# Genetic Parameters for a Weighted Analysis of Survivability in Dairy Cattle

^{1}

^{2}

^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

_{d}) would be the sum of all periods (∑w) minus the effective counts achieved by the individual in the previous periods. The weight of the record is as follows:

_{d}/(m − w

_{d}),

_{d}is the effective number of cases in the culling period.

^{2}i = 1 − k∗cii,

^{2}

_{i}is the BV reliability of individual I;

^{2})/h

^{2};

_{ii}is the diagonal element for individual i from the inverse matrix of the system.

^{2}= 0.15 in all three presented examples. There was also the same difference in BV among the three methods, at approximately 1.73. For other individuals, the BV reliabilities and differences between individuals varied depending on how the weights were used. Weighting changed the overall BV layout and equalised BV reliability. Higher weights (w2) yielded better results, and only w2 will be considered in the rest of the manuscript.

#### Statistical Model

^{−17}(convergence criterion). Genetic parameters and effects in the model, including BV, were estimated.

## 3. Results

#### 3.1. The Least-Squares Method Fixed Effects

#### 3.2. Components of Variance and Genetic Parameters

^{2}

_{G}, heritability h

^{2}, variance ratio: k = s

^{2}

_{r}/s

^{2}

_{G}(residual variance/genetic variance)), where the mean errors were approximately 4% of the estimated value. The phenotypic variance s

^{2}

_{P}in Table 4 corresponded to the standard deviation in Table 3 when fixed effects were considered. In the case without weights, the values of standard deviations were approximately equivalent (without milk: 31.28 compared to 31.26, and with regression to milk: 31.01 compared to 30.99), but not in the case with weights (w2) (55.90 compared to 45.26 and 54.46 compared to 43.88). The inclusion of weights multiplies all variance components. Proportionally, the components for the individual’s permanent environment and the genetic component increased the most. The heritability coefficients for the unweighted calculations were 1.3% and 1.5%, which are consistent with those obtained in studies by van Pelt et al. [13] and Pritchard et al. [21]. For calculations with weights, the heritability was higher—5.8%. Higher heritability of a trait results in a more accurate estimation of breeding value (an additive component of heritability). In the case of weights, an individual’s permanent environment is a significant component, which is essentially zero in cases without weights. Repeatability was 50.5% in the weighted case. An important indicator is the variance ratio k because it indicates the additive component of inheritance. The solution of the system of BLUP equations depends on the ratio of variances. In the statistical models with weights, k was 8.536–9.169, which is several times smaller (more favourable) than in the cases without weights.

#### 3.3. Variability of Effects in the Solution

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Breeding values frequency distributions: calculations without SURV and SURVm weights and according to level (w2) SURVw2 and SURVmw2 weight.

Method of Weighting * | Without Weight | w1 | w2 | ||||||
---|---|---|---|---|---|---|---|---|---|

Individual | Failure to Survive a Stage | BV | r^{2} | Weight | BV | r^{2} | Weight | BV | r^{2} |

A | 1 | −0.73 | 0.02 | 9 | −6.96 | 0.11 | 16 | −10.09 | 0.15 |

B | 2 | −0.60 | 0.04 | 8 | −5.42 | 0.12 | 13 | −7.63 | 0.15 |

C | 3 | −0.47 | 0.05 | 7 | −3.89 | 0.12 | 10 | −4.95 | 0.15 |

D | 4 | −0.34 | 0.07 | 6 | −2.37 | 0.13 | 8 | −2.76 | 0.15 |

E | 5 | −0.21 | 0.09 | 5 | −0.85 | 0.14 | 6 | −0.42 | 0.15 |

F | 6 | −0.09 | 0.10 | 4 | 0.69 | 0.14 | 5 | 1.07 | 0.15 |

G | 7 | 0.03 | 0.12 | 3 | 2.25 | 0.14 | 3 | 3.75 | 0.15 |

H | 8 | 0.15 | 0.13 | 2 | 3.84 | 0.15 | 2 | 5.34 | 0.15 |

I | 9 | 0.26 | 0.15 | 1 | 5.49 | 0.15 | 1 | 6.99 | 0.15 |

J | Survival 9 | 1.99 | 0.15 | 1 | 7.22 | 0.15 | 1 | 8.71 | 0.15 |

*****h

^{2}= 0.02, survival = 100, non-survival = 0, w1 = weight by number of un-survived periods, w2 = weight by effective number of cases, BV = breeding values, r

