The INTERGEN package consists of a collection of routines written in Fortran 90/95 that are useful for animal and plant breeding genetic analysis. It includes different modules for managing sparse and dense matrix allocation and operations, MME solvers such as FSPAK and preconditioned conjugate gradient (PCG), Gibbs and Metropolis-Hastings sampling algorithms for variance component estimation, and Hash and IJA storage computational strategies (Misztal, 1999). Initially, it was developed to conduct Bayesian research involving complex structures, such as genotype-by-environment interactions utilizing reaction norm models with unknown covariates in multibreed populations with uncertain paternity. This encompasses considerations for heterogeneity of residual variances, robustness analysis, and reduced animal models (Cardoso and Tempelman, 2003; Cardoso and Tempelman, 2004). INTERGEN has since been updated to include genetic merit estimation, incorporating (or not) genomic information using BLUP, and normal priors for fixed effects (Sorensen and Gianola, 2007). More recently, various algorithms have been implemented to enhance the statistical modeling capabilities of the package, allowing it to handle diverse management situations commonly encountered in livestock, genomic information, iterative algorithms for solving MME without memory allocation, and approximate accuracy estimation. The package comprises three binary programs written in Fortran 90/95: intergen, intergeniod, and intergenacc. In this section, we will briefly describe the differences among the binaries. The following sections will provide additional details about the functionalities currently available in the package.

The software intergen is the most comprehensive among the three options available for analyzing genetic data using Bayesian and frequentist methods. It employs computational techniques to manage sparse operations and solve the MME for single-trait, multi-trait, and random regression models, which are commonly used in genetic analysis. The software can estimate variance components (Bayesian only) and derive solutions. INTERGEN enables users to define complex models with unlimited random and fixed effects. The MME are constructed and allocated in memory for Bayesian and BLUP analysis.

The software shares most components (i.e., modules, subroutines, and functions) for building equations between the Bayesian and BLUP methods. The most computationally demanding step in the analysis involves executing matrix operations, including ordering, symbolic and numerical factorization, and inversion. Among these, factorization and sparse inversion operations are the most resource-intensive (Junqueira et al., 2022b). In BLUP analysis, the MME solutions can be derived by exact inversion of the coefficient matrix or by using iterative techniques like PCG. All operations in the software are executed by allocating the equations in memory. Therefore, the computational capacity of the software depends on the RAM available in the user’s computer. Computational limitations arise in large-scale analyses when all equations cannot be allocated in memory, particularly in multi-trait genomic analyses, and when prediction error variance (PEV) is required from the coefficient matrix inverse. For example, when analyzing over 25,000 genotyped individuals in multi-trait models, the software intergeniod is necessary for solving mixed-model equations using iteration on data (IOD), as RAM would reach its limits on most servers.

The intergeniod software utilizes the IOD technique with PCG to solve the MME and obtain solutions for all the fixed and random effects specified in the model. This software is recommended whenever BLUE and BLUP solutions for large datasets are required and allocating memory for all equations is not feasible due to memory constraints. Schaeffer and Kennedy (1986) were responsible for the first IOD implementation in animal breeding. These authors implemented IOD with successive overrelaxation (SOR), which requires sorting the equations in a predefined order to solve them efficiently when allocated in memory. However, it is not flexible enough to handle different models without modifying the algorithm (Lidauer et al., 1999; Strandén and Lidauer, 1999). On the other hand, PCG is generic enough to handle any model with unlimited covariates and cross-classified effects. Below is a generic version of the PCG solver algorithm. Assume an MME denoted as Ax = b in which A is the left-hand side matrix, x is the vector of unknown solutions, and b is the right-hand side matrix. In the PCG, some vectors need to be initialized before starting the loop process, which is supposed to stop when reaching the convergence criteria defined by the user (default is 10¹²). The initial values of the scalars and vectors are n = 0, w = 0, x = 0, r₀ = b − Ax₀, d₀ = M⁻¹r₀, f₀ = r'₀d₀.

