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

Abstract 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.


| INTRODUC TI ON
In animal breeding strategies, there are two types of breeding goal to ensure the ability of animal to fit the demands of producers, consumers, and markets. One is the fundamental or sustainable breeding goal, which is not dependent on market trends and focuses on properties such as reproduction, feed efficiency, and longevity. The other is the flexible breeding goal that reflects temporal market or economical demands and focuses on properties such as the qualities of milk and meat products. When we are confronted with the need to improve a new trait because of changing market demands, such as fatty acid composition, which is a meat quality trait and is related to the sweet aroma (Sakuma et al., 2012) and flavor of beef (Suzuki, Yokota, Shioura, Shimazu, & Iida, 2013) in Japanese Black cattle, we add the new trait to conventional selection traits and improve multiple traits simultaneously. In such a case, the most important thing is to confirm the genetic correlations between the traits by estimating the genetic parameters fitted by the multiple-trait model (MTM; Henderson & Quass, 1976) to prevent negative effects on the conventional economical traits. However, there are not only 'correlations' but there could also be 'causal relationships' between the traits. For example, in the case of the relationship between gestation length (GL) and calving difficulty (CD), GL affects CD, because a longer GL causes a bigger infant or harmonization. In contrast, the opposite relationship is not possible; CD never affects GL under biological conditions. When causal relationships exist between traits, fitting a structural equation model (SEM) allows for inferences of the magnitude of the causal effects between traits (Haavelmo, 1943;Wright, 1921). SEMs have been applied in various fields, such as economics, psychology, and sociology. In recent years, Gianola and Solensen (2004) proposed adaptations of the SEM in the context of quantitative genetics and mixed-effect models. Ever since their proposal, several studies have applied the SEM in the context of animal breeding in dairy goats (de los Campos, Gianola, Boettcher, & Moroni, 2006), dairy cattle (de los Campos, Gianola, & Heringstad, 2006;Heringstad, Wu, & Gianola, 2009;Konig, Wu, Gianola, Heringstad, & Simianer, 2008;Wu, Heringstad, Chang, de los Campos, & Gianola, 2007;etc.), pigs (Ibanez-Escriche, Lopez de Maturana, Noguera, & Varona, 2010;Varona, Sorensen, & Thompson, 2007), and beef cattle (Inoue, Hosono, & Tanimoto, 2017;Inoue et al., 2016).
Fitting the SEM requires choosing a causal structure with prior biological or temporal knowledge. However, the inductive causation (IC) algorithm (Pearl, 2000;Verma & Pearl, 1990), which is one of the Bayesian causal inference, can be used to recover an underlying causal structure from observed associations between traits. The IC algorithm can output the causal relationships with a directed acyclic graph (DAG), which is a set of variables connected by directed edges, or a partially oriented graph. Searching for the causal structure in the algorithm is based on a conditional independence (Dawid, 1980) between variables. Valente, Rosa, de los Campos, Gianola, and Silva (2010) adapted the IC algorithm to a mixed-model context and showed that applying this method to the posterior distribution of the residual (co) variance matrix of a standard MTM recovered the expected network in simulated data. They also presented an application of the methodology to real field data of European quail. After their work, several groups have tried to explore the causal structure from the real field data. This review will first describe an overview of the IC algorithm and SEM, and then introduce some papers that apply these techniques to the context of animal breeding.

| THE IC ALG ORITHM
The IC algorithm, which was proposed by Pearl (2000) and Verma and Pearl (1990), allows to search for causal networks. The IC algorithm performs a series of statistical decisions based on partial correlations between traits and consists of the following three steps (an example is shown in Figure 1):

| Step 1
If all partial correlations of two traits conditional on each possible set of the other traits are different from 0, the two traits are connected by an undirected edge (e.g., y1 − y2). For example, when there are four target traits for inferring the causal structure (y1, y2, y3, and y4), as shown on the left graph of Step 1 in Figure 1, if all partial correlations of two arbitrary traits (e.g., y1 and y2) conditional on every set of the other traits (i.e., ρ y1y2|y3 , ρ y1y2|y4 , and ρ y1y2|y3y4 ) and correlation between y1 and y2 (ρ y1y2 ) are different from 0 (significant), the two traits are connected by an undirected edge (y1 − y2). The graph with undirected edges is finally generated by iterating the same process for the remaining possible set of traits (the right graph of Step 1 in Figure 1).

