The beta-binomial model for estimating heritabilities of binary traits

Calculations of individual narrow-sense heritability and family mean heritability of a binary trait in stochastically simulated sib trials in completely randomized block experiments showed that in some situations estimates of realized heritabilities obtained from the mixed linear threshold model could be improved by application of a proposed beta-binomial model. The proposed model adopts the beta-binomial as the conjugate-prior for the distribution of probabilities of observing the binary trait in a genetic entry. Estimation of the beta parameters allows an estimation of selection response and, by linkage to a threshold model for the individual observations, the desired heritabilities can be obtained. The average bias in the betabinomial estimates of heritability and family mean heritability was less than 2%. Improvements over existing procedures were especially manifest at heritabilities above 0.3 and at low overall probabilities of observing the trait (p < 0.30). The lowest root mean square errors were consistently obtained with the algorithm proposed by Harville and Mee (1984). The beta-binomial framework, although restricted to a single random additive genetic effect, further facilitates general analysis, estimation of selection response, and calculation of reliable family mean heritability. Intraclass correlations can be estimated directly from the beta-binomial parameters.     

Confidence intervals for heritability for two-factor mating design single environment linear models

Summary Precision measurement is an essential part of heritability estimate interpretation. Approximate standard errors are commonly used as measures of precision for heritability on a progeny mean basis (H). Their derivation, however, is not inferred from the distribution theory for H. F-distribution based exact confidence intervals have been derived for some onefactor mating design H estimators. Extension of the confidence interval results from one-factor to twofactor mating designs is reported in this paper. Functions of heritability on a full-sib or half-sib progeny mean basis from nested or factorial mating design parameters were distributed according to the F-distribution. Exact confidence intervals were derived for heritability on a full-sib progeny mean basis. Exact confidence intervals for heritability on a half-sib progeny basis were adapted from previous results. Maize (Zea mays L.) data were used to estimate confidence intervals. Complete equations were given for interpolation in F-tables.Oregon Agricultural Experiment Station Technical Paper No. 7659     

Estimation of heritability from varietal trials data

We present the estimation of heritabilities of an observed trait in situations where evaluation of several pure breeding lines is performed in a trial at a single location and in trials from several locations. For the single location situation, we evaluate exact confidence intervals, the probability of invalid estimates, and the percentage points of the distribution of heritability. Simulations were performed to numerically verify the results. Additionally, approximations to the bias and standard error of the estimate were obtained and are presented along with their simulated values and coefficients of skewness and kurtosis. For trials in several locations, explicit expressions for exact values of confidence limits are not available. Further, one would require knowledge of one more parameter, represented by the ratio of genotype x environment (G x E) interaction variance to error variance, in addition to the number of genotypes, replication and true heritability value. Approximations were made for bias and the standard error of estimates of heritability. The evaluation of the distribution of heritability and its moments was recognized as a problem of the linear function of an independent chi-square. The methods have been illustrated by data from experiments on grain and straw yield of 64 barley genotypes evaluated at three locations.     

Selection of sires to reduce sampling variance in the estimates of heritability by half-sib correlation

Summary Standard methods to estimate heritability by half-sib correlation are biased if selection has operated in the parental generation. In this paper a simple method to correct for selection of animals used as sires is described. By selection of both the top and the bottom ranking sires, the sampling variances of the corrected estimates of heritability are substantially reduced. Algebraic expressions to predict the sampling variance of the estimates of heritability using selected sires are derived. Theoretical predictions were checked by Monte-Carlo simulation. The results may have application in the design of experiments to estimate heritabilities.     

Heritability of phenotypic udder traits to improve resilience to mastitis in Texel ewes

There are no estimates of the heritability of phenotypic udder traits in suckler sheep, which produce meat lambs, and whether these are associated with resilience to mastitis. Mastitis is a common disease which damages the mammary gland and reduces productivity. The aims of this study were to investigate the feasibility of collecting udder phenotypes, their heritability and their association with mastitis in suckler ewes. Udder and teat conformation, teat lesions, intramammary masses (IMM) and litter size were recorded from 10 Texel flocks in Great Britain between 2012 and 2014; 968 records were collected. Pedigree data were obtained from an online pedigree recording system. Univariate quantitative genetic parameters were estimated using animal and sire models. Linear mixed models were used to analyse continuous traits and generalised linear mixed models were used to analyse binary traits. Continuous traits had higher heritabilities than binary with teat placement and teat length heritability (h²) highest at 0.35 (SD 0.04) and 0.42 (SD 0.04), respectively. Udder width, drop and separation heritabilities were lower and varied with udder volume. The heritabilities of IMM and teat lesions (sire model) were 0.18 (SD 0.12) and 0.17 (SD 0.11), respectively. All heritabilities were sufficiently high to be in a selection programme to increase resilience to mastitis in the population of Texel sheep. Further studies are required to investigate genetic relationships between traits and to determine whether udder traits predict IMM, and the potential benefits from including traits in a selection programme to increase resilience to chronic mastitis.     

