Realized Sampling Variances of Estimates of Genetic Parameters and the Difference between Genetic and Phenotypic Correlations

A data set of 1572 heritability estimates and 1015 pairs of genetic and phenotypic correlation estimates, constructed from a survey of published beef cattle genetic parameter estimates, provided a rare opportunity to study realized sampling variances of genetic parameter estimates. The distribution of both heritability estimates and genetic correlation estimates, when plotted against estimated accuracy, was consistent with random error variance being some three times the sampling variance predicted from standard formulae. This result was consistent with the observation that the variance of estimates of heritabilities and genetic correlations between populations were about four times the predicted sampling variance, suggesting few real differences in genetic parameters between populations. Except where there was a strong biological or statistical expectation of a difference, there was little evidence for differences between genetic and phenotypic correlations for most trait combinations or for differences in genetic correlations between populations. These results suggest that, even for controlled populations, estimating genetic parameters specific to a given population is less useful than commonly believed. A serendipitous discovery was that, in the standard formula for theoretical standard error of a genetic correlation estimate, the heritabilities refer to the estimated values and not, as seems generally assumed, the true population values.     

Statistical analysis of the Michaelis-Menten equation

An application of the method of maximum likelihood (ML) is described for analysing the results of enzyme kinetic experiments in which the Michaelis-Menten equation is obeyed. Accurate approximate solutions to the ML equations for the parameter estimates are presented for the case in which the experimental errors are of constant relative magnitude. Formulae are derived that approximate the standard errors of these estimates. The estimators are shown to be asymptotically unbiased and the standard errors observed in simulated data rapidly approach the theoretical lower bound as the sample size increases. The results of a large-scale Monte Carlo simulation study indicate that for data with a constant coefficient of variation, the present method is superior to other published methods, including the conventional transformations to linearity and the nonparametric technique proposed by Eisenthal and Cornish-Bowden (1974, Biochemical Journal 139, 715-720). Finally, the present results are extended to the analysis of simple receptor binding experiments using the general approach described by Munson and Rodbard (1980, Analytical Biochemistry 107, 220-239).     

A comparison of regression interval mapping and multiple interval mapping for linked QTL

Regression interval mapping and multiple interval mapping are compared with regard to mapping linked quantitative trait loci (QTL) in inbred-line cross experiments. For that purpose, a simulation study was performed using genetic models with two linked QTL. Data were simulated for F(2) populations of different sizes and with all QTL and marker alleles fixed for alternative alleles in the parental lines. The criteria for comparison are power of QTL identification and the accuracy of the QTL position and effect estimates. Further, the estimates of the relative QTL variance are assessed. There are distinct differences in the QTL position estimates between the two methods. Multiple interval mapping tends to be more powerful as compared to regression interval mapping. Multiple interval mapping further leads to more accurate QTL position and QTL effect estimates. The superiority increased with wider marker intervals and larger population sizes. If QTL are in repulsion, the differences between the two methods are very pronounced. For both methods, the reduction of the marker interval size from 10 to 5 cM increases power and greatly improves QTL parameter estimates. This contrasts with findings in the literature for single QTL scenarios, where a marker density of 10 cM is generally considered as sufficient. The use of standard (asymptotic) statistical theory for the computation of the standard errors of the QTL position and effect estimates proves to give much too optimistic standard errors for regression interval mapping as well as for multiple interval mapping.     

Estimating blood phenotype probabilities and their products.

We consider two methods of estimating phenotype probabilities for a number of standard genetic markers like the ABO, MNSs, and PGM markers. The first method is based on the maximum likelihood estimates of the allele probabilities, and the second (multinomial) method uses the phenotype proportions in the sample. The latter is easy to use, the estimates are always unbiased, and simple formulae for variances are available. The former method, although giving more efficient estimates, requires the assumption of panmixia so that the Hardy-Weinberg law can be used. The two methods are compared theoretically, where possible, or by simulation. Under panmixia, the maximum likelihood estimates can be substantially more efficient than the multinomial estimates. The estimates are also compared in the codominant allele case for nonpanmictic populations. The question of efficiency is of importance when estimating the probability of obtaining a given set of phenotypes, i.e., the product of individual phenotype estimators. This problem is discussed briefly.     

