# Estimation of Dairy Cow Survival in the First Three Lactations for Different Culling Reasons Using the Kaplan–Meier Method

^{1}

^{2}

^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

_{i}), i.e., the probability of culling in the time interval per unit time [39]:

**γ**

_{i}), i.e., the probability (per unit time) that a cow that survived to the start of the interval would be culled in this interval:

_{i}is the probability density function, P

_{i}is the cumulative proportion of surviving cases at the beginning of the interval and the beginning of the subsequent interval (P

_{i+}

_{1}), h

_{i}is the time interval width).

_{j}is the number of cows at risk at time t

_{j}, d

_{j}is the number of cows culled at time t

_{j}, Π is the product of all cases lower than or equal to t.

_{i}is the number of cows culled at the ith time moment, r

_{i}is the number of observations from both groups being compared, in which the survival times are at least as long as or longer than the ith time moment, A(i) is the proportion of observations from Group 2, r

_{2}is the total number of culled cows from Group 2.

_{1}, x

_{2},…x

_{k}) is the resulting hazard given k predictors and the appropriate survival time, e is the natural logarithm base, $\sum}_{i=1}^{k}{a}_{i}{x}_{i$ is the linear combination of independent variables (x

_{1}, x

_{2}…x

_{k}), and model parameters (a

_{1}, a

_{2}, …a

_{k}), h

_{0}(t) is the baseline hazard dependent on duration only.

_{i}) was determined as:

_{i}is the estimated regression coefficient.

_{i}indicates the change in the risk of survival time shortening for one unit increase in the independent variable when adjusted for the remaining independent variables included in the model (it is assumed that they are constant when the independent variable increases by one unit). When HR > 1, the risk of survival time shortening increases, when HR < 1, the risk decreases. The values close to unity are interpreted as no risk. These values were calculated only for statistically significant regression coefficients.

_{i}is the value of the variable that is a clustering criterion for the ith case, k is the number of cases (culling categories) within the cluster, $\overline{x}$ is the mean value of this variable within the cluster.

**x =**(

**x**,

_{1}**x**, …,

_{2}**x**),

_{p}**y =**(

**y**,

_{1}**y**,…,

_{2}**y**), p is the number of variables defining the p-dimensional space.

_{p}## 3. Results

#### 3.1. Survival Tables and the Kaplan–Meier Survival Curves

#### 3.2. The Cox Proportional Hazards Model Parameters Affecting Cow Survival and Hazard Ratio (HR) Coefficients

#### 3.3. Cluster Analysis

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The Kaplan–Meier survival probability until the first lactation. UD—udder diseases, RD—reproductive disorders, MDSD—metabolic and digestive system diseases, RSD—respiratory system diseases, LSD—locomotor system diseases, ACC—accidents, OTH—others, LMY—low milk yield.

**Figure 2.**The Kaplan–Meier survival probability until the second lactation. UD—udder diseases, RD—reproductive disorders, CD—contagious diseases, MDSD—metabolic and digestive system diseases, RSD—respiratory system diseases, LSD—locomotor system diseases, ACC—accidents, OTH—others, LMY—low milk yield.

**Figure 3.**The Kaplan–Meier survival probability until the third lactation. UD—udder diseases, RD—reproductive disorders, CD—contagious diseases, MDSD—metabolic and digestive system diseases, RSD—respiratory system diseases, LSD—locomotor system diseases, ACC—accidents, OTH—others, LMY—low milk yield.

**Figure 5.**The results of the cluster analysis using Ward’s method for the first and second lactations.

**Figure 6.**The results of the cluster analysis using Ward’s method for the first, second, and third lactations.

**Table 1.**Culling categories between 2017 and 2018 according to the breeding documentation in the original dataset and the selected subset.