| Step 2
In the undirected graph obtained by step 1, if partial correlations of two nonadjacent traits (e.g., y1 and y3) with a common adjacent trait (e.g., y2 in y1 − y2 − y3) are dependent conditional on any possible set including the adjacent trait (y2), that is, all partial correlations (ρ y1y3 , ρ y1y3|y2 , ρ y1y3|y4 , and ρ y1y3|y2y4 ) are not significantly different from 0, arrowheads pointing to the common adjacent trait (y2) are added (y1 → y2 ← y3 in the graph of Step 2 in Figure 1). This kind F I G U R E 1 An example of the steps for the inductive causation algorithm. y1 to y4 represent traits, '-' and '→ (←)' represent undirected and directed edges, respectively of 'V' structure is called an 'unshielded collider.' The same process is repeated for the remaining possible set of traits.

| Step 3
In the partially oriented graph obtained from step 2, as many undirected edges as possible are oriented without creating a new unshielded collider or cycle. In the case of the example in Figure 1, y2 and y4 are connected by an undirected edge. If an arrowhead pointing to y2 from y4 is added to the undirected edge (y2 ← y4), new unshielded colliders are created (y1 → y2 ← y4 and y3 → y2 ← y4); therefore, a new directed edge from y4 to y2 cannot be added. In contrast, arrowheads pointing to y4 from y2 can be added to the undirected edge (y2 → y4) because no new unshielded colliders are created. Finally, the graph of the causal structure is completed, shown as the graph in Step 3 of In the context of the mixed-model analysis, after fitting the MTM to the traits of interest, the IC algorithm can be applied to samples from the posterior distribution of the residual (co)variance matrix obtained from the multiple-trait analysis because the residual (co) variances are considered to be information from the joint distributions of the traits conditional on genetic effects, which corrects the search for confounding because of such effects when they are correlated (Valente et al., 2010). The program ICPS written in R (R Development Core Team, 2009), which can carry out the IC algorithm analysis, is provided by .

| S TRUC TUR AL EQUATION MODEL
In the case of existing causal relationships between traits, as mentioned in the previous section, fitting a SEM (Haavelmo, 1943;Wright, 1921) can infer the causal coefficients between the traits.
A SEM with a causal structure and random additive genetic effects, which was first proposed by Gianola and Solensen (2004), has been reviewed Wu, Heringstad, & Gianola, 2010) and the model with t traits can be written as follows: where y i is a t × 1 vector of phenotypic records on animal i; Λ is a t × t matrix of structural coefficients, which is filled by 0 except for the off-diagonal elements that correspond to the causal structure; is a vector of fixed effects; u i is a vector of random additive genetic effects; e i is a vector of random residual effects; and X i is a known incidence matrix. The joint distribution of vectors u i and e i in the equation is as follows: where G 0 is the additive genetic (co)variance matrix and 0 is the diagonal residual variance matrix.
By reducing the above SEM for y, the model is transformed, as indicated below (Gianola & Solensen, 2004;Varona et al., 2007): In addition, the joint distribution of u * i and e * i is as follows: with , and R * 0 are the similar vectors as described above (G * 0 is the additive genetic (co)variance matrix and R * 0 is the residual (co)variance matrix). Like this, the SEM can be transformed into a MTM, which ignores causal relationships between traits. Therefore, the likelihood between MTM and SEM is equivalent (Varona et al., 2007). However, genetic effects estimated from the two models have different meaning. The MTM only infers 'overall' genetic effects, which include all 'direct' and 'indirect' genetic effects on each trait, as illustrated later. Here, the indirect genetic effects are those that are mediated by other phenotype traits and the direct genetic effects are those that are not. In contrast, the SEM can infer the direct genetic effects, which are separated from the causal effects (corresponding to the indirect genetic effects; Valente, Rosa, Gianola, Wu, & Weigel, 2013). For instance, assuming the causal structure between two traits shown in Figure 2a, the SEM can be represented as follows: where y 1 and y 2 are the phenotypes; u 1 and u 2 are the additive genetic effects which affect directly on y 1 and y 2 , respectively (direct genetic effects), but u 1 is also the indirect genetic effect on y 2 ; e 1 and e 2 are the residual effects; and 21 is the causal effect from y 1 to y 2 . In addition, the equation described above can be transformed as follows: y 1 = u 1 + e 1 y 2 = 21 y 1 + u 2 + e 2 y 1 = u 1 + e 1 = u * 1 + e * 1 y 2 = 21 (u 1 + e 1 ) + u 2 + e 2 = ( 21 u 1 + u 2 ) + ( 21 e 1 + e 2 ) = u * 2 + e * 2 where u * 1 and u * 2 , e * 1 and e * 2 are the additive genetic and residual effects in the MTM, respectively. This equation shows that the genetic effects in the SEM affect not only the corresponding phenotypes directly but also the other phenotypes indirectly via the causal structure. In addition, it suggests that the genetic effects in the MTM represent the overall effects, which also include the direct and indirect effects via the causal effects (u * 2 = 21 u 1 + u 2 ).