R法遗传参数估值置信区间的计算方法和重复估计次数的研究

探讨R法遗传参数估值置信区间的计算方法和重复估计次数(NORE)对参数估值的影响,利用4种模型通过模拟产生数据集。基础群中公、母畜数分别为200和2000头,BLUP育种值选择5个世代。利用多变量乘法迭代(MMI)法,结合先决条件的共扼梯度(PCG)法求解混合模型方程组估计方差组分。用经典方法、Box-Cox变换后的经典方法和自助法计算参数估值的均数、标准误和置信区间。结果表明,重复估计次数较多时,3种方法均可;重复估计次数较少时,建议使用自助法。简单模型下需要较少的重复估计,但对于复杂模型则需要较多的重复估计。随模型中随机效应数的增加,直接遗传力高估。随着PCG和MMI轮次的增大,参数估值表现出低估的趋势。     

Methodology and Accuracy of Estimation of Quantitative Trait Loci Parameters in a Half-Sib Design Using Maximum Likelihood

Maximum likelihood methods were developed for estimation of the six parameters relating to a marker-linked quantitative trait locus (QTL) segregating in a half-sib design, namely the QTL additive effect, the QTL dominance effect, the population mean, recombination between the marker and the QTL, the population frequency of the QTL alleles, and the within-family residual variance. The method was tested on simulated stochastic data with various family structures under two genetic models. A method for predicting the expected value of the likelihood was also derived and used to predict the lower bound sampling errors of the parameter estimates and the correlations between them. It was found that standard errors and confidence intervals were smallest for the population mean and variance, intermediate for QTL effects and allele frequency, and highest for recombination rate. Correlations among standard errors of the parameter estimates were generally low except for a strong negative correlation (r = -0.9) between the QTL's dominance effect and the population mean, and medium positive and negative correlations between the QTL's additive effect and, respectively, recombination rate (r = 0.5) and residual variance (r = -0.6). The implications for experimental design and method of analysis on power and accuracy of marker-QTL linkage experiments were discussed.     

Estimating the initial relative infection rate for a stochastic epidemic model

Consider the problem of making inference about the initial relative infection rate of a stochastic epidemic model. A relatively complete analysis of infectious disease data is possible when it is assumed that the latent and infectious periods are non-random. Here two related martingale-based techniques are used to derive estimates and associated standard errors for the initial relative infection rate. The first technique requires complete information on the epidemic, the second only the total number of people who were infected and the population size. Explicit expressions for the estimates are obtained. The estimates of the parameter and its associated standard error are easily computed and compare well with results of other methods in an application to smallpox data. Asymptotic efficiency differences between the two martingale techniques are considered.     

Marker-Based Estimation of Heritability in Immortal Populations

Heritability is a central parameter in quantitative genetics, from both an evolutionary and a breeding perspective. For plant traits heritability is traditionally estimated by comparing within- and between-genotype variability. This approach estimates broad-sense heritability and does not account for different genetic relatedness. With the availability of high-density markers there is growing interest in marker-based estimates of narrow-sense heritability, using mixed models in which genetic relatedness is estimated from genetic markers. Such estimates have received much attention in human genetics but are rarely reported for plant traits. A major obstacle is that current methodology and software assume a single phenotypic value per genotype, hence requiring genotypic means. An alternative that we propose here is to use mixed models at the individual plant or plot level. Using statistical arguments, simulations, and real data we investigate the feasibility of both approaches and how these affect genomic prediction with the best linear unbiased predictor and genome-wide association studies. Heritability estimates obtained from genotypic means had very large standard errors and were sometimes biologically unrealistic. Mixed models at the individual plant or plot level produced more realistic estimates, and for simulated traits standard errors were up to 13 times smaller. Genomic prediction was also improved by using these mixed models, with up to a 49% increase in accuracy. For genome-wide association studies on simulated traits, the use of individual plant data gave almost no increase in power. The new methodology is applicable to any complex trait where multiple replicates of individual genotypes can be scored. This includes important agronomic crops, as well as bacteria and fungi.     