Figure 1

Note that only a few vectors must be allocated in memory in this solver. In each iteration, the algorithm updates the solution vector x_n, the residual vector r_n, and the search direction d_n. The matrix-vector product q_n = Ad_n-1 is used to compute the step size (α_n). The f_n is a scalar representing the inner product of the residual and its preconditioned form. The direction vector is then updated using β_n, ensuring the conjugacy of successive directions. The most complex computation is related to derivation q = Ad as it requires estimating all elements of the coefficient matrix without allocating them in memory and multiplying them by the vector d. Tsuruta et al. (2001) stated that the preconditioner matrix (M) influences the convergence rate of PCG. The more straightforward construction—and more computationally efficient in terms of memory savings—of this square and symmetric matrix can become a vector if only diagonals of the coefficient matrix are utilized. Other possibilities rely on block diagonal structure, resulting in faster convergence rates at the expense of using more memory.

The intergenacc package is the third binary designed to estimate the approximate accuracies of estimated breeding values using the MME method. This software analyzes large-scale datasets in which accuracies are required, and dimensionality and memory limitations prevent inverting the coefficient matrix of MME. It estimates the accuracies of random effects, such as animal or maternal effects, adjusted for one fixed effect and one additional random effect. This correction method is consistent across algorithms. The method implemented has been detailed in several papers (Misztal and Wiggans, 1988; Misztal et al., 1991; Misztal et al., 1993; Misztal et al., 2013). The current version of intergenacc supports single-trait models, with or without genomic information, and can estimate additive and maternal accuracies in separate runs.

As previously mentioned, the INTERGEN package offers various useful features for conducting genetic evaluations of animals and plants. This section presents a brief overview of these functionalities. For more detailed guidance on using the software, please refer to the manual or contact the research group directly.

The software allows five different types of effects to be used to build the MME matrices. These effects include cross (cross-classified), cov (covariate), unknowncov (cross-classified covariate for reaction norms via random regression), rnorm (reaction norms via random regression), and ram (reduced animal model). In the EFFECT section of the parameter file, these effects are listed along with the number of levels and any indication of a nested effect for random regression modeling. Additionally, in this section, users can indicate if they wish to save samples of each MCMC cycle in an external file during Bayesian analysis. The software can handle an unlimited number of effects.

The intergen software accommodates six types of residual (co)variance structures. It can fit both single- and multi-trait models within Bayesian framework, and allows Gaussian and heavier-tailed alternatives like Student’s t or Slash densities for estimating variance components. Users can specify three residual densities, allowing for either homoscedastic (i.e., homogeneous) or heteroskedastic error specifications. Below are further details on each of the allowed specifications:

Gaussian homoscedastic: normal distribution with homogeneous variance e ∼ N ( 0 , σ e 2 ) , with e as the vector of residual effects and the σ²_e residual variance common to all elements of MME. Currently, this is the only option allowed in the intergeniod.

Gaussian heteroskedastic: normal distribution with heterogeneous variance e ∼ N ( 0 , σ e ( i ) 2 ) , with e as the vector of residual effects and σ²_e(i) the residual variance defined for subclass i.

Student’s t homoscedastic: the Student’s t distribution (or simply t distribution) is a continuous probability distribution that generalizes the normal distribution with heavier tails with controlled parameters determined by v as the degree of freedom. This distribution was presented as an alternative for mitigating the effects of preferential treatment and deviant observations (Strandén and Gianola, 1998). Under the assumption of homogeneous residual variance distribution, it can be defined as e ∼ p ( e ∣ σ e 2 , w i ) = N ( 0 , σ e 2 w i ) , in which p(w_i|v_e) represents the lack of fit of the marginal density of e to the Gaussian distribution. Thus, the distribution of the phenotypes becomes y~t(0, v_e).

Student’s t heteroskedastic: this distribution is similar to the Student’s t homoscedastic, and the difference relies on the assumptions around the presence of heterogenous residual variance among the subclasses. Thus, we fit σ²_ei in e ∼ p ( e i ∣ σ e i 2 , w i ) = N ( 0 , σ e i 2 w i ) .

Slash homoscedastic: extending the derivation of Student’s t distribution, Slash arises when p(w_i|v_e) = v_ew^{ve− 1}, in which i = 1,2, …, n, v_e > 0, 0 < w_i < 1 such that σ E 2 = v e v e − 1 σ e 2 .

