Statistical analysis of test-day milk yields using random regression models for the comparison of feeding groups during the lactation period

Abstract

Random regression models are widely used in the field of animal breeding for the genetic evaluation of daily milk yields from different test days. These models are capable of handling different environmental effects on the respective test day, and they describe the characteristics of the course of the lactation period by using suitable covariates with fixed and random regression coefficients. As the numerically expensive estimation of parameters is already part of advanced computer software, modifications of random regression models will considerably grow in importance for statistical evaluations of nutrition and behaviour experiments with animals. Random regression models belong to the large class of linear mixed models. Thus, when choosing a model, or more precisely, when selecting a suitable covariance structure of the random effects, the information criteria of Akaike and Schwarz can be used. In this study, the fitting of random regression models for a statistical analysis of a feeding experiment with dairy cows is illustrated under application of the program package SAS. For each of the feeding groups, lactation curves modelled by covariates with fixed regression coefficients are estimated simultaneously. With the help of the fixed regression coefficients, differences between the groups are estimated and then tested for significance. The covariance structure of the random and subject-specific effects and the serial correlation matrix are selected by using information criteria and by estimating correlations between repeated measurements. For the verification of the selected model and the alternative models, mean values and standard deviations estimated with ordinary least square residuals are used.     

Statistical analysis of test-day milk yields using random regression models for the comparison of feeding groups during the lactation period

Random regression models are widely used in the field of animal breeding for the genetic evaluation of daily milk yields from different test days. These models are capable of handling different environmental effects on the respective test day, and they describe the characteristics of the course of the lactation period by using suitable covariates with fixed and random regression coefficients. As the numerically expensive estimation of parameters is already part of advanced computer software, modifications of random regression models will considerably grow in importance for statistical evaluations of nutrition and behaviour experiments with animals. Random regression models belong to the large class of linear mixed models. Thus, when choosing a model, or more precisely, when selecting a suitable covariance structure of the random effects, the information criteria of Akaike and Schwarz can be used. In this study, the fitting of random regression models for a statistical analysis of a feeding experiment with dairy cows is illustrated under application of the program package SAS. For each of the feeding groups, lactation curves modelled by covariates with fixed regression coefficients are estimated simultaneously. With the help of the fixed regression coefficients, differences between the groups are estimated and then tested for significance. The covariance structure of the random and subject-specific effects and the serial correlation matrix are selected by using information criteria and by estimating correlations between repeated measurements. For the verification of the selected model and the alternative models, mean values and standard deviations estimated with ordinary least square residuals are used.     

Integration of the effects of animal and dietary factors on total dry matter intake of dairy cows fed silage-based diets

An empirical regression model for the prediction of total dry matter intake (DMI) of dairy cows was developed and compared with four published intake models. The model was constructed to include both animal and dietary factors, which are known to affect DMI. For model development, a data set based on individual cow data from 10 change-over and four continuous milk production studies was collected (n = 1554). Relevant animal (live weight (LW), days in milk (DIM), parity and breed) and dietary (total and concentrate DMI, concentrate composition, forage digestibility and fermentation quality) data were collected. The model factors were limited to those that are available before the diets are fed to animals, that is, standardized energy corrected milk (sECM) yield, LW, DIM and diet quality (total diet DMI index (TDMI index)). As observed ECM yield is a function of both the production potential of the cow and diet quality, ECM yield standardized for DIM, TDMI index and metabolizable protein concentration was used in modelling. In the individual data set, correlation coefficients between sECM and TDMI index or DIM were much weaker (0.16 and 0.03) than corresponding coefficients with observed ECM (0.65 and 0.46), respectively. The model was constructed with a mixed model regression analysis using cow within trial as a random factor. The following mixed model was estimated for DMI prediction: DMI (kg DM/day) = -2.9 (±0.56)+0.258 (±0.011) × sECM (kg/day) + 0.0148 (±0.0009) × LW (kg) -0.0175 (±0.001) × DIM -5.85 (±0.41) × exp (-0.03 × DIM) + 0.09 (±0.002) × TDMI index. The mixed DMI model was evaluated with a treatment mean data set (207 studies, 992 diets), and the following relationship was found: Observed DMI (kg DM/day) = -0.10 (±0.33) + 1.004 (±0.019) × Predicted DMI (kg DM/day) with an adjusted residual mean square error of 0.362 kg/day. Evaluation of the residuals did not result in a significant mean bias or linear slope bias, and random error accounted for proportionally >0.99 of the error. In conclusion, the DMI model developed is considered robust because of low mean prediction error, accurate and precise validation, and numerically small differences in the parameter values of model variables when estimated with mixed or simple regression models. The Cornell Net Carbohydrate and Protein System was the most accurate of the four other published DMI models evaluated using individual or treatment mean data, but in most cases mean and linear slope biases were relatively high, and, interestingly, there were large differences in both mean and linear slope biases between the two data sets.     

