Best linear unbiased allele-frequency estimation in complex pedigrees

Many types of genetic analyses depend on estimates of allele frequencies. We consider the problem of allele-frequency estimation based on data from related individuals. The motivation for this work is data collected on the Hutterites, an isolated founder population, so we focus particularly on the case in which the relationships among the sampled individuals are specified by a large, complex pedigree for which maximum likelihood estimation is impractical. For this case, we propose to use the best linear unbiased estimator (BLUE) of allele frequency. We derive this estimator, which is equivalent to the quasi-likelihood estimator for this problem, and we describe an efficient algorithm for computing the estimate and its variance. We show that our estimator has certain desirable small-sample properties in common with the maximum likelihood estimator (MLE) for this problem. We treat both the case when parental origin of each allele is known and when it is unknown. The results are extended to prediction of allele frequency in some set of individuals S based on genotype data collected on a set of individuals R. We compare the mean-squared error of the BLUE, the commonly used naive estimator (sample frequency) and the MLE when the latter is feasible to calculate. The results indicate that although the MLE performs the best of the three, the BLUE is close in performance to the MLE and is substantially easier to calculate, making it particularly useful for large complex pedigrees in which MLE calculation is impractical or infeasible. We apply our method to allele-frequency estimation in a Hutterite data set.     

Comparative methods based on species mean values

Comparative methods that use simple linear regression based on species mean values introduce three difficulties with respect to the standard regression model. First, species values may not be independent because they form part of a hierarchically structured phylogeny. Second, variation about the regression line includes two sources of error: 'biological error' due to deviations of the true species mean values from the regression line and sampling error associated with the estimation of these mean values [B. Riska, Am. Natural. 138 (1991) 283]. Third, sampling error in the independent variable results in an attenuated estimate of the regression slope. We consider estimation and hypothesis testing using two statistical models which explicitly justify the use of the species mean values, without the need to account for phylogenetic relationships. The first (random-effects) is based on an evolutionary model whereby species evolve to fill a bivariate normal niche space, and the second (fixed-effects) is concerned with describing a relationship among the particular species included in a study, where the only source of error is in the estimation of species mean values. We use a modification of the maximum-likelihood method to obtain an unbiased estimate of the regression slope. For three real datasets we find a close correspondence between this slope and that obtained by simply regressing the species mean values on each other. In the random effects model, the P-value also approximates that based on the regression of species mean values. In the fixed effects model, the P-value is typically much lower. Simulated examples illustrate that the maximum-likelihood approach is useful when the accuracy in estimating the species mean values is low, but the traditional method based on a regression of the species mean values may often be justified provided that the evolutionary model can be justified.     

Estimation of Individual Growth Curves by Empirically Weighted Least Squares

The simultaneous estimation of individual growth curves and a mean growth curve is accomplished by weighted least squares. A polynomial curve is fitted for each individual and the polynomial parameters are linear functions of parameters corresponding to covariates. A simple, computationally efficient variance-covariance estimator is derived. The resultant estimate is used in the weighted least squares estimation. The results are compared to empirical Bayes estimation.     

A Procedure for the Common Estimation of Parameters Corresponding to Several Treatment Groups

In this article we give a procedure for the common estimation of parameters corresponding to several treatment groups. Thereby we assume that the distribution functions of the groups belong to the same family and differ only in the parameter values. The procedure allows the common estimation of some of these parameters. The parameters themselves will be estimated by the maximum likelihood method; the estimators will be calculated iteratively by the Newton-Raphson method. To prove if the common estimation is possible, we propose as a suitable test the maximum likelihood ratio test. Finally we show the application of our procedure in the case of the probit analysis.     

