数量性状主基因＋多基因混合遗传分析中鉴定多基因存在的IECM算法

为了进行２对主基因＋多基因混合遗传分析中的主基因存在的鉴定和多基因存在的鉴定以及多世代的联合遗传分析的分布参数估计,在ＥＣＭ算法和剖分成分分布方差为主基因变异组分、多基因变异组分和误差变异组分三部分基础上,提出了计算简便的迭代ＥＣＭ算法,简称ＩＥＣＭ算法,以利用 Ｐ_１、Ｆ_１、Ｐ_２和 Ｆ_（２：３）家系世代鉴定多基因存在为例阐明该算法．它的 ＣＭ步包含迭代ＣＭ_１、迭代ＣＭ_２和迭代ＣＭ_３步,在固定其它参数的情况下分别求分布平均数、多基因方差组分和误差方差的极大似然估计．通过１１３８－２ｘ邳县天鹅蛋杂交组合的Ｐ_１、Ｐ_２、Ｆ_１和Ｆ_（２：３）家系群体研究了大豆豆秆黑潜蝇的遗传规律．结果表明,它受 １对主基因的控制并有多基因的修饰．     

An Approximate Model of Polygenic Inheritance

The finite polygenic model approximates polygenic inheritance by postulating that a quantitative trait is determined by n independent, additive loci. The 3(n) possible genotypes for each person in this model limit its applicability. CANNINGS, THOMPSON, and SKOLNICK suggested a simplified, nongenetic version of the model involving only 2n + 1 genotypes per person. This article shows that this hypergeometric polygenic model also approximates polygenic inheritance well. In particular, for noninbred pedigrees, trait means, variances, covariances, and marginal distributions match those of the ordinary finite polygenic model. Furthermore as n -> &, the trait values within a pedigree collectively tend toward multivariate normality. The implications of these results for likelihood evaluation under the polygenic threshold and mixed models of inheritance are discussed. Finally, a simple numerical example illustrates the application of the hypergeometric polygenic model to risk prediction under the polygenic threshold model.     

利用P1、F1、P2、F2和F2:3家系五世代联合分离分析的拓展

在王建康等[3]的基础上拓展了利用P1、F1、P2、F2和F2：3家系5群体的2对主基因（B）和2对主基因 多基因（E）2类模型,为使拓展模型成为可能并提高分布参数估计值的精度,用IECM算法估计样本似然函数分布参数,通过重新分析3个大豆杂交组合抗豆秆黑潜蝇主茎虫量遗传资料证实了通过孟德尔氏遗传分析法所获得的结果,并得到存在多基因的统计学依据。     

Development of a variational scheme for model inversion of multi-area model of brain. Part II: VBEM method

In Part I and Part II of these two companion papers (henceforth called Part I and Part II), we develop and evaluate a variational Bayesian expectation maximization (VBEM) method for model inversion of our multi-area extended neural mass model (MEN). In this paper, we develop the VBEM method to estimate posterior distributions of parameters of MEN. We choose suitable prior distributions for the model parameters in order to use properties of a conjugate-exponential model in implementing VBEM. Consequently, VBEM leads to analytically tractable forms. The proposed VBEM algorithm starts with initialization and consists of repeated iterations of a variational Bayesian expectation step (VB E-step) and a variational Bayesian maximization step (VB M-step). Posterior distributions of the model parameters are updated in the VB M-step. Distribution of the hidden state is updated in the VB E-step. We develop a variational extended Kalman smoother (VEKS) to infer the distribution of the hidden state in the VB E-step and derive the forward and backward passes of VEKS, analogous to the Kalman smoother. In Part I, we evaluate and validate the VBEM method using simulation studies.     