Random regression models using different functions to model test-day milk yield of Brazilian Holstein cows

We analyzed 152,145 test-day records from 7317 first lactations of Holstein cows recorded from 1995 to 2003. Our objective was to model variations in test-day milk yield during the first lactation of Holstein cows by random regression model (RRM), using various functions in order to obtain adequate and parsimonious models for the estimation of genetic parameters. Test-day milk yields were grouped into weekly classes of days in milk, ranging from 1 to 44 weeks. The contemporary groups were defined as herd-test-day. The analyses were performed using a single-trait RRM, including the direct additive, permanent environmental and residual random effects. In addition, contemporary group and linear and quadratic effects of the age of cow at calving were included as fixed effects. The mean trend of milk yield was modeled with a fourth-order orthogonal Legendre polynomial. The additive genetic and permanent environmental covariance functions were estimated by random regression on two parametric functions, Ali and Schaeffer and Wilmink, and on B-spline functions of days in milk. The covariance components and the genetic parameters were estimated by the restricted maximum likelihood method. Results from RRM parametric and B-spline functions were compared to RRM on Legendre polynomials and with a multi-trait analysis, using the same data set. Heritability estimates presented similar trends during mid-lactation (13 to 31 weeks) and between week 37 and the end of lactation, for all RRM. Heritabilities obtained by multi-trait analysis were of a lower magnitude than those estimated by RRM. The RRMs with a higher number of parameters were more useful to describe the genetic variation of test-day milk yield throughout the lactation. RRM using B-spline and Legendre polynomials as base functions appears to be the most adequate to describe the covariance structure of the data.     

Isolating the cow-specific part of residual energy intake in lactating dairy cows using random regressions

The ability to properly assess and accurately phenotype true differences in feed efficiency among dairy cows is key to the development of breeding programs for improving feed efficiency. The variability among individuals in feed efficiency is commonly characterised by the residual intake approach. Residual feed intake is represented by the residuals of a linear regression of intake on the corresponding quantities of the biological functions that consume (or release) energy. However, the residuals include both, model fitting and measurement errors as well as any variability in cow efficiency. The objective of this study was to isolate the individual animal variability in feed efficiency from the residual component. Two separate models were fitted, in one the standard residual energy intake (REI) was calculated as the residual of a multiple linear regression of lactation average net energy intake (NEI) on lactation average milk energy output, average metabolic BW, as well as lactation loss and gain of body condition score. In the other, a linear mixed model was used to simultaneously fit fixed linear regressions and random cow levels on the biological traits and intercept using fortnight repeated measures for the variables. This method split the predicted NEI in two parts: one quantifying the population mean intercept and coefficients, and one quantifying cow-specific deviations in the intercept and coefficients. The cow-specific part of predicted NEI was assumed to isolate true differences in feed efficiency among cows. NEI and associated energy expenditure phenotypes were available for the first 17 fortnights of lactation from 119 Holstein cows; all fed a constant energy-rich diet. Mixed models fitting cow-specific intercept and coefficients to different combinations of the aforementioned energy expenditure traits, calculated on a fortnightly basis, were compared. The variance of REI estimated with the lactation average model represented only 8% of the variance of measured NEI. Among all compared mixed models, the variance of the cow-specific part of predicted NEI represented between 53% and 59% of the variance of REI estimated from the lactation average model or between 4% and 5% of the variance of measured NEI. The remaining 41% to 47% of the variance of REI estimated with the lactation average model may therefore reflect model fitting errors or measurement errors. In conclusion, the use of a mixed model framework with cow-specific random regressions seems to be a promising method to isolate the cow-specific component of REI in dairy cows.     

