A Bayesian threshold-normal mixture model for analysis of a continuous mastitis-related trait

A Bayesian approach to estimating COVID-19 incidence and infection fatality rates

Naive estimates of incidence and infection fatality rates (IFR) of coronavirus disease 2019 suffer from a variety of biases, many of which relate to preferential testing. This has motivated epidemiologists from around the globe to conduct serosurveys that measure the immunity of individuals by testing for the presence of SARS-CoV-2 antibodies in the blood. These quantitative measures (titer values) are then used as a proxy for previous or current infection. However, statistical methods that use this data to its full potential have yet to be developed. Previous researchers have discretized these continuous values, discarding potentially useful information. In this article, we demonstrate how multivariate mixture models can be used in combination with post-stratification to estimate cumulative incidence and IFR in an approximate Bayesian framework without discretization. In doing so, we account for uncertainty from both the estimated number of infections and incomplete deaths data to provide estimates of IFR. This method is demonstrated using data from the Action to Beat Coronavirus erosurvey in Canada.

Invited review: Low milk somatic cell count and susceptibility to mastitis

An enduring controversy exists about low milk cell counts and susceptibility to mastitis. The concentration of milk leukocytes, or somatic cell count (SCC), is a well-established direct indicator of mammary gland inflammation that is highly correlated with the presence of a mammary infection. The SCC is also used as a trait for the selection of dairy ruminants less prone to mastitis. As selection programs favor animals with less SCC, and as milk cells contribute to the defense of the mammary gland, the idea that susceptibility to mastitis could possibly be increased in the long term has been put forward and is still widely debated. Epidemiological and experimental studies aimed at relating SCC to susceptibility to mastitis have yielded results that seem contradictory at first sight. Nevertheless, by taking into account the immunobiology of milk and mammary tissue cells and their role in the defense against infection, along with recent studies on SCC-based divergent selection of animals, the issue can be settled. Apparent SCC-linked susceptibility to mastitis is a phenotypic trait that may be linked to immunomodulation but not to selection.

Genetic resistance to infections in sheep

This paper considers genetic resistance to infectious disease in sheep, with appropriate comparison with goats, and explores how such variation may be used to assist in disease control. Many studies have attempted to quantify the extent to which host animals differ genetically in their resistance to infection or in the disease side-effects of infection, using either recorded animal pedigrees or information from genetic markers to quantify the genetic variation. Across all livestock species, whenever studies are sufficiently well powered, then genetic variation in disease resistance is usually seen and such evidence is presented here for three infections or diseases of importance to sheep, namely mastitis, foot rot and scrapie. A further class of diseases of importance in most small ruminant production systems, gastrointestinal nematode infections, is outside the scope of this review. Existence of genetic variation implies the opportunity, at least in principle, to select animals for increased resistance, with such selection ideally used as part of an integrated control strategy. For each of the diseases under consideration, evidence for genetic variation is presented, the role of selection as an aid to disease control is outlined and possible side effects of selection in terms of effects in performance, effects on resistance to other diseases and potential parasite/pathogen coevolution risks are considered. In all cases, the conclusion is drawn that selection should work and it should be beneficial, with the main challenge being to define cost effective selection protocols that are attractive to sheep farmers.

One hundred years of statistical developments in animal breeding

Statistical methodology has played a key role in scientific animal breeding. Approximately one hundred years of statistical developments in animal breeding are reviewed. Some of the scientific foundations of the field are discussed, and many milestones are examined from historical and critical perspectives. The review concludes with a discussion of some future challenges and opportunities arising from the massive amount of data generated by livestock, plant, and human genome projects.

Survival, growth and sexual maturation in Atlantic salmon exposed to infectious pancreatic necrosis: a multi-variate mixture model approach

Background

Outbreaks of infectious pancreatic necrosis (IPN) in Atlantic salmon can result in reduced growth rates in a fraction of the surviving fish (runts). Genetic and environmental variation also affects growth rates within different categories of healthy animals and runts, which complicates identification of runts. Mixture models are commonly used to identify the underlying structures in such data, and the aim of this study was to develop Bayesian mixture models for the genetic analysis of health status (runt/healthy) of surviving fish from an IPN outbreak.

Methods

Five statistical models were tested on data consisting of 10 972 fish that died and 3959 survivors with recorded growth data. The most complex models (4 and 5) were multivariate normal-binary mixture models including growth, sexual maturity and field survival traits. Growth rate and liability of sexual maturation were treated as two-component normal mixtures, assuming phenotypes originated from two potentially overlapping distributions, (runt/normal). Runt status was an unobserved binary trait. These models were compared to mixture models with fewer traits (Models 2 and 3) and a classical linear animal model for growth (Model 1).

