The analysis of contingency tables is a powerful statistical tool used in experiments with categorical variables. This study improves parts of the theory underlying the use of contingency tables. Specifically, the linkage disequilibrium parameter as a measure of two-way interactions applied to three-way tables makes it possible to quantify Simpsons paradox by a simple formula. With tests on three-way interactions, there is only one that determines whether the partial interactions of all variables agree or whether there is at least one variable whose partial interactions disagree. To date, there has been no test available that determines whether the partial interactions of a certain variable agree or disagree, and the presented work closes this gap. This work reveals the relation of the multiplicative and the additive measure of a three-way interaction. Another contribution addresses the question of which cells in a contingency table are fixed when the first- and second-order marginal totals are given. The proposed procedure not only detects fixed zero counts but also fixed positive counts. This impacts the determination of the degrees of freedom. Furthermore, limitations of methods that simulate contingency tables with given pairwise associations are addressed.

The identification of chromosome sections affecting quantitative traits (QTLs) is essential in the field of quantitative genetics. In certain types of populations there is high linkage disequilibrium between markers. This causes the effects of adjoining markers to be highly correlated. Especially when there are more parameters p than observations n (p»n), this may cause inflated estimates of marker effects and genetic variability when the effects of all markers are estimated simultaneously in multi-marker models and may also hamper inferences on the position of putative QTLs. In such situations basis functions (B-splines) offer the opportunity to model the genetic effects of any number of markers p by a limited number of basis function effects. Moreover, a covariance matrix between marker effects can be employed, which, in a Bayesian setting, can be interpreted as a summary of prior knowledge on the genetic marker map and predominant linkage phases in the population. Therefore Bayesian multi-marker models and B-spline models were compared in a simulation study for the example of a backcross population. We simulated 12 different scenarios where 12 QTLs were either placed independently or linked on a genome with 20 equally-sized chromosomes. While linkage was changed, effects of those QTLs remained the same in all scenarios. Markers were equally spaced with distances at 1 cM (p»n) and 5 cM (p≈n). For each scenario average results of 200 experiments with 500 individuals were examined with respect to the size of estimated marker effects, resulting genetic variance and the ability to identify QTL-carrying chromosomes. B-spline basis functions decreased computation time, improved the precision of estimated marker effects, genetic variances and genetic predictions compared to uncorrelated marker effects. Further, including the correlation matrix in the analysis in particular showed no inflated estimates of genetic variance. In conclusion correlations combined with B-splines offer a suitable way for adapting the number of model parameters to the size of the genome, irrespective of marker density.

The conjecture: “If the case (v,k) = (8, 3) is the only one where the trivial balanced incomplete block design (BIBD) is elementary” is formulated. It is supported by the theorem: Theorem The conjecture is correct if at least one of the following conditions holds: v < 26, k < 6, for v > 8 if v is prime or a prime power.

The availability of different single nucleotide polymorphism (SNP) chips and the development of imputation algorithms allow for multistage dairy cattle breeding schemes applying various genomic selection strategies. These SNP genotypes yield genomically estimated breeding values (GEBV) with different accuracies at different costs. Thus, the optimum allocation of investments to different selection paths and strategies to maximize the genetic gain per year (ΔG(a)) and its sensitivity to changes in cost and accuracies of GEBV is of great interest. This is even more relevant under the constraints of limited financial resources. With deterministic methods, optimum multistage breeding plans maximizing ΔG(a) were identified in which selection could take place on GEBV derived from high-density (GEBV(HD)) and low-density (GEBV(LD)) SNP genotypes. To account for the uncertainty of cost and accuracies of GEBV, these parameters were varied in a semi-continuous manner. Overall breeding costs were limited to the crucial expenses of a traditional breeding program with 50 progeny-tested young bulls per year. Results clearly show that, in an optimal selection strategy, selection on GEBV(LD) is predominantly used for the identification of future bull dams but the main part of ΔG(a) is still generated from selection of sires. The low selection intensity in the path dam to sire induced a higher sensitivity of ΔG(a) to changes in cost and accuracies of GEBV(LD) compared with the same changes of GEBV(HD). On the contrary, the genetic gain generated from selection of males was only affected by changes in accuracies of GEBV(HD) but almost unaffected by any changes in cost. Thus, changes in cost and accuracies of GEBV(LD) put the most pressure on the breeding scheme structure to maintain a high ΔG(a). Furthermore, genomic selection of bull dams produced by far the majority of breeding cost but the lowest genetic gain.