A Fisher scoring algorithm for the weighted regression method of QTL mapping

An improved weighted least square (LS) method for quantitative trait loci (QTL) mapping using the estimating equation (EE) algorithm was developed recently. The method is more efficient than both the LS and the weighted LS methods and slightly less efficient than the mixture model maximum likelihood (ML) method. The iteration process of the EE algorithm is implicit. We developed a Fisher-scoring algorithm for the weighted LS method. The iteration process is explicit and easy to program. In addition, the method automatically provides an approximate variance-covariance matrix for the estimated QTL parameters as a by-product of the iteration process. As a consequence, a W-test statistic can be used for testing the significance of QTL. To compare the Fisher scoring algorithm with the expectation maximization (EM)-based ML method, we also developed a slightly simplified method to compute the variance-covariance matrix of the estimated parameters under the EM algorithm.     

Computational characterization of double reduction in autotetraploid natural populations

Population genetic theory has been well developed for diploid species, but its extension to study genetic diversity, variation and evolution in autopolyploids, a class of polyploids derived from the genome doubling of a single ancestral species, requires the incorporation of multisomic inheritance. Double reduction, which is characteristic of autopolyploidy, has long been believed to shape the evolutionary consequence of organisms in changing environments. Here, we develop a computational model for testing and estimating double reduction and its genomic distribution in autotetraploids. The model is implemented with the expectation–maximization (EM) algorithm to dissect unobservable allelic recombinations among multiple chromosomes, enabling the simultaneous estimation of allele frequencies and double reduction in natural populations. The framework fills an important gap in the population genetic theory of autopolyploids.     

Robust Clustering Using Exponential Power Mixtures

Summary Clustering is a widely used method in extracting useful information from gene expression data, where unknown correlation structures in genes are believed to persist even after normalization. Such correlation structures pose a great challenge on the conventional clustering methods, such as the Gaussian mixture (GM) model, k‐means (KM), and partitioning around medoids (PAM), which are not robust against general dependence within data. Here we use the exponential power mixture model to increase the robustness of clustering against general dependence and nonnormality of the data. An expectation–conditional maximization algorithm is developed to calculate the maximum likelihood estimators (MLEs) of the unknown parameters in these mixtures. The Bayesian information criterion is then employed to determine the numbers of components of the mixture. The MLEs are shown to be consistent under sparse dependence. Our numerical results indicate that the proposed procedure outperforms GM, KM, and PAM when there are strong correlations or non‐Gaussian components in the data.     

Analysis of single gene multitrait effects in livestock by the use of Gibbs sampling

The paper presents a method of multivariate data analysis described by a model which involves fixed effects, additive polygenic individual effects and the effects of a major gene. To find the estimates of model parameters, the maximization of likelihood function method is applied. The maximum of likelihood function is computed by the use of the Gibbs sampling approach. In this approach, following the conditional posterior distributions, values of all unknown parameters are generated. On the basis of the obtained samples the marginal posterior densities as well as the estimates of fixed effects, gene frequency, genotypic values, major gene, polygenic and error (co)variances are calculated. A numerical example, supplemented to theoretical considerations, deals with data simulated according to the considered model.     

Evidence of a major gene from Bayesian segregation analyses of liability to osteochondral diseases in pigs

Bayesian segregation analyses were used to investigate the mode of inheritance of osteochondral lesions (osteochondrosis, OC) in pigs. Data consisted of 1163 animals with OC and their pedigrees included 2891 animals. Mixed-inheritance threshold models (MITM) and several variants of MITM, in conjunction with Markov chain Monte Carlo methods, were developed for the analysis of these (categorical) data. Results showed major genes with significant and substantially higher variances (range 1.384-37.81), compared to the polygenic variance (sigmau2). Consequently, heritabilities for a mixed inheritance (range 0.65-0.90) were much higher than the heritabilities from the polygenes. Disease allele frequencies range was 0.38-0.88. Additional analyses estimating the transmission probabilities of the major gene showed clear evidence for Mendelian segregation of a major gene affecting osteochondrosis. The variants, MITM with informative prior on sigmau2, showed significant improvement in marginal distributions and accuracy of parameters. MITM with a "reduced polygenic model" for parameterization of polygenic effects avoided convergence problems and poor mixing encountered in an "individual polygenic model." In all cases, "shrinkage estimators" for fixed effects avoided unidentifiability for these parameters. The mixed-inheritance linear model (MILM) was also applied to all OC lesions and compared with the MITM. This is the first study to report evidence of major genes for osteochondral lesions in pigs; these results may also form a basis for underpinning the genetic inheritance of this disease in other animals as well as in humans.     