e-Cow: an animal model that predicts herbage intake, milk yield and live weight change in dairy cows grazing temperate pastures, with and without supplementary feeding

This animal simulation model, named e-Cow, represents a single dairy cow at grazing. The model integrates algorithms from three previously published models: a model that predicts herbage dry matter (DM) intake by grazing dairy cows, a mammary gland model that predicts potential milk yield and a body lipid model that predicts genetically driven live weight (LW) and body condition score (BCS). Both nutritional and genetic drives are accounted for in the prediction of energy intake and its partitioning. The main inputs are herbage allowance (HA; kg DM offered/cow per day), metabolisable energy and NDF concentrations in herbage and supplements, supplements offered (kg DM/cow per day), type of pasture (ryegrass or lucerne), days in milk, days pregnant, lactation number, BCS and LW at calving, breed or strain of cow and genetic merit, that is, potential yields of milk, fat and protein. Separate equations are used to predict herbage intake, depending on the cutting heights at which HA is expressed. The e-Cow model is written in Visual Basic programming language within Microsoft Excel^R. The model predicts whole-lactation performance of dairy cows on a daily basis, and the main outputs are the daily and annual DM intake, milk yield and changes in BCS and LW. In the e-Cow model, neither herbage DM intake nor milk yield or LW change are needed as inputs; instead, they are predicted by the e-Cow model. The e-Cow model was validated against experimental data for Holstein–Friesian cows with both North American (NA) and New Zealand (NZ) genetics grazing ryegrass-based pastures, with or without supplementary feeding and for three complete lactations, divided into weekly periods. The model was able to predict animal performance with satisfactory accuracy, with concordance correlation coefficients of 0.81, 0.76 and 0.62 for herbage DM intake, milk yield and LW change, respectively. Simulations performed with the model showed that it is sensitive to genotype by feeding environment interactions. The e-Cow model tended to overestimate the milk yield of NA genotype cows at low milk yields, while it underestimated the milk yield of NZ genotype cows at high milk yields. The approach used to define the potential milk yield of the cow and equations used to predict herbage DM intake make the model applicable for predictions in countries with temperate pastures.     

The intake and digestion of maize silage-based diets by dairy cows and sheep

Six lactating cows, 6 dry cows and 6 wether sheep were fed ad libitum on diets of maize silage, maize silage plus lucerne, or maize silage plus lucerne plus wheat. Faeces and urine collections allowed for the determination of digestibility of dry matter, organic matter and nitrogen, and balances of nitrogen and water.

Voluntary feed intakes were highest and digestibility values were lowest in lactating cows. The addition of lucerne reduced organic matter digestibility in dry cows, but not in lactating cows or sheep. The addition of wheat decreased intake in dry cows and sheep, but not in lactating cows. Production of milk, protein, solids-not-fat and total solids increased with dietary quality, but there was a depression in milk fat content as a result of wheat supplementation.

The ranking of the 3 diets on the basis of feed intake differed with each class of livestock, but lactating cows and sheep gave the same ranking on the basis of organic matter digestibility.     

