypic selection index.” Xu and Muir  = B′i−1Q(i−1)ibi, where, Bi−1 is an upper triangular matrix 1b(i−1)2…b(i−1)ni−1] is the DMLPSI vector of coefficients at Stage i − 1, Q(i−1)i is as defined above, and bi is the DMLPSI vector of coefficients at Stage i.Let and Muir  showed tSi(i−1) = Qi(i−1)Bi−1 and S′i(i−1) = S(i−1)i = B′i−1Q(i−1)i be the transpose of matrix Si(i−1), and assume that matrices Qii, Qi(i−1), and Λi and vector w are known. To minimize the mean squared difference between H = w′g and iI = b′iωi 2]), under the restriction S(i−1)ibi = B′i−1Q(i−1)ibi = 0, it is necessary to minimize the functionbi and ξ′ = [ξ1 ξ2 … ξi−1], where ξ is a vector of Lagrange multipliers. In matrix notation, the derivative results of bi and ξ areLet E[(H − iI)2] at Stage i can be written asKi = [Ii − Φi], Φi = Qii−1Si(i−1)[S(i−1)iQii−1Si(i−1)]−1S(i−1)i, βi = Qii−1Λiw  = 0 is not imposed, βi = bi, and at Stage 1 (i = 1), Si(i−1) = S1(0) = Q1(0)B0 = 0, Φ1 = Q11−1S1(0)(S(0)1Q11−1S1(0))−1S1(0) = 0, and Ki = K1 = [I − 0] = I, where I is an identity matrix of the same size as Φ1. Thus, b1 = K1β1 = Q11−1Λ1w = β1, where Q11−1 is the inverse of matrix Q11−1, Λ1 = C1, and cov = C1. The method to obtain Equation from where the DMLPSI vector of coefficients that minimizes Equation , Qiitten asbi=Kiβii are similar to those of Equation βi for bi = Kiβi. The only difference between the DMLPSI and OMLPSI is the way the selection intensities are obtained. Xu and Muir , in the single‐stage context, is related to the height of the ordinate of the normal curve z(τ*) and the proportion selected (p) as k = [z(τ*)]/p. Kempthorne and Nordskog (k = [z(τ*)]/p is a left truncated normal distribution ]/p is also called the “hazard function,” the “hazard rate,” or the “inverse Mill's ratio” of the normal distribution ]/p /σ1 and T2 = (I2 − m2)/σ with mean zero and variance 1, where m1 and m2 are the means, and σ1 and σ2 are the standard deviations of I1 and I2, respectively. In this case, the method of selection is to retain animals or plants with T1 ≥ c1 at Stage 1 and T1 + T2 ≥ c2 at Stage 2, where c1 and c2 – T1 are truncation points for I1 and I2, respectively. Thus, according to Young  can now be written asc1 and c2 – τ1 are truncation points for I1 and I2, respectively. It is evident that when the number of stages increases, the number of variables in fT1,T2, and the number of integrals in p also increases.Young  showed tEquation . Thus, sto Young , the selp) before selection is carried out and then, based on the fixed p value, to determine the proportion iq  for each stage under the restrictionS is the number of stages. In a two‐stage selection context, p = q1q2. It is assumed that p is known but iq is unknown. One way of solving the equation p = q1q2 is by trial and error, depending on the importance of each trait  of coded marker values ; thenjth characteristic  of the candidates for selection. We assumed that αj has multivariate normal distribution with mean 0 and covariance matrix Gσα2, where σα2 is the additive genomic variance of αj; G = LL′|κ is the matrix of additive genomic relationship, κ=∑l=1m2ql(1−ql) in an F2 population; lq is the frequency of allele A; and 1 − lq is the frequency of allele a for the lth marker .Let 4.19.2β′ = [β1 β2 … βt] is the vector of regression coefficients, α′ = [α1α2 … αt], and e is the residual with normal distribution, null expectation, and variance σe2. It is possible to show that the covariance between β′α and e equals zero.The net genetic merit is related to the vector of genomic breeding values Equation  as(18)4.19.3H is the conditional expectation of H given α, that isw′Γ = cov′ is the covariance between H and α, Γ−1 is the inverse matrix of the genomic covariance matrix Γ = var(α) = cov, w is as defined in Equation GI = w′α is the LGSI.The BLP of 4.19.4u^j be a predictor of the uj vector of QTL effects associated with the markers that affect the jth trait =u^j. This implies that the expectation of H, given the estimator of the LGSI,α^j=Lu^j is the vector of GEBVs associated with trait jth.Let 4.19.5By Equations RG=kw'Γw is the the upper boundary for the unconstrained LPSI selection response. It is possible to show that, in the constext of the constrained LPSI selection response, there is a similar result −1 = Γ−1 − Γ−1(Γ−1 + R−1)−1Γ−1 is the inverse of matrix P , then w′Γw ≥ w′(Γ−1 + R−1)−1w ≥ 0, from where in the asymptotic context, GR ≥ R.Assuming that C and P can be written as C = Γ + Ξ and P = (Γ + Ξ) + R, respectively, where Ξ = C − Γ. Thus, the inverse of matrix P isb = P−1(Γ + Ξ)w, and the LPSI selection response (R) isIn the non‐asymptotic context, when the number of markers and genotypes is low, matrices GR and R are related in three possible ways:R>RG if w′Ξw>w′[(Γ+Ξ)−1+R−1]−1wR=RG if w′Ξw=w′[(Γ+Ξ)−1+R−1]−1wRG>R if w′Ξw<w′[(Γ+Ξ)−1+R−1]−1wThis last equation indicates that in the non‐asymptotic context, GR may be equal to or larger than R depending on the size of w′Ξw and w′[(Γ + Ξ)−1 + R−1]−1w.By Points 2 and 3, in the non‐asymptotic context, 5H  developed by Cerón‐Rojas and Crossa , and N is the number of QTL affecting the jth trait of interest. Because the QTL effect values are not observable, the hG values are also not observable; however, we can use a linear combination of the markers linked to the QTL (hs) that affect the thj trait to predict hG ashs is a predictor of hG, kb is the regression coefficient of the linear regression model, kl is the coded value of the kth markers , and M is the number of selected markers linked to the QTL that affect the jth trait. The above equation is called the “marker score”  = S be the covariance matrix of marker score and Γ as defined in Equation O=varys=PSSS and Π=vargs=CSSS, Equation O=varyz and Π=vargz, Equation v′ = [w′ 0′] is the vector of economic weights, where 0 is a null vector of size t × 1. The other parameters were previously defined.Let var, whereas CR = R (the LPSI selection response). In addition, when the number of markers and genotypes increases, matrix Γ tends to matrix C and at the limit Γ = C, from where βy = 0, βC = w, and Equation RG=kw′Γw (Equation By Equation RC=kβ′OβEquation . That is5.5EC) and correlation (ρHIC) are, respectively,k is the selection intensity. Note that the denominator of ρHIC , whereas the index based on the DMLPSI will be called “decorrelated combined multi‐stage LGSI” (DCMLGSI) because, at stage two, both indices use GEBVs and phenotypic information jointly to predict the net genetic merit.5.8I^G=w1α^1+w2α^2+·+wtα^t, where