Estimation of dominance components in noninbred populations by using additive animal model residuals

In the case of noninbred and unselected populations with linkage equilibrium, the additive and dominance genetic effects are uncorrelated and the variance-covariance matrix of the second component is simply a product of its variance by a matrix that can be computed from the numerator relationship matrix A. The aim of this study is to present a new approach to estimate the dominance part with a reduced set of equations and hence a lower computing cost. The method proposed is based on the processing of the residual terms resulting from the BLUP methodology applied to an additive animal model. Best linear unbiased prediction of the dominance component d is almost identical to the one given by the full mixed model equations. Based on this approach, an algorithm for restricted maximum likelihood (REML) estimation of the variance components is also presented. By way of illustration, two numerical examples are given and a comparison between the parameters estimated with the expectation maximization (EM) algorithm and those obtained by the proposed algorithm is made. The proposed algorithm is iterative and yields estimates that are close to those obtained by EM, which is also iterative.     

The use of the expectation-maximization (EM) algorithm for maximum likelihood estimation of gametic frequencies of multilocus polymorphic codominant systems based on sampled population data

Estimation of gametic frequencies in multilocus polymorphic systems based on the numerical distribution of multilocus genotypes in a population sample ("analysis without pedigrees") is difficult because some gametes are not recognized in the data obtained. Even in the case of codominant systems, where all alleles can be recognized by genotypes, so that direct estimation of the frequencies of genes (alleles) is possible ("complete data"), estimation of the frequencies of multilocus gametes based on the data on multilocus genotypes is sometimes impossible, whether population data or even family data are used for studying genotypic segregation or analysis of linkage ("incomplete data"). Such "incomplete data" are analyzed based on the corresponding genetic models using the expectation-maximization (EM) algorithm. In this study, the EM algorithm based on the random-marriage model for a nonsubdivided population was used to estimate gametic frequencies. The EM algorithm used in the study does not set any limitations on the number of loci and the number of alleles of each locus. Locus and alleles are identified by numeration making possible to arrange loops. In each combination of alleles for a given combination of m out of L loci (L is the total number of loci studied), all alleles are assigned value 1, and the remaining alleles are assigned value 0. The sum of zeros and unities for each gamete is its gametic value (h), and the sum of the gametic values of the gametes that form a given genotype is the genotypic value (g) of this genotype. Then, gametes with the same h are united into a single class, which reduces the number of the estimated parameters. In a general case of m loci, this procedure yields m + 1 classes of gametes and 2m + 1 classes of genotypes with genotypic values g = 0, 1, 2, ..., 2m. The unknown frequencies of the m + 1 classes of gametes can be represented as functions of the gametic frequencies whose maximum likelihood estimations (MLEs) have been obtained in all previous EM procedures and the only unknown frequency (Pm(m)) that is to be estimated in the given EM procedure. At the expectation step, the expected frequencies (Fm(g) of the genotypes with genotypic values g are expressed in terms of the products of the frequencies of m + 1 classes of gametes. The data on genotypes are the numbers (ng) of individuals with genotypic values g = 0, 1, 2, 3, ..., 2m. The maximization step is the maximization of the logarithm of the likelihood function (LLF) for ng values. Thus, the EM algorithm is reduced, in each case, to solution of only one equation with one unknown parameter with the use of the ng values, i.e., the numbers of individuals after the corresponding regrouping of the data on the individuals' genotypes. Treatment of the data obtained by Kurbatova on the MNSs and Rhesus systems with alleles C, Cw, c, D, d, E, e with the use of Weir's EM algorithm and the EM algorithm suggested in this study yielded similar results. However, the MLEs of the parameters obtained with the use of either algorithm often converged to a wrong solution: the sum of the frequencies of all gametes (4 and 12 gametes for MNSs and Rhesus, respectively) was not equal to 1.0 even if the global maximum of LLF was reached for each of them (as it was for MNSs with the use of Weir's EM algorithm), with each parameter falling within admissible limits (e.g., [0, min(PN,Ps)] for PNs). The chi 2 function is suggested to be used as a goodness-of-fit function for the distribution of genotypes in a sample in order to select acceptable solutions. However, the minimum of this function only guarantee the acceptability of solutions if all limitations on the parameters are met: the sum of estimations of gametic frequencies is 1.0, each frequency falls within the admissible limits, and the "gametic algebra" is complied with (none of the frequencies is negative).     