Merging and characterising phenotypic data on conventional and rare traits from dairy cattle experimental resources in three countries

This study set out to demonstrate the feasibility of merging data from different experimental resource dairy populations for joint genetic analyses. Data from four experimental herds located in three different countries (Scotland, Ireland and the Netherlands) were used for this purpose. Animals were first lactation Holstein cows that participated in ongoing or previously completed selection and feeding experiments. Data included a total of 60 058 weekly records from 1630 cows across the four herds; number of cows per herd ranged from 90 to 563. Weekly records were extracted from the individual herd databases and included seven traits: milk, fat and protein yield, milk somatic cell count, liveweight, dry matter intake and energy intake. Missing records were predicted with the use of random regression models, so that at the end there were 44 weekly records, corresponding to the typical 305-day lactation, for each cow. A total of 23 different lactation traits were derived from these records: total milk, fat and protein yield, average fat and protein percentage, average fat-to-protein ratio, total dry matter and energy intake and average dry matter intake-to-milk yield ratio in lactation weeks 1 to 44 and 1 to 15; average milk somatic cell count in lactation weeks 1 to 15 and 16 to 44; average liveweight in lactation weeks 1 to 44; and average energy balance in lactation weeks 1 to 44 and 1 to 15. Data were subsequently merged across the four herds into a single dataset, which was analysed with mixed linear models. Genetic variance and heritability estimates were greater (P < 0.05) than zero for all traits except for average milk somatic cell count in weeks 16 to 44. Proportion of total phenotypic variance due to genotype-by-environment (sire-by-herd) interaction was not different (P > 0.05) from zero. When estimable, the genetic correlation between herds ranged from 0.85 to 0.99. Results suggested that merging experimental herd data into a single dataset is both feasible and sensible, despite potential differences in management and recording of the animals in the four herds. Merging experimental data will increase power of detection in a genetic analysis and augment the potential reference population in genome-wide association studies, especially of difficult-to-record traits.     

Multiple trait model combining random regressions for daily feed intake with single measured performance traits of growing pigs

A random regression model for daily feed intake and a conventional multiple trait animal model for the four traits average daily gain on test (ADG), feed conversion ratio (FCR), carcass lean content and meat quality index were combined to analyse data from 1 449 castrated male Large White pigs performance tested in two French central testing stations in 1997. Group housed pigs fed ad libitum with electronic feed dispensers were tested from 35 to 100 kg live body weight. A quadratic polynomial in days on test was used as a regression function for weekly means of daily feed intake and to escribe its residual variance. The same fixed (batch) and random (additive genetic, pen and individual permanent environmental) effects were used for regression coefficients of feed intake and single measured traits. Variance components were estimated by means of a Bayesian analysis using Gibbs sampling. Four Gibbs chains were run for 550 000 rounds each, from which 50 000 rounds were discarded from the burn-in period. Estimates of posterior means of covariance matrices were calculated from the remaining two million samples. Low heritabilities of linear and quadratic regression coefficients and their unfavourable genetic correlations with other performance traits reveal that altering the shape of the feed intake curve by direct or indirect selection is difficult.     

Prediction of water intake and excretion flows in Holstein dairy cows under thermoneutral conditions

The increase in the worldwide demand for dairy products, associated with global warming, will emphasize the issue of water use efficiency in dairy systems. The evaluation of environmental issues related to the management of animal dejections will also require precise biotechnical models that can predict effluent management in farms. In this study, equations were developed and evaluated for predicting the main water flows at the dairy cow level, based on parameters related to cow productive performance and diet under thermoneutral conditions. Two datasets were gathered. The first one comprised 342 individual measurements of water balance in dairy cows obtained during 18 trials at the experimental farm of Méjussaume (INRA, France). Predictive equations of water intake, urine and fecal water excretion were developed by multiple regression using a stepwise selection of regressors from a list of seven candidate parameters, which were milk yield, dry matter intake (DMI), body weight, diet dry matter content (DM), proportion of concentrate (CONC) and content of crude protein (CP) ingested with forage and concentrate (CPf and CPc, g/kg DM). The second dataset was used for external validation of the developed equations and comprised 196 water flow measurements on experimental lots obtained from 43 published papers related to water balance or digestibility measurements in dairy cows. Although DMI was the first predictor of the total water intake (TWI), with a partial r² of 0.51, DM was the first predictive parameter of free water intake (FWI), with a partial r² of 0.57, likely due to the large variability of DM in the first dataset (from 11.5 to 91.4 g/100 g). This confirmed the compensation between water drunk and ingested with diet when DM changes. The variability of urine volume was explained mainly by the CPf associated with DMI (r.s.d. 5.4 kg/day for an average flow of 24.0 kg/day) and that of fecal water was explained by the proportion of CONC in the diet and DMI. External validation showed that predictive equations excluding DMI as predictive parameters could be used for FWI, urine and fecal water predictions if cows were fed a well-known total mixed ration. It also appeared that TWI and FWI were underestimated when ambient temperature increased above 25°C and possible means of including climatic parameters in future predictive equations were proposed.     