CLUSTERING POPULATIONS BY MIXED LINEAR MODELS

INTRODUCTIONInevolutionstudiesandplant(oranimal)breedingresearch,clusteranalysesarewidelyusedforgroupingpopulations.Therehavebeenalotofmethodsdevelopedforgroupingpopulations(SneathandSokal,1973;Eve-ritt,1993).Variousmethodsaredifferedinthewaysfordistance(…     

Estimation of Linear Regression Model with Random Coefficients Ensuring Almost Non-Negativity of Variance Estimators

For a linear regression model with random coefficients, this paper considers the estimation of the mean of coefficient vector which, in turn, involves the estimation of variances of random coefficients. The conventional estimation methods for it sometimes provides negative estimates. In order to circumvent this kind of difficulty, a proposal is forwarded and is examined in the light of existing ones.     

No BLUE among phylogenetic estimators

 Multivariate analysis is a branch of statistics that successfully exploits the powerful tools of linear algebra to obtain a fairly comprehensive theory of estimation. The purpose of this paper is to explore to what extent a linear theory of estimation can be developed in the context of coalescent models used in the analysis of DNA polymorphism. We consider a large class of coalescent models, of which the neutral infinite sites model is one example. In the process, we discover several limitations of linear estimators that are quite distinct from those in the classical theory. In particular, we prove that there does not exist a uniformly BLUE (best linear unbiased estimator) for the scaled mutation parameter, under the assumptions of the neutral model of evolution. In fact, we show that no linear estimator performs uniformly better than the Watterson (1975) method based on the total number of segregating sites. For certain coalescent models, the segregating-sites estimator is actually optimal. The general conclusion is the following. If genealogical information is useful for estimating the rate of evolution, then there is no optimal linear method. If there is an optimal linear method, then no information other than the total number of segregating sites is needed. Received: 29 July 1998 / Revised version: 9 October 1998     

基于叶面积指数反演的区域冬小麦单产遥感估测

利用定量遥感技术反演的叶面积指数（LAI）在中国北方黄淮海地区典型县市进行冬小麦单产预测研究.为提高数据质量和减少估产误差,利用Savitzky-Golay滤波技术降低云对NDVI数据的影响及数据缺失;通过冬小麦实测LAI进行时序内插,模拟得到实测点每日冬小麦LAI,继而获得实测点主要生育时期平均LAI;在此基础上,建立了冬小麦主要生育时期平均LAI与作物单产关系模型,改变目前利用生育时期内某一时间点LAI代替整个生育时期LAI的方法;在模型择优基础上,得到最佳遥感估产关键期--开花期LAI与单产统计模型;最后,利用MODIS-NDVI经验模型反演得到的开花期平均LAI进行2008年冬小麦单产预测.结果表明：与地面实测的冬小麦单产相比,研究区估产平均相对误差为1.21%,RMSE达到257.33 kg·hm^-2,可以满足大范围估产的要求.利用上述方法可以在研究区冬小麦收获前20～30 d进行准确的单产估计.     

Adaptive Methods of Trend Detection and Their Application in Analysing Biosignals

In the study estimations of the linear trend and its slope of discrete deterministic and stochastic signals are introduced, which are based on adaptive methods of mean estimation. Because of its simple recursive nature both the trend and the slope estimation may be realized in real time and thus they may be used for on-line procedures e.g. in monitoring systems of intensive care units. The adaptive estimation functions are self-learning, that means they are able to adapt automatically to the new conditions after structure changes of the time series and thus may be also used for detecting points of structure fracture. The properties of the adaptive trend- and slope estimations are demonstrated by some characteristic simulation examples and the possibilities of their applications for analysing biological signals are investigated in three different medical cases.     

Effects of variance-function misspecification in analysis of longitudinal data

The approach of generalized estimating equations (GEE) is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within-subject correlations. The variance is often assumed to be a known function of the mean. This article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses. Our numerical studies indicate that (1) correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified; (2) misspecification of the variance function impacts much more on estimators for within-cluster covariates than for cluster-level covariates; and (3) if the variance function is misspecified, correct choice of the correlation structure may not necessarily improve estimation efficiency. We illustrate impacts of different variance functions using a real data set from cow growth.     

