Contrasting models for lactation curve analysis

Short communication: The lag response of daily milk yield to heat stress in dairy cows

Previous studies suggest that there exists a lag relationship between daily milk yield and heat stress. The values of heat stress indicators (e.g., temperature-humidity index and ambient temperature) before test day have a simple correlation with daily milk yield on test day. However, the simple correlation might not be the best description because daily milk yield and heat stress indicators have a nature of time series in common, and their correlations are cross correlations that could be affected by autocorrelations. We hope to give a more reliable estimation on the lag relationship of daily milk yield via excluding autocorrelations with transfer function modeling. In this study, we found a lag relationship between daily milk yield and heat stress indicators based on transfer function modeling. Heat stress indicators included ambient temperature and temperature-humidity index. The daily milk yield data from 123 cows were obtained during a consecutive 63-d period (July 10-September 10, 2016). The mean daily milk yield (MY) and the maximum daily ambient temperature (TA_max) satisfied the stationary hypothesis, and the cross correlation between them was calculated. Before excluding autocorrelation, MY at 0 to 4 d after test day had significant cross correlations with TA_max on test day. After excluding the influence of autocorrelations, MY at 1 to 3 d after the test day had significant cross correlations with TA_max on test day. This result suggested that MY would respond to TA_max 1 d after the test day. In addition, the strength of cross correlations between MY and TA_max decreased from 1 to 3 d in sequence, implying a declining lag response of MY that would last for 3 d. The transfer function model for MY and TA_max is written as: MY_t = 16.90 + 0.74MY_{t- 1} - 0.25TA_max_{t- 1} + N_t, where N_t is white noise. This model can be used to track and predict the dynamic response of MY to TA_max.

Average information residual maximum likelihood in practice

Gilmour, Thompson, and Cullis (Biometrics, 1995, 51, 1440) presented the average information residual maximum likelihood (REML) algorithm for efficient variance parameter estimation in the linear mixed model. That paper dealt specifically with traditional variance component models, but the algorithm was quickly applied to more general models and implemented in several REML packages including ASReml (Gilmour et al., Biometrics, 2015, 51, 1440). This paper outlines the theory with respect to these more general models, describes the main issues encountered in fitting these models and how they have been addressed in the ASReml software. The issues covered are the basics steps in the implementation of the algorithm, keeping parameters within the parameter space, maximizing sparsity, avoiding issues associated with unstructured variance matrices by using the factor-analytic structure and handling singularities in marker-based relationship matrices and current work.

A new standard model for milk yield in dairy cows based on udder physiology at the milking-session level

Milk production in dairy cow udders is a complex and dynamic physiological process that has resisted explanatory modelling thus far. The current standard model, Wood’s model, is empirical in nature, represents yield in daily terms, and was published in 1967. Here, we have developed a dynamic and integrated explanatory model that describes milk yield at the scale of the milking session. Our approach allowed us to formally represent and mathematically relate biological features of known relevance while accounting for stochasticity and conditional elements in the form of explicit hypotheses, which could then be tested and validated using real-life data. Using an explanatory mathematical and biological model to explore a physiological process and pinpoint potential problems (i.e., “problem finding”), it is possible to filter out unimportant variables that can be ignored, retaining only those essential to generating the most realistic model possible. Such modelling efforts are multidisciplinary by necessity. It is also helpful downstream because model results can be compared with observed data, via parameter estimation using maximum likelihood and statistical testing using model residuals. The process in its entirety yields a coherent, robust, and thus repeatable, model.

Dissecting high-dimensional phenotypes with bayesian sparse factor analysis of genetic covariance matrices

Quantitative genetic studies that model complex, multivariate phenotypes are important for both evolutionary prediction and artificial selection. For example, changes in gene expression can provide insight into developmental and physiological mechanisms that link genotype and phenotype. However, classical analytical techniques are poorly suited to quantitative genetic studies of gene expression where the number of traits assayed per individual can reach many thousand. Here, we derive a Bayesian genetic sparse factor model for estimating the genetic covariance matrix (G-matrix) of high-dimensional traits, such as gene expression, in a mixed-effects model. The key idea of our model is that we need consider only G-matrices that are biologically plausible. An organism's entire phenotype is the result of processes that are modular and have limited complexity. This implies that the G-matrix will be highly structured. In particular, we assume that a limited number of intermediate traits (or factors, e.g., variations in development or physiology) control the variation in the high-dimensional phenotype, and that each of these intermediate traits is sparse - affecting only a few observed traits. The advantages of this approach are twofold. First, sparse factors are interpretable and provide biological insight into mechanisms underlying the genetic architecture. Second, enforcing sparsity helps prevent sampling errors from swamping out the true signal in high-dimensional data. We demonstrate the advantages of our model on simulated data and in an analysis of a published Drosophila melanogaster gene expression data set.

