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.     

Diallel analysis for sex-linked and maternal effects

Genetic models including sex-linked and maternal effects as well as autosomal gene effects are described. Monte Carlo simulations were conducted to compare efficiencies of estimation by minimum norm quadratic unbiased estimation (MINQUE) and restricted maximum likelihood (REML) methods. MINQUE(1), which has 1 for all prior values, has a similar efficiency to MINQUE(), which requires prior estimates of parameter values. MINQUE(1) has the advantage over REML of unbiased estimation and convenient computation. An adjusted unbiased prediction (AUP) method is developed for predicting random genetic effects. AUP is desirable for its easy computation and unbiasedness of both mean and variance of predictors. The jackknife procedure is appropriate for estimating the sampling variances of estimated variances (or covariances) and of predicted genetic effects. A t-test based on jackknife variances is applicable for detecting significance of variation. Worked examples from mice and silkworm data are given in order to demonstrate variance and covariance estimation and genetic effect prediction.     

Estimating sampling error of evolutionary statistics based on genetic covariance matrices using maximum likelihood

We explore the estimation of uncertainty in evolutionary parameters using a recently devised approach for resampling entire additive genetic variance–covariance matrices ( G ). Large‐sample theory shows that maximum‐likelihood estimates (including restricted maximum likelihood, REML) asymptotically have a multivariate normal distribution, with covariance matrix derived from the inverse of the information matrix, and mean equal to the estimated G . This suggests that sampling estimates of G from this distribution can be used to assess the variability of estimates of G , and of functions of G . We refer to this as the REML‐MVN method. This has been implemented in the mixed‐model program WOMBAT. Estimates of sampling variances from REML‐MVN were compared to those from the parametric bootstrap and from a Bayesian Markov chain Monte Carlo (MCMC) approach (implemented in the R package MCMCglmm). We apply each approach to evolvability statistics previously estimated for a large, 20‐dimensional data set for Drosophila wings. REML‐MVN and MCMC sampling variances are close to those estimated with the parametric bootstrap. Both slightly underestimate the error in the best‐estimated aspects of the G matrix. REML analysis supports the previous conclusion that the G matrix for this population is full rank. REML‐MVN is computationally very efficient, making it an attractive alternative to both data resampling and MCMC approaches to assessing confidence in parameters of evolutionary interest.     

Evolution of floral display in Eichhornia paniculata (Pontederiaceae): direct and correlated responses to selection on flower size and number

Trade-offs between flower size and number seem likely to influence the evolution of floral display and are an important assumption of several theoretical models. We assessed floral trade-offs by imposing two generations of selection on flower size and number in a greenhouse population of bee-pollinated Eichhornia paniculata. We established a control line and two replicate selection lines of 100 plants each for large flowers (S+), small flowers (S-), and many flowers per inflorescence (N+). We compared realized heritabilities and genetic correlations with estimates based on restricted-maximum-likelihood (REML) analysis of pedigrees. Responses to selection confirmed REML heritability estimates (flower size, h2 = 0.48; daily flower number, h2 = 0.10; total flower number, h2 = 0.23). Differences in nectar, pollen, and ovule production between S+ and S- lines supported an overall divergence in investment per flower. Both realized and REML estimates of the genetic correlation between daily and total flower number were r = 1.0. However, correlated responses to selection were inconsistent in their support of a trade-off. In both S- lines, correlated increases in flower number indicated a genetic correlation of r = -0.6 between flower size and number. In contrast, correlated responses in N+ and S+ lines were not significant, although flower size decreased in one N+ line. In addition, REML estimates of genetic correlations between flower size and number were positive, and did not differ from zero when variation in leaf area and age at first flowering were taken into account. These results likely reflect the combined effects of variation in genes controlling the resources available for flowering and genes with opposing effects on flower size and number. Our results suggest that the short-term evolution of floral display is not necessarily constrained by trade-offs between flower size and number, as is often assumed.     