SNP-based heritability estimation using a Bayesian approach

Heritability is a central element in quantitative genetics. New molecular markers to assess genetic variance and heritability are continually under development. The availability of molecular single nucleotide polymorphism (SNP) markers can be applied for estimation of variance components and heritability on population, where relationship information is unknown. In this study, we evaluated the capabilities of two Bayesian genomic models to estimate heritability in simulated populations. The populations comprised different family structures of either no or a limited number of relatives, a single quantitative trait, and with one of two densities of SNP markers. All individuals were both genotyped and phenotyped. Results illustrated that the two models were capable of estimating heritability, when true heritability was 0.15 or higher and populations had a sample size of 400 or higher. For heritabilities of 0.05, all models had difficulties in estimating the true heritability. The two Bayesian models were compared with a restricted maximum likelihood (REML) approach using a genomic relationship matrix. The comparison showed that the Bayesian approaches performed equally well as the REML approach. Differences in family structure were in general not found to influence the estimation of the heritability. For the sample sizes used in this study, a 10-fold increase of SNP density did not improve precision estimates compared with set-ups with a less dense distribution of SNPs. The methods used in this study showed that it was possible to estimate heritabilities on the basis of SNPs in animals with direct measurements. This conclusion is valuable in cases when quantitative traits are either difficult or expensive to measure.     

Maximum likelihood estimation of human craniometric heritabilities

This study presents univariate narrow-sense heritability estimates for 33 common craniometric dimensions, calculated using the maximum likelihood variance components method on a skeletal sample of 298 pedigreed individuals from Hallstatt, Austria. Quantitative genetic studies that use skeletal cranial measurements as a basis for inferring microevolutionary processes in human populations usually employ heritability estimates to represent the genetic variance of the population. The heritabilities used are often problematic: most come from studies of living humans, and/or they were calculated using statistical techniques or assumptions violated by human groups. Most bilateral breadth measures in the current study show low heritability estimates, while cranial length and height measures have heritability values ranging between 0.102-0.729. There appear to be differences between the heritabilities calculated from crania and those from anthropometric studies of living humans, suggesting that the use of the latter in quantitative genetic models of skeletal data may be inappropriate. The univariate skeletal heritability estimates seem to group into distinct regions of the cranium, based on their relative values. The most salient group of measurements is for the midfacial/orbital region, with a number of measures showing heritabilities less than 0.30. Several possible reasons behind this pattern are examined. Given the fact that heritabilities calculated on one population should not be applied to others, suggestions are made for the use of the data presented.     

Simultaneous inference for ratios of linear combinations of general linear model parameters

Consider a general linear model with p -dimensional parameter vector beta and i.i.d. normal errors. Let K(1), ..., K(k ), and L be linearly independent vectors of constants such that L(T)beta not equal 0. We describe exact simultaneous tests for hypotheses that Ki(T)beta/L(T)beta equal specified constants using one-sided and two-sided alternatives, and describe exact simultaneous confidence intervals for these ratios. In the case where the confidence set is a single bounded contiguous set, we describe what we claim are the best possible conservative simultaneous confidence intervals for these ratios - best in that they form the minimum k -dimensional hypercube enclosing the exact simultaneous confidence set. We show that in the case of k = 2, this "box" is defined by the minimum and maximum values for the two ratios in the simultaneous confidence set and that these values are obtained via one of two sources: either from the solutions to each of four systems of equations or at points along the boundary of the simultaneous confidence set where the correlation between two t variables is zero. We then verify that these intervals are narrower than those previously presented in the literature.     

