The Mathematical description of the lactation curve of ruminants: issues and perspectives

Test day records of milk production traits in the main livestock species reared for milk production were analyzed with several empirical mathematical models, with aim to identify suitable functions to fit different types of lactation curve shape. In the first chapter the modelling of extended lactations in dairy cows were analyzed. Several empirical mathematical models currently used to fit lactations with a function developed specifically for long patterns were compared. The second experimental contribution has approached a common problem when small ruminant lactation curves are modelled. In the specific case of the Sarda goat, the partial overlapping of altitude of location of flock with the partition into three different genetic subpopulation represents a further peculiarity. The third study was a investigation on the peculiar situation of the Massese breed ewes, where the intensification of reproductive cycle results in an alternate sequence of long and short lactations for each ewe. A frequent issue in modelling lactation curves is represented by relevant occurrence of atypical shapes, i.e. those that lacks of the peak yield and consist of a monotonically decreasing pattern

Multivariate meta-analysis of QTL mapping studies

A large number of quantitative trait loci (QTLs) for milk production and quality traits in dairy cattle has been reported in literature. The large amount of information available could be exploited by meta-analyses to draw more general conclusions from results obtained in different experimental conditions (animals, statistical methodologies). QTL meta-analyses have been carried out to estimate the distribution of QTL effects in livestock and to find consensus on QTL position. In this study, multivariate dimension reduction techniques are used to analyse a database of dairy cattle QTL published results, in order to extract latent variables able to characterise the research. A total of 92 papers by 72 authors were found on 25 scientific Journals for the period January 1995-February 2008. More than thirty parameters were picked up from the articles. To overcome the problem of different map location, the flanking markers were mapped on release 4.1 of the Bos taurus genome sequence (www.ensembl. org). Their position was retrieved from public databases and, when absent, was calculated in silico by blasting (http://blast.wustl.edu/) the markers’ nucleotide sequence against the genomic sequence. Records were discarded if flanking markers or P-values were not available. After these edits, the final archive consisted of 1,162 records. Seven selected variables were analysed both with the Factor Analysis (FA), combined with the varimax rotation technique, and Principal Component Analysis (PCA). FA was able to explain 68% of the original variability with 3 latent factors: the first factor extracted was highly associated (factor loading of 0.98) to marker location along the chromosome and could be considered as a marker map index; the second factor showed factor loadings of 0.74 and 0.84 related to the variable number of animals involved and year of the experiment, respectively, and it can be regarded as an indicator of the dimension of the study; the third factor was correlated to the significance level of the statistical test (0.78), number of families (0.63), and, negatively, to the marker density (-0.43). It can be named as index of power of the experiment. Same patterns can be observed in the eigenvectors of PCA. Four PCs were able to explain about 80% of the original variance. The first two PCs basically underlined accurately the same structure found with the first two factors in FA, whereas PC3 and PC4 summarized the structure of F3. The score that each QTL gets on each Factor or PC could be useful to classify the original QTL records and make them more comparable once that the redundancy of information has been removed

The Mathematical description of lactation curves in dairy cattle

This review gives an overview of the mathematical modelling of lactation curves in dairy cattle. Over the last ninety years, the development of this field of study has followed the main requirements of the dairy cattle industry. Non-linear parametric functions have represented the preferred tools for modelling average curves of homogeneous groups of animals, with the main aim of predicting yields for management purposes. The increased availability of records per individual lactations and the genetic evaluation based on test day records has shifted the interest of modellers towards more flexible and general linear functions, as polynomials or splines. Thus the main interest of modelling is no longer the reconstruction of the general pattern of the phenomenon but the fitting of individual deviations from an average curve. Other specific approaches based on the modelling of the correlation structure of test day records within lactation, such as mixed linear models or principal component analysis, have been used to test the statistical significance of fixed effects in dairy experiments or to create new variables expressing main lactation curve traits. The adequacy of a model is not an absolute requisite, because it has to be assessed according to the specific purpose it is used for. Occurrence of extended lactations and of new productive and functional traits to be described and the increase of records coming from automatic milking systems likely will represent some of the future challenges for the mathematical modelling of the lactation curve in dairy cattle

Modelling extended lactation curves for milk production traits in Italian Holsteins

Test day records of milk production traits (milk yield, fat and protein percentage, and somatic cell score) of 45,132 Italian Holstein cows were analyzed with seven mathematical models in order to assess the main features of lactations of different length. Lactations curves were grouped according to parity (1, 2, and 3) and lactation length (1<350d; 2=from 351 to 450d; 3=from 451 to 650d; 4=651 to 1000d). Models with a larger number of parameters showed better fitting performances for all classes of length for milk yield, whereas poor fitting was observed for fat and protein percentages and SCS in the 651-1000d class. In lactation with length>650d, peak yield was about 31, 37, and 39 kg for first, second, and third parity respectively; peak was predicted at around 60 and 40 days for younger and older animals respectively. The asymptotic level of production was below 10 kg

Analysis of genetic correlations between multivariate measures of lactation persistency and somatic cell score in Italian Simmental cattle