Lactation 1 | Lactation 2 | Lactation 3 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Class | Original Dataset | Subset | Original Dataset | Subset | Original Dataset | Subset | ||||||

n | % | n | % | n | % | n | % | n | % | n | % | |

LMY ^{1} | 2849 | 3.72 | 195 | 3.97 | 5416 | 3.38 | 427 | 3.40 | 7323 | 3.15 | 608 | 3.12 |

UD ^{2} | 9973 | 13.03 | 686 | 13.98 | 22,811 | 14.23 | 1862 | 14.80 | 35,205 | 15.12 | 2990 | 15.36 |

RD ^{3} | 33,619 | 43.92 | 1878 | 38.28 | 69,526 | 43.36 | 5090 | 40.47 | 97,116 | 41.71 | 7741 | 39.76 |

CD ^{4} | 121 | 0.16 | 2 | 0.04 | 251 | 0.16 | 13 | 0.10 | 352 | 0.15 | 30 | 0.15 |

MDSD ^{5} | 5654 | 7.39 | 435 | 8.87 | 12,621 | 7.87 | 1085 | 8.63 | 19,598 | 8.42 | 1746 | 8.97 |

RSD ^{6} | 533 | 0.70 | 41 | 0.84 | 1060 | 0.66 | 87 | 0.69 | 1485 | 0.64 | 126 | 0.65 |

LSD ^{7} | 8062 | 10.53 | 591 | 12.05 | 16,807 | 10.48 | 1438 | 11.43 | 25,347 | 10.89 | 2250 | 11.56 |

ACC ^{8} | 8177 | 10.68 | 577 | 11.76 | 16,455 | 10.26 | 1360 | 10.81 | 24,109 | 10.36 | 2109 | 10.83 |

OTH ^{9} | 7552 | 9.87 | 501 | 10.21 | 15,383 | 9.59 | 1215 | 9.66 | 22,274 | 9.57 | 1870 | 9.60 |

^{1}low milk yield,

^{2}udder diseases,

^{3}reproductive disorders,

^{4}contagious diseases,

^{5}metabolic and digestive system diseases,

^{6}respiratory system diseases,

^{7}locomotor system diseases,

^{8}accidents,

^{9}others (the “others” category included those culling reasons that were difficult to classify to the remaining categories or more than one culling reason existed).

**Table 2.**Cumulative proportions of cows surviving (P1 and P2) until the age interval (w1 and w2) and the percentage decrease (DP).

Reason | Until the First Lactation | Until the Second Lactation | Until the Third Lactation | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