A Novel Targeted Learning Method for Quantitative Trait Loci Mapping

We present a novel semiparametric method for quantitative trait loci (QTL) mapping in experimental crosses. Conventional genetic mapping methods typically assume parametric models with Gaussian errors and obtain parameter estimates through maximum-likelihood estimation. In contrast with univariate regression and interval-mapping methods, our model requires fewer assumptions and also accommodates various machine-learning algorithms. Estimation is performed with targeted maximum-likelihood learning methods. We demonstrate our semiparametric targeted learning approach in a simulation study and a well-studied barley data set.     

Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola

As an emergent infectious disease outbreak unfolds, public health response is reliant on information on key epidemiological quantities, such as transmission potential and serial interval. Increasingly, transmission models fit to incidence data are used to estimate these parameters and guide policy. Some widely used modelling practices lead to potentially large errors in parameter estimates and, consequently, errors in model-based forecasts. Even more worryingly, in such situations, confidence in parameter estimates and forecasts can itself be far overestimated, leading to the potential for large errors that mask their own presence. Fortunately, straightforward and computationally inexpensive alternatives exist that avoid these problems. Here, we first use a simulation study to demonstrate potential pitfalls of the standard practice of fitting deterministic models to cumulative incidence data. Next, we demonstrate an alternative based on stochastic models fit to raw data from an early phase of 2014 West Africa Ebola virus disease outbreak. We show not only that bias is thereby reduced, but that uncertainty in estimates and forecasts is better quantified and that, critically, lack of model fit is more readily diagnosed. We conclude with a short list of principles to guide the modelling response to future infectious disease outbreaks.     

A simulation study on the accuracy of position and effect estimates of linked QTL and their asymptotic standard deviations using multiple interval mapping in an F2 scheme

Approaches like multiple interval mapping using a multiple-QTL model for simultaneously mapping QTL can aid the identification of multiple QTL, improve the precision of estimating QTL positions and effects, and are able to identify patterns and individual elements of QTL epistasis. Because of the statistical problems in analytically deriving the standard errors and the distributional form of the estimates and because the use of resampling techniques is not feasible for several linked QTL, there is the need to perform large-scale simulation studies in order to evaluate the accuracy of multiple interval mapping for linked QTL and to assess confidence intervals based on the standard statistical theory. From our simulation study it can be concluded that in comparison with a monogenetic background a reliable and accurate estimation of QTL positions and QTL effects of multiple QTL in a linkage group requires much more information from the data. The reduction of the marker interval size from 10 cM to 5 cM led to a higher power in QTL detection and to a remarkable improvement of the QTL position as well as the QTL effect estimates. This is different from the findings for (single) interval mapping. The empirical standard deviations of the genetic effect estimates were generally large and they were the largest for the epistatic effects. These of the dominance effects were larger than those of the additive effects. The asymptotic standard deviation of the position estimates was not a good criterion for the accuracy of the position estimates and confidence intervals based on the standard statistical theory had a clearly smaller empirical coverage probability as compared to the nominal probability. Furthermore the asymptotic standard deviation of the additive, dominance and epistatic effects did not reflect the empirical standard deviations of the estimates very well, when the relative QTL variance was smaller/equal to 0.5. The implications of the above findings are discussed.     

Likelihood calculations to evaluate experimental designs to estimate genetic variances

Mixed model analyses via restricted maximum likelihood, fitting the so-called animal model, have become standard methodology for the estimation of genetic variances. Models involving multiple genetic variance components, due to different modes of gene action, are readily fitted. It is shown that likelihood-based calculations may provide insight into the quality of the resulting parameter estimates, and are directly applicable to the validation of experimental designs. This is illustrated for the example of a design suggested recently to estimate X-linked genetic variances. In particular, large sample variances and sampling correlations are demonstrated to provide an indication of 'problem' scenarios. Using simulation, it is shown that the profile likelihood function provides more appropriate estimates of confidence intervals than large sample variances. Examination of the likelihood function and its derivatives are recommended as part of the design stage of quantitative genetic experiments.     

Accuracy of predicting the genetic risk of disease using a genome-wide approach

Background

The prediction of the genetic disease risk of an individual is a powerful public health tool. While predicting risk has been successful in diseases which follow simple Mendelian inheritance, it has proven challenging in complex diseases for which a large number of loci contribute to the genetic variance. The large numbers of single nucleotide polymorphisms now available provide new opportunities for predicting genetic risk of complex diseases with high accuracy.