Numerical algorithms for estimation and calculation of parameters in modeling pest population dynamics and evolution of resistance

Computational simulation models can provide a way of understanding and predicting insect population dynamics and evolution of resistance, but the usefulness of such models depends on generating or estimating the values of key parameters. In this paper, we describe four numerical algorithms generating or estimating key parameters for simulating four different processes within such models. First, we describe a novel method to generate an offspring genotype table for one- or two-locus genetic models for simulating evolution of resistance, and how this method can be extended to create offspring genotype tables for models with more than two loci. Second, we describe how we use a generalized inverse matrix to find a least-squares solution to an over-determined linear system for estimation of parameters in probit models of kill rates. This algorithm can also be used for the estimation of parameters of Freundlich adsorption isotherms. Third, we describe a simple algorithm to randomly select initial frequencies of genotypes either without any special constraints or with some pre-selected frequencies. Also we give a simple method to calculate the “stable” Hardy–Weinberg equilibrium proportions that would result from these initial frequencies. Fourth we describe how the problem of estimating the intrinsic rate of natural increase of a population can be converted to a root-finding problem and how the bisection algorithm can then be used to find the rate. We implemented all these algorithms using MATLAB and Python code; the key statements in both codes consist of only a few commands and are given in the appendices. The results of numerical experiments are also provided to demonstrate that our algorithms are valid and efficient.     

An Estimation Procedure for the Joint Distribution of Spatial Direction and Thickness of Flat Bodies Using Vertical Sections Part I: Theoretical Considerations

In soil micromorphology fissures are considered in vertical sections. To get information about the properties of the soil the joint distribution of spatial direction and width is of interest. The fissures are mathematically generalized to flat bodies which form a stationary weighted surface process with the weight “thickness”. Because of stationarity a joint distribution of spatial direction and thickness exists in a “typical point” of the surface process. A suitable parametric family of distributions is assumed. The corresponding parameters can be estimated from measurements on the vertical sections. But on the sections only the visible thickness and the visible angle of a fissure can be measured. Therefore the joint distribution of these variables is expressed by the joint spatial distribution of spatial direction and thickness. This derived distribution depends on the same parameters. The Chi-Square method is proposed for the parameter estimation. The estimation procedure is demonstrated using the Bingham-Mardia distribution for the direction and the lognormal distribution for the thickness and by defining a way to correlate the mean thickness and the direction.     

黄土高原北部典型灌丛枝条生物量估算模型

于2015年8月末在陕西神木县六道沟小流域采集200个柠条和210个沙柳枝条,测定枝条的基径(D)、长度(H)、含水量(W₀)、鲜质量(W_F)和干质量(W),选用指数函数和异速生长方程建立了4种由枝条形态指标估算枝条生物量的简易模型,并对模型的拟合效果进行验证. 结果表明: 对于柠条和沙柳灌丛,基于D、H二者组合变量(D²H)的异速生长方程是估算枝条生物量的最优模型,该模型经线性转化后可以消除生物量数据的异方差性,且拟合效果最优,决定系数(R²)最大,平均误差(ME)、平均绝对误差(MAE)、总相对误差(TRE)、平均系统误差(MSE)和平均绝对百分误差(MPSE)整体上最小,基本满足生态学研究的精度要求.     

MIXED MODEL APPROACHES FOR ESTIMATING GENETIC VARIANCES AND COVARIANCES

The limitations of methods for analysis of variance(ANOVA)in estimating genetic variances are discussed. Among the three methods(maximum likelihood ML, restricted maximum likelihood REML, and minimum norm quadratic unbiased estimation MINQUE)for mixed linear models, MINQUE method is presented with formulae for estimating variance components and covariances components and for predicting genetic effects. Several genetic models, which cannot be appropriately analyzed by ANOVA methods, are introduced in forms of mixed linear models. Genetic models with independent random effects can be analyzed by MINQUE(1)method whieh is a MINQUE method with all prior values setting 1. MINQUE(1)method can give unbiased estimation for variance components and covariance components, and linear unbiased prediction (LUP) for genetic effects. There are more complicate genetic models for plant seeds which involve correlated random effects. MINQUE(0/1)method, which is a MINQUE method with all prior covariances setting 0 and all prior variances setting 1, is suitable for estimating variance and covariance components in these models. Mixed model approaches have advantage over ANOVA methods for the capacity of analyzing unbalanced data and complicated models. Some problems about estimation and hypothesis test by MINQUE method are discussed.     