Analysis of feed intake and energy balance of high-yielding first lactating Holstein cows with fixed and random regression models

At the dairy research farm Karkendamm, the individual roughage intake was measured since 1 September 2005 using a computerised scale system to estimate daily energy balances as the difference between energy intake and calculated energy requirements for lactation and maintenance. Data of 289 heifers with observations between the 11th and 180th day of lactation over a period of 487 days were analysed. Average energy-corrected milk yield, feed intake, live weight and energy balance were 31.8kg, 20.6kg, 584 kg and 13.6 MJ NEL (net energy lactation), respectively, per day. Fixed and random regression models were used to estimate repeatabilities, correlations between cow effects and genetic parameters. The resulting genetic correlations in different lactation stages demonstrate that feed intake and energy balance at the beginning and the middle of lactation are genetically different traits. Heritability of feed intake is low with h2=0.06 during the first days after parturition and increases in the middle of lactation, whereas the energy balance shows the highest heritability with h2=0.34 in the first 30 days of lactation. Genetic correlations between energy balance and feed intake and milk yield, respectively, illustrate that energy balance depends more on feed intake than on milk yield. Genetic correlation between body condition score and energy balance decreases rapidly within the first 100 days of lactation. Hence, to avoid negative effects on health and reproduction as consequences of strong energy deficits at the beginning of lactation, the energy balance itself should be measured and used as a selection criterion in this lactation stage. Since the number of animals is rather small for a genetic analysis, the genetic parameters have to be evaluated on a more comprehensive dataset.     

Lucerne silage in high yielding dairy cows: effect of parity and analysis of reasons in mixed-parity group housing using random regression test-day models

A study with high-yielding dairy cows was re-analysed in order to test the suitability of lucerne silage separately for primi- and multiparous cows as an alternative to grass silage in maize-based total mixed rations (TMR). Lactation curves were fitted using random regression test-day models for energy corrected milk (ECM) and dry matter intake (DMI) as well as for number and duration of feeder visits (NFV and DFV, respectively). Existing models for ECM and DMI were extended by animal-specific random effects, which were formulated in their dependency on days in milk. For NFV and DFV random regression models were applied for the very first time. The chosen approach of statistical analysis permitted comparisons of the lactation curves as well as of least square means for sub-periods to answer nutritional questions. Whilst primiparous cows had generally lower DMI and ECM as compared to multiparous cows, only in primiparous cows a negative effect of lucerne TMR on ECM was observed, especially in early lactation. Nutritional factors should be rejected because of very similar ECM between the various TMR in multiparous cows. Traits of feeding behaviour indicated that particle size could contribute to the decreased ECM. Even more impact on the lower ECM should be addressed to domination behaviour of multiparous cows. The resulting restlessness of primiparous cows caused a reduced intake per minute spent at the feeder. Further studies should focus on optimising the proportion and chopping length of lucerne in the diet and to improve flock management to maximise feed intake of primiparous cows. Generally, statistical analysis of lactation data became a very complex issue. It seems inevitable that nutritionists and statisticians team up to address this problem.     