For some purposes, identity-by-descent (IBD) probabilities for entire chromosome segments are required. Making use of pedigree information, length of the segment and the assumption of no crossing-over, a generalization of a previously published graph theory oriented algorithm accounting for nonzero IBD of common ancestors is given, which can be viewed as method of path coefficients for entire chromosome segments. Furthermore, rules for setting up a gametic version of a segmental IBD matrix are presented. Results from the generalized graph theory oriented method, the gametic segmental IBD matrix and the segmental IBD matrix for individuals are identical.

Sex differences in birth weight contribute to within-litter variability, which itself is connected to piglet survival. Therefore, we studied whether the sex difference in piglet birth weight is a genetically variable sex dimorphism. For that purpose a linear mixed model including sex-specific additive genetic effects was set up. A hypothesis testing problem was defined to detect whether these genetic effects significantly differ between sexes. In a second step, the effect of sex-linked genes was studied explicitly by partitioning the additive genetic effects into autosomal and gonosomal effects. Furthermore, a definition of heritability for the sex difference of a randomly chosen pair of littermates with opposite sex was given. The proposed models were applied separately to a Landrace and Large White data set. Significant genetic variability for the sex dimorphism was found in Landrace (P = 0.03) but not in Large White (P = 0.10). Heritability estimates were at 3 to 5% depending on the model. The X-chromosomal genetic variation was not significant (P > 0.18) at all, whereas the Y-chromosome significantly (P < 0.01) contributed to the genetic variation in Landrace with a corresponding SD of 34 g. It can be concluded that the sex dimorphism of piglet birth weight is genetically variable and a potential target of genetic improvement.

Reduction in the variability of piglet birth weight within litter and increased piglet survival are key objective in schemes aiming to improve sow prolificacy. In previous studies, variation in birth weight was described by the sample standard deviation of birth weights within one litter, and the genetic impact has been proved. In this study, we additionally considered the sex effect on piglet’s birth weight and on its variability. The sample variance of birth weights per litter separated by sex was assigned as a trait of the sow. Different transformations of the trait were fitted by linear and generalized linear mixed models. Based on 1111 litters from Landrace sows, the estimates of heritability for the different measures ranged from 11 to 12%. We analysed the influence of including birth weight of stillborn piglets on the variability of birth weight within litter. With omitted stillborns, the heritability was estimated approximately 2% higher than that in investigations of all born piglets, and the impact of sex on birth weight variability was increased. Because the proportion of intrapartum deaths is rather high, it is recommended to consider the total number of piglets born per litter when analysing birth weight variation.

The development of different muscles and adipose tissues during growth was investigated in commercial Japanese Black (JB) cattle and compared with breeds of the largest variation to be found in Europe. Animals, reared under typical conditions for Japanese and European beef production systems, gained similar body weights but different carcass composition at 24 months of age. The carcass of JB contained more adipose tissue and the least proportion of muscle. The longissimus muscle of JB developed extraordinary amounts of 23.3% intramuscular fat (IMF) at 24 months of age, compared from 0.6% to 4.7% in European breeds. The relationships between IMF content in the longissimus muscle and different adipose tissue weights indicate that a large amount of “waste fat” is accreted with every percent of IMF. However in JB, the good ability of IMF deposition is associated with relatively least development of “waste fat”, as a result of unique breed characteristics combined with special feeding system.

Genetics affects not only the weight of piglets at birth but also the variability of birth weight within litter. Previous studies on this topic assigned the sample standard deviation of piglet birth weights within litter as an observation to the sow. However, the contribution of the difference in mean birth weight per sex on the within-litter variance has been neglected so far. This work deals with the genetic effect on within-litter variance when different statistical models with different distributional assumptions are used and considers the sex effect and appropriate weights per trait. Traits were formed from the pooled sample variance of male and female birth weights within litter. A linear model approach was fitted to the logarithmized within-litter variance and the sample standard deviation. A generalized linear model with gamma-distributed residuals and log-link function was applied to the untransformed sample variance. Models were compared by analysing data from 9439 litters from Landrace and Large White of a commercial breeding programme. The estimates of heritability for different traits ranged from 7% to 11%. Although the generalized linear mixed model is preferred from a mathematical view, the rank correlations between breeding values of the linear mixed models and the generalized linear mixed model were relatively high, i.e. 94% to 98%. By residual diagnostics, a linear mixed model using the weighted and pooled within-litter standard deviation was identified as most suitable.