Genome-wide eQTLs and heritability for gene expression traits in unrelated individuals

Background

While the possible sources underlying the so-called ‘missing heritability’ evident in current genome-wide association studies (GWAS) of complex traits have been actively pursued in recent years, resolving this mystery remains a challenging task. Studying heritability of genome-wide gene expression traits can shed light on the goal of understanding the relationship between phenotype and genotype. Here we used microarray gene expression measurements of lymphoblastoid cell lines and genome-wide SNP genotype data from 210 HapMap individuals to examine the heritability of gene expression traits.

Results

Heritability levels for expression of 10,720 genes were estimated by applying variance component model analyses and 1,043 expression quantitative loci (eQTLs) were detected. Our results indicate that gene expression traits display a bimodal distribution of heritability, one peak close to 0% and the other summit approaching 100%. Such a pattern of the within-population variability of gene expression heritability is common among different HapMap populations of unrelated individuals but different from that obtained in the CEU and YRI trio samples. Higher heritability levels are shown by housekeeping genes and genes associated with cis eQTLs. Both cis and trans eQTLs make comparable cumulative contributions to the heritability. Finally, we modelled gene-gene interactions (epistasis) for genes with multiple eQTLs and revealed that epistasis was not prevailing in all genes but made a substantial contribution in explaining total heritability for some genes analysed.

Conclusions

We utilised a mixed effect model analysis for estimating genetic components from population based samples. On basis of analyses of genome-wide gene expression from four HapMap populations, we demonstrated detailed exploitation of the distribution of genetic heritabilities for expression traits from different populations, and highlighted the importance of studying interaction at the gene expression level as an important source of variation underlying missing heritability.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2164-15-13) contains supplementary material, which is available to authorized users.     

WHAT DO WE KNOW ABOUT THE HERITABILITY OF DEVELOPMENTAL INSTABILITY? ANSWERS FROM A BAYESIAN MODEL

In spite of over half a century of research, little is known about the genetic basis of developmental instability (DI). The estimation of the heritability of DI is seriously hampered by the fact that fluctuating asymmetry-FA, that is, the observable outcome of DI-only poorly reflects DI. This results in an underestimation of the heritability of DI. Current methods transforming heritabilities in FA into those of DI fail to take all sources of variation into account and yield incorrect confidence bands that are usually based on unrealistic assumptions. Therefore, a Bayesian latent variable model is developed and explored. Simulations show that with sample sizes currently applied in empirical studies, extremely wide posterior distributions are obtained and data do not allow to distinguish between high (0.5) and low (0.1) heritabilities of DI at all. Even sample sizes of 5000 result in very wide posteriors in many cases. Furthermore, for smaller samples (250 and 1250), up to 70% of the estimates of the heritability of DI were below the mean expected value because of the high skewness of its distribution. Knowing that in only one study, sample sizes were above 5000, there is a need for larger studies to evaluate the evolutionary potential of DI. Designs with relatively low numbers of sires (1-2% of total number of offspring) appear most efficient. Because such high sample sizes are hard to obtain for many study organisms, more insights are required about how data from different traits can be combined in a single analysis. In addition, new designs and methods, such as QTL analyses and micro-array techniques, should be applied to gain a better understanding of the genetic basis of DI.     

The Heritability of Difference Scores when Environments are Correlated

The statistical analysis of difference scores (contrasts) is a fundamental problem in all learning, feeding, and training experiments and tests, and in longitudinal studies of growth and development. Outgoing from the analogy between the mathematical models of classical psychological test theory and quantitative genetics, as well as between the parameters reliability and heritability of these models, the present paper derives the formulas of the heritability of difference scores in general cases where it is not assumed that environmental deviations on distinct tests and measurements are uncorrelated. Contrary to the assertion, made by FELDMAN and LEWONTIN , heritabilities in the broad sense can be used as ideal weighting factors in long-range personnel index selection. Longitudinal studies of twins and the cotwin method are powerful experimental designs to estimate heritabilities of differences.     

