# Genetics of Lifetime Reproductive Performance in Italian Heavy Draught Horse Mares

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Subject and Organization of the Study

#### 2.2. Training Dataset and Analysis

^{®}statistical software; SAS

^{®}version 9.4, SAS Institute Inc., Cary, NC, USA) allowing the estimates of the no. of foals produced at the 6th breeding season depending on the basis of (i) the previous no. of foals after either 3, 4, or 5 breeding seasons, and (ii) the age at first foaling (3 or 4 years; Table 1). A lifetime foaling rate (LFR) was then obtained for each mare by dividing the no. of actual and predicted foals at the 6th reproductive events for the no. of opportunities of doing so (i.e., 6). The predictive ability of coefficients or equations was analyzed by comparing the actuals and the predicted values expressed for each mare using the following statistics:

- the percentage squared bias (PSB; [26]), obtained from the formula:PSB = 100 (
**y**−**ŷ**)’ (**y**−**ŷ**)/(**y**’**y**)**y**is a vector of actual and**ŷ**is a vector of predicted values; - the mean absolute deviation of residuals (MAD; [27]) calculated from the formula:MAD = Σ |(
**y**−**ŷ**)|/n**y**−**ŷ**| are absolute differences between actual (**y**) and predicted values (**ŷ**), respectively, and n is the number of observations; - the standard deviation of residuals obtained as
**y**−**ŷ**, where**y**and**ŷ**are vectors of actual and predicted values, respectively.

#### 2.3. Full Dataset and Analysis

^{®}statistical software; SAS

^{®}version 9.4, SAS Institute Inc., Cary, NC, USA) were carried out. The latter one was run to establish which non-genetic effects could be taken into account in the genetic model. Some preliminary genetic models with increasing complexity were also run to see the variations in heritability when different animal-based factors were included (Table S1). The final and best-fitted model (AIC criterion [29]) was the one with the greatest complexity. Among the non-genetic fixed effects that accounted for a significant part of the total variance there were the above-mentioned environmental unit by birth year (EU-BY, 125 levels, including 2 to 82 records; Table 1), and the age first foaling (AF; 2 levels, 3 or 4 years; Table 1). The final model also included the effect of individual inbreeding as covariate, calculated on the whole studbook data updated at December 2019 using a recursive algorithm able to recover the incomplete lineages in pedigree [30]. Moreover, both additive and non-additive genetic effects were considered, the latter in terms of dominance effect. The matrix notation for the final single trait animal model genetic analysis can be written as follows:

**y**=

**X**

**β**+

**Z**+

_{h}h**Z**+

_{a}a**Z**+

_{d}d**e**

**y**is an N x 1 vector of observations,

**β**is the vector of systematic fixed effects of order

**p**(AF and the linear covariate for inbreeding),

**h**is the vector for the random effect of order

**q**for EU-BY,

**a**is the vector of random animal effect of order

**q**(6801 animals in pedigree file, i.e., tracing back up to 12th generation for mares with records; Table 1),

**d**is the vector of random dominance effect (9413 levels) and

**e**is the vector of residual effects. Furthermore,

**X**,

**Z**and

_{h}, Z_{a}**Z**are the corresponding incidence matrices with the appropriate dimension. The assumptions about the structure of (co)variance were as follows:

_{d}**A**is the numerator additive relationship matrix,

**D**is the dominance relationship matrix, and

**I**is an identity matrix.

^{2}is heritability of the trait, ${\mathsf{\sigma}}_{\mathrm{a}}^{2}$ and ${\mathsf{\sigma}}_{\mathrm{p}}^{2}$ are the additive genetic and phenotypic variances of the trait, Var(${\mathsf{\sigma}}_{\mathrm{a}}^{2}$), Var(${\mathsf{\sigma}}_{\mathrm{p}}^{2}$) are their respective predicted error variances, and Cov(${\mathsf{\sigma}}_{\mathrm{a}}^{2}{,\mathsf{\sigma}}_{\mathrm{p}}^{2}$) is the predicted error (co)variance.