α^j=Lu^j is the vector of GEBV of trait jth) is used to predict H because at this stage, both indices select immature seedlings (or embryos) based only on GEBV information. The OCMLGSI and DCMLGSI maximized selection responses are, respectively,kO1 and kD1 (the selection intensities for each index) are the only difference between RO1 and RD1. The maximized correlation between H = w′g and GI = w′α is ρHI=w′Γww′Cw, where w′Γw is the standard deviation of the variance of GI = w′α, σH=w′Cw is the standard deviation of H = w′g, and C is the covariance matrix of g.The LGSI and LPSI are part of the OCMLGSI and DCMLGSI. At stage‐one the estimator of the LGSI , where βi′=[βi1βi2…βiti].The main difference between the multi‐stage LPSIs described earlier (OMLPSI and DMLPSI) and OCMLGSI and DCMLGSI is that, in addition to Equation  in the pEquation , and Iig′ = [G1 G2 … Gt] be a vector of true unobservable random breeding values for t traits, and let α, y, and ω1 be as defined earlier. The covariance between αj and Gj  is equal to the variance of αj   = Γ, and the covariance matrices between ωi  and g for S stages isω1 = α, cov = Γ is the genomic covariance matrix, and from Stage 2 to S, cov = Ci is the ith genotypic covariance submatrix of ΩΓ. Note that while the size of matrix Γ is t × t, the size of matrix Ci will be in × t, where in is the size of vector ωi. For example, if g′ = [g1g2g3g4], ω′1 = [α1α2α3α4], and ω′1 = [Y1Y2], then Γ will be of size 4×4, and C2 will be of size 2×4, where 2 is the number of rows and 4 is the number of columns in matrix C2. By the foregoing results, the covariance matrix of η′ = [α′ y′] = [ω′1 ω′2 … ω′S] isQii and Λi are constructed as∑ and ΩΓ, respectively, until Stage i.Let Dekkers, ; the cov5.10y′ = [ω′2ω′3 … ω′S] and α is as defined earlier; βG and βy are vectors of coefficients of genomic and phenotypic weight values, respectively, β′2 = [β′Gβ′y], and t′* = [α′ y′]. The only difference between OCMLGSI and DCMLGSI is the way vector β′2 = [β′Gβ′y] is obtained.At Stage 2, OCMLGSI and DCMLGSI is5.11β2) that maximizes the response, correlation, and expected genetic gain per trait ; Λ2=ΓC2, C2=cov, ν′∗=0′w′ is a vector where 0 is a null vector of size t × 1, and w is the vector of economic weights. The maximized selection response, and the maximized correlation between H and OCMLGSI, respectively, arekO2 is the OCMLGSI intensity at stage two. The total selection response for Stages 1 and 2 is ROt=RO1+RO2.The OCMLGSI vector of coefficients  into Equation β2 = Q22−1Λ2ν* by Equation kO2) by the DCMLGSI selection intensity (kD2). Finally, note that at Stage 2 B′1 = [w′ 0′S−1], where w = b1 is the vector of economic weights and 0′S−1 is a null vector of size S − 1. The size of matrix Q12 depend on the number of traits selected at Stage 2.The DCMLGSI vector of coefficients at Stage 2 isEquation ; and I2b2=K2β2b2=K2β26β = P−1Cw), in the multi‐stage context, the constrained optimum multi‐stage LPSI (COMLPSI) vector of coefficients is a linear combination of the OMLPSI vector of coefficients  × r of predetermined proportional gains, where jd  is the jth element of vector d′ = [d1d2 … rd], and let U′ be a matrix (t − r) × t  of 1s and 0s, where 1 indicates that the traits are restricted, and 0 indicates that the traits are unrestricted. The size of matrix U′ depends on the number of restricted traits.a Mallard  matrix 2] under the restriction M′β = 0 assuming that P, C, U′, N′, and w are known; that is, we need to minimize the functionβ and ξ′ = [ξ1 ξ2 … ξr−1], where ξ is a vector of Lagrange multipliers. The derivative results from β and ξ areE[(I − H)2] under the restriction M′β = 0 isK = [It − D], D = P−1M(M′P−1M)M′, and It is an identity matrix of size t × t. It is possible to show that when N = U, the restriction M′β = 0 changes to Ψ′β = 0, from where Equation KR = [I − D], D = P−1Ψ(Ψ′P−1Ψ)−1Ψ′, P−1 is the inverse of matrix P, and I is an identity matrix t × t. If N = U and U is a null matrix, then bR = β (the LPSI vector of coefficients). This means that the CLPSI is the most general index and it includes the RLPSI and LPSI as particular cases.Let  Mallard  matrix o = 0 isβC=KβwhU′C′β = θ*d, Tallis (bR = KRβ (Equation δ* = P−1Ψ(Ψ′P−1Ψ)−1d, d′ = [d1d2 … rd], and θ* = [β′Ψ(Ψ′P−1Ψ)−1d]/[d′(Ψ′P−1Ψ)−1d]. When θ* = 0, bT = bR, and if θ* = 0 and U′ is a null matrix, bT = β  = w′Cβ] is not defined. In addition, this index maximizes neither the correlation between I and H (ρIH) nor the selection response 2], under the restrictions Mi′ βi = 0, where Mi′ = N′U′Λi′ and Λi′ is as defined earlier. Supposing that matrices Qii, U, and Λi′ and vectors d and w are known, then at Stage i it is necessary to minimize the functionβi and the vector of Lagrange multipliers ξ′ = [ξ1 ξ2 … ξr−1]. The COMLPSI vector of coefficients at Stage i isβi is as defined in Equation KOi=[Ii−FOi], FOi=Qii−1Mi(M′iQii−1Mi)−1M′i, and Ii is an identity matrix of the same size as Qii. When N = U, the vector of coefficients of Equation N = U is a null matrix, Equation According to the single‐stage CLPSI Equation  and OMLPge i isβCi=KOiβge i isβCi=KOiβ6.4Xie and Xu  extendediI−1 = δ′i−1ωi−1 and iI = δ′iωi be the CDMLPSI at Stages i − 1 and i, respectively, and let J′i−1 = [I1I2 … iI−1] be a vector of CDMLPSIs values until Stage i − 1 so that the covariance between iI = δ′iωi and J′i−1 is null. To obtain δi, we need to minimize the mean squared difference between H = w′g and iI = δ′iωi {E[(H − iI)2]} under the joint restrictions M′iδi = 0 and cov = B′(i−1)Q(i−1)iδi = 0, where M′i = N′U′Λ′i, and Λ′i is as defined earlier. In this case, Bi−1 is an upper triangular matrix  = Qi(i−1)B(i−1) and S′i(i−1) = S(i−1)i; then at Stage i, the CDMLPSI vector of coefficients that maximizes Equations βi is as defined in Equation KDi=[Ii−FDi], FDi=Qii−1Vi(V′iQii−1Vi)−1V′i, Vi=MiSi(i−1), Vi′=M"iS(i−1)i, and Ii is an identity matrix of the same size as Qii. When matrix Si(i−1) is null, Equations N = U, Equation 36 imposes null restrictions, giving us a null restricted DMLPSI index similar to Equation In a manner similar to the DMLPSI, let uations δi=KDiβiKOi=[Ii−FOi] and KDi=[Ii−FDi] transform the OMLPSI vector of coefficients (βi) into the COMLPSI and the CDMLPSI vectors of coefficients, respectively. In addition, the CDMLPSI expected genetic gain per trait, selection response, and the correlation at Stage i are the same as those of the OMLPSI when replacing βi with δi=KDiβi.According to Equations 6.5KG = [It − DG], where DG = UN(N′U′ΓUN)−1N′U′Γ. Matrices