Maximum-likelihood variance components analysis of heritabilities of cranial nonmetric traits

Nonmetric traits of the cranium are often used to support hypotheses of the history and divergence of human populations. These studies rely on the assumption that nonmetric traits are heritable, yet few skeletal series exist with associated pedigree information that allow for the calculation of additive genetic variance, or heritability. In addition, traits for which heritabilities have been published represent dichotomous present/absent forms instead of the range of expression that can be observed for many nonmetric characters. In the present study I use a maximum-likelihood variance components analysis to calculate univariate narrow-sense heritability estimates on the skeletal series from Hallstatt, Austria, for 9 sutural bones, 27 multilevel traits, and dichotomized present/absent forms for 19 of these multilevel characters. Most of the trait heritabilities do not differ significantly from a model of h2 = 0, and they have large standard errors. In a heuristic comparison of multilevel versus dichotomous trait forms, most of the nonmetric characters showed no differences in heritability between the two methods used for parsing the phenotypic variation, although where differences were noted, the presence-absence version had higher heritabilities. These results have implications not only for the use of particular nonmetric traits in population studies but also for the practice of character dichotomization in data collection.     

The quantitative genetics of sexually selected traits,preferred traits and preference: a review and analysis of the data

The maintenance of variation in sexually selected traits is a puzzle that has received increasing attention in the past several decades. Traits that are related to fitness, such as life‐history or sexually selected traits, are expected to have low additive genetic variance (and hence, heritability) due to the rapid fixation of advantageous alleles. However, previous analyses have suggested that the heritabilities of sexually selected traits are on average higher than nonsexually selected traits. We show that the heritabilities of sexually selected traits are not significantly different from those of nonsexually selected traits overall or when separated into the three trait categories: behavioural, morphological and physiological. In contrast with previous findings, the heritability of preference is quite low (h² = 0.25 ± 0.06) and is in the same range as life‐history traits. We distinguish preferred traits as a category of sexually selected traits and find that the heritability of the former is not significantly different than sexually selected traits overall (0.48 ± 0.04 vs. 0.46 ± 0.03). We test the hypothesis that the heritability of sexually selected traits is negatively correlated with the strength of sexual selection. As predicted, there is a significant negative correlation between the heritabilities of sexually selected traits and the strength of selection. This suggests that heritabilities do indeed decrease as sexual selection increases but sexual selection is not strong enough to cause heritabilities of sexually selected traits to deviate from the same type of nonsexually selected traits.     

Comparison of REML and Gibbs sampling estimates of multi-trait genetic parameters in Scots pine

Multi-trait (co)variance estimation is an important topic in plant and animal breeding. In this study we compare estimates obtained with restricted maximum likelihood (REML) and Bayesian Gibbs sampling of simulated data and of three traits (diameter, height and branch angle) from a 26-year-old partial diallel progeny test of Scots pine (Pinus sylvestris L.). Based on the results from the simulated data we can conclude that the REML estimates are accurate but the mode of posterior distributions from the Gibbs sampling can be overestimated depending on the level of the heritability. The mean and median of the posteriors were considerably higher than the expected values of the heritabilities. The confidence intervals calculated with the delta method were biased downwardly. The highest probablity density (HPD) interval provides a better interval estimate, but could be slightly biased at the lower level. Similar differences between REML and Gibbs sampling estimates were found for the Scots pine data. We conclude that further simulation studies are needed in order to evaluate the effect of different priors on (co)variance components in the genetic individual model.     

Simultaneous confidence intervals for a success probability and intraclass correlation,with an application to screening mammography

This paper provides asymptotic simultaneous confidence intervals for a success probability and intraclass correlation of the beta‐binomial model, based on the maximum likelihood estimator approach. The coverage probabilities of those intervals are evaluated. An application to screening mammography is presented as an example. The individual and simultaneous confidence intervals for sensitivity and specificity and the corresponding intraclass correlations are investigated. Two additional examples using influenza data and sex ratio data among sibships are also considered, where the individual and simultaneous confidence intervals are provided.     

Reliability of Confidence Interval Estimators under Various Nested Design Parental Sample Sizes

The coverage probabilities of several confidence limit estimators of genetic parameters, obtained from North Carolina I designs, were assessed by means of Monte Carlo simulations. The reliability of the estimators was compared under three different parental sample sizes. The coverage of confidence intervals set on the Normal distribution, and using standard errors either computed by the “delta” method or derived using an approximation for the variance of a variance component estimated by means of a linear combination of mean squares, was affected by the number of males and females included in the experiment. The “delta” method was found to provide reliable standard errors of the genetic parameters only when at least 48 males were each mated to six different females randomly selected from the reference population. Formulae are provided for obtaining “delta” method standard errors, and appropriate statistical software procedures are discussed. The error rates of confidence limits based on the Normal distribution and using standard errors obtained by an approximation for the variance of a variance component varied widely. The coverage of F-distribution confidence intervals for heritability estimates was not significantly affected by parental sample size and consistently provided a mean coverage near the stated coverage. For small parental sample sizes, confidence intervals for heritability estimates should be based on the F-distribution.     