P1 | w1 | P2 | w2 | DP | P1 | w1 | P2 | w2 | DP | P1 | w1 | P2 | w2 | DP | |

LMY ^{1} | 0.9708 | 24 | 0.6364 | 31 | 33.44 | 0.8000 | 36 | 0.0742 | 60 | 72.58 | 0.8325 | 40 | 0.0742 | 71 | 75.83 |

UD ^{2} | 0.9718 | 24 | 0.6429 | 31 | 32.89 | 0.8280 | 36 | 0.0742 | 60 | 75.38 | 0.8613 | 40 | 0.0742 | 71 | 78.71 |

RD ^{3} | 0.9973 | 25 | 0.9276 | 31 | 6.97 | 0.9001 | 40 | 0.2229 | 60 | 67.72 | 0.9012 | 40 | 0.2620 | 70 | 63.92 |

CD ^{4} | - | - | - | - | - | 0.9999 | 38 | 0.2778 | 47 | 72.21 | 0.9999 | 40 | 0.3771 | 70 | 62.28 |

MDSD ^{5} | 0.9711 | 24 | 0.5455 | 31 | 42.56 | 0.8000 | 36 | 0.0553 | 60 | 74.47 | 0.8207 | 40 | 0.0553 | 71 | 76.54 |

RSD ^{6} | 0.9705 | 24 | 0.6098 | 31 | 36.07 | 0.7772 | 36 | 0.0394 | 58 | 73.78 | 0.8206 | 40 | 0.0394 | 71 | 78.12 |

LSD ^{7} | 0.9722 | 24 | 0.6201 | 31 | 35.21 | 0.8280 | 36 | 0.0872 | 60 | 74.08 | 0.8613 | 40 | 0.0872 | 71 | 77.41 |

ACC ^{8} | 0.9721 | 24 | 0.6521 | 31 | 32.00 | 0.8532 | 36 | 0.1020 | 60 | 75.12 | 0.8613 | 40 | 0.1020 | 71 | 75.93 |

OTH ^{9} | 0.9705 | 24 | 0.6751 | 31 | 29.54 | 0.8557 | 36 | 0.1153 | 60 | 74.04 | 0.8676 | 40 | 0.1153 | 71 | 75.23 |

^{1}low milk yield,

^{2}udder diseases,

^{3}reproductive disorders,

^{4}contagious diseases,

^{5}metabolic and digestive system diseases,

^{6}respiratory system diseases,

^{7}locomotor system diseases,

^{8}accidents,

^{9}others (the “others” category included those culling reasons that were difficult to classify to the remaining categories or more than one culling reason existed), P1—the cumulative proportion of cows surviving until the interval in which culling started; P2—the cumulative proportion of cows surviving until the interval in which culling ended; w1—the month of life of a cow, in which culling started for a given category; w2—the month of life of a cow, in which culling ended for a given lactation; DP—the rate of decrease = (P1 − P2) · 100%; the values determined from Kaplan–Meier curves.

**Table 3.**p-values for comparisons between survival curves for cows from individual culling categories.

First Lactation | ||||||||
---|---|---|---|---|---|---|---|---|

Category | UD ^{1} | RD ^{2} | CD ^{3} | MDSD ^{4} | RSD ^{5} | LSD ^{6} | ACC ^{7} | OTH ^{8} |

LMY ^{9} | 0.8505 | 0.0000 | 0.0265 | 0.7102 | 0.7330 | 0.7049 | 0.2597 | |

UD | 0.0000 | 0.0005 | 0.6206 | 0.4351 | 0.7622 | 0.1799 | ||

RD | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |||

MDSD | 0.4333 | 0.0067 | 0.0003 | 0.0000 | ||||

RSD | 0.8199 | 0.5470 | 0.3106 | |||||

LSD | 0.3071 | 0.0440 | ||||||

ACC | 0.3150 | |||||||

Until the Second Lactation | ||||||||

LMY | 0.7113 | 0.0000 | 0.2995 | 0.0001 | 0.0789 | 0.5368 | 0.1325 | 0.0236 |

UD | 0.0000 | 0.3344 | 0.0000 | 0.0378 | 0.6205 | 0.0534 | 0.1591 | |

RD | 0.0037 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

CD | 0.1073 | 0.1750 | 0.3661 | 0.4543 | 0.5868 | |||

MDSD | 0.8175 | 0.0000 | 0.0000 | 0.0000 | ||||

RSD | 0.0936 | 0.0762 | 0.0171 | |||||

LSD | 0.1800 | 0.0356 | ||||||

ACC | 0.2776 | |||||||

Until the Third Lactation | ||||||||

LMY | 0.0110 | 0.0000 | 0.0026 | 0.0247 | 0.2958 | 0.0329 | 0.0402 | 0.0022 |

UD | 0.0000 | 0.0098 | 0.0000 | 0.0136 | 0.3608 | 0.5817 | 0.2602 | |

RD | 0.8454 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

CD | 0.0003 | 0.0015 | 0.0087 | 0.0107 | 0.0205 | |||

MDSD | 0.9918 | 0.0000 | 0.0000 | 0.0000 | ||||

RSD | 0.0237 | 0.0273 | 0.0062 | |||||

LSD | 0.9185 | 0.1517 | ||||||

ACC | 0.1325 |

^{1}udder diseases,

^{2}reproductive disorders,

^{3}contagious diseases,

^{4}metabolic and digestive system diseases,

^{5}respiratory system diseases,

^{6}locomotor system diseases,

^{7}accidents,

^{8}others,

^{9}low milk yield.

**Table 4.**HR values for the individual parameters of the Cox proportional hazards model according to culling categories and lactation groups (1—until the first lactation, 2—until the second lactation, 3—until the third lactation; HR values for statistically significant parameters are marked in bold).

