# Modelling Extended Lactations in Polish Holstein–Friesian Cows

^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

- 1.
- The Wilmink [11] model (WIL) fitted to the records from the whole lactation$$\begin{array}{c}\hfill y\left(t\right)=a+b\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t+c\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{e}^{-0.05\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t}\end{array}$$
- t —day in milk (DIM);
- $a,b,c$ —parameters to be fitted;
- $y\left(t\right)$ —milk, fat, protein or lactose yield, or urea content in milk at DIM t.

- 2.
- Wilmink model (WIL305) fitted to TD records from 5 to 305 DIM in lactation and:
- (a)
- Linear function (LIN) fitted to TD records from 306 to 400 DIM in lactation, assuming that at least one TD record beyond 305 DIM occurred. To fit LIN, the last TD record before 305 DIM and all TD records beyond 305 DIM were used;
- (b)
- Squared function (SQRT) fitted to TD records from 306 to 400 DIM in lactation. In this case, at least two TD records beyond 305 DIM were required. To fit SQRT, the last TD record before 305 DIM and all TD records beyond 305 DIM were used.

- 1.
- Mean Error (ME $=\frac{\sum {e}_{i}}{n}$);
- 2.
- Mean Squared Error (MSE $=\frac{\sum {e}_{i}^{2}}{n}$);
- 3.
- Mean Absolute Error (MAE $=\frac{\sum |{e}_{i}|}{n}$);
- 4.
- Pearson’s correlation (R) between the measured (${y}_{i}$) and estimated (${\widehat{y}}_{i}$) milk yields;
- 5.
- Quotient between the error sum of squares and the observed sum of squares (Q $=\frac{\sum {e}_{i}^{2}}{\sum {y}_{i}^{2}}$), with lower values indicating closer similarity between the true (${y}_{i}$) and estimated (${\widehat{y}}_{i}$) values.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Example of lactation curve with standard shape fitted using Wilmink (WIL and WIL305), linear (LIN) and squared (SQRT) functions.

**Figure 2.**Examples of different first lactation curve shapes fitted using Wilmink (WIL305) function.

**Table 1.**Number of test-day (TD) records and lactations and means with standard deviations (SD) for TD milk, fat, protein and lactose yield, and milk urea content, by lactation.

Number of | Milk (kg) | Fat (kg) | Protein (kg) | Lactose (kg) | Urea (mg/L) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Lactation | TD Records | Lactations | Mean | SD | Mean | SD | Mean | SD | Mean | SD | Mean | SD |

1 | 1,918,079 | 247,729 | 24.24 | 7.77 | 0.97 | 0.30 | 0.81 | 0.25 | 1.19 | 0.39 | 219.66 | 86.08 |

2 | 1,253,570 | 193,902 | 26.70 | 10.00 | 1.08 | 0.39 | 0.90 | 0.31 | 1.28 | 0.50 | 218.87 | 86.68 |

3 | 1,298,760 | 127,590 | 27.57 | 10.45 | 1.12 | 0.42 | 0.92 | 0.32 | 1.31 | 0.52 | 216.09 | 86.35 |

4–6 | 1,918,079 | 133,609 | 26.77 | 10.26 | 1.10 | 0.42 | 0.89 | 0.31 | 1.26 | 0.51 | 212.32 | 85.89 |

Total | 6,955,768 | 702,830 | 25.99 | 9.50 | 1.05 | 0.38 | 0.87 | 0.29 | 1.25 | 0.47 | 217.43 | 86.30 |

Parameter ${}^{\mathit{a}}$ | ||
---|---|---|

Curve Shape | b | c |

Standard lactation curve | <0 | <0 |

Reversed curve | >0 | >0 |

Continuously increasing curve | >0 | <0 |

Continuously decreasing (atypical) curve | <0 | >0 |

**Table 3.**Goodness of fit of WIL and WIL305 functions to test-day (TD) records until 305 day in milk.

No. of TD | |||||||
---|---|---|---|---|---|---|---|

Trait | Function | Records | ME ${}^{\mathit{a}}$ | MSE ${}^{\mathit{b}}$ | MAE ${}^{\mathit{c}}$ | R ${}^{\mathit{d}}$ | Q ^{e} |

Milk (kg) | WIL ${}^{f}$ | 6,133,988 | 5.3 × 10${}^{-2}$ | 8.41 | 2.06 | 0.950 | 1.03 |

WIL305 ${}^{g}$ | 6,133,988 | −2.18 × 10${}^{-9}$ | 7.63 | 1.95 | 0.955 | 0.93 | |

Fat (kg) | WIL | 6,104,472 | 2.90 × 10${}^{-3}$ | 0.024 | 0.106 | 0.913 | 1.796 |