Breed differences in adult animals are determined during fetal development. If interventions are to be developed that influence growth of muscle and fat, it is important to know at which time during gestation breed differences appear and are fixed. The objective of this study was to characterize fetal development in cattle of different breeds. Pregnant cows of 4 cattle breeds with different growth impetus and muscularity were slaughtered under normal processing conditions and the fetuses were removed. German Angus, a typical beef cattle; Galloway, a smaller, environmentally resistant beef type; Holstein Friesian, a dairy type; and Belgian Blue, an extreme type for muscle growth were used. Fetuses of each breed were investigated at 3, 6, and 9 mo of gestation. Fetuses were weighed and dissected into carcass, organs, and muscles. Body fat weight was obtained using the Soxhlet extraction method. Fetal weight increased most rapidly in the third trimester of gestation mainly due to the accelerated muscle and fat deposition. The organ weight to body weight (BW) ratios decreased and the muscle and fat weight to BW ratios increased. At 3 mo of gestation, Galloway fetuses had the significantly smallest BW, half-carcass weight, leg weight, organ weight, muscle weight and shortest leg length. In contrast, Holstein fetuses had the significantly greatest BW, liver, kidney, and lung weights and significantly longest leg length among the 4 breeds, but no differences between Holstein Friesian and Belgian Blue were detected in half-carcass and leg weight. Indeed, Belgian Blue fetuses had the significantly greatest half-carcass weight, leg weight, and muscle weight at 9 mo of gestation, and Galloway had a significantly greater body fat to BW ratio than Holstein Friesian and Belgian Blue. These differences were not evident at 3 and 6 mo of gestation. These data show that the profound increase of tissue and organ weights occurred in later gestation in cattle fetuses even though breed differences were evident as early as 3 mo of gestation. Depending on the tissue of interest, impacting fetal growth likely needs to occur early in gestation before the appearance of breed-specific differences.

Non-parametric procedures are sometimes in use even in cases where the corresponding parametric procedure is preferable. This is mainly due to the fact that in practical applications of statistical methods too much attention is paid to any violation of the normality assumption-normal distribution is, however, primarily supposed in order to easily derive the exact distribution of the statistic used within parametric approaches.As concerns the case of two independent samples and the comparison of (the expectations of) two populations the t-test and its non-parametric counterpart, the Wilcoxon (Mann-Whitney) test are of particular interest. These both serve as the illustrative example, given here.The Wilcoxon test compares the two distributions, indeed, but may in cases where we are interested in comparing expectations lead to significance even if the expectations are equal; this because any higher moments in the two populations may differ. On the other hand, the t-test is so robust against non-normality that there is nearly no need to use the Wilcoxon test in comparing expectations.Most results for continuous distributions have been obtained in a research group in Dummerstorf-Rostock some years ago and have been published by Herrendbrfer [1980. Robustheit I, Arbeitsmaterial zum Forschungsthema Robustheit, Probleme der angewandten Statistik, vol. 4. Forschungszentrum Dummerstorf-Rostock, Heft], Heffendorfer et al. [1983. Robustness of statistical methods. 11, Methods for the one-sample problem. Biometfical J. 25, 327-343], Rasch [1995. Mathernatische Statistik. Johann Ambrosius Barth, Leipzig-Heidelberg], Rasch and Tiku [1984. Robustness of Statistical Methods and Nonparametric Statistics. VEB Deutscher Verlag der Wissenschaften, Berlin (D, Reidel Publ. Co., Dordrecht, Lancaster, Boston, Tokyo, 1985)], and Rasch and Guiard [2004. The robustness of parametric statistical methods. Psychol. Sci. 46(2), 175-208]. Most of the results are based on extensive simulation experiments with 10 000 runs each.In the present paper we discuss the two-sample problem firstly by summarising the results for continuous distributions from some preprints in German language; however, secondly we offer new results. All above we investigate the t-test's and the Wilcoxon test's robustness in cases of the distribution not being continuous.All the results are that in most practical cases the two-sample t-test is so robust that it can be recommended in nearly all applications. (c) 2007 Published by Elsevier B.V.

