A comparison of methods to calculate a total merit index using stochastic simulation

Background

Modern dairy cattle breeding goals include several production and more and more functional traits. Estimated breeding values (EBV) that are combined in the total merit index usually come from single-trait models or from multivariate models for groups of traits. In most cases, a multivariate animal model based on phenotypic data for all traits is not feasible and approximate methods based on selection index theory are applied to derive the total merit index. Therefore, the objective of this study was to compare a full multitrait animal model with two approximate multitrait models and a selection index approach based on simulated data.

Methods

Three production and two functional traits were simulated to mimic the national Austrian Brown Swiss population. The reference method for derivation of the total merit index was a multitrait evaluation based on all phenotypic data. Two of the approximate methods were variations of an approximate multitrait model that used either yield deviations or de-regressed breeding values. The final method was an adaptation of the selection index method that is used in routine evaluations in Austria and Germany. Three scenarios with respect to residual covariances were set up: residual covariances were equal to zero, or half of or equal to the genetic covariances.

Results

Results of both approximate multitrait models were very close to those of the reference method, with rank correlations of 1. Both methods were nearly unbiased. Rank correlations for the selection index method showed good results when residual covariances were zero but correlations with the reference method decreased when residual covariances were large. Furthermore, EBV were biased when residual covariances were high.

Conclusions

We applied an approximate multitrait two-step procedure to yield deviations and de-regressed breeding values, which led to nearly unbiased results. De-regressed breeding values gave even slightly better results. Our results confirmed that ignoring residual covariances when a selection index approach is applied leads to remarkable bias. This could be relevant in terms of selection accuracy. Our findings suggest that the approximate multitrait approach applied to de-regressed breeding values can be used in routine genetic evaluation.     

Analysis of precore/core covariances associated with viral kinetics and genotypes in hepatitis B e antigen-positive chronic hepatitis B patients

Hepatitis B virus (HBV) is one of the most common DNA viruses that can cause aggressive hepatitis, cirrhosis and hepatocellular carcinoma. Although many people are persistently infected with HBV, the kinetics in serum levels of viral loads and the host immune responses vary from person to person. HBV precore/core open reading frame (ORF) encoding proteins, hepatitis B e antigen (HBeAg) and core antigen (HBcAg), are two indicators of active viral replication. The aim of this study was to discover a variety of amino acid covariances in responses to viral kinetics, seroconversion and genotypes during the course of HBV infection. A one year follow-up study was conducted with a total number of 1,694 clones from 23 HBeAg-positive chronic hepatitis B patients. Serum alanine aminotransferase, HBV DNA and HBeAg levels were measured monthly as criteria for clustering patients into several different subgroups. Monthly derived multiple precore/core ORFs were directly sequenced and translated into amino acid sequences. For each subgroup, time-dependent covariances were identified from their time-varying sequences over the entire follow-up period. The fluctuating, wavering, HBeAg-nonseroconversion and genotype C subgroups showed greater degrees of covariances than the stationary, declining, HBeAg-seroconversion and genotype B. Referring to literature, mutation hotspots within our identified covariances were associated with the infection process. Remarkably, hotspots were predominant in genotype C. Moreover, covariances were also identified at early stage (spanning from baseline to a peak of serum HBV DNA) in order to determine the intersections with aforementioned time-dependent covariances. Preserved covariances, namely representative covariances, of each subgroup are visually presented using a tree-based structure. Our results suggested that identified covariances were strongly associated with viral kinetics, seroconversion and genotypes. Moreover, representative covariances may benefit clinicians to prescribe a suitable treatment for patients even if they have no obvious symptoms at the early stage of HBV infection.     

Influence of mom and dad: quantitative genetic models for maternal effects and genomic imprinting

The expression of an imprinted gene is dependent on the sex of the parent it was inherited from, and as a result reciprocal heterozygotes may display different phenotypes. In contrast, maternal genetic terms arise when the phenotype of an offspring is influenced by the phenotype of its mother beyond the direct inheritance of alleles. Both maternal effects and imprinting may contribute to resemblance between offspring of the same mother. We demonstrate that two standard quantitative genetic models for deriving breeding values, population variances and covariances between relatives, are not equivalent when maternal genetic effects and imprinting are acting. Maternal and imprinting effects introduce both sex-dependent and generation-dependent effects that result in differences in the way additive and dominance effects are defined for the two approaches. We use a simple example to demonstrate that both imprinting and maternal genetic effects add extra terms to covariances between relatives and that model misspecification may over- or underestimate true covariances or lead to extremely variable parameter estimation. Thus, an understanding of various forms of parental effects is essential in correctly estimating quantitative genetic variance components.     