SIMS(DAIRY): a modelling framework to identify sustainable dairy farms in the UK. Framework description and test for organic systems and N fertiliser optimisation

Multiple demands are placed on farming systems today. Society, national legislation and market forces seek what could be seen as conflicting outcomes from our agricultural systems, e.g. food quality, affordable prices, a healthy environmental, consideration of animal welfare, biodiversity etc., Many of these demands, or desirable outcomes, are interrelated, so reaching one goal may often compromise another and, importantly, pose a risk to the economic viability of the farm. SIMS(DAIRY), a farm-scale model, was used to explore this complexity for dairy farm systems. SIMS(DAIRY) integrates existing approaches to simulate the effect of interactions between farm management, climate and soil characteristics on losses of nitrogen, phosphorus and carbon. The effects on farm profitability and attributes of biodiversity, milk quality, soil quality and animal welfare are also included. SIMS(DAIRY) can also be used to optimise fertiliser N. In this paper we discuss some limitations and strengths of using SIMS(DAIRY) compared to other modelling approaches and propose some potential improvements. Using the model we evaluated the sustainability of organic dairy systems compared with conventional dairy farms under non-optimised and optimised fertiliser N use. Model outputs showed for example, that organic dairy systems based on grass-clover swards and maize silage resulted in much smaller total GHG emissions per l of milk and slightly smaller losses of NO(3) leaching and NO(x) emissions per l of milk compared with the grassland/maize-based conventional systems. These differences were essentially because the conventional systems rely on indirect energy use for 'fixing' N compared with biological N fixation for the organic systems. SIMS(DAIRY) runs also showed some other potential benefits from the organic systems compared with conventional systems in terms of financial performance and soil quality and biodiversity scores. Optimisation of fertiliser N timings and rates showed a considerable scope to reduce the (GHG emissions per l milk too).

Factor-analytic models for genotype x environment type problems and structured covariance matrices

Background

Analysis of data on genotypes with different expression in different environments is a classic problem in quantitative genetics. A review of models for data with genotype × environment interactions and related problems is given, linking early, analysis of variance based formulations to their modern, mixed model counterparts.

Results

It is shown that models developed for the analysis of multi-environment trials in plant breeding are directly applicable in animal breeding. In particular, the 'additive main effect, multiplicative interaction' models accommodate heterogeneity of variance and are characterised by a factor-analytic covariance structure. While this can be implemented in mixed models by imposing such structure on the genetic covariance matrix in a standard, multi-trait model, an equivalent model is obtained by fitting the common and specific factors genetic separately. Properties of the mixed model equations for alternative implementations of factor-analytic models are discussed, and extensions to structured modelling of covariance matrices for multi-trait, multi-environment scenarios are described.

Conclusion

Factor analytic models provide a natural framework for modelling genotype × environment interaction type problems. Mixed model analyses fitting such models are likely to see increasing use due to the parsimonious description of covariance structures available, the scope for direct interpretation of factors as well as computational advantages.

Body Trait Profiles in Holstein-Friesians Modeled Using Random Regression

Legendre polynomial and cubic spline functions were used in random regression models to model the change in body traits over the course of the first lactation for daughters of 954 sires. Both functions estimated similar genetic variances for d 50 to 250 across lactation for the majority of traits. The heritability of the traits was similar to other studies using univariate models as well as random regression models. There was little difference between the 2 functions in their predictive power for each of the body type traits, as measured by the absolute difference between the predicted and actual type traits and the proportion of the total phenotypic variance explained by the model. Overall, the Legendre polynomial appeared to model these traits slightly better. Plots of the fixed curves and daily sire solutions obtained from the random regression models showed that there were differences in how the traits and sires changed across lactation. The daily sire solutions were then used to predict differences in liveweight of sires' daughters across first lactation and showed that the daughters of some sires grew faster during first lactation than others. The spatial differences in the body traits that are displayed by this study could be an important indicator of the physical and biological changes that cows are undergoing in their first lactation. Information from these sire profiles could be harnessed to indicate production and functional traits later in life.

Multivariate character process models for the analysis of two or more correlated function-valued traits

Various methods, including random regression, structured antedependence models, and character process models, have been proposed for the genetic analysis of longitudinal data and other function-valued traits. For univariate problems, the character process models have been shown to perform well in comparison to alternative methods. The aim of this article is to present an extension of these models to the simultaneous analysis of two or more correlated function-valued traits. Analytical forms for stationary and nonstationary cross-covariance functions are studied. Comparisons with the other approaches are presented in a simulation study and in an example of a bivariate analysis of genetic covariance in age-specific fecundity and mortality in Drosophila. As in the univariate case, bivariate character process models with an exponential correlation were found to be quite close to first-order structured antedependence models. The simulation study showed that the choice of the most appropriate methodology is highly dependent on the covariance structure of the data. The bivariate character process approach proved to be able to deal with quite complex nonstationary and nonsymmetric cross-correlation structures and was found to be the most appropriate for the real data example of the fruit fly Drosophila melanogaster.