N and U were defined in Equation It is an identity matrix t × t. Matrix KG is idempotent and projects the LGSI vector of coefficients (w = β) into a space that is smaller than the original space of w. Thus, while w is the vector of coefficients of the unconstrained LGSI,w for bG = KGw.In a manner similar to the single‐stage CLPSI context Equation , it can 7H = w′g and making selections. The other indices associated with ESIM  can be seen in Cerón‐Rojas and Crossa  in the LPSI context, in ESIM w is fixed but unknown and can be estimated in each selection cycle as a linear combination of the ESIM vector of coefficients. Moreover, in ESIM, the breeding values (g) and the phenotypic records (y) are two sets of variables that should have maximum correlation (λj), from where the CCA theory is the base of ESIM. The ESIM index can be written as I = b′Ey, where b′E = [b1b2 … bt] is the unknown index vector of coefficients, and t is the number of traits. In the CCA context, the vector of ESIM coefficients (b′E) is called the “canonical vector,” whereas λj is called the “canonical correlation” , the CCA theory allows us to write ESIM and H = w′g as IE=b′Ejy and HE=w′Ejg, respectively, where wEj=C−1PbEj, and bEjand λj are obtained from For the bEj is the jth vector  of the multi‐trait heritability matrix P−1C and maximizes the correlation ρHI = σHI/σHσI described in Equation bE1) of matrix P−1C in IE=b′E1y and the first λ1 in the ESIM selection response.Note that 7.2(P−1C)IbEj=λj2IbEj, where I = F−1F is an identity matrix of size t × t (t = number of traits), and F = diag{f1f1 … ft} is a diagonal matrix with values equal to any real number, except zero values. Thus, another way of writing Equation βEj=FbEj. Matrices P−1C and F(P−1C)F−1 are similar, and both have the same eigenvalues but different eigenvectors F−1 is called “similarity transformation,” and matrix F is called the “transforming matrix” σHE2, where σHE2 is the variance of EH.The ESIM variance prediction error is EI in predicting EH is the ratio of (1−λ12)σHE2 over σHE2 ; thus, the greater λ12 is, the more effective EI is in predicting EH.The relative effectiveness of EI on EH or the total variance of EH explained by EI is σIE2=λ12σHE2, and the relative mean squared effect can be measured by λ12.The mean squared effect of Cerón‐Rojas and Crossa  showed t7.5r of t (r < t) traits by predicting only the genetic gains of (t − r) of them. In Equation j to μj + jd, where jd is a predetermined change in μj. In the RESIM, jd = 0 , where r is the number of predetermined proportional gains.Similar to the CLPSI Equation , the obj7.6KE = [It − QE] and QE = P−1M(M′P−1M)M′ are the same as those obtained in Equation It is an identity matrix t × t, and λP2 and bp are the eigenvalue and the eigenvector of matrix KEP−C, respectively. When N = U, bp = bR (the vector of coefficients of RESIM), and when U′ is a null matrix, bp = bE (the vector of coefficients of ESIM). That is, PPG‐ESIM is more general than RESIM and ESIM and includes the latter two indices as particular cases.According to Cerón‐Rojas et al. , the PPG7.7β and bC = Kβ, respectively) are in the space generated by the eigenvectors of Equations β and bC can be written asj* and ja* are constants that can be estimated by least squares, and bEj and bPj are the eigenvectors of Equations β and bC are in the space generated by the eigenvectors of Equations β and bC.Cerón‐Rojas et al. , 2016b d7.8I^=β^′y. In the above equations, P^(P^−1) and C^ are REML estimators of the phenotypic (P) and genotypic (C) covariance matrices, whereas σ^I=β^′P^β^ is the estimator of the standard deviation of I^, and w′C^w is the estimator of the standard deviation of H.Cerón‐Rojas and Crossa  have desP^ (P^−1) and C^ are REML estimators, by the invariance property of maximum likelihood estimators  and genetic (C^) covariance matrices among the rate of lay , age at sexual maturity , and egg weight (kg), were P^=240.57−95.622.07−95.62167.204.582.074.5822.80 and C^=29.86−17.90−4.13−17.9018.561.49−4.131.499.24, respectively. Furthermore, the number of genotypes and the vector of economic weights were n = 3,330 and w′ = [19.54 −3.56 17.01], respectively, and the selection intensity was 10% (k = 1.755) for both indices.We compare the ESIM efficiency versus LPSI efficiency using a real data set from commercial egg poultry lines obtained from Akbar et al. . The estβ^=w′P^−1C^=1.82−1.383.25, and the LPSI estimated selection response, expected genetic gain per trait, and correlation was R^=kβ^′P^β^=74.91, E^′=kβ′^C^β^′P^β^=[2.70−2.200.84], and ρ^HI=β^′P^β^w′C^w=0.362, respectively.The estimated LPSI vector of coefficients was b^′E1b^E1=1, the best way of comparing ESIM results versus LPSI results is when the LPSI coefficient vector is normalized ; however, it can be shown that the normalization process only affects the estimated LPSI selection response because in that case, R^=74.91 is divided by β^′β^. For example, for this data set result, β^′β^=15.76; then, the estimated LPSI selection response using β^∗=β^/β^′β^ will be R^=74.9115.74=4.75, while the rest of the estimated LPSI parameters will be the same. When 0<β^′β^<1 and 1<R^, the values of R^ will increase, but when 1<β^′β^, the values of R^ will decrease, as in this example. The product β^′β^ does not affect ρ^HI because it is invariant to scale change. Also, β^′β^ does not affect E^ because β^′β^ appears in the numerator and denominator of both estimated parameters.Because in the ESIM context b^E1 as described earlier  in order to achieve expected genetic gain per trait values in RL and SM.In ESIM, the sign and proportion of the expected genetic gain values for traits RL, SM, and egg weight should be according to the breeder's interest. For example, if the breeder's interest is that the expected genetic gain per trait for RL be positive and negative for SM, the sign and proportion of the values of the first eigenvector should be modified using a linear combination of P^−1C^. We need to find the eigenvalues and eigenvectors of the equation [(P^−1C^)(P^−1C^)′−μ^jI]b^Ej=0, where μ^j=λ^Ej4=4.75, the estimated ESIM selection response was higher than the estimated LPSI response for this data set.The estimated ESIM selection response and expected genetic gain per trait were E^′ is a good result but E^′E is wrong. We can change the sign and proportion of E^′E by transforming b^E1 as β^E1=Fb^E1 using a convenient matrix F such asNow suppose that the breeder's interest is to increase RL and decrease SM; then F=−900010001.