Category | LMY ^{1} | UD ^{2} | RD ^{3} | CD ^{4} | MDSD ^{5} | RSD ^{6} | LSD ^{7} | ACC ^{8} | OTH ^{9} | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lactation | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

CS ^{10} | 1.04 | 1.07 | 0.98 | 0.95 | 1.08 | 0.98 | 1.15 | 0.99 | 0.99 | 1.52 | 1.22 | 0.89 | 1.03 | 1.00 | 0.45 | 0.84 | 1.04 | 0.99 | 1.01 | 1.04 | 0.87 | 0.95 | 1.00 | 0.73 | 1.04 | 1.00 |

HF ^{11} | 1.26 | 1.00 | 0.99 | 1.00 | 0.99 | 0.98 | 0.95 | 0.99 | 0.98 | 0.98 | 0.99 | 1.00 | 0.16 | 0.55 | 0.77 | 1.03 | 0.99 | 0.99 | 0.94 | 0.97 | 0.97 | 0.98 | 0.97 | 0.99 | ||

AFC ^{12} | 0.10 | 0.82 | 0.89 | 0.05 | 0.78 | 0.85 | 0.40 | 0.82 | 0.86 | 0.87 | 0.96 | 0.09 | 0.77 | 0.88 | 0.03 | 0.59 | 0.86 | 0.22 | 0.77 | 0.85 | 0.32 | 0.77 | 0.84 | 0.24 | 0.80 | 0.86 |

AFB ^{13} | 0.87 | 0.99 | 0.98 | 1.01 | 1.03 | 1.02 | 0.97 | 1.00 | 0.99 | 1.78 | 0.98 | 0.99 | 0.99 | 0.99 | 0.98 | 0.95 | 0.94 | 0.92 | 1.03 | 1.03 | 0.98 | 1.04 | 1.02 | 0.87 | 1.01 | 1.02 |

LL ^{14} | 0.92 | 0.99 | 1.00 | 0.91 | 0.99 | 1.00 | 0.96 | 1.00 | 1.00 | 0.96 | 1.00 | 0.90 | 0.99 | 1.00 | 0.88 | 0.99 | 1.00 | 0.94 | 0.99 | 1.00 | 0.96 | 0.99 | 1.00 | 0.94 | 0.99 | 1.00 |

CI ^{15} | 0.99 | 1.00 | 0.99 | 1.00 | 0.99 | 0.99 | 1.01 | 0.99 | 0.99 | 1.00 | 0.99 | 0.99 | 0.99 | 1.00 | 1.00 | 1.00 | 0.99 | 1.00 | ||||||||

PSI ^{16} | 0.98 | 1.02 | 1.02 | 1.02 | 0.99 | 1.02 | 1.01 | 0.98 | 1.00 | 0.98 | 1.16 | 1.00 | 1.02 | 1.02 | 1.17 | 0.99 | 1.00 | 1.02 | 1.01 | 1.01 | 1.00 | 1.02 | 1.02 | 0.97 | 1.00 | 1.02 |

CSI ^{17} | 0.96 | 1.01 | 1.01 | 0.87 | 1.11 | 1.03 | 0.67 | 0.97 | 1.09 | 0.85 | 0.01 | 1.22 | 1.05 | 1.13 | 1.02 | 1.11 | 0.71 | 0.71 | 0.87 | 1.13 | 1.59 | 0.88 | 1.05 | 0.95 | 1.19 | 1.22 |

BFSI ^{18} | 0.94 | 1.00 | 1.00 | 1.05 | 0.99 | 1.01 | 1.01 | 1.01 | 0.99 | 1.05 | 2.63 | 0.98 | 1.00 | 0.99 | 0.98 | 0.95 | 1.06 | 1.11 | 1.05 | 0.99 | 0.97 | 1.02 | 1.00 | 1.11 | 0.98 | 0.98 |

SMYSI ^{19} | 1.10 | 1.00 | 1.00 | 1.02 | 0.99 | 1.00 | 1.08 | 1.01 | 1.00 | 0.88 | 2.07 | 0.97 | 1.01 | 1.00 | 1.06 | 1.01 | 1.11 | 1.05 | 1.01 | 1.01 | 0.95 | 1.03 | 1.02 | 0.91 | 0.98 | 0.99 |