Results

Assuming growth as a mixture trait improved the predictive ability of the statistical model considerably (Model 2 vs. 1). The final models (4 and 5) yielded the following results: estimated (underlying) heritabilities were moderate for growth in healthy fish (0.32 ± 0.04 and 0.35 ± 0.05), runt status (0.39 ± 0.07 and 0.36 ± 0.08) and sexual maturation (0.33 ± 0.05), and high for field survival (0.47 ± 0.03 and 0.48 ± 0.03). Growth in healthy animals, runt status and survival showed consistent favourable genetic associations. Sexual maturation showed an unfavourable non-significant genetic correlation with runt status, but favourable genetic correlations with other traits. The estimated fraction of healthy fish was 81-85%. The estimated breeding values for runt status and (normal) growth were consistent for the most complex models (4 and 5), but showed imperfect correlations with estimated breeding values from the simpler models.

Conclusions

Modelling growth in IPN survivors as a mixture trait improved the predictive ability of the model compared with a classical linear model. The results indicated considerable genetic variation in health status among survivors. Mixture modelling may be useful for the genetic analysis of diseases detected mainly through indicator traits.

The genetic analysis of tolerance to infections: a review

Tolerance to infections is defined as the ability of a host to limit the impact of a given pathogen burden on host performance. Uncoupling resistance and tolerance is a challenge, and there is a need to be able to separate them using specific trait recording or statistical methods. We present three statistical methods that can be used to investigate genetics of tolerance-related traits. Firstly, using random regressions, tolerance can be analyzed as a reaction norm slope in which host performance (y-axis) is regressed against an increasing pathogen burden (x-axis). Genetic variance in tolerance slopes is the genetic variance for tolerance. Variation in tolerance can induce genotype re-ranking and changes in genetic and phenotypic variation in host performance along the pathogen burden trajectory, contributing to environment-dependent genetic responses to selection. Such genotype-by-environment interactions can be quantified by combining random regressions and covariance functions. To apply random regressions, pathogen burden of individuals needs to be recorded. Secondly, when pathogen burden is not recorded, the cure model for time-until-death data allows separating two traits, susceptibility and endurance. Susceptibility is whether or not an individual was susceptible to an infection, whereas endurance denotes how long time it took until the infection killed a susceptible animal (influenced by tolerance). Thirdly, the normal mixture model can be used to classify continuously distributed host performance, such as growth rate, into different sub-classes (e.g., non-infected and infected), which allows estimation of host performance reduction specific to infected individuals. Moreover, genetics of host performance can be analyzed separately in healthy and affected animals, even in the absence of pathogen burden and survival data. These methods provide novel tools to increase our understanding on the impact of parasites, pathogens, and production diseases on host traits.

Genetic correlation between heart ratio and body weight as a function of ascites frequency in broilers split up into sex and health status

Ascites or pulmonary hypertension syndrome is a metabolic disorder in broilers. Male broilers have a higher BW and are therefore expected to be more prone to developing ascites than females. As genetic parameters might be affected by the ascites incidence, genetic parameters might differ between male and female broilers. The aims of this study were to estimate the heritability for the ratio of right ventricular weight to total ventricular weight (RATIO) and BW in male and female broilers, the genetic correlation between RATIO and BW separately for male and female broilers, and the genetic correlations between BW for ascitic and nonascitic broilers. Data were available from 7,856 broilers (3,819 males and 4,037 females). The broilers in the experiment were kept under a cold temperature regimen and increased CO(2) levels. In this study, we showed that the incidence of ascites is higher in male than in female broilers. Heritability estimates for BW at 7 wk of age were higher for male (0.22) than for female (0.17) broilers, and for RATIO heritability, estimates were higher for female (0.44) than for male (0.32) broilers. The genetic correlations between RATIO and BW measured at different ages changed from slightly positive at 2 wk of age to moderately negative at 7 wk. The change in genetic correlation was more extreme for male (from 0.01 to -0.62) than for female (from 0.13 to -0.24) broilers. The difference in ascites incidence between male and female broilers is the most likely reason for the difference in genetic correlations. The genetic correlation between BW traits measured in broilers with fluid in the abdomen and without fluid in the abdomen decreased from 0.91 at 2 wk to 0.69 at 7 wk. We conclude that under circumstances with ascites, data from male and female broilers should be analyzed separately.

Genetic evaluation of mastitis liability and recovery through longitudinal analysis of transition probabilities

Background

Many methods for the genetic analysis of mastitis use a cross-sectional approach, which omits information on, e.g., repeated mastitis cases during lactation, somatic cell count fluctuations, and recovery process. Acknowledging the dynamic behavior of mastitis during lactation and taking into account that there is more than one binary response variable to consider, can enhance the genetic evaluation of mastitis.