Inheritance of body size in the Barnacle Goose under different environmental conditions

Heritabilities, genetic variances and covariances for body size traits, i.e. tarsus length, head length and body mass, were estimated under different environmental conditions in a Barnacle Goose (Branta leucopsis) population. Under poor growth conditions, that is, when average body size of fully grown offspring in a given cohort was small, the offspring-parent regressions and full-sib analyses yielded heritability estimates not significantly different from zero. By contrast, when growth conditions were normal or good the heritability estimates were generally significantly positive. Comparisons of genetic covariance estimates indicated that they also differed across the analysed environmental conditions. This result, together with similar results obtained in studies of passerine birds, suggests that genotype-environment interactions might be frequent within the range of environments normally encountered by birds in natural populations. If general, such results might question the validity of assuming approximate constancy of additive genetic variances and covariances over time and environments in evolutionary models.     

Genetic models with reduced penetrance related to the Y chromosome

Classical statistical genetics models of a quantitative trait depending on an autosomal gene indicate that father-to-daughter and mother-to-son correlations should be the same. If phenotypes are not sex-dependent, father-to-son and mother-to-daughter correlations also share this common value. On the other hand, if the gene is sex-linked, then the father-to-son correlation is zero. Such models do not explain genetic variation in pulmonary artery pressure (PAP) of cattle--important because cattle with high PAP are known to develop brisket disease, pulmonary heart disease, and congestive heart failure when taken to high altitudes. Data on 966 calves at a ranch in Colorado showed positive correlation (0.2) between sire PAP and male calf PAP but slightly negative correlation (-0.01) between sire PAP and female calf PAP; the dam-to-male calf and dam-to-female calf correlations are both about 0.1. The model presented here postulates an autosomal gene with reduced penetrance (i.e., the trait may remain at a normal level even when the genotype suggests abnormality) and that, in males, the rate of penetrance is related to an abnormality in the Y chromosome and is therefore passed on from father to son. Then under plausible selective breeding assumptions, the pairwise correlation between fathers and daughters can become zero or negative. Explicit formulas are computed for the model covariances, and numerical computations indicate that plausible parameter values can be chosen for the model.     

Maximum Likelihood Methods for Fitting the Burr Type XII Distribution to Life Test Data

This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log-likelihood and conditional log-likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log-likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.     

Improved least squares topology testing and estimation

Generalized least squares (GLS) methods provide a relatively fast means of constructing a confidence set of topologies. Because they utilize information about the covariances between distances, it is reasonable to expect additional efficiency in estimation and confidence set construction relative to other least squares (LS) methods. Difficulties have been found to arise in a number of practical settings due to estimates of covariance matrices being ill conditioned or even noninvertible. We present here new ways of estimating the covariance matrices for distances that are much more likely to be positive definite, as the actual covariance matrices are. A thorough investigation of performance is also conducted. An alternative to GLS that has been proposed for constructing confidence sets of topologies is weighted least squares (WLS). As currently implemented, this approach is equivalent to the use of GLS but with covariances set to zero rather than being estimated. In effect, this approach assumes normality of the estimated distances and zero covariances. As the results here illustrate, this assumption leads to poor performance. A 95% confidence set is almost certain to contain the true topology but will contain many more topologies than are needed. On the other hand, the results here also indicate that, among LS methods, WLS performs quite well at estimating the correct topology. It turns out to be possible to improve the performance of WLS for confidence set construction through a relatively inexpensive normal parametric bootstrap that utilizes the same variances and covariances of GLS. The resulting procedure is shown to perform at least as well as GLS and thus provides a reasonable alternative in cases where covariance matrices are ill conditioned.     