Slash heteroskedastic: in situations in which it is assumed heterogeneity of residual variances, e ∼ p ( e i ∣ σ e i 2 , w i ) = N ( 0 , σ e i 2 w i ) with w_i fitted as Slash homoscedastic.

In the case of Student’s t and Slash distributions, the Metropolis-Hastings algorithm is applied on every cycle of MCMC. Previous publications provide more details on the derivation of the heavy-tailed alternatives and their implications for prior, posterior, and marginal probabilities (Cardoso and Tempelman, 2003; Rosa et al., 2003; Cardoso and Tempelman, 2004; Cardoso et al., 2005; Cardoso et al., 2007).

Intergen currently accepts ten different (co)variance matrices for the random effects. These matrices can be calculated internally or imported into the software. The (co)variance structures defined at this stage are then added to the (off-)diagonal elements of the coefficient matrix. BLUP and MCMC models currently accept the following random types: diagonal, add_sire, add_animal, add_an_upg, add_an_upginb, add_an_ms, add_an_mb, diag_mb, user_file, user_file_i, and norm_prior.

For Bayesian analysis, additional parameters are required to define the hyperparameters of the distributions correctly. A pedigree file is needed as input for all types except diagonal types (i.e., diagonal, diag_mb, and norm_prior). The random types add_an_ms and add_an_mb handle multiple sires and breeds, respectively. Further statistical details on model parameterization can be found in Cardoso and Tempelman (2003) and Cardoso and Tempelman (2004). Furthermore, these random types can fit genetic groups, assumed as fixed, by including dummy parents in the pedigree file. The multi-sire and breed inferences are only available in Bayesian analysis. In such cases, the user should define the hyperparameter values for prior distributions (Dirichlet, Inverse Gamma, or Wishart). The software will run defined cycles of Metropolis-Hastings for each MCMC iteration and adjust the variance of the prior distributions based on the acceptance rates during the burn-in phase of MCMC.

The norm_prior allows the inclusion of prior knowledge (i.e., μ and σ²) on the fixed MME matrices of the model (Sorensen and Gianola, 2007). The value μ can be understood as the best estimate a priori, and σ² the variance associated with the uncertainty of that estimate. Thus, with lower values of σ² higher is the confidence of μ. In this scenario, the traditional mixed model equation is then adjusted as follows:

in which C is the coefficient matrix of MME, b is the vector of BLUE and BLUP solutions, V¹ is the (co)variance matrix for the fixed effects, and m is the vector of a priori means for each fixed effect level.

The MCMC and BLUP allow using three types of random effects: add_animal, add_an_upg, and add_an_upginb. Amongst these, add_animal is the simplest, and the other are extensions. This type utilizes information from the pedigree file, including the individual, parent 1, and parent 2. Iteratively, the inverse of the numerator relationship matrix (A) is built using this information without considering the inbreeding coefficients. The method implemented utilizes the concepts described in Henderson (1976) to directly create the inverse of A. The A matrix has many nonzero elements for a well-structured pedigree, but its inverse has only a few nonzero elements, making its construction fast and computationally efficient following Henderson (1976) rules. The add_an_upg type is the animal model that builds the A matrix with unknown parent groups (Westell et al., 1988), and add_an_upginb is the animal model with unknown parent groups and inbreeding. These models also allow the inclusion of genomic information from an external file. More details about the genomic module will be presented in the next section. The software provides two types of random effects that offer more flexibility: user_file and user_file_i. Using these types, the user can create their own (co)variance matrices externally and then upload them to the software. This functionality allows users to use INTERGEN even when no built-in features are available.

Our in-house genomic module has been under development for over a decade, and we have continuously modified it to enhance its computational efficiency in handling genomic information. While most non-genomic analyses are managed efficiently using sparse formats (Misztal, 1990), genomic analysis demands more efficient programming. The genomic module includes several parallel operations utilizing MKL libraries (Intel, 2024) to optimize computational performance.