Longitudinal analyses of correlated response efficiencies of fillet traits in Nile tilapia

Recent studies with Nile tilapia have shown divergent results regarding the possibility of selecting on morphometric measurements to promote indirect genetic gains in fillet yield (FY). The use of indirect selection for fillet traits is important as these traits are only measurable after harvesting. Random regression models are a powerful tool in association studies to identify the best time point to measure and select animals. Random regression models can also be applied in a multiple trait approach to analyze indirect response to selection, which would avoid the need to sacrifice candidate fish. Therefore, the aim of this study was to investigate the genetic relationships between several body measurements, weight and fillet traits throughout the growth period and to evaluate the possibility of indirect selection for fillet traits in Nile tilapia. Data were collected from 2042 fish and was divided into two subsets. The first subset was used to estimate genetic parameters, including the permanent environmental effect for BW and body measurements (8758 records for each body measurement, as each fish was individually weighed and measured a maximum of six times). The second subset (2042 records for each trait) was used to estimate genetic correlations and heritabilities, which enabled the calculation of correlated response efficiencies between body measurements and the fillet traits. Heritability estimates across ages ranged from 0.05 to 0.5 for height, 0.02 to 0.48 for corrected length (CL), 0.05 to 0.68 for width, 0.08 to 0.57 for fillet weight (FW) and 0.12 to 0.42 for FY. All genetic correlation estimates between body measurements and FW were positive and strong (0.64 to 0.98). The estimates of genetic correlation between body measurements and FY were positive (except for CL at some ages), but weak to moderate (−0.08 to 0.68). These estimates resulted in strong and favorable correlated response efficiencies for FW and positive, but moderate for FY. These results indicate the possibility of achieving indirect genetic gains for FW and by selecting for morphometric traits, but low efficiency for FY when compared with direct selection.     

Genetic parameters for litter size in sheep: natural versus hormone-induced oestrus

The litter size in Suffolk and Texel-sheep was analysed using REML and Bayesian methods. Litters born after hormonal induced oestrus and after natural oestrus were treated as different traits in order to estimate the genetic correlation between the traits. Explanatory variables were the age of the ewe at lambing, period of lambing, a year*flock-effect, a permanent environmental effect associated with the ewe, and the additive genetic effect. The heritability estimates for litter size ranged from 0.06 to 0.13 using REML in bi-variate linear models. Transformation of the estimates to the underlying scale resulted in heritability estimates from 0.12 to 0.17. Posterior means of the heritability of litter size in the Bayesian approach with bi-variate threshold models varied from 0.05 to 0.18. REML estimates of the genetic correlations between the two types of litter size ranged from 0.57 to 0.64 in the Suffolk and from 0.75 to 0.81 in the Texel. The posterior means of the genetic correlation (Bayesian analysis) were 0.40 and 0.44 for the Suffolk and 0.56 and 0.75 for the Texel in the sire and animal model respectively. A bivariate threshold model seems appropriate for the genetic evaluation of prolificacy in the breeds concerned.     

Multiple-trait BLUP: a suitable strategy for genetic selection of <Emphasis Type="Italic">Eucalyptus</Emphasis>

Usually, genetic selection is carried out based on several traits, which can be genetically correlated. In this case, selection bias may occur if these traits are analyzed individually. Thus, the present work aimed to evaluate the applicability and efficiency of multiple-trait best linear unbiased prediction (BLUP) in the genetic selection of Eucalyptus. The data used in this work refer to the evaluation of a partial diallel of Eucalyptus spp. in relation to height, diameter at breast height (DBH), and volume. Variance components and genetic and non-genetic parameters were estimated via residual maximum likelihood (REML). Multiple-trait BLUP led to estimates of mean additive genetic variance higher than the estimates obtained via single-trait BLUP and, consequently, led to higher estimates of narrow-sense individual interpopulational heritabilities and mean accuracies. Partial genetic correlations obtained via multiple-trait BLUP allowed a real understanding of the association between traits, differently from those obtained via single-trait BLUP. Multiple-trait BLUP led to higher gains predicted with the selection for height, DBH, and volume and can be efficiently applied in the genetic selection of Eucalyptus.     

Bayesian adaptive Markov chain Monte Carlo estimation of genetic parameters

Accurate and fast estimation of genetic parameters that underlie quantitative traits using mixed linear models with additive and dominance effects is of great importance in both natural and breeding populations. Here, we propose a new fast adaptive Markov chain Monte Carlo (MCMC) sampling algorithm for the estimation of genetic parameters in the linear mixed model with several random effects. In the learning phase of our algorithm, we use the hybrid Gibbs sampler to learn the covariance structure of the variance components. In the second phase of the algorithm, we use this covariance structure to formulate an effective proposal distribution for a Metropolis-Hastings algorithm, which uses a likelihood function in which the random effects have been integrated out. Compared with the hybrid Gibbs sampler, the new algorithm had better mixing properties and was approximately twice as fast to run. Our new algorithm was able to detect different modes in the posterior distribution. In addition, the posterior mode estimates from the adaptive MCMC method were close to the REML (residual maximum likelihood) estimates. Moreover, our exponential prior for inverse variance components was vague and enabled the estimated mode of the posterior variance to be practically zero, which was in agreement with the support from the likelihood (in the case of no dominance). The method performance is illustrated using simulated data sets with replicates and field data in barley.     