Recombinant inbred lines (RIL) derived from multiple inbred strains can serve as a powerful resource for the genetic dissection of complex traits. The use of such multiple-strain RIL requires a detailed knowledge of the haplotype structure in such lines. Broman (2005) derived the two- and three-point haplotype probabilities for 2(n)-way RIL; the former required hefty computation to infer the symbolic results, and the latter were strictly numerical. We describe a simpler approach for the calculation of these probabilities, which allowed us to derive the symbolic form of the three-point haplotype probabilities. We also extend the two-point results for the case of additional generations of intermating, including the case of 2(n)-way intermated recombinant inbred populations (IRIP).

Livestock populations are usually kept in groups. As a consequence, social interactions among individuals affect productivity, health, and welfare. Current selection methods (individual selection), however, ignore those interactions and yield suboptimal or in some cases even negative responses. In principle, selection between groups instead of individuals offers a solution, but has rarely been adopted in practice for two reasons. First, the relationship between group selection theory and common animal breeding concepts, such as the accuracy of selection, is unclear. Second, application of group selection requires keeping selection candidates in groups, which is often undesirable in practice. This work has two objectives. First, we derive expressions for the accuracy of individual and group selection, which provides a measurement of quality for those methods. Second, we investigate the opportunity to improve traits affected by interactions by using information on relatives kept in family groups, while keeping selection candidates individually. The accuracy of selection based on relatives is shown to be an analogy of the classical expression for traits not affected by interactions. Our results show that selection based on relatives offers good opportunities for effective genetic improvement of traits affected by interactions.

objective of this study was to investigate the phenotypical differences in feed conversion, carcass composition, and meat quality of two cattle breeds representing either the accretion type (Charolais) or the secretion type (Holstein). Eighteen Charolais bulls and 18 German Holstein bulls were raised and fattened under identical conditions on a high energy level with concentrates and hay. The animals were slaughtered at the age of 18 months. Charolais bulls had a body weight (BW) of 750.6 kg and an average daily gain (ADG) of 1377 g. German Holstein bulls were 84.7 kg lighter with an ADG of 1197 g. The energy expense per kg BW gain was 14% lower in the Charolais bulls compared with German Holstein. Charolais bulls gained more energy in protein than the Holstein bulls and needed less energy per kg protein gain in the hot carcass weight (HCW). The Charolais bulls showed a greater carcass weight and a higher yield grade as well as a greater cross sectional area of the M. longissimus dorsi (LD). Holstein bulls showed a greater marbling score, darker color, and more water binding capacity of the LD. The Holstein bulls gained more subcutaneous, intramuscular, and visceral fat during growth, which reflects the ability of the Holstein as a dairy breed to deposit fat as an energy source for milk production and points out the capability of Charolais for extended protein accretion. The genetic background of the recorded different nutrient utilization and turnover will be investigated in further experiments.

To circumvent the obvious waste of resources when producing good marbled beef, research is needed to clarify mechanisms which are involved in intramuscular fat storage in cattle. The possible association between the activity of diacylglycerol acyltransferase (DGAT) and muscle fat content was examined in samples of longissimus dorsi (LD) and semitendinosus (ST) muscle from Holstein and Charolais bulls. The Holstein bulls exhibited higher fat content in both muscles and higher marbling score. In Holstein, DGAT activity was enhanced in the LD muscle, and there was a tentative positive relationship between DGAT activity and the fat content in ST muscle. When muscle DGAT activity was examined as a function of DGAT1 genotype for all animals, regardless of breed, the DGAT activity of LD muscle of the K/K genotype was about five-fold greater than for either the K/A or A/A genotypes. Further investigation on the relationship between DGAT1 genotype and i.m. fat is required before this may be developed as a selection tool for marbling.