Depending on the content of the pedigree, this module can handle single-step GBLUP (Aguilar et al., 2010) or GBLUP (genomic BLUP) (VanRaden, 2008). Assume, for instance, the following model:

in which y is the vector of phenotypes; X and Z are incidence matrices relating levels to phenotypes of the fixed and random effects, respectively; b is the vector of solutions of fixed (or systematic) effects; and a is the vector of random effects solutions. The vector a~N(0, G₀⨂H⁻¹), H⁻¹ is the inverse of the matrix describing the genetic relationships of the size of all individuals included in the pedigree file. This relationship matrix can be built assuming only pedigree in which H^-1 becomes A^-1, only genotypes in which H^-1becomes G^-1 (i.e., GBLUP), or both information jointly, also known as single-step GBLUP (Aguilar et al., 2010). The G₀ matrix contains the additive (co)variances and can have a scalar or a symmetric multidimensional matrix depending on whether it is a single- or multi-trait model. The e~N(0, I⨂R⁻¹), and R contains the residual (co)variances of the model, and the dimension also depends on the number of traits under evaluation. Then, based on the above statistical model, the MME that intergen and intergeniod handle can be described as follows:

The INTERGEN package has a genomic module shared among its three binaries. The current version of the module (version 1.0) can handle GBLUP and ssGBLUP models, depending on the pedigree provided by the user. If the pedigree file used in the analysis is empty (i.e., A and A₂₂ are diagonals), the software will build a GBLUP model by creating a genomic matrix. However, if the pedigree file is not empty, the software builds a ssGBLUP model. The software constructs the genomic and pedigree-based matrices as described below:

in which τ, α, β, and ω are scaling factors used to improve matrices equivalences, with default values as 1.00, 0.95, 0.05, and 1.00, respectively (Misztal et al., 2010; Lourenco et al., 2014; Aguilar et al., 2020), and A₂₂ is the pedigree-based relationship matrix constructed using the ancestral and genotyped individuals.

The INTERGEN package includes three binaries that can be executed on UNIX systems (Linux and OSX) via a terminal command line. Users must create a parameter file to specify the model parameters according to the software guidelines. Once executed, the software generates various intermediate and final files, which are saved in the directory the software is executed.

Before using the software, the user must ensure that the Math Kernel Library (MKL) is installed on their computer or server. The Intel compilers have been free since 2020 and can be downloaded from the Intel website (https://www.intel.com/content/www/us/en/developer/tools/oneapi/fortran-compiler-download.html).

Although all functionalities presented in this paper are operational in the INTERGEN package, we will not provide a comprehensive list of results since they have already been reported in several other publications (Santana et al., 2013; Ribeiro et al., 2015; Ribeiro et al., 2018; Junqueira et al., 2018; Junqueira et al., 2022a). Instead, in this publication, we will present the intergeniod results as the new INTERGEN package software.

The phenotypic, genotypic, and pedigree Angus data used in the analysis were kindly provided by the Programa de Melhoramento de Bovinos de Carne (PROMEBO) of the Associação Nacional de Criadores (ANC) “Herd-Book Collares”. In the following sections, we will describe the data and the procedures used to prepare them for the analysis.

The Angus animals used in this analysis were born between 1979 and 2022 and were primarily raised on pasture in the southern region of Brazil. Various traits were measured, such as weaning weight gain (WWG), weaning conformation (WC), weaning hair coat (WHC), post-weaning gain (PWG), yearling conformation (YC), yearling hair coat (YHC), scrotal circumference (SC), rib-eye area (REA) at yearling age, backfat thickness (BFT) at yearling, rump fat thickness (RFT) at yearling, and intramuscular fat (IMF) at yearling. For more information on each trait, please refer to Table 1.