Genetic relationships between lactation curve traits and Somatic Cell Count are of great interest for dairy cattle breeding. Factor Analysis (MFA) and Principal Component Analysis (PCA) can be used to extract from the correlation matrix of milk test day records new unobservable (latent) variables that can be related to lactation curve shape. Previous researches report that MFA is particularly able to extract two latent variables related with level of production in early lactation (PEL) and lactation persistency (PERS), respectively, whereas PCA yields a leading component related to the average level of production (AVY) for the whole lactation and a second component negatively related with tests of early lactation and positively with tests of the second part of lactation (SLOPE). Aim of this work was to estimate genetic correlations between lactation curve shape traits and Somatic Cell Score (SCS). MFA and PCA were carried out on a data set of 16,020 lactations of Italian Simmental cows, each with six TD records for milk yield recorded with the A4 scheme. Genetic parameters were estimated with a bivariate animal model that included fixed effects of herd-test date, parity*age*lactation stage (only parity*age for lactation curve traits), calving season, and random effects of additive genetic and permanent environment. Heritability estimates were moderate for lactation curve traits (0.15, 0.15, 0.21 and 0.09 for PEL, PERS, AVY and SLOPE, respectively) and low for SCS (0.09). Correlations between lactation curve traits and SCS were favourable, i.e. negative, except for the level of production in early lactation. In particular, the genetic improvement of lactation persistency result in a contemporary reduction of SCS (rg -0.55 and -0.51 with PERS and SLOPE, respectively) whereas the increase of level of production in early lactation can lead to a moderate increase of SCS (rg 0.13). Finally, the two measures of persistency could be used for different selection strategies: the use of PERS may allow for the increase of persistency together with the total lactation yield whereas the use of SLOPE may result in an improvement of the lactation curve shape without modifying total lactation yield

Dynamic Corporate Liquidity

We examine the determinants of corporate liquidity management through the lens of an estimated dynamic model of corporate investment and financing. When external finance is costly, firms can absorb shocks and cover liquidity needs by holding cash and by drawing down credit lines. In contrast to cash, we model credit lines as providing liquidity contingent on economic news, but limited by collateral constraints and covenants. The option to draw down credit lines creates value as it allows firms to take advantage of investment opportunities in an effective way, facilitating firm growth. We find that our estimated model matches well the levels and joint dynamics of cash, credit lines, leverage, equity financing and investment when firms can collateralize roughly one third of their assets. In the cross-section, the model provides novel empirical predictions and rationalizes a wide range of stylized facts regarding credit line usage, covenant violations, and cash holdings

The Sources of Financing Constraints

In order to identify the relevant sources of firms' financing constraints, we ask what financial frictions matter for corporate policies. To that end, we build, solve, and estimate a range of dynamic models of corporate investment and financing, embedding a host of financial frictions. We focus on limited enforcement, moral hazard, and trade-off models. All models share a common technology, but differ in the friction generating financing constraints. Using panel data on Compustat firms for the period 1980-2015 and a more recent dataset on private firms from Orbis, we determine which features of the observed data allow to distinguish among the models, and we assess which model or model combination performs best at rationalizing observed corporate investment and financing policies across various samples. Our tests, based on empirical policy function benchmarks, favor trade-off models for larger Compustat firms, limited commitment models for smaller firms, and moral hazard models for private firms. Our estimates point to significant financing constraints due to agency frictions and highlight the importance of identifying their relevant sources for firm valuation

Pre-selection of most significant SNPS for the estimation of genomic breeding values

The availability of a large amount of SNP markers throughout the genome of different livestock species offers the opportunity to estimate genomic breeding values (GEBVs). However, the estimation of many effects in a data set of limited size represent a severe statistical problem. A pre-selection of SNPS based on single regression may provide a reasonable compromise between accuracy of results, number of independent variables to be considered and computing requirements

Use of partial least squares regression to impute SNP genotypes in Italian Cattle breeds

Background The objective of the present study was to test the ability of the partial least squares regression technique to impute genotypes from low density single nucleotide polymorphisms (SNP) panels i.e. 3K or 7K to a high density panel with 50K SNP. No pedigree information was used. Methods Data consisted of 2093 Holstein, 749 Brown Swiss and 479 Simmental bulls genotyped with the Illumina 50K Beadchip. First, a single-breed approach was applied by using only data from Holstein animals. Then, to enlarge the training population, data from the three breeds were combined and a multi-breed analysis was performed. Accuracies of genotypes imputed using the partial least squares regression method were compared with those obtained by using the Beagle software. The impact of genotype imputation on breeding value prediction was evaluated for milk yield, fat content and protein content. Results In the single-breed approach, the accuracy of imputation using partial least squares regression was around 90 and 94% for the 3K and 7K platforms, respectively; corresponding accuracies obtained with Beagle were around 85% and 90%. Moreover, computing time required by the partial least squares regression method was on average around 10 times lower than computing time required by Beagle. Using the partial least squares regression method in the multi-breed resulted in lower imputation accuracies than using single-breed data. The impact of the SNP-genotype imputation on the accuracy of direct genomic breeding values was small. The correlation between estimates of genetic merit obtained by using imputed versus actual genotypes was around 0.96 for the 7K chip. Conclusions Results of the present work suggested that the partial least squares regression imputation method could be useful to impute SNP genotypes when pedigree information is not available