LHSI ^{20} | 0.97 | 0.98 | 0.99 | 1.09 | 0.96 | 0.99 | 1.17 | 1.01 | 0.97 | 1.55 | 11.30 | 0.94 | 0.97 | 0.95 | 1.05 | 1.00 | 1.14 | 1.13 | 1.08 | 0.96 | 0.85 | 1.06 | 0.98 | 1.07 | 0.95 | 0.93 |

USI ^{21} | 0.91 | 1.00 | 1.00 | 1.05 | 0.92 | 0.98 | 1.37 | 1.03 | 0.96 | 0.88 | 25.26 | 0.87 | 0.98 | 0.92 | 1.04 | 0.91 | 1.30 | 1.26 | 1.10 | 0.92 | 0.73 | 1.11 | 0.98 | 1.09 | 0.91 | 0.88 |

FSI ^{22} | 1.08 | 1.25 | 0.97 | 1.08 | 0.99 | 1.01 | 1.04 | 0.99 | 1.01 | 0.11 | 0.14 | 0.85 | 0.96 | 0.98 | 1.02 | 1.13 | 1.18 | 1.13 | 1.01 | 1.02 | 1.79 | 1.20 | 1.09 | 0.95 | 1.13 | 1.11 |

HCR ^{23} | 0.91 | 0.81 | 1.03 | 0.93 | 1.00 | 0.98 | 1.04 | 1.00 | 0.99 | 7.68 | 6.13 | 1.19 | 1.03 | 1.01 | 1.06 | 0.89 | 0.85 | 0.89 | 0.97 | 0.97 | 0.62 | 0.83 | 0.91 | 1.03 | 0.86 | 0.89 |

CCR ^{24} | 0.95 | 0.98 | 0.99 | 0.97 | 1.00 | 1.00 | 0.95 | 1.00 | 0.99 | 1.14 | 1.21 | 0.98 | 1.00 | 1.02 | 0.98 | 0.97 | 1.02 | 0.97 | 1.02 | 1.01 | 0.89 | 0.98 | 1.00 | 1.03 | 1.01 | 0.99 |

CFI ^{25} | 0.96 | 0.99 | 0.96 | 0.98 | 0.97 | 0.98 | 1.01 | 1.01 | 1.00 | 1.18 | 1.93 | 1.05 | 1.00 | 1.02 | 1.05 | 0.98 | 1.03 | 0.97 | 1.00 | 1.01 | 0.86 | 0.98 | 1.01 | 1.02 | 0.97 | 0.98 |

CCI ^{26} | 0.95 | 1.06 | 1.02 | 1.02 | 0.99 | 1.01 | 1.10 | 0.96 | 1.02 | 0.99 | 1.03 | 0.91 | 1.00 | 0.99 | 0.98 | 0.98 | 1.01 | 1.00 | ||||||||

BVSCC ^{27} | 0.95 | 1.01 | 1.01 | 1.00 | 0.99 | 1.01 | 1.00 | 1.00 | 0.99 | 0.82 | 1.21 | 1.00 | 1.01 | 1.00 | 0.99 | 0.96 | 1.00 | 1.05 | 1.01 | 1.00 | 0.95 | 0.98 | 0.99 | 0.97 | 1.00 | 1.00 |

BVL ^{28} | 1.08 | 1.03 | 1.03 | 1.00 | 1.03 | 1.03 | 1.03 | 1.02 | 1.02 | 1.11 | 0.87 | 1.02 | 1.02 | 1.03 | 1.05 | 1.07 | 1.12 | 1.00 | 1.01 | 1.03 | 1.05 | 1.03 | 1.03 | 0.97 | 1.01 | 1.02 |

Temp ^{29} | 0.83 | 0.53 | 0.47 | 0.79 | 0.63 | 0.71 | 0.79 | 0.81 | 0.94 | 2.72 | 0.42 | 1.46 | 0.47 | 0.49 | 0.38 | 0.88 | 0.62 | 1.13 | 0.51 | 0.59 | 0.81 | 0.61 | 0.58 | 0.81 | 0.70 | 0.67 |