Table 1

Trait	Phenotypes	Geno_Pheno	CG	Mean	Minimum	Maximum	SD
WWG	291,538	9,602	16,938	140.98	20.50	410.00	40.06
WC	268,559	9,732	18,381	3.18	1.00	5.00	1.09
WHC	113,390	8,033	8,093	2.01	1.00	3.00	0.71
PWG	193,849	7,872	14,512	154.74	0.74	693.83	77.69
REA	33,626	5,943	4,326	59.51	15.06	129.70	18.03
BFT	34,421	5,948	4,250	2.97	0.10	23.60	1.63
RFT	30,140	5,940	3,963	3.49	0.10	24.60	2.24
IMF	26,344	5,847	3,430	3.05	0.30	10.28	1.15
YC	186,584	8,215	18,433	3.27	1.00	5.00	1.07
SC	49,937	4,466	4,352	34.53	18.00	50.00	3.71
YHC	91,797	7,647	9,252	1.81	1.00	3.00	0.69

WWG - weaning weight gain; WC - weaning conformation; WHC - weaning hair coat; PWG - post-weaning gain; REA - rib-eye area; BFT - backfat thickness at yearling; RFT - rump fat thickness at yearling; IMF - intramuscular fat at yearling; YC - yearling conformation; SC - scrotal circumference at yearling; YHC - yearling hair coat.

Scores of WC and YC are visual measures of the volume of the carcass, considering body length and rib depth. Each animal was assigned a score between 1 and 5, with 5 being the highest expression of the trait and 1 being the lowest relative to its contemporary group (CG). The CG consisted of animals of the same sex, born in the same year and season, and raised on the same farm under the same management conditions. The phenotypes were collected on the same date, and CG with fewer than two observations were excluded from the analysis. Additionally, any observations that exceeded 3.5 standard deviations from their corresponding CG were removed.

Scores of WHC and YHC range from 1 to 3, with 1 indicating a short hair coat, 2 indicating a medium hair coat, and 3 indicating a long hair coat (Reimann et al., 2018). Traits REA (cm²), BFT (mm), and IMF (%) were measured using an ultrasound device on the region between the 12th and 13th ribs, transversely over the longissimus muscle. The ultrasound measured RFT (mm) between the gluteus medius and biceps femoris muscles on the animal’s rump.

A total of 12,637 animals were genotyped using sixteen different commercial SNP panels. Table S1 contains details about the number of SNP in each panel and the overlapping markers among them. Quality control of the genotypes was carried out using the R/SNPStats package (Clayton, 2023), which removed samples with genotyping call rates below 0.90, heterozygosity values three standard deviations above or below the observed mean, mismatched sex, and duplicate records. Only SNP mapped to autosomes with call rates greater than 0.98, minor allele frequencies greater than 0.03, and a probability of deviation from the Hardy-Weinberg equilibrium greater than 10⁷ were considered for the analyses. Lastly, SNP with the highest minor allele frequencies were retained when observed in the same position or when genotypes were highly correlated (r > 0.98). Excluding SNP with high correlation from the analysis enhances numerical stability, boosts computational efficiency, and avoids overrepresentation of genomic areas.

After quality control, a joint imputation combined the SNP from the fourteen panels. Following the editing and joining of the SNP, 74,227 SNP and 12,637 samples were available for imputation. Missing genotypes were imputed using Fimpute software version 3 (Sargolzaei et al., 2011).

Seven multi-trait ssGBLUP models were analyzed, varying the random terms included. The complete ssGBLUP model can be described as follows:

in which X, Z, P, and W are the incidence matrices associating each level of fixed, additive, and maternal effects to observations. The y is the vector of observations, and the vector of BLUP solutions for additive and maternal effects are correlated in the form [ a m ] ∼ N ( 0 , G 0 ⊗ H − 1 ) , with H^-1 as the relationship matrix constructed combining pedigree and genotypes (Aguilar et al., 2010), and the (co)variance matrix for this correlated effect defined as G 0 = [ σ a 1 2 σ a 1 a 2 σ a 1 m 2 σ a 1 m 2 σ a 2 2 σ a 2 m 1 σ a 2 m 2 σ m 1 2 σ m 1 m 2 s y m σ m 2 2 ] , the σ²_a(m)x is the additive (maternal) variance for each trait x={1,2}, σ_a(m)1a(m)2 is the additive (maternal) covariance between traits and effects. The vector c ∼ N ( 0 , I ⊗ [ σ c 1 2 0 0 σ c 2 2 ] ) is associated with the maternal permanent environment. The residual term is described as e ∼ N ( 0 , I ⊗ [ σ e 1 2 σ e 1 e 2 σ e 2 e 1 σ e 2 2 ] ) with σ²_e1(2) and σ_e1e2 as the residual variance and covariances, respectively.

