Prediction of Complex Traits: Robust Alternatives to Best Linear Unbiased Prediction

Local and Bayesian Survival FDR Estimations to Identify Reliable Associations in Whole Genome of Bread Wheat

Estimating the FDR significance threshold in genome-wide association studies remains a major challenge in distinguishing true positive hypotheses from false positive and negative errors. Several comparative methods for multiple testing comparison have been developed to determine the significance threshold; however, these methods may be overly conservative and lead to an increase in false negative results. The local FDR approach is suitable for testing many associations simultaneously based on the empirical Bayes perspective. In the local FDR, the maximum likelihood estimator is sensitive to bias when the GWAS model contains two or more explanatory variables as genetic parameters simultaneously. The main criticism of local FDR is that it focuses only locally on the effects of single nucleotide polymorphism (SNP) in tails of distribution, whereas the signal associations are distributed across the whole genome. The advantage of the Bayesian perspective is that knowledge of prior distribution comes from other genetic parameters included in the GWAS model, such as linkage disequilibrium (LD) analysis, minor allele frequency (MAF) and call rate of significant associations. We also proposed Bayesian survival FDR to solve the multi-collinearity and large-scale problems, respectively, in grain yield (GY) vector in bread wheat with large-scale SNP information. The objective of this study was to obtain a short list of SNPs that are reliably associated with GY under low and high levels of nitrogen (N) in the population. The five top significant SNPs were compared with different Bayesian models. Based on the time to events in the Bayesian survival analysis, the differentiation between minor and major alleles within the association panel can be identified.

Inference about quantitative traits under selection: a Bayesian revisitation for the post-genomic era

BACKGROUND

Selection schemes distort inference when estimating differences between treatments or genetic associations between traits, and may degrade prediction of outcomes, e.g., the expected performance of the progeny of an individual with a certain genotype. If input and output measurements are not collected on random samples, inferences and predictions must be biased to some degree. Our paper revisits inference in quantitative genetics when using samples stemming from some selection process. The approach used integrates the classical notion of fitness with that of missing data. Treatment is fully Bayesian, with inference and prediction dealt with, in an unified manner. While focus is on animal and plant breeding, concepts apply to natural selection as well. Examples based on real data and stylized models illustrate how selection can be accounted for in four different situations, and sometimes without success.

RESULTS

Our flexible "soft selection" setting helps to diagnose the extent to which selection can be ignored. The clear connection between probability of missingness and the concept of fitness in stylized selection scenarios is highlighted. It is not realistic to assume that a fixed selection threshold t holds in conceptual replication, as the chance of selection depends on observed and unobserved data, and on unequal amounts of information over individuals, aspects that a "soft" selection representation addresses explicitly. There does not seem to be a general prescription to accommodate potential distortions due to selection. In structures that combine cross-sectional, longitudinal and multi-trait data such as in animal breeding, balance is the exception rather than the rule. The Bayesian approach provides an integrated answer to inference, prediction and model choice under selection that goes beyond the likelihood-based approach, where breeding values are inferred indirectly.

CONCLUSIONS

The approach used here for inference and prediction under selection may or may not yield the best possible answers. One may believe that selection has been accounted for diligently, but the central problem of whether statistical inferences are good or bad does not have an unambiguous solution. On the other hand, the quality of predictions can be gauged empirically via appropriate training-testing of competing methods.

Heterosis and Hybrid Crop Breeding: A Multidisciplinary Review

Although hybrid crop varieties are among the most popular agricultural innovations, the rationale for hybrid crop breeding is sometimes misunderstood. Hybrid breeding is slower and more resource-intensive than inbred breeding, but it allows systematic improvement of a population by recurrent selection and exploitation of heterosis simultaneously. Inbred parental lines can identically reproduce both themselves and their F1 progeny indefinitely, whereas outbred lines cannot, so uniform outbred lines must be bred indirectly through their inbred parents to harness heterosis. Heterosis is an expected consequence of whole-genome non-additive effects at the population level over evolutionary time. Understanding heterosis from the perspective of molecular genetic mechanisms alone may be elusive, because heterosis is likely an emergent property of populations. Hybrid breeding is a process of recurrent population improvement to maximize hybrid performance. Hybrid breeding is not maximization of heterosis per se, nor testing random combinations of individuals to find an exceptional hybrid, nor using heterosis in place of population improvement. Though there are methods to harness heterosis other than hybrid breeding, such as use of open-pollinated varieties or clonal propagation, they are not currently suitable for all crops or production environments. The use of genomic selection can decrease cycle time and costs in hybrid breeding, particularly by rapidly establishing heterotic pools, reducing testcrossing, and limiting the loss of genetic variance. Open questions in optimal use of genomic selection in hybrid crop breeding programs remain, such as how to choose founders of heterotic pools, the importance of dominance effects in genomic prediction, the necessary frequency of updating the training set with phenotypic information, and how to maintain genetic variance and prevent fixation of deleterious alleles.