Methodology/Principal Findings

We have derived simple deterministic formulae to predict the accuracy of predicted genetic risk from population or case control studies using a genome-wide approach and assuming a dichotomous disease phenotype with an underlying continuous liability. We show that the prediction equations are special cases of the more general problem of predicting the accuracy of estimates of genetic values of a continuous phenotype. Our predictive equations are responsive to all parameters that affect accuracy and they are independent of allele frequency and effect distributions. Deterministic prediction errors when tested by simulation were generally small. The common link among the expressions for accuracy is that they are best summarized as the product of the ratio of number of phenotypic records per number of risk loci and the observed heritability.

Conclusions/Significance

This study advances the understanding of the relative power of case control and population studies of disease. The predictions represent an upper bound of accuracy which may be achievable with improved effect estimation methods. The formulae derived will help researchers determine an appropriate sample size to attain a certain accuracy when predicting genetic risk.     

Estimating genetic correlations in natural populations in the absence of pedigree information: accuracy and precision of the Lynch method

Usually, genetic correlations are estimated from breeding designs in the laboratory or greenhouse. However, estimates of the genetic correlation for natural populations are lacking, mostly because pedigrees of wild individuals are rarely known. Recently Lynch (1999) proposed a formula to estimate the genetic correlation in the absence of data on pedigree. This method has been shown to be particularly accurate provided a large sample size and a minimum (20%) proportion of relatives. Lynch (1999) proposed the use of the bootstrap to estimate standard errors associated with genetic correlations, but did not test the reliability of such a method. We tested the bootstrap and showed the jackknife can provide valid estimates of the genetic correlation calculated with the Lynch formula. The occurrence of undefined estimates, combined with the high number of replicates involved in the bootstrap, means there is a high probability of obtaining a biased upward, incomplete bootstrap, even when there is a high fraction of related pairs in a sample. It is easier to obtain complete jackknife estimates for which all the pseudovalues have been defined. We therefore recommend the use of the jackknife to estimate the genetic correlation with the Lynch formula. Provided data can be collected for more than two individuals at each location, we propose a group sampling method that produces low standard errors associated with the jackknife, even when there is a low fraction of relatives in a sample.     

Inference from single occasion capture experiments using genetic markers

Accurate estimation of the size of animal populations is an important task in ecological science. Recent advances in the field of molecular genetics researches allow the use of genetic data to estimate the size of a population from a single capture occasion rather than repeated occasions as in the usual capture–recapture experiments. Estimating the population size using genetic data also has sometimes led to estimates that differ markedly from each other and also from classical capture–recapture estimates. Here, we develop a closed form estimator that uses genetic information to estimate the size of a population consisting of mothers and daughters, focusing on estimating the number of mothers, using data from a single sample. We demonstrate the estimator is consistent and propose a parametric bootstrap to estimate the standard errors. The estimator is evaluated in a simulation study and applied to real data. We also consider maximum likelihood in this setting and discover problems that preclude its general use.     

Estimation of evolutionary parameters with phylogenetic trees

An important issue in the phylogenetic analysis of nucleotide sequence data using the maximum likelihood (ML) method is the underlying evolutionary model employed. We consider the problem of simultaneously estimating the tree topology and the parameters in the underlying substitution model and of obtaining estimates of the standard errors of these parameter estimates. Given a fixed tree topology and corresponding set of branch lengths, the ML estimates of standard evolutionary model parameters are asymptotically efficient, in the sense that their joint distribution is asymptotically normal with the variance–covariance matrix given by the inverse of the Fisher information matrix. We propose a new estimate of this conditional variance based on estimation of the expected information using a Monte Carlo sampling (MCS) method. Simulations are used to compare this conditional variance estimate to the standard technique of using the observed information under a variety of experimental conditions. In the case in which one wishes to estimate simultaneously the tree and parameters, we provide a bootstrapping approach that can be used in conjunction with the MCS method to estimate the unconditional standard error. The methods developed are applied to a real data set consisting of 30 papillomavirus sequences. This overall method is easily incorporated into standard bootstrapping procedures to allow for proper variance estimation.     