The effects associated with each trait and the variance components estimated by intergen are presented in Table 2. The analyses were carried out on intergen (benchmarking) and intergeniod using PCG for deriving fixed and random solutions. All traits included additive effect as random. A random maternal effect was included for BW and WWG, while a random maternal permanent effect was included for BW, WWG, WC, and WHC. The fixed effects are CG, age of dam, and linear and quadratic covariates for animal age. The number of animals included in each analysis and the number of rounds to the convergence of each bivariate ssGBLUP model are presented in Table 3. The convergence criteria for analyzing conformation traits were set at 10²⁰, while those for continuous records were established at 10¹².

Table 2

Trait	Variance components
	σ2a	σ2m	σ2c	σ2e	h2	h2m
WWG	150.00	30.00	55.00	350.00	0.26	0.06
WC	0.112	-	0.027	0.613	0.15	-
WHC	0.076	-	0.063	0.238	0.20	-
PWG	110.00	-	-	726.00	0.13	-
REA	20.000	-	-	35.000	0.36	-
BFT	0.160	-	-	0.800	0.17	-
RFT	0.400	-	-	1.000	0.28	-
IMF	0.400	-	-	0.600	0.40	-
YC	0.149	-	-	0.631	0.19	-
SC	2.588	-	-	3.300	0.44	-
YHC	0.096	-	-	0.264	0.27	-

WWG - weaning weight gain; WC - weaning conformation; WHC - weaning hair coat; PWG - post-weaning gain; REA - rib-eye area at yearling; BFT - backfat thickness at yearling; RFT - rump fat thickness at yearling; IMF - intramuscular fat at yearling; YC - yearling conformation; SC - scrotal circumference at yearling; YHC - yearling hair coat; σ²_a - additive variance; σ²_m - maternal variance; σ²_p - permanent environment variance; σ²_c - maternal permanent environment variance; σ²_e - residual variance; h² - additive heritability; h²_m - maternal heritability.

Table 3

Analysis	Pedigree	Rounds
WC-YC	465,138	838
WWG-PWG	523,362	332
WWG-SC	521,230	113
WWG-REA	409,275	487
WWG-IMF	409,124	272
WWG-BFT-RFT	420,092	1,885
WHC-YHC	217,522	1,207

WC - weaning conformation; YC - yearling conformation; WWG - weaning weight gain; IMF - intramuscular fat at yearling; PWG - post-weaning gain; SC - scrotal circumference at yearling; REA - rib-eye area; WHC - weaning hair coat; YHC - yearling hair coat; BFT - backfat thickness at yearling age; RFT - rump fat thickness at yearling age.

All Fortran programs were compiled using an Intel Fortran compiler with maximum optimization. The analyses were performed on a Linux computer (x86-64) with an Intel^® Xeon^® Gold 5218R CPU running at 2.10GHz. To measure the random-access memory (RAM), the resident set size (RSS) and the virtual memory size (VSZ) displayed by the Linux ps command (https://man7.org/linux/man-pages/man1/ps.1.html) were summed up. The resulting value of ps is shown in kilobytes and then converted into gigabytes by dividing it by 1e6.

The INTERGEN package is consistently updated to improve computational efficiency as the volume of genotype data increases. In the future, the package will include several new functionalities, such as estimating (co)variance components using restricted maximum likelihood (REML) (Harville, 1977) and average information REML (AI-REML) (Gilmour et al., 1995), robust modeling (Rosa et al., 2003; Cardoso et al., 2007), an algorithm for building a genomic matrix with proven and young (APY) strategy (Misztal, 2016), constructing the numerator relationship for selfing populations, and a multi-trait model for approximate accuracies (Strabel et al., 2001).

The INTERGEN package has been used since 2016 by the PAMPAPLUS breeding program to estimate the genetic merit of beef cattle Hereford and Braford multibreed populations. In 2024, the PROMEBO breeding program adopted the intergeniod and intergenacc software to calculate the genetic merit of animals for eight different breeds. In these commercial partnerships, genetic evaluations are done each week, and the report that contains breeding values (additive and maternal) and BIF reliabilities is updated in the database for in-farm decisions.