Recording strategies and selection potential of feed intake measured using the X-ray method in rainbow trout

This study examines the way long-term feed intake should be recorded accurately for selective breeding purposes, and estimates selection potential in feed intake using the X-ray method to record individual daily feed intake in rainbow trout (Oncorhynchus mykiss). The analysis showed that the point estimates of daily feed intake displayed low repeatabilities (r = 0.09–0.32). This indicates that a minimum of three repeated records were needed to accurately record average feed intake at a fixed age. To effectively breed for feed intake over the whole growing period, it is necessary to determine average feed intake at different ages, since there were only moderate phenotypic and genetic correlations between average daily feed intake recorded at 140 g, 750 g and 2000 g wet mass. Heritability for average daily feed intake was low (average h² = 0.10), indicating that modest genetic changes can be obtained in response to selection. It was concluded that selection to genetically change long-term feed intake can be successful, yet repeated observations at several life stages are needed to ensure the accuracy of feed intake estimates and the efficiency of selection.     

Census data aggregation decisions can affect population‐level inference in heterogeneous populations

1. Conservation and population management decisions often rely on population models parameterized using census data. However, the sampling regime, precision, sample size, and methods used to collect census data are usually heterogeneous in time and space. Decisions about how to derive population‐wide estimates from this patchwork of data are complicated and may bias estimated population dynamics, with important implications for subsequent management decisions.
2. Here, we explore the impact of site selection and data aggregation decisions on pup survival estimates, and downstream estimates derived from parameterized matrix population models (MPMs), using a long‐term dataset on grey seal (Halichoerus grypus) pup survival from southwestern Wales. The spatiotemporal and methodological heterogeneity of the data are fairly typical for ecological census data and it is, therefore, a good model to address this topic.
3. Data were collected from 46 sampling locations (sites) over 25 years, and we explore the impact of data handling decisions by varying how years and sampling locations are combined to parameterize pup survival in population‐level MPMs. We focus on pup survival because abundant high‐quality data are available on this developmental stage.
4. We found that survival probability was highly variable with most variation being at the site level, and poorly correlated among sampling sites. This variation could generate marked differences in predicted population dynamics depending on sampling strategy. The sample size required for a confident survival estimate also varied markedly geographically.
5. We conclude that for populations with highly variable vital rates among sub‐populations, site selection and data aggregation methods are important. In particular, including peripheral or less frequently used areas can introduce substantial variation into population estimates. This is likely to be context‐dependent, but these choices, including the use of appropriate weights when summarizing across sampling areas, should be explored to ensure that management actions are successful.

     

Multilevel selection 3: modeling the effects of interacting individuals as a function of group size

Bijma et al. (2007a,b) presented a quantitative genetic theory of multilevel selection and showed how to estimate the relevant parameters using standard restricted maximum-likelihood (REML) methodology. Extending their results we develop a wider class of models that provide a more realistic framework for capturing the effects of interacting individuals. These models also make use of standard REML techniques and include the original model as a special case.     

Estimating major gene effects with partial information using Gibbs sampling

A method for estimating major gene effects using Gibbs sampling to infer genotype of individuals with unknown values, was compared with a standard mixed-model analysis. The purpose of this study was to evaluate the effect of including information of individuals with unknown genotypes on the estimates and their error variances (Ve) of the single-gene effects. When genotypes were known for all the individuals, results using the Gibbs method (GS) were similar to those obtained with the mixed model (MM). In the absence of selection, when information from individuals with unknown genotypes was included, GS yielded unbiased estimates of the major gene effects while reducing the Ve associated with them. This reduction in Ve depended on the gene frequency and mode of action of the major locus. For the additive effect, the reduction in Ve ranged from 29 to 69% of the total reduction which would have been obtained if all individuals had had a known genotype. Similarly the reduction in Ve found for the dominance effect ranged from 12 to 58%. Estimates using GS generally had small detectable biases when the polygenic heritability used in the analysis was inflated or estimated simultaneously. However, the benefit of using information from individuals with unknown genotypes was still maintained when comparing the mean square error of the estimates using either GS or MM when genotypes are only known for a subset of the population. When the population has been under selection, the use of Gibbs sampling to incorporate information of individuals without genotypes reduced substantially the bias and mean square error found for MM analysis on partial data. Nevertheless, there was some bias detected using Gibbs sampling. The gene frequency of the major gene in the base population was also well estimated despite its change over generations due to selection.     