MY ^{30} | 0.98 | 0.96 | 0.96 | 0.96 | 1.01 | 0.98 | 1.02 | 1.02 | 0.99 | 1.17 | 0.77 | 1.02 | 0.97 | 0.97 | 1.05 | 1.00 | 0.97 | 1.00 | 0.99 | 0.98 | 1.00 | 0.98 | 0.98 | 1.04 | 1.00 | 0.99 |

Fat ^{31} | 0.79 | 1.12 | 0.92 | 1.00 | 1.23 | 1.21 | 0.89 | 1.36 | 1.05 | 19.41 | 2.10 | 1.01 | 1.27 | 1.30 | 2.05 | 1.64 | 1.17 | 0.90 | 1.12 | 1.20 | 1.08 | 1.18 | 1.20 | 0.84 | 1.35 | 1.22 |

Prot ^{32} | 0.90 | 0.72 | 0.76 | 0.90 | 0.44 | 0.17 | 0.30 | 0.34 | 0.53 | 0.00 | 0.00 | 0.93 | 0.31 | 0.26 | 0.22 | 0.37 | 0.46 | 0.66 | 0.33 | 0.36 | 0.85 | 0.28 | 0.26 | 1.08 | 0.32 | 0.37 |

^{1}low milk yield,

^{2}udder diseases,

^{3}reproductive disorders,

^{4}contagious diseases,

^{5}metabolic and digestive system diseases,

^{6}respiratory system diseases,

^{7}locomotor system diseases,

^{8}accidents,

^{9}others,

^{10}calving season,

^{11}percentage of HF genes,

^{12}age at first calving,

^{13}age at first breeding,

^{14}lactation length,

^{15}calving interval,

^{16}production subindex,

^{17}conformation subindex,

^{18}body frame subindex,

^{19}strength and milk yield subindex,

^{20}leg and hoof subindex,

^{21}udder subindex,

^{22}fertility subindex,

^{23}heifer conception rate,

^{24}cow conception rate,

^{25}the interval from calving to first insemination,

^{26}calving-to-conception interval,

^{27}breeding value for somatic cell count,

^{28}breeding value for longevity,

^{29}temperament,

^{30}milk yield,

^{31}fat percentage,

^{32}protein percentage.

Cluster Number | Culling Category |
---|---|

First lactation | |

1 | Reproductive disorders |

2 | Udder diseases |

Others | |

Low milk yield | |

Accidents | |

3 | Metabolic and digestive system diseases |

Locomotor system diseases | |

Respiratory system diseases | |

First and second lactation | |

1 | Reproductive disorders |

2 | Udder diseases |

Others | |

Low milk yield | |

Accidents | |

Locomotor system diseases | |

3 | Metabolic and digestive system diseases |

Respiratory system diseases | |

Contagious diseases | |

First, second and third lactation | |

1 | Reproductive disorders |

2 | Udder diseases |

Others | |

Low milk yield | |

Contagious diseases | |

3 | Accidents |

Locomotor system diseases | |

4 | Metabolic and digestive system diseases |

Respiratory system diseases |

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**MDPI and ACS Style**

Grzesiak, W.; Adamczyk, K.; Zaborski, D.; Wójcik, J.
Estimation of Dairy Cow Survival in the First Three Lactations for Different Culling Reasons Using the Kaplan–Meier Method. *Animals* **2022**, *12*, 1942.
https://doi.org/10.3390/ani12151942

**AMA Style**

Grzesiak W, Adamczyk K, Zaborski D, Wójcik J.
Estimation of Dairy Cow Survival in the First Three Lactations for Different Culling Reasons Using the Kaplan–Meier Method. *Animals*. 2022; 12(15):1942.
https://doi.org/10.3390/ani12151942

**Chicago/Turabian Style**

Grzesiak, Wilhelm, Krzysztof Adamczyk, Daniel Zaborski, and Jerzy Wójcik.
2022. "Estimation of Dairy Cow Survival in the First Three Lactations for Different Culling Reasons Using the Kaplan–Meier Method" *Animals* 12, no. 15: 1942.
https://doi.org/10.3390/ani12151942