β^′E1=b^′E1F=[1.5310.0260.981], R^E=1.755β^′P^β^=42.44, and E^′E=1.755(β′^C^)/(β′^P^β^)=2.990−1.850.205. However, vector β^′E1 has not been normalized. To normalize β^′E1, we need to divide it by β^′E1β^E1=3.314, but β^′E1β^E1 should only affect R^E=42.44, which should be divided by 3.314; that is, R^E=42.44/3.314=12.806. According to the theory of similar matrices  should not be affected by matrix F.In such a case, 7.10RIndSel is a powerful freely available Java software useful for estimating the LPSI, LGSI, CLPSI, ESIM, etc., parameters and selecting individual candidates as parents for the next selection cycle. It is available in the CIMMYT  Selection indices for plant breedingRIndSel  repository at https://data.cimmyt.org/dataverse/root?q=rindsel. RIndSel is compatible with Windows XP, 7, 8, and 10 and can be installed on 32‐bit and 64‐bit computers. To input the data information, the user must create an Excel file (saved as a comma‐delimited “.csv”) with the data information to be analyzed, from where the software reads that information and estimates the index parameters.RIndSel uses REML and two experimental designs  to estimate the phenotypic and genotypic covariance matrices. This software presents the results of the estimated index parameters in two Excel files  and one text file (outextSmith.txt). File “alloutSmith.csv” contains all the trait adjusted means for all genotypes, and the estimated index values, whereas file “outSmith.csv” contains the genotypes and the trait values selected with the index values after these have been ranked. File “outextSmith.txt” contains all information associated with the estimated index parameters such as the estimated phenotypic and genotypic covariance matrices, the estimated selection response, etc. A complete user manual with instructions on how to use RIndSel in plant breeding is available at https://data.cimmyt.org/dataset.xhtml?persistentId=hdl:11529/10854.In the linear mixed model context , the LSI parameters  are the same.y (Searle et al., The BLP theory provides the mathematical‐statistical framework to develop the LSI theory. Thus, the LSI is a particular case or an application of the BLP to plant and animal breeding selection. This result allows breeders to locate the LSI theory in a general mathematical framework and then confidently use it when making selections. Nevertheless, some problems of the LSI not discussed in this work are associated with the estimation of the LSI parameters. For example, when the phenotypic and genotypic covariance matrices are estimated, the LPSI could not be BLP; indeed, it is not linear in J. Jesus Cerón‐Rojas: Conceptualization; Investigation. Jose Crossa: Conceptualization; Investigation.The authors do not have any conflict of interest.


Congenital heart diseases (CHDs) remain a significant cause of infant morbidity and mortality. Epidemiological studies have explored maternal risk factors for offspring CHDs, but few have used genetic epidemiology methods to improve causal inference.N = 562 CHD cases) were included. We used Mendelian randomisation (MR) analyses to explore the effects of genetically predicted maternal body mass index (BMI), smoking and alcohol on offspring CHDs. We generated genetic risk scores (GRS) using summary data from large-scale genome-wide association studies (GWAS) and validated the strength and relevance of the genetic instrument for exposure levels during pregnancy. Logistic regression was used to estimate the odds ratio (OR) of CHD per 1 standard deviation (SD) higher GRS. Results for the three cohorts were combined using random-effects meta-analyses. We performed several sensitivity analyses including multivariable MR to check the robustness of our findings.Three birth cohorts, including 65,510 mother/offspring pairs  per 1SD higher GRS: 0.95 ), lifetime smoking ) and alcoholic drinks per week ) were close to the null. Sensitivity analyses yielded similar results.Our results do not provide robust evidence of an effect of maternal BMI, smoking or alcohol on offspring CHDs. However, results were imprecise. Our findings need to be replicated, and highlight the need for more and larger studies with maternal and offspring genotype and offspring CHD data.The online version contains supplementary material available at 10.1186/s12916-023-02731-y. Congenital heart diseases (CHDs) are the most common congenital anomaly, affecting around 1% of live births and 10% of stillbirths . CHDs arSeveral modifiable maternal characteristics have been found to be associated with increased risk of CHDs, including maternal pre/early pregnancy body mass index (BMI) –8, smokiMendelian randomisation (MR) uses genetic variants as instrumental variables (IVs) to test causal effects in observational data . The keyThis study is reported using the Strengthening The Reporting of Observational Studies in Epidemiology Using Mendelian Randomisation (STROBE-MR) guidelines  chip, and ALSPAC children were genotyped using Illumina HumanHap550 quad genome-wide SNP genotyping platform. Genotype data for both ALSPAC mothers and children were imputed against the Haplotype Reference Consortium v1.1 reference panel, after performing the QC procedure  ≥ 1%, a call rate ≥ 95%, in Hardy–Weinberg equilibrium (HWE), correct sex assignment, no evidence of cryptic relatedness, and of European descent). The samples of the BiB cohort (mothers and offspring) were processed on three different type of Illumina chips: HumanCoreExome12v1.0, HumanCoreExome12v1.1 and HumanCoreExome24v1.0. Genotype data were imputed against UK10K + 1000 Genomes reference panel, after a similar QC procedure . In MoBa, blood samples were obtained from both parents during pregnancy and from mothers and children  at birth . Genotypp-value threshold used to indicate genome-wide significance after accounting for multiple testing. Of those reaching this threshold we ensured that we only took forward independent SNPs to create the GRSs (described below). For BMI, there were 941 near-independent SNPs in a combined GWAS of ~ 700,000 individuals as reported in Yengo et al. [P < 1 × 10−8 after a conditional and joint multiple SNP analysis to take into account linkage disequilibrium (LD) between SNPs at a given locus). For smoking analyses, there were 126 independent SNPs (genome-wide significant (p < 5 × 10−8) SNPs that achieved independence at LD r2 = 0.001 and a distance of 10,000 kb). The study was a GWAS of a lifetime smoking index , conducted in a sample