## 3. Results

#### 3.1. Validation of the Phenotypic Variable to Measure Lifetime Fertility

#### 3.2. Genetic Analysis of Lifetime Fertility

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Class distribution of lifetime fertility rate (LFR) obtained in the full dataset (n = 3033) combining actual and predicted foals after the 6th breeding season; In (

**a**) distribution of linear LFR obtained by predicting incomplete reproductive career with coefficients (LFR-C) or equations (LFR-E); in (

**b**) normal distribution of arcsine transformed LFR obtained by predicting LFR with coefficients (Arcsine LFR-C) or equations (Arcsine LFR-E).

Item | Training Dataset | Full Dataset |
---|---|---|

Mares with actual records, number (no.) | 1487 | 3033 |

Mares with projected records, no. | - | 1443 |

Environmental units ^{1} by birth year (EU-BY), no. | 97 | 125 |

Mean records in EU-BY, no. | 15.3 | 24.2 |

Age at first known breeding season, months (mo.) | 43.7 ± 6.6 | 43.9 ± 6.5 |

3 years first foaling mares, mo. | 36.6 ± 1.8 | 36.6 ± 1.8 |

4 years first foaling mares, mo. | 49.1 ± 2.5 | 49.0 ± 2.4 |

Animals in the pedigree file, no. | - | 6803 |

Sires of mares with record, no. | 400 | 602 |

Dams of mares with record, no. | 1011 | 1848 |

Daughters/sire | 3.5 | 4.8 |

Daughters/dam | 1.3 | 1.5 |

^{1}Environmental units are intended to be a group of farm-studs in the same geographical area and common rearing system of mares and foals.

**Table 2.**Predictive ability of coefficients or equations projection methods used to estimate the foals production at the 6th breeding season and to obtain the lifetime fertility rate (foal produced divided by the number of opportunities) starting from foals produced after 3, 4, or 5 known breeding seasons or considering the whole predictive ability of projections.

Item | Projection Method | |
---|---|---|

Coefficients (LFR-C) | Equations (LFR-E) | |

Projection from 3 known breeding seasons | ||

- PSB ^{1} | 0.0604 | 0.0148 |

- MAD ^{2} | 0.1813 | 0.0793 |

- SDR ^{3} | 0.1028 | 0.0986 |

Projection from 4 known breeding seasons | ||

- PSB | 0.0437 | 0.0092 |

- MAD | 0.1561 | 0.0606 |

- SDR | 0.0832 | 0.0784 |

Projection from 5 known breeding seasons | ||

- PSB | 0.0319 | 0.0041 |

- MAD | 0.1377 | 0.0350 |

- SDR | 0.0622 | 0.0537 |

Average | ||

- PSB | 0.0453 | 0.0094 |

- MAD | 0.1584 | 0.0583 |

- SDR | 0.0827 | 0.0769 |

**Table 3.**Descriptive statistics for lifetime fertility rate (LFR), normality tests (Kolmogorov-Smirnov D and Anderson Darling A-Sq parameters and significance), skewness, and kurtosis of the 3033 data in the full data set obtained combining actual and predicted number of foals after 6 breeding seasons and considering different prediction methods.

Statistic | LFR-C ^{1} | LRF-E ^{2} | Arcsine LFR-C ^{3} | Arcsine LFR-E ^{4} |
---|---|---|---|---|

Mean ± standard deviation | 0.700 ± 0.142 | 0.699 ± 0.144 | 0.794 ± 0.195 | 0.793 ± 0.197 |

Kolmogorov-Smirnov D | 0.16 (p < 0.01) | 0.14 (p < 0.01) | 0.15 (p < 0.01) | 0.11 (p < 0.01) |

Anderson-Darling A-Sq | 82.9 (p < 0.01) | 78.7 (p < 0.01) | 67.2 (p < 0.01) | 60.1 (p < 0.01) |

Skewness | −0.881 | 0.144 | −0.506 | 0.197 |

Kurtosis | 0.485 | 0.986 | −0.300 | −0.004 |

^{1}LFR-C = predictive method for LFR based on regression coefficients;

^{2}LFR-E = predictive method of LFR based on equations;

^{3}Arcsine LFR-C = arcsine transformation of the LFR-C;

^{4}Arcsine LFR-E = arcsine transformation of the LFR-E.

