# Reparametrization-based estimation of genetic parameters in multi-trait animal model using Integrated Nested Laplace Approximation

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## Abstract

### Key message

**A novel reparametrization-based INLA approach as a fast alternative to MCMC for the Bayesian estimation of genetic parameters in multivariate animal model is presented**.

### Abstract

Multi-trait genetic parameter estimation is a relevant topic in animal and plant breeding programs because multi-trait analysis can take into account the genetic correlation between different traits and that significantly improves the accuracy of the genetic parameter estimates. Generally, multi-trait analysis is computationally demanding and requires initial estimates of genetic and residual correlations among the traits, while those are difficult to obtain. In this study, we illustrate how to reparametrize covariance matrices of a multivariate animal model/animal models using modified Cholesky decompositions. This reparametrization-based approach is used in the Integrated Nested Laplace Approximation (INLA) methodology to estimate genetic parameters of multivariate animal model. Immediate benefits are: (1) to avoid difficulties of finding good starting values for analysis which can be a problem, for example in Restricted Maximum Likelihood (REML); (2) Bayesian estimation of (co)variance components using INLA is faster to execute than using Markov Chain Monte Carlo (MCMC) especially when realized relationship matrices are dense. The slight drawback is that priors for covariance matrices are assigned for elements of the Cholesky factor but not directly to the covariance matrix elements as in MCMC. Additionally, we illustrate the concordance of the INLA results with the traditional methods like MCMC and REML approaches. We also present results obtained from simulated data sets with replicates and field data in rice.

## Keywords

Markov Chain Monte Carlo Markov Chain Monte Carlo Method Multivariate Normal Distribution Cholesky Factor Additive Genetic Effect## Introduction

Estimation of variance components and associated breeding values is an important topic in classic (e.g., Piepho et al. 2008; Oakey et al. 2006; Bauer et al. 2006) and in Bayesian (e.g., Wang et al. 1993; Blasco 2001; Sorensen and Gianola 2002; Mathew et al. 2012) single-trait mixed model context. Similarly, multi-trait models have been proposed in both settings (e.g., Bauer and Léon 2008; Thompson and Meyer 1986; Korsgaard et al. 2003; Van Tassell and Van Vleck 1996; Hadfield 2010). Multi-trait analyses can take into account the correlation structure among all traits and that increases the accuracy of evaluation. However, this gain in accuracy is dependent on the absolute difference between the genetic and residual correlation between the traits (Mrode and Thompson 2005). This evaluation accuracy will increase as the differences between these correlations become high (Schaeffer 1984; Thompson and Meyer 1986). Persson and Andersson (2004) compared single-trait and multi-trait analyses of breeding values and they showed that multi-trait predictors resulted in a lower average bias than the single-trait analysis. Estimation of genetic and residual covariance matrices are the main challenging problem in multi-trait analysis in mixed model framework. However, in Bayesian analysis of multi-trait animal models, inverse-Wishart distribution is the common choice as the prior distribution for those unknown covariance matrices. The use of inverse-Wishart prior distribution for covariance matrix guarantees that the resulting covariance matrix will be positive definite (that is, invertible). However, the use of inverse-Wishart prior distribution is quite restrictive, because then one gives same degrees of freedom for all components in the covariance matrix (Barnard et al. 2000). Moreover, it is often difficult to suggest prior distributions that can be used for common situations. Matrix decomposition approach presented in this paper assigns independent priors for elements in the Cholesky factor.

Markov Chain Monte Carlo (MCMC) methods are a popular choice for Bayesian inference of animal models (Sorensen and Gianola 2002). Often, inference using MCMC methods is challenging for a non-specialist. Although there are various packages available for Bayesian inference which are based on MCMC methods (e.g., MCMCglmm, Hadfield 2010; BUGS, Lunn et al. 2000; Stan, Stan Development Team 2014), most of these packages are not easy to use and computationally expensive. Among these packages, MCMCglmm seems to be easy to implement and computationally inexpensive. As an alternative to MCMC methods one can use the recently implemented non-sampling-based Bayesian inference method, Integrated Nested Laplace Approximation (INLA, Rue et al. 2009). INLA methodology is comparatively easy to implement, but less flexible than MCMC methods (Holand et al. 2013).