The objective of this study was to investigate the growth- and breed-related changes of marbling characteristics in cattle. Four cattle breeds with different growth impetus and muscularity were reared and slaughtered under experimental conditions. German Angus, as a typical beef cattle; Galloway, as a smaller, environmentally resistant beef cattle; Holstein-Friesian, as a dairy-type cattle; and double-muscled Belgian Blue, as an extreme type for muscle growth, were used. These 4 breeds were expected to have differences in muscle development and i.m. fat deposition. Between 5 and 15 bulls of each breed were slaughtered at 2, 4, 6, 12, or 24 mo of age. Marbling characteristics were determined and classified in LM and semitendinosus muscle by computerized image analysis. Among breeds, differences appeared in the quantity, structure, and distribution of the marbling flecks in both muscles. The deposition of fat in the double-muscled Belgian Blue bulls remained substantially inferior to that of the other breeds, up to the age of 24 mo. Marbling in German Angus bulls particularly showed larger (P < 0.05) marbling fleck areas. Galloway cattle had the greatest (P < 0.05) number and the most regular (P < 0.05) distribution of the marbling flecks in young animals. Furthermore, for marbling characteristics in Holstein-Friesian animals, a great number and slightly finer structure were observed compared with the other breeds investigated. Postnatal growth-related changes of marbling in LM were characterized by as much as a 40-fold increase in the number of marbling flecks from 2 to 24 mo of age but also by up to a 4-fold enlargement in the area of the marbling flecks. The structure of marbling flecks was determined by 2 development trends. On the one hand, the marbling flecks became larger (P < 0.05), and the structure became coarser, which was reflected by an increasing (P < 0.01) proportion of long marbling flecks as well as an increasing (P < 0.01) maximum skeleton line length. On the other hand, continually new small, round marbling flecks appeared. This caused a decrease (P < 0.01) in the proportion of the 3 largest marbling fleck areas. The distribution of the marbling flecks became more regular (P < 0.05) with increasing proportion and number of marbling flecks. The results suggest that hyperplasia of adipocytes plays an important role in marbling during growth of muscle in cattle.

The objective of this study was to investigate the changes in muscle fiber bundles of cattle of different breeds during growth. Different numbers of muscle fibers are surrounded by connective tissue to form bundles macroscopically visible as meat fibers or meat grain, a common meat quality trait. To determine the influence of breed and age on morphological characteristics of muscle fiber bundles, 4 cattle breeds with different growth impetus and muscularity were reared and slaughtered under experimental conditions. German Angus, a typical beef cattle; Galloway, a smaller beef type; Holstein Friesian, a dairy type; and double-muscled Belgian Blue, an extreme type for muscle growth, were used. Between 5 and 15 bulls of each breed were slaughtered at 2, 4, 6, 12, or 24 mo of age, and slices of semitendinosus muscle were removed. Muscle structure characteristics were determined by computerized image analysis. During growth, the muscle cross-sectional area enlarged (P < 0.001) about 5-fold in double-muscled Belgian Blue bulls and about 4-fold in the other breeds. This was a result of the enlargement (P < 0.001) of primary bundles and muscle fibers. The bundle size was similar (P > or = 0.15) in bulls of German Angus and Galloway in all age groups and was doubled (P < 0.001) in double-muscled Belgian Blue animals from 4 mo of age on. The Holstein Friesian bulls had the smallest (P < 0.001) muscle fiber bundles at 24 mo of age. The number of muscle fibers per bundle and the number of bundles per muscle remained nearly constant (P > 0.05) during growth. This supports the existing view that the structure of the muscle is already fixed in prenatal life. The double-muscled Belgian Blue bulls showed a more than 2.5-fold greater (P < 0.001) number of muscle fibers per primary bundle compared with the other breeds investigated. The larger muscle fiber bundles led to a smaller amount of connective tissue per muscle area in double-muscled cattle. The coarser grain of meat in double-muscled Belgian Blue bulls and in older animals was not related to greater shear force values.

The generation of special crosses between different inbred lines such as recombinant inbred strains (RIS) and intermated recombinant inbred populations (IRIP) is being used to improve the power of QTL detection techniques, in particular fine mapping. These approaches acknowledge the fact that recombination of linked loci increases with every generation, caused by the accumulation of crossovers appearing between the loci at each meiosis. This leads to an expansion of the map distance between the loci. While the amount of the map expansion of RIS and IRIP is known for infinite inbred generations, it is not known for finite numbers of generations. This gap was closed here. Since the recursive evaluation of the map expansion factors turned out to be complex, a useful approximation was derived.

Sixty longissimus (L) muscle chops were selected according to marbling score in order to develop a technique for the quantitative description of marbling fat by means of computer image analysis (CIA) and study its relationship with intramuscular fat content and shear force variation in pork. L muscle samples were taken from gilts belonging to three genetic lines differing in carcass leanness, namely Large White (LW), Meishan-derived dam line (M) and Synthetic Genex 3000 (SG). SG gilts had leaner loins (P<0.001) than LW and M. However, the SG loins were more marbled (P<0.001) and had higher IMF content. In addition, CIA analysis showed that marbling fat was finer and better distributed in the SG line. CIA marbling characteristics were significantly correlated with both IMF content (P<0.001) and shear force values (P<0.05) demonstrating the reliability of this technique to assess the contribution of marbling characteristics to variations in pork eating quality.