**Table 4.**Results of genetic analysis carried out on the lifetime fertility rate (LFR) obtained combining actual and predicted number of foals after 6 breeding seasons and considering different prediction methods for incomplete reproductive career (by coefficients; LFR-C; by equations; LFR-E) and arcsine transformation of both the LFR-C (Arcsine LFR-C) and the LFR-E (Arcsine LFR-E).

Item | LFR-C | LRF-E | Arcsine LFR-C | Arcsine LFR-E |
---|---|---|---|---|

Herd Variance ^{1} | 0.141 | 0.160 | 0.259 | 0.276 |

Genetic Variance ^{1} | 4.848 | 4.693 | 8.975 | 8.829 |

Dominance Variance ^{1} | 1.628 | 1.358 | 2.896 | 2.573 |

Residual Variance ^{1} | 13.426 | 14.259 | 25.660 | 25.463 |

Phenotypic Variance ^{1} | 20.043 | 19.469 | 37.790 | 37.141 |

Heritability | 0.242 | 0.241 | 0.238 | 0.237 |

SE Heritability | 0.043 | 0.042 | 0.042 | 0.042 |

AIC | −3290 | −3380 | −1370 | −1424 |

^{1}Multiplied by 10

^{3}.

**Table 5.**Rank correlation coefficients between standardized EBVs obtained for different expression of lifetime fertility rate (LFR

^{1}) in Italian Heavy Draught Horse mares with records (n = 3033) or stallions with at a minimum accuracy of 0.65 (n = 77).

Comparison | Mares with Actual or Predicted LFR | Stallions with ≥9 Daughters with Actual or Predicted LFR |
---|---|---|

LFR-C ^{2} vs. LFR-E ^{3} | 0.997 | 0.993 |

LFR-C vs. Arcsine LFR-C ^{4} | 0.996 | 0.996 |

LFR-E vs. Arcsine LFR-E ^{5} | 0.996 | 0.995 |

Arcsine LFR-C vs. Arcsine LFR-E | 0.997 | 0.993 |

^{1}LFR was obtained combining actual and predicted number of foals after 6th breeding season and considering different prediction methods for incomplete reproductive career;

^{2}LFR-C = prediction method for LFR based on regression coefficients;

^{3}LFR-E = prediction method of LFR based on equations;

^{4}Arcsine LFR-C = arcsine transformation of the LFR-C;

^{5}Arcsine LFR-E = arcsine transformation of the LFR-E.

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**MDPI and ACS Style**

Mantovani, R.; Folla, F.; Pigozzi, G.; Tsuruta, S.; Sartori, C.
Genetics of Lifetime Reproductive Performance in Italian Heavy Draught Horse Mares. *Animals* **2020**, *10*, 1085.
https://doi.org/10.3390/ani10061085

**AMA Style**

Mantovani R, Folla F, Pigozzi G, Tsuruta S, Sartori C.
Genetics of Lifetime Reproductive Performance in Italian Heavy Draught Horse Mares. *Animals*. 2020; 10(6):1085.
https://doi.org/10.3390/ani10061085

**Chicago/Turabian Style**

Mantovani, Roberto, Fabio Folla, Giuseppe Pigozzi, Shogo Tsuruta, and Cristina Sartori.
2020. "Genetics of Lifetime Reproductive Performance in Italian Heavy Draught Horse Mares" *Animals* 10, no. 6: 1085.
https://doi.org/10.3390/ani10061085