of 462,690 current, former and never smokers in UK Biobank [P < 5 × 10−8), measured as the number of alcoholic drinks per week [We selected SNPs from the largest and most relevant GWAS of European ancestry participants for each exposure  and SNP is the genotype dosage of exposure-increasing alleles at that locus . Selected SNPs were extracted from the imputed genotype data in dosage format using QCTOOL (v2.0) and VCF tools (v 0.1.12b) in ALSPAC and BiB, respectively. PLINK (v1.9) was then used to construct the GRS for each exposure coded so that an increased score was associated with increased exposure. In MoBa, we constructed the GRSs from the QC’d data in PLINK format. Further information on GRS construction for each cohort is shown in Additional File In the ALSPAC cohort, cases were obtained from a range of data sources, including health record linkage and questionnaire data up until age 25 following European surveillance of congenital anomalies (EUROCAT) guidelines . In BiB,As noted above, the SNP selection and weights for the GRS were taken from GWAS in women and men , 30, 32.Analyses were performed in R version 4.0.2 . A pre-specified analysis plan was uploaded to the Open Science Framework . We undeR2. For smoking and alcohol, instrument strength was assessed using the area under the ROC curve and pseudo-R2 by the Nagelkerke method [The key assumptions of MR are (i) relevance assumption, (ii) independence assumption and (iii) exclusion restriction. To explore the relevance of the GRS to each exposure in pregnancy, we undertook linear (BMI) and logistic  regression to derive the difference in mean BMI and OR of pregnancy smoking and pregnancy alcohol consumption per 1SD higher GRS in each cohort. For BMI, instrument strength was assessed with F-statistics and e method . To minie method . We alsoe method . Methodse method –54. In se method .Analyses included 65,510 mother–offspring pairs, of which 562 offspring had CHD  per 1SD higher GRS: 1.01 , with no statistical evidence of between study heterogeneity Fig. B. The smThe maternal GRS proxying drinks per week also had a pooled OR close to the null, but with CIs ranging from 2% reduced odds to 15% increased odds ) Fig. D–G.R2 and AUC suggested potential weak instrument bias for alcohol and smoking. Weak instruments in this study would be expected to bias results toward the confounded association. Weak instruments and imprecise associations also limit clear interpretation of our sensitivity analyses to explore bias due to GRS influencing CHDs via other paths independently of the exposure of interest. We tried to identify all cohorts with maternal genetic data and offspring, CHD measures and to the best of our knowledge, this is the first MR study of these maternal exposures on offspring CHD risk. Despite the relatively large sample, our inconclusive findings highlight the importance of existing and new cohorts, many of which have genomic data, linking to health care records to obtain information on CHDs and other rare outcomes, for example through electronic health records.Using MR across three birth cohorts, we found no strong evidence for an effect of genetically predicted maternal BMI, smoking or alcohol on risk of offspring CHD. However, for all three exposures, confidence intervals were wide and the pseudo This MR study complements our previous negative paternal control study . Our MR There are several strengths of the current study. We attempted to identify all studies with relevant data knowing that MR is statistically inefficient, and CHD is a rare outcome. We explored potential bias due to other paths from the GRS to CHDs by examining associations of each GRS with the other two exposures and with other risk factors that might influence CHD and undertook multivariable MR where such associations were found. We also adjusted for offspring genotype in a subsample of the pooled data cohort, which is important in attempting to separate the influence of a path from GRS to CHDs via foetal genotype rather than solely from the mothers’ exposure .N = 65,510, N = 562 CHD cases) the effect estimates were imprecise due to CHD being a rare condition. In knowing this, we explored a collaborative base of birth cohorts and searched the literature for any additional cohorts that might contribute but found none that were eligible for inclusion. Furthermore, the GRS for smoking and alcohol may have been biased by the weak instruments. These limitations importantly contribute to our main and sensitivity analysis results. We were not able to clearly differentiate between horizontal  and vertical pleiotropy (i.e. where the relation of the smoking GRS with BMI is downstream of the GRS effect on smoking). We want to adjust for horizontal pleiotropy but not vertical pleiotropy, as the latter would be adjusting away part of the mechanism by which, for example smoking might influence CHD. Adjustment for offspring genotypes was only possible in a subsample of the main analysis, making results more imprecise and prone to selection bias. However, it is encouraging that our results do not notably differ in these analyses. MR results may be biased by population stratification confounding. We tried to mitigate against that by adjusting for ancestry PCs and exploring the consistency of the main results with results removing BiB. Largely these were consistent but even more imprecise. Whilst we included all participants, including those from non-European ancestries, both MoBa (the largest contributing study) and ALSPAC participants are mostly of White European origin, and the GWAS data used to construct the GRSs were restricted to European participants. Therefore, our results may not generalise to other populations. By examining gene associations, without estimating causal effects, as we have done here , the three IV assumptions that we discuss need to be considered. However, with this approach, the fourth assumption, which is often ignored even in MR studies that do attempt to estimate causal effects, is not required. The fourth assumption which often receives less attention states that the effect of the exposure on the outcome may differ for different people [The key limitation of this study is that despite a relatively large sample size  and therefore could have missed the potential effects of these exposures on specific CHD subtypes. Nevertheless, we believe there is value for prospective parents, clinicians and policy makers in knowing the effects of any CHD. MoBa cohort only had