Methods

Genetic evaluation of mastitis was carried out by modeling the dynamic nature of somatic cell count (SCC) within the lactation. The SCC patterns were captured by modeling transition probabilities between assumed states of mastitis and non-mastitis. A widely dispersed SCC pattern generates high transition probabilities between states and vice versa. This method can model transitions to and from states of infection simultaneously, i.e. both the mastitis liability and the recovery process are considered. A multilevel discrete time survival model was applied to estimate breeding values on simulated data with different dataset sizes, mastitis frequencies, and genetic correlations.

Results

Correlations between estimated and simulated breeding values showed that the estimated accuracies for mastitis liability were similar to those from previously tested methods that used data of confirmed mastitis cases, while our results were based on SCC as an indicator of mastitis. In addition, unlike the other methods, our method also generates breeding values for the recovery process.

Conclusions

The developed method provides an effective tool for the genetic evaluation of mastitis when considering the whole disease course and will contribute to improving the genetic evaluation of udder health.

Genetic analysis of somatic cell score in danish holsteins using a liability-normal mixture model

Mixture models are appealing for identifying hidden structures affecting somatic cell score (SCS) data, such as unrecorded cases of subclinical mastitis. Thus, liability-normal mixture (LNM) models were used for genetic analysis of SCS data, with the aim of predicting breeding values for such cases of mastitis. Here, putative mastitis statuses and breeding values for liability to putative mastitis were inferred solely from SCS observations. In total, there were 395,906 test-day records for SCS from 50,607 Danish Holstein cows. Four different statistical models were fitted: A) a classical (nonmixture) random regression model for test-day SCS; B1) an LNM test-day model assuming homogeneous (co)variance components for SCS from healthy (IMI-) and infected (IMI+) udders; B2) an LNM model identical to B1, but assuming heterogeneous residual variances for SCS from IMI- and IMI+ udders; and C) an LNM model assuming fully heterogeneous (co)variance components of SCS from IMI- and IMI+ udders. For the LNM models, parameters were estimated with Gibbs sampling. For model C, variance components for SCS were lower, and the corresponding heritabilities and repeatabilities were substantially greater for SCS from IMI- udders relative to SCS from IMI+ udders. Further, the genetic correlation between SCS of IMI- and SCS of IMI+ was 0.61, and heritability for liability to putative mastitis was 0.07. Models B2 and C allocated approximately 30% of SCS records to IMI+, but for model B1 this fraction was only 10%. The correlation between estimated breeding values for liability to putative mastitis based on the model (SCS for model A) and estimated breeding values for liability to clinical mastitis from the national evaluation was greatest for model B1, followed by models A, C, and B2. This may be explained by model B1 categorizing only the most extreme SCS observations as mastitic, and such cases of subclinical infections may be the most closely related to clinical (treated) mastitis.

Developments in statistical analysis in quantitative genetics

A remarkable research impetus has taken place in statistical genetics since the last World Conference. This has been stimulated by breakthroughs in molecular genetics, automated data-recording devices and computer-intensive statistical methods. The latter were revolutionized by the bootstrap and by Markov chain Monte Carlo (McMC). In this overview a number of specific areas are chosen to illustrate the enormous flexibility that McMC has provided for fitting models and exploring features of data that were previously inaccessible. The selected areas are inferences of the trajectories over time of genetic means and variances, models for the analysis of categorical and count data, the statistical genetics of a model postulating that environmental variance is partly under genetic control, and a short discussion of models that incorporate massive genetic marker information. We provide an overview of the application of McMC to study model fit, and finally, a discussion is presented on the development of efficient McMC updating schemes for non-standard models.

Genetic Analysis of Somatic Cell Scores in US Holsteins with a Bayesian Mixture Model

The objective of this study was to apply finite mixture models to field data for somatic cell scores (SCS) for estimation of genetic parameters. Data were approximately 170,000 test-day records for SCS from first-parity Holstein cows in Wisconsin. Five different models of increasing level of complexity were fitted. Model 1 was the standard single-component model, and the others were 2-component Gaussian mixtures consisting of similar but distinct linear models. All mixture models (i.e., 2 to 5) included separate means for the 2 components. Model 2 assumed entirely homogeneous variances for both components. Models 3 and 4 assumed heterogeneous variances for either residual (model 3) or genetic and permanent environmental variances (model 4). Model 5 was the most complex, in which variances of all random effects were allowed to vary across components. A Bayesian approach was applied and Gibbs sampling was used to obtain posterior estimates. Five chains of 205,000 cycles were generated for each model. Estimates of variance components were based on posterior means. Models were compared by use of the deviance information criterion. Based on the deviance information criterion, all mixture models were superior to the linear model for analysis of SCS. The best model was one in which genetic and PE variances were heterogeneous, but residual variances were homogeneous. The genetic analysis suggested that SCS in healthy and infected cattle are different traits, because the genetic correlation between SCS in the 2 components of 0.13 was significantly different from unity.