Analysis of response to 20 generations of selection for body composition in mice: fit to infinitesimal model assumptions

Data were analysed from a divergent selection experiment for an indicator of body composition in the mouse, the ratio of gonadal fat pad to body weight (GFPR). Lines were selected for 20 generations for fat (F), lean (L) or were unselected (C), with three replicates of each. Selection was within full-sib families, 16 families per replicate for the first seven generations, eight subsequently. At generation 20, GFPR in the F lines was twice and in the L lines half that of C. A log transformation removed both asymmetry of response and heterogeneity of variance among lines, and so was used throughout. Estimates of genetic variance and heritability (approximately 50%) obtained using REML with an animal model were very similar, whether estimated from the first few generations of selection, or from all 20 generations, or from late generations having fitted pedigree. The estimates were also similar when estimated from selected or control lines. Estimates from REML also agreed with estimates of realised heritability. The results all accord with expectations under the infinitesimal model, despite the four-fold changes in mean. Relaxed selection lines, derived from generation 20, showed little regression in fatness after 40 generations without selection.     

Employing a Monte Carlo Algorithm in Newton-Type Methods for Restricted Maximum Likelihood Estimation of Genetic Parameters

Estimation of variance components by Monte Carlo (MC) expectation maximization (EM) restricted maximum likelihood (REML) is computationally efficient for large data sets and complex linear mixed effects models. However, efficiency may be lost due to the need for a large number of iterations of the EM algorithm. To decrease the computing time we explored the use of faster converging Newton-type algorithms within MC REML implementations. The implemented algorithms were: MC Newton-Raphson (NR), where the information matrix was generated via sampling; MC average information(AI), where the information was computed as an average of observed and expected information; and MC Broyden''s method, where the zero of the gradient was searched using a quasi-Newton-type algorithm. Performance of these algorithms was evaluated using simulated data. The final estimates were in good agreement with corresponding analytical ones. MC NR REML and MC AI REML enhanced convergence compared to MC EM REML and gave standard errors for the estimates as a by-product. MC NR REML required a larger number of MC samples, while each MC AI REML iteration demanded extra solving of mixed model equations by the number of parameters to be estimated. MC Broyden''s method required the largest number of MC samples with our small data and did not give standard errors for the parameters directly. We studied the performance of three different convergence criteria for the MC AI REML algorithm. Our results indicate the importance of defining a suitable convergence criterion and critical value in order to obtain an efficient Newton-type method utilizing a MC algorithm. Overall, use of a MC algorithm with Newton-type methods proved feasible and the results encourage testing of these methods with different kinds of large-scale problem settings.     

Efficiency of variable-intensity and sequential sampling for insect control decisions in cole crops in the Netherlands

A total of 24 commercial fields of cabbages and Brussels sprouts were sampled in a grid fashion with 20–25 equally spaced cells with four plants per cell. Using this data base of 80–100 plants, we conducted computer stimulations to compare the treatment decisions that would be made for the major insect pests using published sequential sampling programs and a newly developed variable-intensity sampling program. Additionally, we compared the number of samples required to make the decision. At low thresholds (10–20%) for both Lepidoptera and cabbage aphids, variable intensity-sampling required a smaller sample size and provided more reliable decisions, while at high thresholds (40–50%) sequential sampling provided more reliable decisions. In both procedures, the occurrence of incorrect decisions was minimal. The number of cases in which a decision would not be reached after a 40-plant sample was lower for variable-intensity sampling. Considering the number of samples required to make a correct decision and the greater need for reliable decisions at lower thresholds, variable-intensity sampling was superior to sequential sampling. Additionally, variable-intensity sampling has the advantage of requiring samples to be taken in a greater area of the field and thus increases the probability of detecting localized infestations. Although variable-intensity sampling was not designed to classify pest populations for treatment decisions but rather to achieve sampling precision around the population mean, our present studies indicate that it can also be an effective method to aid in treatment decisions.     

An under-appreciated difficulty: sampling of plant populations for analysis using molecular markers

Results of studies using molecular markers for determining demographic and genetical population parameters especially in plants or sessile animals under field conditions are strongly dependent on the sampling strategy adopted. There are two critical decisions to make when determining this strategy: (i) what is the unit to be sampled?, (ii) how should units to be sampled in the field be selected? For the first decision, there are two conceptually different approaches: sampling ramets of clonal plants as units (to get information about within-genet parameters, such as genet sizes or numbers) and sampling genets of clonal or non-clonal plants as units (to get information of the genetic structure of the population). For the second decision, it is critically important to make the goal of the study explicit. We argue that in this case fully random sampling is needed only when an estimate of the true value of the population parameter is needed; if a comparison between populations is the goal, however, other sampling schemes may be adopted. The efficiency of different types of sampling strategies to recover relative values in a spatially extended population is studied by means of a spatially explicit simulation model. The results show that a regular pattern of sampling is best for obtaining information on genet sizes or inbreeding coefficients; in contrast, random or hierarchical sampling strategies are better for obtaining information on parameters that are based on comparison of pairs of individuals, such as distribution of genet sizes or autocorrelation in genetic structure. A set of recommendations is provided for designing a good sampling strategy.     