cases diagnosed antenatally or around the time of birth (first year of life) obtained from a single source  which would increase the chances of outcome misclassification by assigning CHD cases which were diagnosed later in life as non-CHD cases. This misclassification is likely to be random with respect to the GRS  and would be expected to bias results towards the null, meaning we may have missed some associations. A previous study using MoBa data found a larger proportion of CHD cases than the present study (1.39% vs 0.83%) via ascertainment of linked health records . We wereUK Our Future Health) gain consent to collect health data on CHDs . (iv) Continue to update the cohorts used in this study and update our results. For example, there are plans to continue running GWAS assays on mothers, fathers and offspring in MoBa who are currently not genotyped which will more than double the sample available in that study. (v) To the best of our knowledge, there are currently no publicly available GWAS summary statistics for CHD. To date, the largest GWAS for CHD in a European population included ~ 4000 cases [Identifying modifiable causal risk factors for the development of CHD is important for developing preventive interventions to reduce the risk of CHDs. Improvements in surgery over the last two decades mean that most patients with CHD now live into adulthood. Nonetheless, prevention remains important. Many patients require repeat procedures through childhood and adolescence to accommodate their growth, which produces a burden on them, their family and society. Despite trying to identify all relevant studies our results are inconclusive. They highlight the need for more data with maternal genetic and offspring CHD data. We think this is possible over the coming years as running GWAS is relatively cheap, and most cohort studies increasingly have these data. The following could considerably increase the sample size for MR in this field and result in key advances in preventing CHDs: (i) Add data on CHDs through electronic record linkage to existing cohorts; this was done recently in ALSPAC nearly 30 years after the original pregnancies . Many of00 cases . As thesThe analysis steps taken in this paper aimed to explore the presence of a causal effect of maternal BMI, smoking and alcohol on offspring CHDs. In summary, we found no robust evidence of an effect for maternal genetically determined BMI or smoking on offspring CHD. We did observe a weak relationship between genetically predicted maternal alcohol intake on offspring CHDs, but this may be explained by weak instrument bias. Despite a large sample size, our results produced imprecise estimates. We have highlighted the need for future larger studies that employ a range of causal methods to further interrogate maternal gestational risk factors for offspring CHDs.Additional file 1. STROBE-MR checklist of recommended items to address in reports of Mendelian randomization studies.Additional file 2:Text S1. Genetic data methods; Table S1. Further information on the genome-wide association studies used to generate genetic risk scores; Text S2. Genetic risk score generation; Text S3. Defining congenital heart disease; Table S2. Subcategories of CHD; Text S4. Describing the pregnancy phenotype data: maternal BMI, smoking, alcohol, education, parity, diabetes separated by each cohort; Text S5. Genetic risk scores for multivariable Mendelian randomisation (MVMR); Table S3. Exploring associations between the BMI GRS and risk factors for CHDs. We also include the association of the BMI GRS with BMI  for comparison; Table S4. Exploring associations between the smoking GRS (lifetime smoking index) and risk factors for CHDs. We also include the association of the smoking GRS with smoking  for comparison; Table S5. Exploring associations between the alcohol GRS (drinks per week) and risk factors for CHDs. We also include the association of the alcohol GRS with alcohol  for comparison; Figure S1. Showing the main results and results from additional analyses for the MR analyses of genetically predicted maternal BMI and offspring CHDs; Figure S2. Showing the main results and results from additional analyses for the MR analyses of genetically predicted maternal smoking (using a genetic risk score of a lifetime smoking index) and offspring CHDs; Figure S3. Showing the main results and results from additional analyses for the MR analyses of genetically predicted maternal alcohol consumption (using a genetic risk score of drinks per week) and offspring CHDs.


The present study estimated the heritability, phenotypic and genetic correlations for different pelt quality measures, body weight and length traits in mink. Body weight and length measured on live animals in November of the first year of life were reliable indicators of dried pelt size, without negative impacts on pelt quality traits. In addition, body length in November and harvest had moderate positive genetic correlations with dried pelt quality, which made this trait an appealing trait to select for increased pelt size. The estimated genetic parameters for these traits can be used to improve fur characteristics in Canadian mink populations.p < 0.05) of random effects  and fixed effects  for each trait. Subsequently, bivariate models were used to estimate the genetic and phenotypic parameters using ASReml 4.1. Heritability (±SE) estimates were 0.41 ± 0.06 for DPS, 0.23 ± 0.10 for DNAP, 0.12 ± 0.04 for DQU, 0.28 ± 0.06 for LQU, 0.44 ± 0.07 for LNAP, 0.29 ± 0.10 for Nov_BW, 0.28 ± 0.09 for Nov_BL, 0.41 ± 0.07 for HW and 0.31 ± 0.06 for HL. DPS had high positive genetic correlations (±SE) with Nov_BW (0.89 ± 0.10), Nov_BL (0.81 ± 0.07), HW (0.85 ± 0.05) and HL (0.85 ± 0.06). These results suggested that body weight and length measured on live animals in November of the first year were reliable indicators of dried pelt size. DQU had favorable genetic correlations with Nov_BL (0.55 ± 0.24) and HL (0.46 ± 0.20), and nonsignificant genetic correlations with DNAP (0.13 ± 0.25), Nov_BW (0.25 ± 0.25) and HW (0.06 ± 0.20), which made body length traits an appealing trait for selection for increased pelt size. High positive genetic correlation (±SE) was observed between LNAP and DNAP (0.82 ± 0.22), which revealed that nap size measurement on live animals is a reliable indicator trait for dried pelt nap size. However, nonsignificant (p > 0.05) low genetic correlation (±SE) was obtained between LQU and DQU (0.08 ± 0.45), showing that indirect