Effects of varying the level of palm oil on feed intake, milk yield and composition and postpartum weight changes of Red Sokoto goats

Twenty-five pregnant Red Sokoto goats (average liveweight, 33.14 ± 1.75 kg) were used from the last month of pregnancy until 118 day of lactation to evaluate the effect of varying the level of palm (Elaeis guineensis, Jacq.) oil (PO) in concentrate supplement on lactation performance. The goats were fed one of five iso-nitrogenous (16% CP) supplements containing 0% PO (control), 4% PO, 8% PO, 12% PO or 16% PO to a basal diet of Wooly finger grass (Digitaria smutsii, Stent) hay. Average consumption of concentrate was 400 g/goat/day, representing 48% of total dry matter intake. Daily dry matter intake decreased linearly with increasing levels of palm oil. The 4% PO concentrate elicited the highest milk production and was the most cost-effective, while improving daily milk production by 29% compared with the control. Milk composition and postpartum weight changes of the goats were not significantly affected by the concentrate supplements but milk fat percent was generally increased by inclusion level of palm oil in the supplement. It is concluded from this study that the concentrate supplement containing 4% palm oil can increase milk yield in Red Sokoto goats without adversely affecting dry matter intake.     

Application of the covariance function approach with an iterative two-stage algorithm to the estimation of parameters of a random regression test day model for dairy production traits

The covariance function approach with an iterative two-stage algorithm of LIU et al. (2000) was applied to estimate parameters for the Polish Black-and-White dairy population based on a sample of 338 808 test day records for milk, fat, and protein yields. A multiple trait sire model was used to estimate covariances of lactation stages. A third-order Legendre polynomial was subsequently fitted to the estimated (co)variances to derive (co)variances of random regression coefficients for both additive genetic and permanent environment effects. Daily and 305-day heritability estimates obtained are consistent with several studies which used both fixed and random regression test day models. Genetic correlations between any two days in milk (DIM) of the same lactation as well as genetic correlations between the same DIM of two lactations were within a biologically acceptable range. It was shown that the applied estimation procedure can utilise very large data sets and give plausible estimates of (co)variance components.     

Feeding behavior improves prediction of dairy cow voluntary feed intake but cannot serve as the sole indicator

Low-cost feeding-behavior sensors will soon be available for commercial use in dairy farms. The aim of this study was to develop a feed intake model for the individual dairy cow that includes feeding behavior. In a research farm, the individual cows’ voluntary feed intake and feeding behavior were monitored at every meal. A feed intake model was developed based on data that exist in commercial modern farms: ‘BW,’ ‘milk yield’ and ‘days in milking’ parameters were applied in this study. At the individual cow level, eating velocity seemed to be correlated with feed intake (R²=0.93 to 0.94). The eating velocity coefficient varied among individuals, ranging from 150 to 230 g/min per cow. The contribution of feeding behavior (0.28) to the dry matter intake (DMI) model was higher than the contribution of BW (0.20), similar to the contribution of fat-corrected milk (FCM)/BW (0.29) and not as large as the contribution of FCM (0.49). Incorporating feeding behavior into the DMI model improved its accuracy by 1.3 (38%) kg/cow per day. The model is ready to be implemented in commercial farms as soon as companies introduce low-cost feeding-behavior sensors on commercial level.     

Developing and validating a model to predict the dry matter intake of grazing lactating beef cows

