An application of change-point recursive models to the relationship between litter size and number of stillborns in pigs

Invited review: Recursive models in animal breeding: Interpretation, limitations, and extensions

Structural equation models allow causal effects between 2 or more variables to be considered and can postulate unidirectional (recursive models; RM) or bidirectional (simultaneous models) causality between variables. This review evaluated the properties of RM in animal breeding and how to interpret the genetic parameters and the corresponding estimated breeding values. In many cases, RM and mixed multitrait models (MTM) are statistically equivalent, although subject to the assumption of variance-covariance matrices and restrictions imposed for achieving model identification. Inference under RM requires imposing some restrictions on the (co)variance matrix or on the location parameters. The estimates of the variance components and the breeding values can be transformed from RM to MTM, although the biological interpretation differs. In the MTM, the breeding values predict the full influence of the additive genetic effects on the traits and should be used for breeding purposes. In contrast, the RM breeding values express the additive genetic effect while holding the causal traits constant. The differences between the additive genetic effect in RM and MTM can be used to identify the genomic regions that affect the additive genetic variation of traits directly or causally mediated for another trait or traits. Furthermore, we presented some extensions of the RM that are useful for modeling quantitative traits with alternative assumptions. The equivalence of RM and MTM can be used to infer causal effects on sequentially expressed traits by manipulating the residual (co)variance matrix under the MTM. Further, RM can be implemented to analyze causality between traits that might differ among subgroups or within the parametric space of the independent traits. In addition, RM can be expanded to create models that introduce some degree of regularization in the recursive structure that aims to estimate a large number of recursive parameters. Finally, RM can be used in some cases for operational reasons, although there is no causality between traits.

A cross-specific multiplicative binomial recursive model for the analysis of perinatal mortality in a diallel cross among three varieties of Iberian pig

Perinatal piglet mortality is an important factor in pig production from economic and animal welfare perspectives; however, the statistical analysis of mortality is difficult because of its categorical nature. Recent studies have suggested that a binomial model for the survival of each specific piglet with a logit approach is appropriate and that recursive relationships between traits are useful for taking into account non-genetic relationships with other traits. In this study, the recursive binomial model is expanded in two directions: (1) the recursive phenotypic dependence among traits is allowed to vary among groups of individuals or crosses, and (2) the binomial distribution is replaced by the multiplicative binomial distribution to account for over or underdispersion. In this study, five recursive multiplicative binomial models were used to obtain estimates of the Dickerson crossbreeding parameters in a diallel cross among three varieties of Iberian pigs [Entrepelado (EE), Torbiscal (TT), and Retinto (RR)]. Records (10,255) from 2110 sows were distributed as follows: EE (433 records, 100 sows), ER (2336, 527), ET (942, 177), RE (806, 196), RR (870, 175), RT (2450, 488), TE (193, 36), TR (1993, 359), and TT (232, 68). Average litter size [Total Number Born (TNB)] and number of stillborns (SB) were 8.46 ± 2.27 and 0.25 ± 0.72, respectively. The overdispersion was evident with all models. The model with the best fit included a linear recursive relationship between TNB and the logit of [Formula: see text] of the multiplicative binomial distribution, and it implies that piglet mortality increases with litter size. Estimates of direct effects showed small differences among populations. The analysis of maternal effects indicated that the dams whose mothers were EE had a larger SB, while dams with RR mothers reduced the probability of born dead. The posterior estimates of heterosis suggested a reduction in SB when the sow is crosbred. The multiplicative binomial distribution provides a useful alternative to the binomial distribution when there is overdispersion in the data. Recursive models can be used for modeling non-genetic relationships between traits, even if the phenotypic dependency between traits varies among environments or groups of individuals. Piglet perinatal mortality increased with TNB and is reduced by maternal heterosis.

Application of Bayesian causal inference and structural equation model to animal breeding

Optimized breeding goals and management practices for the improvement of target traits requires knowledge regarding any potential functional relationships between them. Fitting a structural equation model (SEM) allows for inferences about the magnitude of causal effects between traits to be made. In recent years, an adaptation of SEM was proposed in the context of quantitative genetics and mixed models. Several studies have since applied the SEM in the context of animal breeding. However, fitting the SEM requires choosing a causal structure with prior biological or temporal knowledge. The inductive causation (IC) algorithm can be used to recover an underlying causal structure from observed associations between traits. The results of the papers, which are introduced in this review, showed that using the IC algorithm to infer a causal structure is a helpful tool for detecting a causal structure without proper prior knowledge or with uncertain relationships between traits. The reports also presented that fitting the SEM could infer the effects of interventions, which are not given by correlations. Hence, information from the SEM provides more insights into and suggestions on breeding strategy than that from a multiple-trait model, which is the conventional model used for multitrait analysis.