Measuring and comparing the accuracy of species distribution models with presence–absence data

Species distribution models have been widely used to predict species distributions for various purposes, including conservation planning, and climate change impact assessment. The success of these applications relies heavily on the accuracy of the models. Various measures have been proposed to assess the accuracy of the models. Rigorous statistical analysis should be incorporated in model accuracy assessment. However, since relevant information about the statistical properties of accuracy measures is scattered across various disciplines, ecologists find it difficult to select the most appropriate ones for their research. In this paper, we review accuracy measures that are currently used in species distribution modelling (SDM), and introduce additional metrics that have potential applications in SDM. For the commonly used measures (which are also intensively studied by statisticians), including overall accuracy, sensitivity, specificity, kappa, and area and partial area under the ROC curves, promising methods to construct confidence intervals and statistically compare the accuracy between two models are given. For other accuracy measures, methods to estimate standard errors are given, which can be used to construct approximate confidence intervals. We also suggest that as general tools, computer‐intensive methods, especially bootstrap and randomization methods can be used in constructing confidence intervals and statistical tests if suitable analytic methods cannot be found. Usually, these computer‐intensive methods provide robust results.     

Statistical methods in regression and calibration analysis of chromosome aberration data

The method of iteratively reweighted least squares for the regression analysis of Poisson distributed chromosome aberration data is reviewed in the context of other fit procedures used in the cytogenetic literature. As an application of the resulting regression curves methods for calculating confidence intervals on dose from aberration yield are described and compared, and, for the linear quadratic model lambda = beta 0 + beta 1 chi + beta 2 chi 2 a confidence interval for the ratio beta 1/ beta 2 is given. Emphasis is placed on the rationale, interpretation and the limitations of various methods from a statistical point of view.     

The lipoprotein subfraction profile: heritability and identification of quantitative trait loci

The HDL and LDL subclass profile is an emerging cardiovascular risk factor. Yet, the biological and genetic mechanisms controlling the lipoprotein subclass distribution are unclear. Therefore, we aimed 1) to determine the heritability of the entire spectrum of LDL and HDL subclass features and 2) to identify gene loci influencing the lipoprotein subfraction pattern. Using NMR spectroscopy, we analyzed the lipoprotein subclass distribution in 1,275 coronary artery disease patients derived from the Regensburg Myocardial Infarction Family Study. We calculated heritabilities, performed a microsatellite genome scan, and calculated linkage. HDL and LDL subclass profiles showed heritabilities ranging from 23% to 67% (all P < 10(-3)) of traits using univariate calculation. After multivariate adjustment, we found heritabilities of 27-48% (all P < 0.05) for HDL and 21-44% for LDL traits. The linkage analysis revealed a significant logarithm of the odds (LOD) score (3.3) for HDL particle concentration on chromosome 18 and a highly suggestive signal for HDL particle size on chromosome 12 (2.9). After multivariate adjustment, we found a significant maximum LOD score of 3.7 for HDL size. Our study is the first to analyze heritability and linkage for the entire spectrum of LDL and HDL subclass features. Our findings may lead to the identification of genes controlling the lipoprotein subclass distribution.     

Group differences in the heritability of items and test scores

It is important to understand potential sources of group differences in the heritability of intelligence test scores. On the basis of a basic item response model we argue that heritabilities which are based on dichotomous item scores normally do not generalize from one sample to the next. If groups differ in mean ability, the functioning of items at different ability levels may result in group differences in the heritability of items, even when these items function equivalently across groups and the heritability of the underlying ability is equal across groups. We illustrate this graphically, by computer simulation, and by focusing on several problems associated with a recent study by Rushton et al. who argued that the heritability estimates of items of Raven''s Progressive Matrices test in North-American twin samples generalized to other population groups, and hence that the population group differences on this test of general mental ability (or intelligence) had a substantial genetic component. Our results show that item heritabilities are strongly dependent on the group on which the heritabilities were based. Rushton et al.''s results were artefactual and do not speak to the nature of population group differences in intelligence test performance.