Canonical transformation is a common matrix decomposition technique in multi-trait animal models to simultaneously diagonalize the genetic covariance matrix and make residual covariance matrix to identity matrix (see e.g., Ducrocq and Chapuis 1997). After transformation, best linear unbiased prediction (BLUP) values can be calculated independently for each trait using univariate animal model and then back transformed to obtain benefits of multi-trait analysis. However, common requirement in canonical transformation is that covariance matrices need to be known before the transformation. Therefore, it cannot be applied for variance component estimation—with unknown genetic and residual covariance matrices. Here, as an improvement, we introduce another kind of decomposition approach, where elements of the transformation matrix are estimated simultaneously together with the other mixed model parameters allowing us to apply this transformation also for the case of variance component estimation. This kind of modified Cholesky decomposition approach is required to perform multi-trait analysis in INLA (see Bøhn 2014). The closely related decomposition approach has been presented in Pourahmadi (1999, 2000, 2011) and Gao et al. (2015). Also our approach is somewhat related to factor analytic (FA) models (e.g. Meyer 2009; Cullis et al. 2014) and which was first introduced in a breeding context by Piepho (1997, 1998).

In this paper, we illustrate this approach to estimate genetic and residual covariance matrices with INLA and compare the obtained estimates with those from REML (Patterson and Thompson 1971) and MCMC approaches using simulated and real data sets. With the recent development of new low-cost high-throughput DNA sequencing technologies, it is now possible to obtain thousands of single nucleotide polymorphism (SNP) markers covering the whole genome, at the same time, in many animal and plant breeding programs often the detailed pedigree information is available. So we present results obtained using the marker data (real dataset) along with estimates obtained using pedigree information (simulated data) in this study. We also outline a more simple approach to simulate correlated traits based on the additive relationship matrix.

## Models and methods

### Model

*n*. Then the multi-trait mixed linear model for

*n*traits can be written as:

### Reparametrization of trivariate animal model in INLA

To extend this method for more than three traits (say, *n* traits) can be done by modifying the terms of Eq. (9), so that the additive genetic Cholesky factor \({\mathbf {W}_{a}}\) is a Kronecker product of \(n\times n\) lower triangular matrix with **I** and \({\mathbf {\Sigma }}_{{X}_{a}}\) is the additive genetic block matrix containing *n* blocks. For example, number of dependency parameters required for a \(4 \times 4\) Cholesky factor is already \(n(n-1)/2=4\times 3/2=6\).

As an additional supplementary material we provide the R scripts we used for the INLA analysis.

### Back transformation in INLA

INLA analysis returns the marginal posterior distributions of the hyperparameters (\({\mathbf {\sigma }}^2_{a_{i}`s},{\mathbf {\sigma }}^2_{e_{i}`s}\)) and the dependency parameters (\(\kappa_{i,j}`s,\alpha _{i,j}`s\)) for the reparametrized model (Eq. 7). So one need to perform the back transformation after the INLA analysis in order to obtain (co)variance components in the original scale. Let \({\mathbf {\sigma }}^2_{u_{i}}\), \({\mathbf {\sigma }}^2_{\epsilon _{i}}\), where \(i = 1, 2, 3\) and \({\mathbf {\sigma }}_{u_{ij}}\), \({\mathbf {\sigma }}_{\epsilon _{ij}}\), where \(i,j = 1, 2, 3\) be the (genetic and residual) variance and (genetic and residual) covariance components, respectively, in the original scale. First, calculate the approximated posterior marginal distribution for the hyperparameters (\({\mathbf {\sigma }}^2_{a_{i}`s},{\mathbf {\sigma }}^2_{e_{i}`s}\)) and the dependency parameters (\(\kappa_{i,j}`s,\alpha _{i,j}`s\)) by sampling from their joint distribution using the ‘inla.hyperpar.sample’ (Martins et al. 2013) function. Then, following Eq. (9) the genetic variance components can be calculated using the posterior distributions as, \({\mathbf {\sigma }}^2_{{u}_{1}}={\mathbf {\sigma }}^2_{{a}_{1}}\), \({\mathbf {\sigma }}^2_{{u}_{2}}=\kappa ^2_{1,2}\sigma ^2_{{a}_1} +{\mathbf {\sigma }}^2_{{a}_2}\) and \({\mathbf {\sigma }}^2_{{u}_{3}}={\mathbf {\kappa }}^2_{1,3}\sigma ^2_{{a}_1}+{\mathbf {\kappa }}^2_{2,3}\sigma ^2_{{a}_2}+{\mathbf {\sigma }}^2_{{a}_3}\). Similarly the genetic covariance components can be obtained as, \({\mathbf {\sigma }}_{{u}_{12}}={\mathbf {\kappa }}_{1,2}\sigma ^2_{{a}_1}\), \({\mathbf {\sigma }}_{{u}_{13}}={\mathbf {\kappa }}_{1,3}\sigma ^2_{{a}_1}\), and \({\mathbf {\sigma }}_{u_{23}}={\mathbf {\kappa }}_{1,2}\kappa _{1,3}\sigma ^2_{{a}_1}+{\mathbf {\kappa }}_{2,3}\sigma ^2_{{a}_2}\). The same procedure can be used to calculate the residual (co)variance components using Eq. (11). R scripts for the back transformation can be found in the supplementary material.

