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### Comparison of Three Different Lactation Curve Models for Prediction of Monthly Test Day Milk Yields in First Lactation Murrah Buffaloes

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Nitesh Kumar Singh Rajbir Singh Atul Gupta Aashaq Hussain Dar Kurat ul Ain

Vol 7(6), 125-130

DOI- http://dx.doi.org/10.5455/ijlr.20170417042041

The present study was carried out using data on 11 monthly test-day milk yield (MTDMY) records during first lactation of 191 Murrah buffaloes calved during 2003- 2014 at Government Livestock Farm, Hisar and Livestock Research Centre, SVPUA&T, Meerut. The least square means of the MTDMY ranged from 4.625±0.145 to 10.636±0.214 kg. The relative efficiency of 3 lactation curve models via Inverse polynomial function (IPF), Gamma- type function (GF) and Mixed log function (MLF) were compared. Inverse polynomial functions described the highest coefficient of determination (99.49%) and comparatively lower root mean square errors 0.233 kg. Whereas, least coefficient of determination (97.24%) was observed in GF having comparatively lower root mean square errors 0.117 kg. Highest coefficient of determination (R2- value) suggesting IPF the best mathematical model for prediction of monthly test-day milk yields in Murrah buffaloes.

Keywords : Gamma Type Function Inverse Polynomial Function Lactation Curve Mixed Log Function Monthly Test Day Milk Yields Murrah Buffalo

**Introduction**

The graphical representation of milk yield against time can be defined as a lactation curve. The shape of lactation curve also indicates towards needed changes in dairy management. A lactation curve of milk yield against weekly/ monthly test days since calving can be partitioned by considering initial milk yield, the increasing slope, the peak and the decreasing slope. A mathematical function can represent the lactation curve if the accuracy of fitting the function is very good. Many workers have tried to standardize the lactation curve of cattle and buffalo by fitting different function (Nelder, 1966; Wood, 1967; Wilmink, 1987; Ali and Schaffer, 1987; Guo and Swalve, 1995); however the investigation on mathematical properties of functions used to evaluate the goodness of fit of lactation curve models is usually scanty in Murrah buffaloes. Hence, the present investigation was conducted to fit and compare the lactation curve models for describing the shape of the lactation curve in Murrah buffaloes for prediction of monthly test-day milk yields.

**Materials and Methods**

The data of 11 monthly test-day milk yields (MTDMYs) records during first lactation of 191 Murrah buffaloes calved during 2003- 2014 at Government Livestock Farm, Hisar and Livestock Research Centre, SVPUA&T, Meerut were collected from the history sheets and daily milk record registers. Culling, disposal in middle of lactation, abortion, stillbirth and other pathological conditions which affect the lactation yield were considered as abnormalities and hence, such records were excluded from the analysis. The records of animals with milk production less than 500 kg and lactation length less than 100 days were discarded from the present investigation. To ensure the normal distribution, the outliers were removed and data within the range of mean ± 2SD were considered for the study.

**Statistical Methodology**

Three lactation curve, inverse polynomial function (IPF), the gamma type function (GF) and mixed log function (MLF) were fitted on MTDMY records to find the best model amongst these three models. The mathematical models for these three lactation curve functions are as follows-

- Inverse polynomial function (Nelder, 1966)
- The gamma type function (Wood, 1967)
- Mixed log function (Guo and Swalve, 1995)

In above models, the parameters viz. Y_{t }=Average daily yield in the t^{th} week of lactation; a = initial milk yield after calving; b = Inclining slope parameter up to peak yield; c = Declining slope parameter and t = length of time since calving and these parameters characterize the shape of curve. The most suitable model amongst above mentioned models was identified on the basis of the highest R^{2}– value and lowest RMSE value.

**Results and Discussion**

The least square means of the MTDMYs have been presented in Table 2. The average first lactation monthly test day milk yield increased from 5.373±0.177 kg in MTDMY1 (6^{th} day) to a peak yield of 10.636±0.214 kg in MTDMY3 (65^{th} day) and subsequently declined to 4.625±0.145 kg in MTDMY11 (305^{th} day). Similar estimates were reported by Geetha (2005) and Katneni (2007) in Murrah buffaloes. However comparatively higher estimate of peak MTDMY3 (65^{th}day) as 7.15±0.15 kg and 8.05±0.11 kg while minimum least square means as 3.91±0.17 kg on MTDMY1 (5^{th} day) and 4.17±0.12 kg on MTDMY11 (305^{th} day) were reported by Geetha (2005) and Katneni (2007) in first lactation of Murrah buffaloes. Various lactation curve parameters (a, b and c) were estimated to fit into the mathematical models and to obtain the predicting equation for MTDMYs. The three lactation curve functions were developed after fitting the values of lactation curve parameters for predicting MTDMYs (Table 1).