Evolutionary potential of a widespread clonal grass under changing climate

Adaptive responses are probably the most effective long‐term responses of populations to climate change, but they require sufficient evolutionary potential upon which selection can act. This requires high genetic variance for the traits under selection and low antagonizing genetic covariances between the different traits. Evolutionary potential estimates are still scarce for long‐lived, clonal plants, although these species are predicted to dominate the landscape with climate change. We studied the evolutionary potential of a perennial grass, Festuca rubra, in western Norway, in two controlled environments corresponding to extreme environments in natural populations: cold–dry and warm–wet, the latter being consistent with the climatic predictions for the country. We estimated genetic variances, covariances, selection gradients and response to selection for a wide range of growth, resource acquisition and physiological traits, and compared their estimates between the environments. We showed that the evolutionary potential of F. rubra is high in both environments, and genetic covariances define one main direction along which selection can act with relatively few constraints to selection. The observed response to selection at present is not sufficient to produce genotypes adapted to the predicted climate change under a simple, space for time substitution model. However, the current populations contain genotypes which are pre‐adapted to the new climate, especially for growth and resource acquisition traits. Overall, these results suggest that the present populations of the long‐lived clonal plant may have sufficient evolutionary potential to withstand long‐term climate changes through adaptive responses.     

Specific interactions between host and parasite genotypes do not act as a constraint on the evolution of antiviral resistance in Drosophila

Genetic correlations between parasite resistance and other traits can act as an evolutionary constraint and prevent a population from evolving increased resistance. For example, previous studies have found negative genetic correlations between host resistance and life-history traits. In invertebrates, the level of resistance often depends on the combination of the host and parasite genotypes, and in this study, we have investigated whether such specific resistance also acts as an evolutionary constraint. We measured the resistance of different genotypes of the fruit fly Drosophila melanogaster to different genotypes of a naturally occurring pathogen, the sigma virus. Using a multitrait analysis, we examine whether genetic covariances alter the potential to select for general resistance against all of the different viral genotypes. We found large amounts of heritable variation in resistance, and evidence for specific interactions between host and parasite, but these interactions resulted in little constraint on Drosophila evolving greater resistance.     

Comparative methods with sampling error and within-species variation: contrasts revisited and revised

Comparative methods analyses have usually assumed that the species phenotypes are the true means for those species. In most analyses, the actual values used are means of samples of modest size. The covariances of contrasts then involve both the covariance of evolutionary changes and a fraction of the within-species phenotypic covariance, the fraction depending on the sample size for that species. Ives et al. have shown how to analyze data in this case when the within-species phenotypic covariances are known. The present model allows them to be unknown and to be estimated from the data. A multivariate normal statistical model is used for multiple characters in samples of finite size from species related by a known phylogeny, under the usual Brownian motion model of change and with equal within-species phenotypic covariances. Contrasts in each character can be obtained both between individuals within a species and between species. Each contrast can be taken for all of the characters. These sets of contrasts, each the same contrast taken for different characters, are independent. The within-set covariances are unequal and depend on the unknown true covariance matrices. An expectation-maximization algorithm is derived for making a reduced maximum likelihood estimate of the covariances of evolutionary change and the within-species phenotypic covariances. It is available in the Contrast program of the PHYLIP package. Computer simulations show that the covariances are biased when the finiteness of sample size is not taken into account and that using the present model corrects the bias. Sampling variation reduces the power of inference of covariation in evolution of different characters. An extension of this method to incorporate estimates of additive genetic covariances from a simple genetic experiment is also discussed.     

Evolution of Zygotic Linkage Disequilibrium in a Finite Local Population

One crucial feature of zygotic linkage disequilibrium (LD) analysis is its direct use of diploid genotyping data, irrespective of the type of mating system. Previous theories from an evolutionary perspective mainly focus on gametic LD, but the equivalent development for zygotic LD is not available. Here I study the evolution of zygotic LD and the covariances between gametic and zygotic LDs or between distinct zygotic LDs in a finite local population under constant immigration from a continent population. I derive the analytical theory under genetic hitchhiking effects or in a neutral process. Results indicate that zygotic LDs (diploid level) are more informative than gametic LD (haploid level) in indicating the effects of different evolutionary forces. Zygotic LDs may be greater than or comparable to gametic LD under the epistatic selection process, but smaller than gametic LD under the non epistatic selection process. The covariances between gametic and zygotic LDs are strongly affected by the mating system, linkage distance, and genetic drift effects, but weakly affected by seed and pollen flow and natural selection. The covariances between different zygotic LDs are generally robust to the effects of gene flow, selection, and linkage distance, but sensitive to the effects of genetic drift and mating system. Consistent patterns exist for the covariances between the zygotic LDs for the two-locus genotypes with one common genotype at one locus or without any common genotype at each locus. The results highlight that zygotic LDs can be applied to detecting natural population history.     