A Bayesian Framework for Robust Quantitative Trait Locus Mapping and Outlier Detection

We introduce a Bayesian framework for simultaneous feature selection and outlier detection in sparse high-dimensional regression models, with a focus on quantitative trait locus (QTL) mapping in experimental crosses. More specifically, we incorporate the robust mean shift outlier handling mechanism into the multiple QTL mapping regression model and apply LASSO regularization concurrently to the genetic effects and the mean-shift terms through the flexible extended Bayesian LASSO (EBL) prior structure, thereby combining QTL mapping and outlier detection into a single sparse model representation problem. The EBL priors on the mean-shift terms prevent outlying phenotypic values from distorting the genotype-phenotype association and allow their detection as cases with outstanding mean shift values following the LASSO shrinkage. Simulation results demonstrate the effectiveness of our new methodology at mapping QTLs in the presence of outlying phenotypic values and simultaneously identifying the potential outliers, while maintaining a comparable performance to the standard EBL on outlier-free data.

A Multiple-Trait Bayesian Lasso for Genome-Enabled Analysis and Prediction of Complex Traits

A multiple-trait Bayesian LASSO (MBL) for genome-based analysis and prediction of quantitative traits is presented and applied to two real data sets. The data-generating model is a multivariate linear Bayesian regression on possibly a huge number of molecular markers, and with a Gaussian residual distribution posed. Each (one per marker) of the [Formula: see text] vectors of regression coefficients (T: number of traits) is assigned the same T-variate Laplace prior distribution, with a null mean vector and unknown scale matrix Σ. The multivariate prior reduces to that of the standard univariate Bayesian LASSO when [Formula: see text] The covariance matrix of the residual distribution is assigned a multivariate Jeffreys prior, and Σ is given an inverse-Wishart prior. The unknown quantities in the model are learned using a Markov chain Monte Carlo sampling scheme constructed using a scale-mixture of normal distributions representation. MBL is demonstrated in a bivariate context employing two publicly available data sets using a bivariate genomic best linear unbiased prediction model (GBLUP) for benchmarking results. The first data set is one where wheat grain yields in two different environments are treated as distinct traits. The second data set comes from genotyped Pinus trees, with each individual measured for two traits: rust bin and gall volume. In MBL, the bivariate marker effects are shrunk differentially, i.e., "short" vectors are more strongly shrunk toward the origin than in GBLUP; conversely, "long" vectors are shrunk less. A predictive comparison was carried out as well in wheat, where the comparators of MBL were bivariate GBLUP and bivariate Bayes Cπ-a variable selection procedure. A training-testing layout was used, with 100 random reconstructions of training and testing sets. For the wheat data, all methods produced similar predictions. In Pinus, MBL gave better predictions that either a Bayesian bivariate GBLUP or the single trait Bayesian LASSO. MBL has been implemented in the Julia language package JWAS, and is now available for the scientific community to explore with different traits, species, and environments. It is well known that there is no universally best prediction machine, and MBL represents a new resource in the armamentarium for genome-enabled analysis and prediction of complex traits.