**Table1: **Different lactation curve model with parameters for prediction of MTDMYs of first lactation in Murrah buffaloes

S. No. |
Lact. Curve Models |
Parameters of lactation curve models |
R^{2} – value (%) |
RMSE (Kg) |

1 |
IPF | Y_{t} = t (0.266326+ -0.06904*t + 0.023202*t^{2})^{-1} |
99.49 | 0.233 |

2 |
GF | Y_{t} = 7.630442* t 0.877062e-0.238661*t |
97.24 | 0.117 |

3 |
MLF | Y_{t} = 19.98221-13.9892*t ^{½ }+ 12.82659log t + e_{t} |
98.08 | 0.081 |

Inverse polynomial function explained best fit for MTDMYs in Murrah buffaloes with highest coefficient of determination (R^{2}-value) and comparatively lower root mean square errors of 99.49 % and 0.233 kg respectively. Kumar and Bhat (1979) also reported R^{2}– value of inverse polynomial function near to present finding in Indian buffaloes, whereas Dave (1971) reported lower R^{2}–value of inverse polynomial function. The gamma type function gave high R^{2}-values as 97.24% and comparatively lower root mean square errors as 0.117 kg which was in conformity with the results of Katneni (2007) and Singh *et al.* (2015) who also reported almost similar R^{2}-value in Murrah breed of buffaloes. A similar result of R^{2}-values of gamma type function was reported by Aziz *et al.* (2006) in Egyptian buffaloes. However our results were in contrary with the results of Sahoo *et al.* (2014) who reported lower R^{2}-value in Murrah buffaloes for this function. Mixed log function gave good fit for MTDMYs under the present study with high first lactation coefficient of determination (R^{2}-value) as 98.08% and moderate root mean square error to be 0.081 kg. Almost similar R^{2}-value of this function has been reported by Kumar (2007), Sahoo *et al.* (2014) and Singh *et al.* (2015) in Murrah buffalo. The peak MTDMY predicted by IPF, GF and MLF were 11.19 kg, 9.91 kg and 9.84 kg, respectively. The IPF predicted MTDMY higher than observed values for MTDMY3, MTDMY4, MTDMY5, MTDMY6 & MTDMY11, whereas lower values for rest of the MTDMYs were predicted. Kumar and Bhat (1979) also reported R^{2}–value of inverse polynomial function near to present finding in Indian buffaloes, whereas Dave (1971) reported lower R^{2}–value of IPF. The GF predicted MTDMY higher than observed values except for MTDMY1, MTDMY4, MTDMY5, MTDMY6 & MTDMY7 and lesser values for rest of the MTDMYs. MLF predicted higher MTDMYs than observed values for MTDMY1, MTDMY4, MTDMY5, MTDMY6, MTDMY7 andMTDMY8. The actual and predicted MTDMY records of different functions are presented in Table 2.

**Table 2: **Predicted MTDMYs and error (kg) from different lactation curve functions

MTDMY |
Obs. Value |
IPF |
GF |
MLF |
|||

Predicted |
Error |
Predicted |
Error |
Predicted |
Error |
||

MTDMY 1 |
5.67 | 4.54 | -1.13 | 6.01 | 0.34 | 5.99 | 0.32 |

MTDMY 2 |
9.69 | 9.05 | -0.64 | 8.70 | -1.00 | 9.09 | -0.60 |

MTDMY 3 |
10.04 | 11.19 | 1.15 | 9.77 | -0.27 | 9.84 | -0.20 |

MTDMY 4 |
9.70 | 11.07 | 1.37 | 9.91 | 0.21 | 9.79 | 0.09 |

MTDMY 5 |
9.12 | 9.98 | 0.86 | 9.49 | 0.37 | 9.35 | 0.22 |

MTDMY 6 |
8.46 | 8.73 | 0.27 | 8.77 | 0.31 | 8.70 | 0.24 |

MTDMY 7 |
7.72 | 7.61 | -0.11 | 7.91 | 0.19 | 7.93 | 0.21 |

MTDMY 8 |
7.04 | 6.67 | -0.37 | 7.01 | -0.04 | 7.09 | 0.04 |

MTDMY 9 |
6.21 | 5.90 | -0.31 | 6.12 | -0.09 | 6.20 | -0.01 |

MTDMY 10 |
5.35 | 5.27 | -0.08 | 5.29 | -0.07 | 5.28 | -0.07 |

MTDMY 11 |
4.58 | 4.75 | 0.17 | 4.53 | -0.05 | 4.34 | -0.24 |

*IPF: inverse polynomial function; GF: gamma type function; MLF: mixed log function; MTDMY: monthly test day milk yield*

The fitting of observed and predicted monthly test day milk yields by all the three lactation curve functions for first lactation in Murrah buffaloes are presented in Figures 1, 2 & 3. For the first lactation the peak monthly test day milk yield (MTDMY) was found to be 11.19 kg in 3rd month of lactation by Inverse polynomial function. The monthly test day milk yields were predicted with higher degree of accuracy by all the models with the maximum accuracy obtained by inverse polynomial function and least with gamma function.

**Fig. 1: **Observed and predicted MTDMYs for first lactation by IPF

**Fig. 2: **Observed and predicted MTDMYs for first lactation by GF

**Fig. 3: **Observed and predicted MTDMYs for first lactation by MLF

**Conclusions**

On the basis of results obtained from present study, it can be concluded that inverse polynomial function gave best fit lactation curve model among three models with highest coefficient of determination and comparatively lower root square error whereas gamma type function proved to be least fit as it explained lowest coefficient of determination with lower root square error. Hence, amongst these three lactation curve models, Inverse polynomial function (IPF) can be used for prediction of MTDMYs in Murrah buffaloes. The trend of best fit inverse polynomial function indicated that this function might be most suitable to buffaloes starting their lactation at a very low level of production, reach their peak yield very early and then start declining at an early stage.

**Acknowledgements**

The authors extend their sincere thanks to COVAS, SVPUAT, Meerut and Director of Government Livestock Farm, Hisar for providing the facilities for conducting the present investigation.

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