WIL305 | 6,104,472 | −1.92 × 10${}^{-9}$ | 0.022 | 0.101 | 0.919 | 1.669 | |

Protein (kg) | WIL | 6,104,679 | 3.44 × 10${}^{-3}$ | 0.010 | 0.072 | 0.938 | 1.140 |

WIL305 | 6,104,679 | 9.78 × 10${}^{-10}$ | 0.009 | 0.068 | 0.945 | 1.020 | |

Lactose (kg) | WIL | 6,104,668 | 1.87 × 10${}^{-3}$ | 0.020 | 0.102 | 0.951 | 1.068 |

WIL305 | 6,104,668 | −4.32 × 10${}^{-10}$ | 0.019 | 0.096 | 0.956 | 0.969 | |

Urea (mg/L) | WIL | 6,091,233 | 0.308 | 2544.67 | 36.93 | 0.813 | 4.64 |

WIL305 | 6,091,233 | −1.84 × 10${}^{-8}$ | 2339.26 | 35.40 | 0.829 | 4.27 |

^{e}Q—quotient between the error sum of squares and the observed sum of squares, ${}^{f}$ WIL—Wilmink function fitted to TD records from 5 to the last DIM in lactation, ${}^{g}$ WIL305—Wilmink function fitted to TD records from 5 to 305 DIM in lactation.

**Table 4.**Goodness of fit of WIL and WIL305 functions to test-day (TD) milk yields until 305 day in milk, by different shape of lactation curve.

No. of | ||||||||
---|---|---|---|---|---|---|---|---|

Function | Curve Shape | Lactations | TD Records | ME ${}^{\mathit{a}}$ | MSE ${}^{\mathit{b}}$ | MAE ${}^{\mathit{c}}$ | R ${}^{\mathit{d}}$ | Q ^{e} |

WIL ${}^{f}$ | Standard lactation curve | 459,499 | 4,039,702 | 0.0730 | 8.68 | 2.10 | 0.95 | 0.97 |

Continuously decreasing (atypical) curve | 221,172 | 1,908,990 | 0.0108 | 7.78 | 1.98 | 0.95 | 1.15 | |

Reversed curve | 17,399 | 144,244 | 0.0366 | 9.12 | 2.09 | 0.91 | 1.61 | |

Continuously increasing curve | 4760 | 41,052 | 0.0940 | 9.45 | 2.12 | 0.93 | 1.27 | |

WIL305 ${}^{g}$ | Standard lactation curve | 441,583 | 3,878,838 | −1.86 × 10${}^{-9}$ | 7.85 | 1.99 | 0.96 | 0.88 |

Continuously decreasing (atypical) curve | 222,451 | 1,922,683 | −4.12 × 10${}^{-9}$ | 7.02 | 1.87 | 0.95 | 1.01 | |

Reversed curve | 29,743 | 252,491 | 8.67 × 10${}^{-9}$ | 8.72 | 2.04 | 0.92 | 1.42 | |

Continuously increasing curve | 9053 | 79,976 | −5.38 × 10${}^{-9}$ | 8.56 | 2.04 | 0.93 | 1.04 |

^{e}Q—quotient between the error sum of squares and the observed sum of squares, ${}^{f}$ WIL—Wilmink function fitted to TD records from 5 to the last DIM in lactation, ${}^{g}$ WIL305—Wilmink function fitted to TD records from 5 to 305 DIM in lactation.

No. of TD | |||||||
---|---|---|---|---|---|---|---|

Trait | Function | Records | ME ${}^{\mathit{a}}$ | MSE ${}^{\mathit{b}}$ | MAE ${}^{\mathit{c}}$ | R ${}^{\mathit{d}}$ | Q ^{e} |