## Example analyses

### Simulated dataset with high heritability

### Simulated dataset with low heritability

### Field data

*Oryza sativa*) dataset (Spindel et al. 2015) and we selected three traits, grain yield (YLD), flowering time (FL) and plant height (PH) from 2012 dry season for the analysis. The population was genotyped with 73,147 markers using genotyping-by-sequencing method and we selected 323 lines where both the phenotypic and genotypic informations were available (see Spindel et al. 2015 for more details). So we used the available marker information for the estimation of genetic (co)variance components and the realized genomic relationship matrix (\(\mathbf {M}\)) was obtained from the marker information using R-package ‘rrBLUP’ (Endelman 2011). For the real data analysis, we considered the marker data instead of the pedigree information, so in model (1) the vector of random effects (\(\mathbf {u}\)) were assumed to follow a normal distribution according to Eq. (5) as

## Analyses and results

### Simulated data with replicates

In multi-trait analysis using iterative algorithms, it is often difficult to find suitable starting values for the parameters of interest. However, by performing test-runs using single-trait data one could find suitable starting values for the variance components. The (co)variance components were estimated using MCMCglmm, R-INLA and ASReml-R packages. For MCMC analysis using MCMCglmm package, we considered a total chain length of 50,000 iterations with a burning period of 10,000 iterations. The MCMCglmm package assign inverse-Wishart prior distribution for the random and residual covariance matrices. In our MCMC analysis, we used identity matrix as the scaling matrix of the prior distribution (ones for the variances and zeros for the covariances) assigned for the genetic covariance matrix (\({\mathbf {G}}_{0}\)) and for the residual covariance matrix (\({\mathbf {R}}_{0}\)) between the three traits. Moreover, we specified the degree of belief parameter (d) as 1 for the inverse-Wishart prior distribution. By default MCMCglmm uses the scaling matrix values as the starting values. For the REML analysis we used ones as the variances and zeros as the covariances for the genetic covariance matrix (\({\mathbf {G}}_{0}\)) as the starting values; whereas, for the residual covariance matrix (\({\mathbf {R}}_{0}\)) we used half of the phenotypic variance matrix of the data as initial values (ASReml-R default). The total computation time for the simulated dataset using MCMCglmm package was around 10 min and INLA took around 4 min, whereas the time for ASReml-R package was around 1 min. The INLA approach we used in the current study was not able to analyze bigger datasets (around 1000 lines), mainly due to the lack of memory on our system. We used a Linux system with 8GB RAM for our calculations. However, it is possible to analyze such large datasets using computers with more memory size or arguably one can use the option ‘inla.remote()’ to run R-INLA on a remote server with more memory size.

*Y*-axis scale in those plots corresponds to the differences between the true simulated values and the estimated values, whereas the

*X*-axis corresponds to different estimation methods. In order to calculate the estimation errors, for MCMC we used posterior mode, whereas for INLA we used the posterior mean estimates. From Figs. 1 and 2, it can be concluded that, different methods were able to provide similar estimates. We also plotted the box plots for the estimation errors for the variance (Fig. 3) and covariance (Fig. 4) for the dataset with low heritability. However, for the low heritability dataset the MCMC and INLA approaches provided variance estimates closer to true values than the REML method. The narrow-sense heritability estimates for the simulated datasets using 50 simulation replicates are shown in Table 4. Here we did not account the covariances between the traits in order to calculate the heritability. The narrow-sense heritability (\(h^2\)) was calculated for each trait separately as \({h}^2 = V_a/(V_a+V_e)\), where \(V_a\) and \(V_e\) are the additive genetic variance and error variance of the particular trait, respectively.