selection based on live grading might not lead to the satisfactory improvement of dried pelt overall quality. The estimated genetic parameters for live grading, dried pelt quality, and body weight and body length traits may be incorporated into breeding programs to improve fur characteristics in Canadian mink populations.Understanding the genetics of fur characteristics and skin size is important for developing effective breeding programs in the mink industry. Therefore, the objectives of this study were to estimate the genetic and phenotypic parameters for pelt quality traits including live grading overall quality (LQU), live grading nap size (LNAP), dried pelt size (DPS), dried pelt nap size (DNAP) and overall quality of dried pelt (DQU), and body length and weight traits, including November body weight (Nov_BW), November body length (Nov_BL), harvest weight (HW) and harvest length (HL) in American mink. Dried pelt quality traits on 1195 mink and pelt quality traits on live animals on 1680 were collected from mink raised at two farms, in Nova Scotia and Ontario. A series of univariate analyses were implemented in ASReml 4.1 software to identify the significance ( Neogale vison) is the most favored fur-bearing animal being raised under an intensive production system. Mink farming was initiated in Canada in 1866 [American mink  and dried skins (pelt traits). Live grading is subjective and performed by certified technicians on the farm, while evaluation of fur characteristics is performed on dried pelt using sorting machines by certified fur grading specialists in auction houses. Although the price of pelts is highly determined based on pelt quality traits, these traits are not available until post-harvest. Alternatively, live grading and body weight and length measurements can be used as potential indicators of dried pelt characteristics ,11. TherCanada is a major producer and exporter of mink pelt in the world with a production record of 1.76 million pelts in 2018 ; howeverhttps://www.nfacc.ca/pdfs/codes/mink_code_of_practice.pdf accessed on 25 March 2018).The present study was approved by the Dalhousie University Animal Care and Use Committee . Moreover, the mink used in this research work were cared for following the Code of Practice for the Care and Handling of Farmed Mink guidelines  at Dalhousie University, Faculty of Agriculture  and Millbank Fur farm . The animals consisted of five color types, including dark, demi, mahogany, pastel and stardust. All mink were raised under standard farming conditions and had ad libitum access to food and water. The diets were formulated based on the nutrient requirements of animals in each production period. No formal breeding program was used in CCFAR or Millbank Fur Farm. Before each breeding season, weak and infertile animals were culled from the herd in December, and those with an adequate score for live grading, disease history, and reproductive records were kept for breeding. The pedigree file included 25,688 animals  and was traced through 16 generations. n = 1608) for mink in CCFAR at the end of November in 2018, 2019 and 2021 when the coat was in prime condition. Live quality grading of pelage was performed on these mink using the North American Fur Auctions (NAFA) live animal grading procedure by their certified technician. Live grading traits included the overall quality of fur (LQU) and nap size (LNAP). LQU was scored into three categories from 1 (poor) to 3 (best). LNAP was measured as the length of guard hair protruding from the underfur and scored into five categories from 1 (long) to 5 (short). Pelt quality traits were assessed on live animals (n = 1195); for mink in Millbank, pelting was carried out in a pelting facility located in Millbank Fur Farm  and mink from CCFAR were sent to custom pelting facilities . The dried raw pelts were shipped to the North American Fur Auctions-NAFA  and Saga Furs  auction houses for fur quality evaluation and sale. Dried pelt quality traits included dried pelt size (DPS), dried pelt nap size (DNAP) and overall quality of dried pelt (DQU). Evaluation of DPS and DNAP was performed using sorting machines on dried skins. All skins were stretched with differential weights to adjust skin lengths into categories. DPS was measured from the tip of the nose to the base of the tail. DPS was classified into nine categories of 47.1–53 cm (category 1), 53.1–59 cm (category 2), 59.1–65 cm (category 3), 65.1–71 cm (category 4), 71.1–77 cm (category 5), 77.1–83 cm (category 6), 83.1–89 cm (category 7), 89.1–95 cm (category 8) and 95.1–101 cm (category 9). DNAP was scored into eight categories: 8 (extra short nap), 7 (short nap), 6 (short-medium open), 5 (short-medium nap), 4 (short-medium to medium nap), 3 (medium nap), 2 (medium nap to medium-long nap) and 1 (medium-long nap). DQU was assessed by professional fur grading experts in auction houses and classified into four categories: Mink were euthanized in December 2018 and 2019 ((1) Bronze: weakest grade assigned to the pelts, no damages, but weakest in terms of underfur, guard hair or general appearance. These pelts were very flat, coarse, weak and loose; (2) Silver: these pelts were complete and prime and had silky appearances. However, they had weaker underfur and poorer or uneven coverage or coarser guard hair; (3) Silver to golden: higher quality than silver pelts but not as good golden quality furs; (4) Golden: these pelts had very high-quality with good and even guard hair coverage and dense underfur. These pelts were fully prime and had smooth appearances and silky textures. n = 1734) in mid-November 2018 and 2019. Nov_BL was measured from the snout to the tail base of each mink. Following euthanasia in December 2018 and 2019, harvest body weight (HW) and harvest body length (HL) were collected from 2162 mink. HW was obtained by measuring the weight of a whole body of animals, and HL was the length of body measured from the snout to the base of tail on a whole body of mink. November body weight (Nov_BW) and November body length (Nov_BL) were measured on live animals at approximately seven months of age (p < 0.05) of nongenetic factors, including fixed effects of farm (CCFAR and Millbank Fur farm), year , sex , color type  and age (1 and 2 years), and random effects of additive genetics, maternal genetics and common litter were tested for studied traits using univariate models implemented in ASReml 4.1 [p < 0.05) effects were included in subsequent mixed model analyses. The following