| INTERVENTI ON S IN THE S EM
As shown in the previous section, the SEM can infer direct and in- If the causal effects can be inferred by fitting the SEM to the causal structure, giving an artificial intervention, such as management practice, to the variation in causing traits can phenotypically improve the following traits. In this way, the application of causal inference and fitting the SEM are effective not only for breeding programs but also for planning for management practices in livestock production systems . Therefore, when causal relationships exist between target traits, application of the SEM to the traits should be considered.

| APPLYING THE I C ALG ORITHM AND S EM FOR ANIMAL B REED ING
Ever since Gianola and Solensen (2004) proposed adaptations of the SEM in the context of quantitative genetics and mixed-effect models, several studies have applied the SEM to different species and traits in the context of animal breeding. de los Campos, Gianola, Boettcher, et al. (2006) and de los Campos, Gianola, and Heringstad (2006) studied the relationship between somatic cell score (SCS) and milk yield (MY) in dairy cows and dairy goats. This work is the first application of the SEM to field data. Similarly, in dairy cattle, there are many reports exploring the relationships between SCS or clinical mastitis and MY in Norwegian Red cows (Wu et al., 2007; between claw disorders and MY in Holstein cows (Konig et al., 2008); between health and fertility traits in Norwegian Red cows (Heringstad et al., 2009); among GL, CD, and SB in US Holstein cows (Lopez de Maturana et al., 2009; and between SCS and MY in Canadian Holstein cows (Jamrozik, Bohmanova, & Schaeffer, 2010;. Some of the authors proposed extensions for the SEM, such as accounting for population heterogeneity (Wu et al., 2007), the Gaussian-threshold Bayesian hierarchical model , the heterogeneous causal model , and the random regression model . In swine, Varona et al. (2007) used a recursive model to analyze the relationship between litter . y 1 and y 2 are phenotypes; u 1 and u 2 are the additive genetic effects, which affect directly on y 1 and y 2 , respectively, but u 1 is also the indirect genetic effect on y 2 ; e 1 and e 2 are the residual effects; λ 21 is the causal coefficient; and c is a constant. Dashed arc represents genetic correlation between the additive genetic effects using change-point detection techniques (Carlin, Gelfand, & Smith, 1992;Chib, 1998) and fitted the model to the litter size and number of SBs in Large White pigs. However, these studies assumed that causal structures were known a priori.
Recently, some studies fitted SEMs based on a data-driven causal structure search using the IC algorithm. The IC algorithm fitted to the posterior distribution of residual (co)variance matrices that are considered phenotypic correlations conditional on additive genetic correlations between traits; this approach was proposed by Valente et al. (2010). They used simulated data and showed that this approach recovered the correct causal network. Then, Valente, Rosa, Silva, Teixeira, and Torres (2011)

| CON CLUS IONS
The results of the papers that are introduced in this review showed that the application of the IC algorithm to infer a causal structure is one of the most helpful tools for detecting an underlying causal structure without proper prior knowledge or with uncertain relationships between traits. The reports also indicated that visualization of the causal structure among traits on a graph and fitting the SEM to the causal structure 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 strategies than does that from a MTM, which is a conventional model usually used for multiple-trait analysis. These new procedures could be utilized in the field of animal breeding.