^{2}= reliability of breeding values.

Statistical Model | ||||
---|---|---|---|---|

without Weight | with Weight (w2) | |||

Cow with records | 298,290 | |||

Individuals in the pedigree | 660,476 | |||

HYS, records | Number 15,919, Mean size 113.93, Range 15 to 650 | |||

Milk deviation in the herd (SD) | Average 0.99 (0.21), Range 0.06 to 4.60 | |||

Correlation survival rate x milk | 0.13 | 0.22 | ||

Period | Frequency % | Discarding % (SD) | Weighted Frequency % | Weighted Discarding % (SD) |

Average | 12.29 (32.8) | 49.01 (50.0) | ||

1 | 16.45 | 3.64 (18.7) | 14.78 | 37.68 (48.5) |

2 | 15.85 | 8.69 (28.2) | 18.83 | 55.31 (49.7) |

3 | 14.47 | 18.67 (39.0) | 22.55 | 69.66 (46.0) |

4 | 11.77 | 3.43 (18.2) | 8.48 | 22.11 (41.5) |

5 | 11.36 | 12.74 (33.3) | 10.82 | 46.71 (49.9) |

6 | 9.92 | 25.61 (43.6) | 11.67 | 63.26 (48.2) |

7 | 7.38 | 5.35 (22.5) | 4.75 | 14.49 (35.2) |

8 | 6.98 | 16.56 (37.2) | 4.73 | 28.41 (45.1) |

9 | 5.83 | 28.60 (45.2) | 3.39 | 28.60 (45.2) |

Number of survival records | 1,813,636 | |||

Average weight (w2) | 1.72 | |||

Sum of weights (w2) | 3,119,557 |

Effects Considered | Without Weight | With Weight (w2) | ||
---|---|---|---|---|

SD | R^{2} | SD | R^{2} | |

Simple records | 32.81 | 49.99 | ||

Effects of the herd–year–season least-squares method | 32.14 | 0.05 | 48.13 | 0.08 |

Herd–year–season + period | 31.26 | 0.10 | 45.26 | 0.18 |

Herd–year–season + period + milk | 30.99 | 0.12 | 43.88 | 0.23 |

Herd–year–season + period + milk + sire | 30.94 | 0.12 | 43.72 | 0.24 |

^{2}= coefficient of determination, w2 = weight by effective number of cases.

Quantity | Statistical Model | |||
---|---|---|---|---|

SURV | SURVm | SURVSw2 | SURVmw2 | |

s^{2}_{r} | 965.97 (1.02) | 947.02 (1.00) | 1545.40 (1.35) | 1529.40 (1.34) |

s^{2}_{G} | 12.46 (0.02) | 14.60 (0.03) | 181.26 (7.56) | 166.81 (6.94) |

s_{PE} | 0.00 | 0.00 | 1397.70 (6.76) | 1269.60 (6.22) |

s^{2}_{P} | 978.43 (1.02) | 961.62 (1.00) | 3124.36 (5.19) | 2965.81 (4.78) |

h^{2} | 0.013 (0.0000) | 0.015 (0.0000) | 0.058 (0.0024) | 0.056 (0.0023) |

r | 0.013 (0.0000) | 0.015 (0.0000) | 0.505 (0.0009) | 0.484 (0.0009) |

k = s^{2}_{r}/s^{2}_{G} | 77.52 (0.1558) | 64.54 (0.1302) | 8.536 (0.3598) | 9.169 (0.3856) |

^{2}

_{r}= residual variance, s

^{2}

_{G}= genetic variance, s

_{PE}= permanent environmental variance, h

^{2}= heritability, r = repeatability, k = variance ratio. Mixed model—see Equation (5).