Estimated genetic and residual correlation coefficients (\(\rho\)), between each traits (\(T_1\) to \(T_3\)) for both simulated dataset using REML, INLA and MCMC estimates

Correlation | REML | MCMC | INLA | True |
---|---|---|---|---|

High heritability dataset | ||||

\(\rho _{T_{1},T_{2}(a)}\) | 0.31(0.28, 0.32) | 0.32(0.28, 0.33) | 0.30(0.27, 0.32) | 0.33 |

\(\rho _{T_{1},T_{3}(a)}\) | 0.41(0.37, 0.43) | 0.40(0.37, 0.43) | 0.40(0.38, 0.43) | 0.38 |

\(\rho _{T_{2},T_{3}(a)}\) | 0.45(0.44, 0.48) | 0.45(0.44, 0.47) | 0.47(0.44, 0.48) | 0.46 |

\(\rho _{T_{1},T_{2}(e)}\) | 0.51(0.50, 0.53) | 0.52(0.51, 0.53) | 0.52(0.51, 0.53) | 0.50 |

\(\rho _{T_{1},T_{3}(e)}\) | 0.48(0.46, 0.50) | 0.49(0.46, 0.51) | 0.49(0.47, 0.50) | 0.50 |

\(\rho _{T_{2},T_{3}(e)}\) | 0.50(0.48, 0.51) | 0.51(0.48, 0.52) | 0.50(0.48, 0.52) | 0.50 |

Low heritability dataset | ||||

\(\rho _{T_{1},T_{2}(a)}\) | −0.19(−0.41, −0.03) | −0.40(−0.50, −0.31) | −0.38(−0.49, −0.29) | −0.34 |

\(\rho _{T_{1},T_{3}(a)}\) | 0.65(0.64, 0.66) | 0.44(0.39, 0.47) | 0.47(0.43, 0.51) | 0.42 |

\(\rho _{T_{2},T_{3}(a)}\) | 0.49(0.48, 0.50) | 0.43(0.39, 0.44) | 0.42(0.40, 0.45) | 0.45 |

\(\rho _{T_{1},T_{2}(e)}\) | −0.22(−0.24, −0.19) | −0.19(−0.21, −0.17) | −0.20(−0.21, −0.17) | −0.21 |

\(\rho _{T_{1},T_{3}(e)}\) | 0.02(0.01, 0.04) | 0.03(0.01, 0.04) | 0.03(0.02, 0.04) | 0.04 |

\(\rho _{T_{2},T_{3}(e)}\) | 0.11(0.09, 0.12) | 0.10(0.08, 0.11) | 0.10(0.08, 0.11) | 0.10 |

The additive genetic variance (\(\sigma ^2_{a})\) and the error variance (\(\sigma ^2_{e}\)) components obtained using a univariate INLA analysis (INLA-U) using the simulated dataset with negative covariance

Variance parameter | INLA-U | INLA-M | True |
---|---|---|---|

\(\sigma ^2_{a1}\) | 5.03 | 4.92 | 5.00 |

\(\sigma ^2_{a2}\) | 5.99 | 6.45 | 7.00 |

\(\sigma ^2_{a3}\) | 10.15 | 10.34 | 10.00 |

\(\sigma ^2_{e1}\) | 20.03 | 20.16 | 20.00 |

\(\sigma ^2_{e2}\) | 28.91 | 28.93 | 28.00 |

\(\sigma ^2_{e3}\) | 35.08 | 35.30 | 35.00 |

Heritability and breeding values are of great interest to breeders in order to plan an efficient breeding program. In our study, we also calculated the correlation coefficients between the estimated and true breeding values using different estimation methods. We used average over 50 simulation replicates for both datasets to calculate the correlation coefficients. For the high heritability dataset, the correlation coefficients were 0.85, 0.84 and 0.83 for REML, MCMC and INLA methods, respectively. However, for the low heritability dataset the correlation coefficients were relatively low being 0.64, 0.65 and 0.58 for REML, MCMC and INLA, respectively.

### Field data

The additive genetic variance (\(\sigma ^2_{a}\)) and the error variance (\(\sigma ^2_{e}\)) for the field data obtained from the REML analysis and the posterior mode estimates obtained from the MCMCglmm package along with R-INLA posterior mean estimates are presented