Stochastic compartmental modeling and parameter estimation with application to cancer treatment follow-up studies

In this paper we use marginal probabilities to derive expressions for the means, variances and covariances ofm-compartment systems. We also present an efficient algorithm for the estimation of the parameters of the system using time series data when measurements are available fromk of them compartments. An application of the analysis and parameter estimation procedure for a model representing the results of a cancer treatment follow-up study is given. Supported in part by NSF Grant Number DCR74-17282.     

Effect of various parameter combinations on genetic gain in computer simulated two-character selection

Summary In a simulation study, the effect of various parameter combinations such as linkage, dominance, heritability, and economic weights on the individual trait means was investigated using additive genetic, genotypic and the phenotypic index of Elston (1963). The characters responded differently to these indices under various parameter combinations, indicating favourable and unfavourable effects of the mentioned parameters. Linkage was found to reduce the rate of progress through selection. Depression of genetic gain was greater where the genes governing a character showed dominance and/or heritability coefficients were low. It was, however, noticed that depression of genetic gain due to low heritability of a character could be avoided by assigning higher economic weight to that character. This suggests that desirable changes in the means of characters available for selection can be manipulated by choosing appropriate economic weights. The additive genetic index, where only the additive genetic variances and covariances go into its construction, does not seem to be affected by intra-allelic interactions since they add to variances and covariances due to dominance deviations and these have nothing to do with the additive genetic variances and covariances. It seems that from such studies, if conducted extensively incorporating still more parameters, conclusions may be drawn on the most suitable selection model for simultaneous selection under a given set of parameters available in real biological systems.     

Constraints on the coevolution of contemporary human males and females

Because autosomal genes in sexually reproducing organisms spend on average half their time in each sex, and because the traits that they influence encounter different selection pressures in males and females, the evolutionary responses of one sex are constrained by processes occurring in the other sex. Although intralocus sexual conflict can restrict sexes from reaching their phenotypic optima, no direct evidence currently supports its operation in humans. Here, we show that the pattern of multivariate selection acting on human height, weight, blood pressure and glucose, total cholesterol, and age at first birth differs significantly between males and females, and that the angles between male and female linear (77.8 ± 20.5°) and nonlinear (99.1 ± 25.9°) selection gradients were closer to orthogonal than zero, confirming the presence of sexually antagonistic selection. We also found evidence for intralocus sexual conflict demonstrated by significant changes in the predicted male and female responses to selection of individual traits when cross-sex genetic covariances were included and a significant reduction in the angle between male- and female-predicted responses when cross-sex covariances were included (16.9 ± 15.7°), compared with when they were excluded (87.9 ± 31.6°). We conclude that intralocus sexual conflict constrains the joint evolutionary responses of the two sexes in a contemporary human population.     

Trade‐offs between growth and reproduction: an analysis of the quantitative genetic evidence

The assumption of a trade‐off between development time and fecundity, resulting from a positive correlation between body size and fecundity and between body size and development time, is a common feature of life history models. The present paper examines the evidence for such a trade‐off as indicated by genetic correlations between traits. The genetic covariances between traits are derived using a model in which maturation occurs when the organism achieves a genetically variable size threshold, and fecundity is an allometric function of body size with one genetically variable parameter (excluding body size itself). This model predicts that the heritabilities of the life history traits (growth rate, development time, fecundity) will not necessarily be less than the heritability of adult size (i.e. morphological traits). It is shown that if growth rate is genetically correlated with adult size then it is not possible, in general, to predict the sign of the genetic correlation between development time and fecundity. For particular cases the signs of the covariances between traits can be predicted. These predictions are tested using data drawn from the literature.     

Effect of selection on genetic parameters of correlated traits

Summary Changes in genetic parameters of correlated traits due to the buildup of linkage (gametic phase) disequilibrium from repeated truncation selection on a single trait are studied. After several generations of selection, an equilibrium is approached where there are no further changes in genetic parameters and limiting values are reached. Formulae are derived under an infinitesimal model for these limiting values of genetic variances and covariances, heritabilities, and genetic correlations between traits directly and indirectly selected. Changes from generation zero to the limit in all these parameters become greater as heritability of the trait under direct selection increases and, to a lesser extent, as intensity of selection increases. Change in heritability of a trait under indirect selection also increases as the absolute value of the correlation between the trait under indirect and the trait under direct selection increases. The change is maximum when the initial value of heritability is close to 0.5 and insignificant when the initital value is close to zero or one. Change in the genetic correlation between the trait under direct selection and the trait under indirect selection is maximum when its initial value is close to ±0.6 and insignificant when its initial value is close to zero or ±1. Heritability of the trait indirectly selected and genetic correlation between that trait and the trait directly selected always decrease in absolute value, whereas genetic correlation between two traits indirectly selected can either decrease or increase in absolute value. It is suggested that use be made of formulae at selection equilibrium in the prediction of correlated responses after several generations of selection.     