**Table 5.**Estimates and solutions of fixed and random effects from mixed-model analyses of unweighted and weighted survival of culling and involuntary culling.

Effects | Statistical Model | ||||
---|---|---|---|---|---|

SURV | SURVm | SURVw2 | SURVmw2 | ||

Herd–year–season | SD | 10.73 | 10.59 | 18.37 | 17.94 |

Min | −88.08 | −88.92 | −105.62 | −100.20 | |

Max | 28.02 | 27.07 | 57.40 | 51.34 | |

Periods | SD | 8.81 | 8.71 | 19.98 | 19.65 |

Min | −13.25 | −13.17 | −30.48 | −30.10 | |

Max | 9.84 | 9.68 | 34.69 | 33.95 | |

Breeding values | SD | 1.75 | 2.18 | 6.20 | 6.04 |

Min | −10.64 | −12.43 | −38.79 | −39.64 | |

Max | 9.76 | 10.63 | 37.46 | 34.95 | |

Permanent environment | SD | - | - | 33.93 | 32.19 |

Min | - | - | −107.96 | −137.79 | |

Max | - | - | 85.18 | 86.68 | |

Milk regression | - | 20.13 | - | 31.22 |

Period | Statistical Model | |||
---|---|---|---|---|

SURV | SURVm | SURVw2 | SURVmw2 | |

1 | 9.43 | 9.52 | 34.69 | 33.95 |

2 | 4.48 | 4.61 | 19.28 | 18.86 |

3 | −5.15 | −4.92 | −6.29 | −6.73 |

4 | 9.84 | 9.68 | 14.80 | 14.85 |

5 | 0.68 | 0.63 | −7.71 | −7.49 |

6 | −11.69 | −11.45 | −30.48 | −30.10 |

7 | 8.31 | 7.97 | 2.63 | 2.89 |

8 | −2.64 | −2.88 | −12.33 | −12.16 |

9 | −13.25 | −13.17 | −14.58 | −14.06 |

Herd–Year–Season | Breeding Values | Permanent Environment | |||||
---|---|---|---|---|---|---|---|

SURVm | SURVw2 | SURVmw2 | SURVm | SURVw2 | SURVmw2 | SURVmw2 | |

SURV | 100 | 88 | 89 | 91 | 92 | 94 | |

SURVm | 88 | 89 | 75 | 89 | |||

SURVw2 | 99 | 96 | 99 |

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**MDPI and ACS Style**

Černá, M.; Zavadilová, L.; Vostrý, L.; Bauer, J.; Šplíchal, J.; Vařeka, J.; Fulínová, D.; Brzáková, M. Genetic Parameters for a Weighted Analysis of Survivability in Dairy Cattle. *Animals* **2023**, *13*, 1188.
https://doi.org/10.3390/ani13071188

**AMA Style**

Černá M, Zavadilová L, Vostrý L, Bauer J, Šplíchal J, Vařeka J, Fulínová D, Brzáková M. Genetic Parameters for a Weighted Analysis of Survivability in Dairy Cattle. *Animals*. 2023; 13(7):1188.
https://doi.org/10.3390/ani13071188

**Chicago/Turabian Style**

Černá, Michaela, Ludmila Zavadilová, Luboš Vostrý, Jiří Bauer, Jiří Šplíchal, Jan Vařeka, Daniela Fulínová, and Michaela Brzáková. 2023. "Genetic Parameters for a Weighted Analysis of Survivability in Dairy Cattle" *Animals* 13, no. 7: 1188.
https://doi.org/10.3390/ani13071188