Fit of standard models of the lactation curve to weekly records of milk production of cows in a single herd

A model to describe growth patterns of the mammary gland during pregnancy and lactation.

In test-day (TD) models, records from individual test days are used to determine lactation production instead of aggregating records. Test-day models have recently gained considerable interest because they are more flexible in handling records from different recording schemes. Compared with only using records of complete lactations, they can reduce the generation interval through frequent genetic evaluations with the latest data. Test-day models can predict total production more accurately by accounting for time-dependent environmental effects. Test-day models may be separated into three groups: First, two-step models under which corrections are carried out at TD level and subsequently corrected TD records are processed in an aggregated form as lactation records. Second, fixed regression models assume that TD records within a lactation are repeated records. Because yields in the course of the lactation follow a curvilinear pattern, this curve can be considered by using suitable covariates. Third, random regression models additionally define the animal’s genetic effect by using regression coefficients and allowing for covariances among them. The difference between random regression and fixed regression models is that the genetic merit of an individual is allowed to differ in the course of the lactation in random regression models. Random regressions are related to the approach of defining covariance functions for longitudinal data. Computationally, TD models are very demanding. For evaluations on a national scale, the size of the equation system can go to hundreds of millions of equations, depending on the size of the database and the specific model defined.

SYMPOSIUM: TEST-DAY MODELS Theoretical Basis and Computational Methods for Different Test-Day Genetic Evaluation Methods

Theoretical basis and computational methods for different test-day genetic evaluation methods.

The Effect of Test Day Models on the Estimation of Genetic Parameters and Breeding Values for Dairy Yield Traits

Lactation curves are fitted to data as a preliminary to estimating summary statistics. Two widely quoted curves are at b e -ct (Wood, 1967) and a(1 - e -bt ) - ct (Cobby and Le Du, 1978), each of which has three parameters. Restriction to either of these curves imposes limitations on the fit to the data and can result in biased estimation of summary statistics. Alternatively, lactation curves can be generated by the use of a non-parametric method which requires only weak assumptions about the signs of derivatives of the curves. Because the non-parametric curves are more flexible, estimates of summary statistics are less likely to be biased than those based on parametric models. Use of the non-parametric curves is particularly advantageous around the time of peak yield, where the curves of Wood and Cobby and Le Du are known to fit data poorly.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0003356100040320

Non-parametric lactation curves

Variability within system models: A case study

given values of p and tat times tj, . .., t,. Their procedure requires interpolation on two functions of the variates and generalized least squares estimation. Also, at some stages in the analysis they make use of a fitted relationship between p and t, whilst at other stages they choose not to use the relationship. Identical results are produced by the simpler approach outlined below. Further, it is shown why a different estimate of 12 is obtained when the equation relating p and t is used. The above equation can be integrated with respect to time, between limits ti-1 and ti, to give

An Example of Fitting a Differential Equation to Data

The acceptance—rejection algorithm is slow to simulate from mixture distributions when the acceptance probability is small. For a mixture of exponential distributions, reformulation as a mixture of general Erlang distributions may increase efficiency. The reformulation which maximises acceptance probability can be found by linear programming. An example is given in which reformulation reduces the average simulation time by a factor of 15.

C. A. Glasbey

Papers

Non-parametric lactation curves

Variability within system models: A case study

An Example of Fitting a Differential Equation to Data

Simulating from a mixture of exponential distributions with some negatively weighted components