Genetic variances and covariances of aerobic metabolic rates in laboratory mice

The genetic variances and covariances of traits must be known to predict how they may respond to selection and how covariances among them might affect their evolutionary trajectories. We used the animal model to estimate the genetic variances and covariances of basal metabolic rate (BMR) and maximal metabolic rate (MMR) in a genetically heterogeneous stock of laboratory mice. Narrow-sense heritability (h²) was approximately 0.38 ± 0.08 for body mass, 0.26 ± 0.08 for whole-animal BMR, 0.24 ± 0.07 for whole-animal MMR, 0.19 ± 0.07 for mass-independent BMR, and 0.16 ± 0.06 for mass-independent MMR. All h² estimates were significantly different from zero. The phenotypic correlation of whole animal BMR and MMR was 0.56 ± 0.02, and the corresponding genetic correlation was 0.79 ± 0.12. The phenotypic correlation of mass-independent BMR and MMR was 0.13 ± 0.03, and the corresponding genetic correlation was 0.72 ± 0.03. The genetic correlations of metabolic rates were significantly different from zero, but not significantly different from one. A key assumption of the aerobic capacity model for the evolution of endothermy is that BMR and MMR are linked. The estimated genetic correlation between BMR and MMR is consistent with that assumption, but the genetic correlation is not so high as to preclude independent evolution of BMR and MMR.     

Longitudinal analysis of pre- and post-treatment measurements with equal baseline assumptions in randomized trials

For continuous variables of randomized controlled trials, recently, longitudinal analysis of pre- and posttreatment measurements as bivariate responses is one of analytical methods to compare two treatment groups. Under random allocation, means and variances of pretreatment measurements are expected to be equal between groups, but covariances and posttreatment variances are not. Under random allocation with unequal covariances and posttreatment variances, we compared asymptotic variances of the treatment effect estimators in three longitudinal models. The data-generating model has equal baseline means and variances, and unequal covariances and posttreatment variances. The model with equal baseline means and unequal variance–covariance matrices has a redundant parameter. In large sample sizes, these two models keep a nominal type I error rate and have high efficiency. The model with equal baseline means and equal variance–covariance matrices wrongly assumes equal covariances and posttreatment variances. Only under equal sample sizes, this model keeps a nominal type I error rate. This model has the same high efficiency with the data-generating model under equal sample sizes. In conclusion, longitudinal analysis with equal baseline means performed well in large sample sizes. We also compared asymptotic properties of longitudinal models with those of the analysis of covariance (ANCOVA) and t-test.     

Theory of inbreeding and covariances between relatives under full-sib mating in diploids.

A generation matrix theory of full-sib mating is developed in which 13 mating "classes" are distinguished according to identity of genes in individuals mated and identity of genotypes as belonging to homozygous, parental, or offspring sets. The 13 times 13 matrix reveals some properties of the full-sib mating system not shown by previous work. The eigenvalues and a set of eigenvectors for the generation matrix, and the general solution for the frequencies of mating classes among descendants of an original mating of genotypes ab times cd, are given. The genotypic array of descendants in an arbitrary generation is also given. A new formula is derived for the coefficient of inbreeding in generation n + m in terms of coefficients of inbreeding in earlier generations. An algorithm is presented for calculating the probability of a given situation of identity of alleles carried by two individuals given only the indices of their own respective generations and the generation of their most recent common ancestor. The application of such probabilities to obtaining covariances between relatives in a full-sib mating system, under the assumptions of independence and non-interaction among loci, is illustrated. All results are shown to agree with previous work in special cases. All possible full sib, generation n - 1 parent-generation n + m offspring, and generation n uncle-generation n + m nephew covariances for 1 less than n + m less than or equal to 8 are obtained using the given algorithm.