Comparing alternative models to empirical data: cognitive models of western scrub-jay foraging behavior

Animals often select one item from a set of candidates, as when choosing a foraging site or mate, and are expected to possess accurate and efficient rules for acquiring information and making decisions. Little is known, however, about the decision rules animals use. We compare patterns of information sampling by western scrub-jays (Aphelocoma californica) when choosing a nut with three decision rules: best of n (BN), flexible threshold (FT), and comparative Bayes (CB). First, we use a null hypothesis testing approach and find that the CB decision rule, in which individuals use past experiences to make nonrandom assessment and choice decisions, produces patterns of behavior that more closely correspond to observed patterns of nut sampling in scrub-jays than the other two rules. This approach does not allow us to quantify how much better CB is at predicting scrub-jay behavior than the other decision rules. Second, we use a model selection approach that uses Akaike Information Criteria to quantify how well alternative models approximate observed data. We find that the CB rule is much more likely to produce the observed patterns of scrub-jay behavior than the other rules. This result provides some of the best empirical evidence of the use of Bayesian information updating by a nonhuman animal.     

Selection of the simplest RNA that binds isoleucine

We have identified the simplest RNA binding site for isoleucine using selection-amplification (SELEX), by shrinking the size of the randomized region until affinity selection is extinguished. Such a protocol can be useful because selection does not necessarily make the simplest active motif most prominent, as is often assumed. We find an isoleucine binding site that behaves exactly as predicted for the site that requires fewest nucleotides. This UAUU motif (16 highly conserved positions; 27 total), is also the most abundant site in successful selections on short random tracts. The UAUU site, now isolated independently at least 63 times, is a small asymmetric internal loop. Conserved loop sequences include isoleucine codon and anticodon triplets, whose nucleotides are required for amino acid binding. This reproducible association between isoleucine and its coding sequences supports the idea that the genetic code is, at least in part, a stereochemical residue of the most easily isolated RNA-amino acid binding structures.     

Variance component estimation techniques compared for two mating designs with forest genetic architecture through computer simulation

Computer simulation was used to compare minimum variance quadratic estimation (MIVQUE), minimum norm quadratic unbiased estimation (MINQUE), restricted maximum likelihood (REML), maximum likelihood (ML), and Henderson's Method 3 (HM3) on the basis of variance among estimates, mean square error (MSE), bias and probability of nearness for estimation of both individual variance components and three ratios of variance components. The investigation also compared three procedures for dealing with negative estimates and included the use of both individual observations and plot means as the experimental unit of the analysis. The structure of data simulated (field design, mating designs, genetic architecture and imbalance) represented typical analysis problems in quantitative forest genetics. Results of comparing the estimation techniques demonstrated that: estimates of probability of nearness did not discriminate among techniques; bias was discriminatory among procedures for dealing with negative estimates but not among estimation techniques (except ML); sampling variance among estimates was discriminatory among procedures for dealing with negative estimates, estimation techniques and unit of observation; and MSE provided no additional information to variance of the estimates. HM3 and REML were the closest competitors under these criteria; however, REML demonstrated greater robustness to imbalance. Of the three negative estimate procedures, two are of practical significance and guidelines for their application are presented. Estimates from individual observations were always preferable to those from plot means over the experimental levels of this study.This is Journal Series NO. R-03768 of the Institute of Food and Agricultural Sciences     

Application of Gibbs sampling for inference in a mixed major gene-polygenic inheritance model in animal populations

The application of Gibbs sampling is considered for inference in a mixed inheritance model in animal populations. Implementation of the Gibbs sampler on scalar components, as used for human populations, appeared not to be efficient, and an approach with blockwise sampling of genotypes was proposed for use in animal populations. The blockwise sampling of genotypes was proposed for use in animal populations. The blockwise sampling by which genotypes of a sire and its final progeny were sampled jointly was effective in improving mixing, although further improvements could be looked for. Posterior densities of parameters were visualised from Gibbs samples; from the former highly marginalised Bayesian point and interval estimates can be obtained.