Non-linear recursive models for growth traits in the Pirenaica beef cattle breed

One of the main goals of selection schemes in beef cattle populations is to increase carcass weight at slaughter. Live weights at different growth stages are frequently used as selection criteria under the hypothesis that they usually have a high and positive genetic correlation with weight at slaughter. However, the presence of compensatory growth may bias the prediction ability of early weights for selection purposes. Recursive models may represent an interesting alternative for understanding the genetic and phenotypic relationship between weight traits during growth. For the purposes of this study, the analysis was performed for three different set of data from the Pirenaica beef cattle breed: weight at 120 days (W120) and at 210 days (W210); W120 and carcass weight at slaughter at 365 days (CW365); W210 and CW365. The number of records for each analysis was 8592, 4648 and 3234, respectively. A pedigree composed of 56323 individuals was also included. The statistical model comprised sex, year-season of birth, herd and slaughterhouse, plus a non-linear recursive dependency between traits. The dependency was modeled as a polynomial up to the 4th degree and models were compared using a Logarithm of Conditional Predictive Ordinates. The results of model comparison suggest that the best models were the 3rd degree polynomial for W120-W210 and W120-CW365 and the 2nd degree polynomial for W210-CW365. The posterior mean estimates for heritabilities ranged between 0.29 and 0.44 and the posterior mean estimates of the genetic correlations were null or very low, indicating that the relationship between traits is fully captured by the recursive dependency. The results imply that the predictive ability of the performance of future growth is low if it is only based on records of early weights. The usefulness of slaughterhouse records in beef cattle breeding evaluation is confirmed.

Mixed effects structural equation models and phenotypic causal networks

Complex networks with causal relationships among variables are pervasive in biology. Their study, however, requires special modeling approaches. Structural equation models (SEM) allow the representation of causal mechanisms among phenotypic traits and inferring the magnitude of causal relationships. This information is important not only in understanding how variables relate to each other in a biological system, but also to predict how this system reacts under external interventions which are common in fields related to health and food production. Nevertheless, fitting a SEM requires defining a priori the causal structure among traits, which is the qualitative information that describes how traits are causally related to each other. Here, we present directions for the applications of SEM to investigate a system of phenotypic traits after searching for causal structures among them. The search may be performed under confounding effects exerted by genetic correlations.

Is structural equation modeling advantageous for the genetic improvement of multiple traits?

Structural equation models (SEMs) are multivariate specifications capable of conveying causal relationships among traits. Although these models offer insights into how phenotypic traits relate to each other, it is unclear whether and how they can improve multiple-trait selection. Here, we explored concepts involved in SEMs, seeking for benefits that could be brought to breeding programs, relative to the standard multitrait model (MTM) commonly used. Genetic effects pertaining to SEMs and MTMs have distinct meanings. In SEMs, they represent genetic effects acting directly on each trait, without mediation by other traits in the model; in MTMs they express overall genetic effects on each trait, equivalent to lumping together direct and indirect genetic effects discriminated by SEMs. However, in breeding programs the goal is selecting candidates that produce offspring with best phenotypes, regardless of how traits are causally associated, so overall additive genetic effects are the matter. Thus, no information is lost in standard settings by using MTM-based predictions, even if traits are indeed causally associated. Nonetheless, causal information allows predicting effects of external interventions. One may be interested in predictions for scenarios where interventions are performed, e.g., artificially defining the value of a trait, blocking causal associations, or modifying their magnitudes. We demonstrate that with information provided by SEMs, predictions for these scenarios are possible from data recorded under no interventions. Contrariwise, MTMs do not provide information for such predictions. As livestock and crop production involves interventions such as management practices, SEMs may be advantageous in many settings.

Bayesian recursive mixed linear model for gene expression analyses with continuous covariates

The analysis of microarray gene expression data has experienced a remarkable growth in scientific research over the last few years and is helping to decipher the genetic background of several productive traits. Nevertheless, most analytical approaches have relied on the comparison of 2 (or a few) well-defined groups of biological conditions where the continuous covariates have no sense (e.g., healthy vs. cancerous cells). Continuous effects could be of special interest when analyzing gene expression in animal production-oriented studies (e.g., birth weight), although very few studies address this peculiarity in the animal science framework. Within this context, we have developed a recursive linear mixed model where not only are linear covariates accounted for during gene expression analyses but also hierarchized and the effects of their genetic, environmental, and residual components on differential gene expression inferred independently. This parameterization allows a step forward in the inference of differential gene expression linked to a given quantitative trait such as birth weight. The statistical performance of this recursive model was exemplified under simulation by accounting for different sample sizes (n), heritabilities for the quantitative trait (h(2)), and magnitudes of differential gene expression (λ). It is important to highlight that statistical power increased with n, h(2), and λ, and the recursive model exceeded the standard linear mixed model with linear (nonrecursive) covariates in the majority of scenarios. This new parameterization would provide new insights about gene expression in the animal science framework, opening a new research scenario where within-covariate sources of differential gene expression could be individualized and estimated. The source code of the program accommodating these analytical developments and additional information about practical aspects on running the program are freely available by request to the corresponding author of this article.