general univariate model was used:y is the vector of phenotypic observations; b is the vector of fixed effects; a is the vector of random additive genetic effects; m is the vector of random maternal genetic effects; c is the vector of common litter effects and e is the vector of residual effects; and X, Z, G and W were the incidence matrices relating the phenotypic observations to fixed, random additive genetic, maternal genetic and common litter effects, respectively. It was assumed that random effects were independent and normally distributed:A is the numerator relationship matrix; I is an identity matrix; The significant influence variance components estimated by bivariate models.Genetic and phenotypic parameters for dried pelt, live grading and body weight and length traits were estimated using animal models. The number of records, mean, range, standard deviation and coefficient of variation are presented in The LQU, HW and Nov_BL had the highest coefficient of variation (CV) among the studied traits . The presence of these variations indicated that there might be potential to improve them through genetic/genomic selection. In the present study, the estimated CVs for DQU, LNAP and LQU were 22.87%, 30.94% and 35.14%, respectively . Thirstrp < 0.05) for all pelt and body weight and length traits (p < 0.05) for DPS, Nov_BW, Nov_BL, HW and HL. This is in agreement with a previous investigation that reported a difference in mature body weight and body length between female and male mink [In order to determine the influence of nongenetic factors on phenotypic variation of the traits, the significance of fixed effects and nongenetic environmental effects were examined using univariate animal models. The fixed effects of farm and year were significant  for DNAP, LNAP, Nov_BW and Nov_BL (p < 0.05) for yearling staple length (0.03) and yearling body weight (0.18) [Estimated variance components for each trait obtained from univariate models are presented in d Nov_BL  and explt (0.18) .p < 0.05) for HL (0.09 ± 0.03). The proportion of common litter effects was not tested for live grading and pelt traits since there were no common dams in different years of data collection for these traits. However, previous studies reported a minor contribution of common litter effects on live grading and pelt traits in mink, ranging from 0.01 to 0.12 [The random common litter effect was only significant  were 0.41 ± 0.06 for DPS, 0.29 ± 0.10 for Nov_BW, 0.28 ± 0.09 for Nov_BL, 0.41 ± 0.07 for HW and 0.31 ± 0.06 for HL . The modAlopex lagopus), the heritability of LNAP is estimated at 0.19 [Heritabilities (±SE) estimated for DNAP, DQU, LNAP and LQU were equal to 0.23 ± 0.10, 0.12 ± 0.04, 0.44 ± 0.07 and 0.28 ± 0.06, respectively . Low to  at 0.19 , which i at 0.19  estimate at 0.19 ,15,25,29Phenotypic and genetic correlations between body weight, body length and pelt traits are shown in p < 0.05). However, DQU had positive genetic correlations with Nov_BL (0.55 ± 0.24) and HL (0.46 ± 0.20), suggesting that body length traits might be good indicator traits to improve both dried pelt size and overall quality. On the other hand, the genetic correlations of DQU with Nov_BW (0.25 ± 0.25) and HW (0.06 ± 0.20) were not significant (p < 0.05). However, Thirstrup et al. [All phenotypic correlations between DQU and body weights and lengths in November and harvest time were not significant  phenotypic or genetic correlations were estimated between DNAP and body weight and length traits, which ranged from 0.29 ± 0.19 to 0.48 ± 0.32 for body weight and from −0.01 ± 0.30 to −0.02 ± 0.19 for body length traits. These results showed that independent genes might be involved in controlling pelt quality, and body weight and length traits suggest that selection for larger body weight and length would not have a negative impact on dried pelt nap size.No significant (p > 0.05). These results suggested that selection for larger body size in November of the first year of life would not negatively affect live grading traits in mink. Similar to our results, Thirstrup et al. [Phenotypic and genetic correlations between live grading and body weight and length traits are shown in p et al.  reportedA strong positive genetic correlation (0.82 ± 0.22) was observed between LNAP and DNAP . This suIndirect selection for pelt traits based on live grading is in demand by mink breeders since they can be measured on selection candidates. In the present study, high positive genetic correlation was estimated between LNAP and its corresponding dried pelt trait (0.82 ± 0.22) and between body weight and length traits with DPS ranging from 0.81 ± 0.07 to 0.89 ± 0.10. However, the genetic and phenotypic correlations between LQU and DQU were poor and nonsignificant. Therefore, selection based solely on live grading may not lead to the maximum improvement for dried pelt characters in mink. Peura et al.  suggesteOur study provides insights into the proportion of genetic and environmental sources of phenotypic variation in dried pelt, live grading and body weight and body length traits in American mink. The absence of live grading records for Millbank farm restricted the number of available records for these traits, which might impact the genetic correlations between live grading traits and pelt traits in our study. The estimated genetic parameters in this study provided the basic knowledge for designing the genetic selection programs for fur quality in Canadian mink populations.Genetic selection for fur quality and skin size can increase the production efficiency and consequently the economic profits of mink farmers. The present study was the first estimation of genetic parameters for fur quality traits in Canadian mink populations. Estimated moderate heritabilities for pelt, live grading and body weight and length traits suggested that improvement of these traits is possible using genetic/genomic selection. The estimated genetic parameters showed the potential of Nov_BW and Nov_BL as good indicators of DPS without negative effects on DQU and DNAP. The presence of moderate positive genetic correlations between body length in November and harvest with DQU makes these traits suitable for select to increase DPS and DQU. In addition, live grading nap size is a reliable indicator of dried pelt nap size. The results established a foundation for a more efficient selective breeding method for the mink industry that can be incorporated into a multi-trait genetic or genomic selection program.