Current techniques for measuring the dry matter intake (DMI) of grazing lactating beef cows are invasive, time consuming and expensive making them impractical for use on commercial farms. This study was undertaken to explore the potential to develop and validate a model to predict DMI of grazing lactating beef cows, which could be applied in a commercial farm setting, using non-invasive animal measurements. The calibration dataset used to develop the model was comprised of 94 measurements recorded on 106 beef or beef–dairy crossbred cows (maternal origin). The potential of body measurements, linear type scoring, grazing behaviour and thermal imaging to predict DMI in combination with known biologically plausible adjustment variables and energy sinks was investigated. Multivariable regression models were constructed for each independent variable using SAS PROC REG and contained milk yield, BW, parity, calving day and maternal origin (dairy or beef). Of the 94 variables tested, 32 showed an association with DMI (P < 0.25) upon multivariable analysis. These variables were incorporated into a backwards linear regression model using SAS PROC REG. Variables were retained in this model if P < 0.05. Five variables; width at pins, full body depth, ruminating mastications, central ligament and rump width score, were retained in the model in addition to milk yield, BW, parity, calving day and maternal origin. The inclusion of these variables in the model increased the predictability of DMI by 0.23 (R² = 0.68) when compared to a model containing milk yield, BW, parity, calving day and maternal origin only. This model was applied to data recorded on an independent dataset; a herd of 60 lactating beef cows two years after the calibration study. The R² for the validation was 0.59. Estimates of DMI are required for measuring feed efficiency. While acknowledging challenges in applicability, the findings suggest a model such as that developed in this study may be used as a tool to more easily and less invasively estimate DMI on large populations of commercial beef cows, and therefore measure feed efficiency.     

A new method to estimate residual feed intake in dairy cattle using time series data

In dairy, the usual way to measure feed efficiency is through the residual feed intake (RFI) method. However, this method is, in its classical form, a linear regression, which, by construction, does not take into account the evolution of the RFI components across time, inducing approximations in the results. We present here a new approach that incorporates the dynamic dimension of the data. Using a multitrait random regression model, the correlations between milk, live weight, DM intake (DMI) and body condition score (BCS) were investigated across the lactation. In addition, at each time point, by a matrix regression on the variance–covariance matrix and on the animal effects from the three predictor traits, a predicted animal effect for intake was estimated, which, by difference with the actual animal effect for intake, gave a RFI estimation. This model was tested on historical data from the Aarhus University experimental farm (1 469 lactations out of 740 cows). Correlations between animal effects were positive and high for milk and DMI and for weight and DMI, with a maximum mid-lactation, stable across time at around 0.4 for weight and BCS, and slowly decreasing along the lactation for milk and weight, DMI and BCS, and milk and BCS. At the Legendre polynomial coefficient scale, the correlations were estimated with a high accuracy (averaged SE of 0.04, min = 0.02, max = 0.05). The predicted animal effect for intake was always extremely highly correlated with the milk production and highly correlated with BW for the most part of the lactation, but only slightly correlated with BCS, with the correlation becoming negative in the second half of the lactation. The estimated RFI possessed all the characteristics of a classical RFI, with a mean at zero at each time point and a phenotypic independence from its predictors. The correlation between the averaged RFI over the lactation and RFI at each time point was always positive and above 0.5, and maximum mid-lactation (> 0.9). The model performed reasonably well in the presence of missing data. This approach allows a dynamic estimation of the traits, free from all time-related issues inherent to the traditional RFI methodology, and can easily be adapted and used in a genetic or genomic selection context.     

Disentangling drivers of spatial autocorrelation in species distribution models

Species distribution models (SDMs) are frequently used to understand the influence of site properties on species occurrence. For robust model inference, SDMs need to account for the spatial autocorrelation of virtually all species occurrence data. Current methods do not routinely distinguish between extrinsic and intrinsic drivers of spatial autocorrelation, although these may have different implications for conservation. Here, we present and test a method that disentangles extrinsic and intrinsic drivers of spatial autocorrelation using repeated observations of a species. We focus on unknown habitat characteristics and conspecific interactions as extrinsic and intrinsic drivers, respectively. We model the former with spatially correlated random effects and the latter with an autocovariate, such that the spatially correlated random effects are constant across the repeated observations whereas the autocovariate may change. We tested the performance of our model on virtual species data and applied it to observations of the corncrake Crex crex in the Netherlands. Applying our model to virtual species data revealed that it was well able to distinguish between the two different drivers of spatial autocorrelation, outperforming models with no or a single component for spatial autocorrelation. This finding was independent of the direction of the conspecific interactions (i.e. conspecific attraction versus competitive exclusion). The simulations confirmed that the ability of our model to disentangle both drivers of autocorrelation depends on repeated observations. In the case study, we discovered that the corncrake has a stronger response to habitat characteristics compared to a model that did not include spatially correlated random effects, whereas conspecific interactions appeared to be less important. This implies that future conservation efforts should primarily focus on maximizing habitat availability. Our study shows how to systematically disentangle extrinsic and intrinsic drivers of spatial autocorrelation. The method we propose can help to correctly identify the main drivers of species distributions.     