Genetic Associations Between Clinical Mastitis and Somatic Cell Score in Early First-Lactation Cows

The objectives of this study were to examine genetic associations between clinical mastitis and somatic cell score (SCS) in early first-lactation cows, to estimate genetic correlations between SCS of cows with and without clinical mastitis, and to compare genetic evaluations of sires based on SCS or clinical mastitis. Clinical mastitis records from 15 d before to 30 d after calving and first test-day SCS records (from 6 to 30 d after calving) from 499,878 first-lactation daughters of 2,043 sires were analyzed. Results from a bivariate linear sire model analysis of SCS in cows with and without clinical mastitis suggest that SCS is a heterogeneous trait. Heritability of SCS was 0.03 for mastitic cows and 0.08 for healthy cows, and the genetic correlation between the 2 traits was 0.78. The difference in rank between sire evaluations based on SCS of cows with and without clinical mastitis varied from -994 to 1,125, with mean 0. A bivariate analysis with a threshold-liability model for clinical mastitis and a linear Gaussian model for SCS indicated that heritability of liability to clinical mastitis is at least as large as that of SCS in early lactation. The mean (standard deviation) of the posterior distribution of heritability was 0.085 (0.006) for liability to clinical mastitis and 0.070 (0.003) for SCS. The posterior mean (standard deviation) of the genetic correlation between liability to clinical mastitis and SCS was 0.62 (0.03). A comparison of sire evaluations showed that genetic evaluation based on SCS was not able to identify the best sires for liability to clinical mastitis. The association between sire posterior means for liability to clinical mastitis and sire predicted transmitting ability for SCS was far from perfect.

A mixture model with random-effects components for clustering correlated gene-expression profiles

MOTIVATION

The clustering of gene profiles across some experimental conditions of interest contributes significantly to the elucidation of unknown gene function, the validation of gene discoveries and the interpretation of biological processes. However, this clustering problem is not straightforward as the profiles of the genes are not all independently distributed and the expression levels may have been obtained from an experimental design involving replicated arrays. Ignoring the dependence between the gene profiles and the structure of the replicated data can result in important sources of variability in the experiments being overlooked in the analysis, with the consequent possibility of misleading inferences being made. We propose a random-effects model that provides a unified approach to the clustering of genes with correlated expression levels measured in a wide variety of experimental situations. Our model is an extension of the normal mixture model to account for the correlations between the gene profiles and to enable covariate information to be incorporated into the clustering process. Hence the model is applicable to longitudinal studies with or without replication, for example, time-course experiments by using time as a covariate, and to cross-sectional experiments by using categorical covariates to represent the different experimental classes.

RESULTS

We show that our random-effects model can be fitted by maximum likelihood via the EM algorithm for which the E(expectation)and M(maximization) steps can be implemented in closed form. Hence our model can be fitted deterministically without the need for time-consuming Monte Carlo approximations. The effectiveness of our model-based procedure for the clustering of correlated gene profiles is demonstrated on three real datasets, representing typical microarray experimental designs, covering time-course, repeated-measurement and cross-sectional data. In these examples, relevant clusters of the genes are obtained, which are supported by existing gene-function annotation. A synthetic dataset is considered too.

AVAILABILITY

A Fortran program blue called EMMIX-WIRE (EM-based MIXture analysis WIth Random Effects) is available on request from the corresponding author.

On the quantitative genetics of mixture characters

Finite mixture models are helpful for uncovering heterogeneity due to hidden structure. Quantitative genetics issues of continuous characters having a finite mixture of Gaussian components as statistical distribution are explored in this article. The partition of variance in a mixture, the covariance between relatives under the supposition of an additive genetic model, and the offspring-parent regression are derived. Formulas for assessing the effect of mass selection operating on a mixture are given. Expressions for the genetic and phenotypic correlations between mixture and Gaussian traits and between two mixture traits are presented. It is found that, if there is heterogeneity in a population at the genetic or environmental level, then genetic parameters based on theory treating distributions as homogeneous can lead to misleading interpretations. Some peculiarities of mixture characters are: heritability depends on the mean values of the component distributions, the offspring-parent regression is nonlinear, and genetic or phenotypic correlations cannot be interpreted devoid of the mixture proportions and of the parameters of the distributions mixed.