Inequality constrained estimation of genetic parameters

Summary A method is presented for computing estimates of genetic parameters under linear inequality constraints such that solutions are within theoretical limits. The method produces biased estimators, yet a small scale numerical study, also presented, shows that the inequality constrained estimators have a small mean squared error of prediction than the best of unbiased estimators. The increase in efficiency of estimation is particularly useful for traits where heritability is near the boundary values of zero or one.     

Classification of Trends Via the Linear State Space Model

A method is presented for classification of trend curves based on the linear state space model. In this approach information about the smoothness of the trend curves is incorporated into the classification model by a nonstationary stochastic trend model and can thereby be used to obtain a better classification. In the case of small data sets the performance of the classification is significantly improved in comparison with the usual cluster analysis. Maximum likelihood estimation can be used to calculate the parameters of this model and to determine the classification. The classification algorithm is described in detail and the results are compared to those of the usual cluster analysis by simulation studies and by an application to tree ring data.     

Robust estimation of mean and dispersion functions in extended generalized additive models

Generalized linear models are a widely used method to obtain parametric estimates for the mean function. They have been further extended to allow the relationship between the mean function and the covariates to be more flexible via generalized additive models. However, the fixed variance structure can in many cases be too restrictive. The extended quasilikelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covariates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this article, we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.     

温室甜瓜营养生长期日蒸腾量估算模型

建立了基于温室环境参数、甜瓜生长发育参数和土壤水分参数的温室甜瓜日蒸腾量估算模型,以研究温室条件下甜瓜蒸腾量的估算方法.根据温室内特定环境对Penman-Monteith方程中空气动力项进行修正,推导出了适于计算温室条件下参考作物蒸腾量的温室环境因子子模型;以甜瓜叶面积指数为自变量构建了作物因子子模型,模型形式为线性函数;以土壤相对有效含水量为自变量构建了土壤水分因子子模型,模型形式为对数函数.采用分期播种法,根据周年不同播期实测数据对模型参数进行估计和分析.采用土壤相对含水量分别为80%、70%、60%的实测蒸腾数据,对模型在充分灌溉和节水灌溉条件下的预测精度进行了检验,模拟值的平均相对误差分别为11.5%、16.2%、16.9%.所建蒸腾模型是对Penman-Monteith公式在温室环境和节水灌溉条件下的有益探索,具有重要推广应用价值.     

The Zurich method for sapwood estimation

We present a new predictor for the estimation of the number of missing sapwood rings in oak. It circumvents a number of problems with predictors used in traditional sapwood estimation procedures such as mean ring width or tree age. Instead, we use the mean ring width of the last 9 heartwood rings and the first sapwood ring, which mirrors the growth level during sapwood formation. We present a model to produce a 95 % prediction interval for the felling year. Our method accounts for the skewed distribution of the number of sapwood rings towards higher values.     

Overestimation of Yield Loss of Tobacco Caused by the Aggregated Spatial Pattern of Meloidogyne incognita

Overestimation of yield loss caused by Meloidogyne incognita on tobacco was calculated as a function of the statistical frequency distribution of sample counts. Sampling frequency distributions were described by a negative binomial model, with parameter k, and the resulting probability generating function was used to calculate discrete damage probabilities. Negative binomial damage predictions were compared to mean-density estimates of damage. Predictions based on mean density alone overestimate yield loss by values ranging from 300% at a k of 0.1 to less than 10% at a k of 1.0. Damage overestimation was described as an exponential function of k and mean density. Preplant sampling data for M. incognita were used to derive a linear model for the estimation of k from mean density, allowing the calculation of yield-loss overestimation based on one parameter, the field mean density. Overestimation of damage ranged from 288% at a density of 50 juveniles/500 cm³ soil, to 5% at a density of 1,000 juvelfiles/500 cm³ soil.