High- and low-purity glycerine supplementation to dairy cows in early lactation: effects on silage intake,milk production and metabolism

This study evaluated the effects of supplemental low- and high-purity glycerine on silage intake, milk yield and composition, plasma metabolites and body condition score (BCS) in dairy cows. A total of 42 cows of the Swedish Red Breed, housed in individual tie stalls, were fed 0.25 kg of low- or high-purity glycerine on top of concentrate, twice daily, during the first 4 weeks of lactation. One-third of the cows acted as controls, receiving no glycerine. Silage was fed for ad libitum intake and concentrate was fed at restricted level of intake, about 6 kg/day for primiparous cows and 7 kg/day for multiparous cows. Feed refusals were weighed daily. Cows were milked twice daily, milk yield was recorded on four occasions per week and milk samples were collected simultaneously. Blood samples were drawn from the coccygeal vessel once a week. Low- and high-purity glycerine had no effect on silage or total dry matter intake (P = 0.38 and P = 0.75, respectively) or on BCS (P = 0.45). Cows fed high-purity glycerine tended to have higher milk yield than control cows (P = 0.06). Milk composition tended to differ among treatments. No main effects of treatment on concentration of glycerine (P = 0.44), glucose (P = 0.78), insulin (P = 0.33), non-esterified fatty acids (P = 0.33) and β-hydroxybutyrate (P = 0.15) in plasma. These data indicate that high-purity glycerine has the potential to increase milk yield, as well as enhance the milk protein concentration and milk fat + protein yield.     

The generalized linear model for spatial data: assessing the effects of environmental covariates on population density in the field

With interest in spatial ecology growing, correlational field studies are likely to become increasingly important. Unfortunately, ecological field data often do not follow the assumptions of classical statistics, so techniques like the popular and powerful multiple linear regression and its variants are often unreliable, and results can be misleading. The generalized linear model (GLM) is a flexible extension of linear regression that has proved especially useful for discrete data. In this paper, the technique is adapted to accommodate spatially correlated, discrete data. Specifically, to demonstrate the approach, Japanese beetle grub [Popillia japonica Newman (Coleoptera, Scarabaeidae)] population density in the field is modeled as a function of soil organic matter content. The response variable (grub counts in small soil samples) was a spatially autocorrelated, discrete random variable. Three classes of GLMs of the association between soil organic matter content and grub density were compared: (i) regression (assuming normally distributed response variable), (ii) GLM assuming negative binomial counts, and (iii) GLM based on the assumption that the counts conformed to Taylor's power law (TPL). Because the grubs were distributed in patches rather than at random, models that explicitly accounted for the spatial autocorrelation of grub counts were constructed, and compared with models that assumed independent observations. The fitted values for the discrete GLMs [viz., (ii) and (iii)] differed noticeably from the fitted values from multiple regression; but fitted values among the negative binomial and TPL GLMs were virtually identical, regardless of whether the spatial covariance was incorporated into a model, whether a spherical or exponential variogram model was used, or whether variance function parameters were estimated over a large or small scale. However, P‐values for the overall significance of the models depended heavily on whether the GLM assumed a discrete or continuous response variable, and whether or not spatial autocorrelation in the response variable was accounted for. On average, P‐values were 45‐fold higher in the spatial GLMs than in the non‐spatial and 23‐fold higher in the discrete GLMs than in the continuous.