Bayesian, Likelihood-Free Modelling of Phenotypic Plasticity and Variability in Individuals and Populations

There is a paradigm shift from the traditional focus on the "average" individual towards the definition and analysis of trait variation within individual life-history and among individuals in populations. This is a result of increasing availability of individual phenotypic data. The shift allows the use of genetic and environment-driven variations to assess robustness to challenge, gain greater understanding of organismal biological processes, or deliver individual-targeted treatments or genetic selection. These consequences apply, in particular, to variation in ontogenetic growth. We propose an approach to parameterise mathematical models of individual traits (e.g., reaction norms, growth curves) that address two challenges: 1) Estimation of individual traits while making minimal assumptions about data distribution and correlation, addressed via Approximate Bayesian Computation (a form of nonparametric inference). We are motivated by the fact that available information on distribution of biological data is often less precise than assumed by conventional likelihood functions. 2) Scaling-up to population phenotype distributions while facilitating unbiased use of individual data; this is addressed via a probabilistic framework where population distributions build on separately-inferred individual distributions and individual-trait interpretability is preserved. The approach is tested against Bayesian likelihood-based inference, by fitting weight and energy intake growth models to animal data and normal- and skewed-distributed simulated data. i) Individual inferences were accurate and robust to changes in data distribution and sample size; in particular, median-based predictions were more robust than maximum- likelihood-based curves. These results suggest that the approach gives reliable inferences using few observations and monitoring resources. ii) At the population level, each individual contributed via a specific data distribution, and population phenotype estimates were not disproportionally influenced by outlier individuals. Indices measuring population phenotype variation can be derived for study comparisons. The approach offers an alternative for estimating trait variability in biological systems that may be reliable for various applications, for example, in genetics, health, and individualised nutrition, while using fewer assumptions and fewer empirical observations. In livestock breeding, the potentially greater accuracy of trait estimation (without specification of multitrait variance-covariance parameters) could lead to improved selection and to more decisive estimates of trait heritability.

A robust Bayesian genome-based median regression model

Current genome-enabled prediction models assumed errors normally distributed, which are sensitive to outliers. We propose a model with errors assumed to follow a Laplace distribution to deal better with outliers. Current genome-enabled prediction models use regressions that fit the expected value (mean) of a response variable with errors assumed normally distributed, which are often sensitive to outliers, either genetic or environmental. For this reason, we propose a robust Bayesian genome median regression (BGMR) model that fits regressions to the medians of a distribution, with errors assumed to follow a Laplace distribution to deal better with outliers. The BGMR model was evaluated under a Bayesian framework with Markov Chain Monte Carlo sampling using a location-scale mixture representation of the Laplace distribution. The BGMR was implemented with two simulated and two real genomic data sets, and we compared its prediction performance with that of a conventional genomic best linear unbiased prediction (GBLUP) model and the Laplace maximum a posteriori (LMAP) method. The prediction accuracies of BGMR were higher than those of the GBLUP and LMAP methods when there were outliers. The BGMR model could be useful to breeders who need to predict and select genotypes based on data with unknown outliers.

Phenotypic Selection in Ornamental Breeding: It's Better to Have the BLUPs Than to Have the BLUEs

Plant breeders always face the challenge to select the best individuals. Selection methods are required that maximize selection gain based on available data. When several crosses have been made, the BLUP procedure achieves this by combining phenotypic data with information on pedigree relationships via an index, known as family-index selection. The index, estimated based on the intra-class correlation coefficient, exploits the relationship among individuals within a family relative to other families in the population. An intra-class correlation coefficient of one indicates that the individual performance can be fully explained based on the family background, whereas an intra-class correlation coefficient of zero indicates the performance of individuals is independent of the family background. In the case the intra-class correlation coefficient is one, family-index selection is considered. In the case the intra-class correlation coefficient is zero, individual selection is considered. The main difference between individual and family-index selection lies in the adjustment in estimating the individual's effect depending on the intra-class correlation coefficient afforded by the latter. Two examples serve to illustrate the application of the BLUP method. The efficiency of individual and family-index selection was evaluated in terms of the heritability obtained from linear mixed models implementing the selection methods by suitably defining the treatment factor as the sum of individual and family effect. Family-index selection was found to be at least as efficient as individual selection in Dianthus caryophyllus L., except for flower size in standard carnation and vase life in mini carnation for which traits family-index selection outperformed individual selection. Family-index selection was superior to individual selection in Pelargonium zonale in cases when the heritability was low. Hence, the pedigree-based BLUP procedure can enhance selection efficiency in production-related traits in P. zonale or shelf-life related in D. caryophyllus L.