(Co)variance parameter | REML | MCMC | INLA |
---|---|---|---|

\(\sigma ^2_{{\text{PH}}_{a}}\) | 22.92 | 21.85 | 24.09 |

\(\sigma ^2_{{\text{FL}}_{a}}\) | 6.50 | 6.56 | 6.58 |

\(\sigma ^2_{{\text{YLD}}_{a}}\) | 34,285.92 | 35,890.51 | 40,218.69 |

\(\sigma ^2_{{\text{PH}}_{e}}\) | 42.34 | 40.91 | 43.68 |

\(\sigma ^2_{{\text{FL}}_{e}}\) | 8.17 | 8.23 | 8.47 |

\(\sigma ^2_{{\text{YLD}}_{e}}\) | 71,698.64 | 73,983.14 | 74,211.42 |

\(\sigma _{{\text{PH}},{\text{FL}}(a)}\) | 4.57 | 4.48 | 4.60 |

\(\sigma _{{\text{PH}},{\text{YLD}}(a)}\) | −285.05 | −237.21 | −263.99 |

\(\sigma _{{\text{FL}},{\text{YLD}}(a)}\) | −158.50 | −120.01 | −161.97 |

\(\sigma _{{\text{PH}},{\text{FL}}(e)}\) | 2.41 | 2.78 | 2.57 |

\(\sigma _{{\text{PH}},{\text{YLD}}(e)}\) | 106.36 | 76.83 | 83.21 |

\(\sigma _{{\text{FL}},{\text{YLD}}(e)}\) | −38.75 | −23.82 | −44.09 |

Narrow-sense heritability estimates (\(h^2\)) for the simulated datasets (averaged over 50 simulation replicates) and the real dataset with the different estimation methods

REML | MCMC | INLA | True | |
---|---|---|---|---|

High heritability dataset | ||||

Trait1 | 0.49 | 0.49 | 0.49 | 0.50 |

Trait2 | 0.61 | 0.62 | 0.61 | 0.60 |

Trait3 | 0.71 | 0.72 | 0.71 | 0.71 |

Low heritability dataset | ||||

Trait1 | 0.15 | 0.21 | 0.20 | 0.20 |

Trait2 | 0.17 | 0.23 | 0.22 | 0.20 |

Trait3 | 0.17 | 0.23 | 0.21 | 0.22 |

Real data | Spindel et al. (2015) | |||

PH | 0.35 | 0.35 | 0.35 | 0.35 |

FL | 0.44 | 0.44 | 0.44 | 0.43 |

YLD | 0.32 | 0.32 | 0.35 | 0.32 |

## Discussion

Multi-trait analysis of mixed models tend to be powerful and provide more accurate estimates than the single-trait analysis because the former method can take into account the underlying correlation structure found in a multi-trait dataset. However, Bayesian and non-Bayesian inference of multi-trait mixed model analysis are complex and computationally demanding. In this study, we explained how to do Bayesian inference of a multivariate animal model using recently developed INLA and the counter part MCMC, while comparing the results with the commonly used REML estimates. Our results show that reparametrization-based INLA approach can be used as a fast alternative to MCMC methods for the Bayesian inference of multivariate animal model. The reparametrization approach, that was here applied for INLA analysis, can be used also more generally together with other tools to speed up the multi-trait animal model computations.

Drawback of the reparametrization-based approach is that priors are assigned for elements in the Cholesky factor instead of the original covariance matrix. Thus, here it is not possible to make a direct comparison between the MCMC and INLA results due to the differences in the prior distributions, however, it is possible to compare both approaches if we choose the same prior distributions. For the MCMC analysis we used inverse-Wishart distributions for the covariance matrices; whereas, for INLA we used Gaussian prior distribution for the elements in the Cholesky factor (i.e., dependence parameters) (\({\mathbf {\kappa }}_{ij}`s,{\mathbf {\alpha }}_{ij}`s\)) and inverse-Gamma distribution for the decomposition variance components (\({\mathbf {\sigma }}^2_{a_{i}`s},{\mathbf {\sigma }}^2_{e_{i}`s}\)). Our results show that the REML estimates are in concordance with MCMCglmm and INLA. We want to emphasize that in our examples, the analyzed data sets were large and we did not encounter any problems. In general, identifiability is a problem in mixed model analyses with small data (Mathew et al. 2012). However, Bayesian methods are in better positions because they can at least find such problems (that posterior distribution has multiple modes) more easily than REML (which provides a single point-estimate).

Nowadays, molecular markers are widely used in animal and plant breeding programs as a valuable tool for genetic improvement. Therefore, we also showed how to estimate genetic parameters in a multivariate animal model using molecular marker information with the reparametrization-based INLA approach and frequentist framework. Finally, our results imply that the reparametrization-based INLA approach can be used as a fast alternative to MCMC methods in order to estimate genetic parameters with a multivariate animal model using pedigree information as well as with molecular marker information.

### Author contribution statement

BM, AH, PK, JL and MS were involved in the conception and design of the study. BM performed the simulations and preprocessing of the data. BM and AH implemented the method, and performed the statistical analyses. BM drafted the manuscript. BM, AH, JL and MS participated in the interpretation of results. All the authors critically revised the manuscript.

## Notes

### Acknowledgments

We thank Jarrod D. Hadfield for helping us with the implementation of our model using MCMCglmm package. We are also grateful to Håvard Rue and Ingelin Steinsland for answering questions about INLA. The authors would like to thank the editor and two anonymous reviewers as well as Karin Woitol for their suggestions and comments which helped us to improve our manuscript.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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