Milk (kg) | WIL ${}^{f}$ | 821,780 | −0.40 | 9.57 | 2.28 | 0.89 | 2.62 |

LIN ${}^{g}$ | 821,780 | 0.10 | 3.67 | 1.23 | 0.96 | 1.00 | |

SQRT ${}^{h}$ | 673,332 | −0.01 | 1.83 | 0.75 | 0.98 | 0.49 | |

Fat (kg) | WIL | 818,006 | −0.0217 | 0.022 | 0.109 | 0.86 | 2.99 |

LIN | 818,006 | 0.0046 | 0.009 | 0.062 | 0.94 | 1.29 | |

SQRT | 669,376 | −0.0005 | 0.005 | 0.039 | 0.97 | 0.66 | |

Protein (kg) | WIL | 818,061 | −0.0257 | 0.013 | 0.086 | 0.88 | 2.61 |

LIN | 818,061 | 0.0037 | 0.005 | 0.047 | 0.95 | 1.01 | |

SQRT | 669,427 | −0.0005 | 0.003 | 0.028 | 0.98 | 0.49 | |

Lactose (kg) | WIL | 818,061 | −0.0140 | 0.023 | 0.111 | 0.89 | 2.75 |

LIN | 818,061 | 0.0050 | 0.009 | 0.060 | 0.96 | 1.05 | |

SQRT | 669,427 | −0.0006 | 0.004 | 0.036 | 0.98 | 0.50 | |

Urea (mg/L) | WIL | 816,018 | −2.30 | 2769.30 | 40.19 | 0.78 | 5.10 |

LIN | 816,018 | −0.24 | 1417.32 | 25.33 | 0.90 | 2.61 | |

SQRT | 667,270 | 0.03 | 792.30 | 16.49 | 0.94 | 1.46 |

^{e}Q—quotient between the error sum of squares and the observed sum of squares, ${}^{f}$ WIL—Wilmink function fitted to TD records from 5 to the last DIM in lactation, ${}^{g}$ LIN—Linear function fitted to TD records from 306 to the last DIM in lactation. ${}^{h}$ SQRT—Squared function fitted to TD records from 306 to the last DIM in lactation.

**Table 6.**Goodness of fit of WIL, LIN and SQRT functions to test-day (TD) milk yields beyond 305 day in milk by number of TD records per cow beyond 305 DIM.

Function | No. of TD Records Per Cow beyond 305 DIM | No. of TD Records | ME ${}^{\mathit{a}}$ | MSE ${}^{\mathit{b}}$ | MAE ${}^{\mathit{c}}$ | R ${}^{\mathit{d}}$ | Q ^{e} |
---|---|---|---|---|---|---|---|

WIL ^{e} | 1 | 148,448 | −1.279 | 10.991 | 2.462 | 0.887 | 3.374 |

2 | 226,912 | −0.540 | 9.402 | 2.254 | 0.889 | 2.636 | |

3 | 240,288 | −0.138 | 9.122 | 2.231 | 0.891 | 2.426 | |

4 | 172,264 | 0.087 | 9.099 | 2.221 | 0.891 | 2.361 | |

5 | 33,670 | 0.171 | 9.945 | 2.322 | 0.891 | 2.451 | |

6 | 198 | −0.056 | 7.730 | 2.096 | 0.914 | 2.226 | |

LIN ${}^{f}$ | 1 | 148,448 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 |

2 | 226,912 | 0.100 | 3.426 | 1.281 | 0.960 | 0.961 | |

3 | 240,288 | 0.133 | 4.573 | 1.548 | 0.947 | 1.216 | |

4 | 172,264 | 0.126 | 5.343 | 1.681 | 0.938 | 1.387 | |

5 | 33,670 | 0.133 | 6.367 | 1.838 | 0.932 | 1.569 | |

6 | 198 | 0.069 | 4.291 | 1.612 | 0.953 | 1.236 | |

SQRT ${}^{g}$ | 2 | 226,912 | 0.000 | 0.000 | 0.000 | 1.000 | 0.000 |

3 | 240,288 | −0.009 | 2.224 | 1.010 | 0.974 | 0.591 | |

4 | 172,264 | −0.023 | 3.250 | 1.257 | 0.963 | 0.843 | |

5 | 33,670 | −0.033 | 4.155 | 1.451 | 0.956 | 1.024 | |

6 | 198 | 0.018 | 3.141 | 1.368 | 0.966 | 0.905 |

^{e}Q—quotient between the error sum of squares and the observed sum of squares, ${}^{f}$ WIL—Wilmink function fitted to TD records from 5 to the last DIM in lactation, ${}^{g}$ LIN—Linear function fitted to TD records from 306 to the last DIM in lactation. ${}^{h}$ SQRT—Squared function fitted to TD records from 306 to the last DIM in lactation.

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**MDPI and ACS Style**

Otwinowska-Mindur, A.; Ptak, E.; Makulska, J.; Jarnecka, O.
Modelling Extended Lactations in Polish Holstein–Friesian Cows. *Animals* **2021**, *11*, 2176.
https://doi.org/10.3390/ani11082176

**AMA Style**

Otwinowska-Mindur A, Ptak E, Makulska J, Jarnecka O.
Modelling Extended Lactations in Polish Holstein–Friesian Cows. *Animals*. 2021; 11(8):2176.
https://doi.org/10.3390/ani11082176

**Chicago/Turabian Style**

Otwinowska-Mindur, Agnieszka, Ewa Ptak, Joanna Makulska, and Olga Jarnecka.
2021. "Modelling Extended Lactations in Polish Holstein–Friesian Cows" *Animals* 11, no. 8: 2176.
https://doi.org/10.3390/ani11082176