The analysis of population survey data on DNA sequence variation

A technique is presented for the partitioning of nucleotide diversity into within- and between-population components for the case in which multiple populations have been surveyed for restriction-site variation. This allows the estimation of an analogue of FST at the DNA level. Approximate expressions are given for the variance of these estimates resulting from nucleotide, individual, and population sampling. Application of the technique to existing studies on mitochondrial DNA in several animal species and on several nuclear genes in Drosophila indicates that the standard errors of genetic diversity estimates are usually quite large. Thus, comparative studies of nucleotide diversity need to be substantially larger than the current standards. Normally, only a very small fraction of the sampling variance is caused by sampling of individuals. Even when 20 or so restriction enzymes are employed, nucleotide sampling is a major source of error, and population sampling is often quite important. Generally, the degree of population subdivision at the nucleotide level is comparable with that at the haplotype level, but significant differences do arise as a result of inequalities in the genetic distances between haplotypes.     

Detection of significant disease risks using a spatial conditional autoregressive model

SUMMARY: The conditional autoregressive (CAR) model is widely used to describe the geographical distribution of a specific disease risk in lattice mapping. Successful developments based on frequentist and Bayesian procedures have been extensively applied to obtain two-stage disease risk predictions at the subregional level. Bayesian procedures are preferred for making inferences, as the posterior standard errors (SE) of the two-stage prediction account for the variability in the variance component estimates; however, some recent work based on frequentist procedures and the use of bootstrap adjustments for the SE has been undertaken. In this article we investigate the suitability of an analytical adjustment for disease risk inference that provides accurate interval predictions by using the penalized quasilikelihood (PQL) technique to obtain model parameter estimates. The method is a first-order approximation of the naive SE based on a Taylor expansion and is interpreted as a conditional measure of variability providing conditional calibrated prediction intervals, given the data. We conduct a simulation study to demonstrate how the method can be used to estimate the specific subregion risk by interval. We evaluate the proposed methodology by analyzing the commonly used example data set of lip cancer incidence in the 56 counties of Scotland for the period 1975-1980. This evaluation reveals a close similarity between the solutions provided by the method proposed here and those of its fully Bayesian counterpart.     

A resolution of the ascertainment sampling problem. II. Generalizations and numerical results

The ascertainment problem arises when families are sampled by a nonrandom process and some assumption about this sampling process must be made in order to estimate genetic parameters. Under classical ascertainment assumptions, estimation of genetic parameters cannot be separated from estimation of the parameters of the ascertainment process, so that any misspecification of the ascertainment process causes biases in estimation of the genetic parameters. Ewens and Shute proposed a resolution to this problem, involving conditioning the likelihood of the sample on the part of the data which is "relevant to ascertainment." The usefulness of this approach can only be assessed by examining the properties (in particular, bias and standard error) of the estimates which arise by using it for a wide range of parameter values and family size distributions and then comparing these biases and standard errors with those arising under classical ascertainment procedures. These comparisons are carried out in the present paper, and we also compare the proposed method with procedures which condition on, or ignore, parts of the data.     

Likelihood-based genetic mark-recapture estimates when genotype samples are incomplete and contain typing errors

Genotypes produced from samples collected non-invasively in harsh field conditions often lack the full complement of data from the selected microsatellite loci. The application to genetic mark-recapture methodology in wildlife species can therefore be prone to misidentifications leading to both ‘true non-recaptures’ being falsely accepted as recaptures (Type I errors) and ‘true recaptures’ being undetected (Type II errors). Here we present a new likelihood method that allows every pairwise genotype comparison to be evaluated independently. We apply this method to determine the total number of recaptures by estimating and optimising the balance between Type I errors and Type II errors. We show through simulation that the standard error of recapture estimates can be minimised through our algorithms. Interestingly, the precision of our recapture estimates actually improved when we included individuals with missing genotypes, as this increased the number of pairwise comparisons potentially uncovering more recaptures. Simulations suggest that the method is tolerant to per locus error rates of up to 5% per locus and can theoretically work in datasets with as little as 60% of loci genotyped. Our methods can be implemented in datasets where standard mismatch analyses fail to distinguish recaptures. Finally, we show that by assigning a low Type I error rate to our matching algorithms we can generate a dataset of individuals of known capture histories that is suitable for the downstream analysis with traditional mark-recapture methods.