Regularized EM algorithm for sparse parameter estimation in nonlinear dynamic systems with application to gene regulatory network inference

Parameter estimation in dynamic systems finds applications in various disciplines, including system biology. The well-known expectation-maximization (EM) algorithm is a popular method and has been widely used to solve system identification and parameter estimation problems. However, the conventional EM algorithm cannot exploit the sparsity. On the other hand, in gene regulatory network inference problems, the parameters to be estimated often exhibit sparse structure. In this paper, a regularized expectation-maximization (rEM) algorithm for sparse parameter estimation in nonlinear dynamic systems is proposed that is based on the maximum a posteriori (MAP) estimation and can incorporate the sparse prior. The expectation step involves the forward Gaussian approximation filtering and the backward Gaussian approximation smoothing. The maximization step employs a re-weighted iterative thresholding method. The proposed algorithm is then applied to gene regulatory network inference. Results based on both synthetic and real data show the effectiveness of the proposed algorithm.     

An EM algorithm for mapping binary disease loci: application to fibrosarcoma in a four-way cross mouse family

Many diseases show dichotomous phenotypic variation but do not follow a simple Mendelian pattern of inheritance. Variances of these binary diseases are presumably controlled by multiple loci and environmental variants. A least-squares method has been developed for mapping such complex disease loci by treating the binary phenotypes (0 and 1) as if they were continuous. However, the least-squares method is not recommended because of its ad hoc nature. Maximum Likelihood (ML) and Bayesian methods have also been developed for binary disease mapping by incorporating the discrete nature of the phenotypic distribution. In the ML analysis, the likelihood function is usually maximized using some complicated maximization algorithms (e.g. the Newton-Raphson or the simplex algorithm). Under the threshold model of binary disease, we develop an Expectation Maximization (EM) algorithm to solve for the maximum likelihood estimates (MLEs). The new EM algorithm is developed by treating both the unobserved genotype and the disease liability as missing values. As a result, the EM iteration equations have the same form as the normal equation system in linear regression. The EM algorithm is further modified to take into account sexual dimorphism in the linkage maps. Applying the EM-implemented ML method to a four-way-cross mouse family, we detected two regions on the fourth chromosome that have evidence of QTLs controlling the segregation of fibrosarcoma, a form of connective tissue cancer. The two QTLs explain 50-60% of the variance in the disease liability. We also applied a Bayesian method previously developed (modified to take into account sex-specific maps) to this data set and detected one additional QTL on chromosome 13 that explains another 26% of the variance of the disease liability. All the QTLs detected primarily show dominance effects.     

利用回交B1和B2及F2群体鉴定数量性状两对主基因+多基因混合遗传模型

ＱＴＬ作图和主基因＋多基因混合遗传分析表明：拓展两对基基因＋多基因混合遗传模型十分必要。本文利用混合分布理论,ＡＩＣ准则在回交Ｂ１和Ｂ２群体或Ｆ２群体中鉴定两对主基因的存在,当主基因存在时估计其遗传参数;同时还改进了利用亲本,Ｆ１和回交Ｂ１和Ｂ２群体,或亲本,Ｆ１和Ｆ２群体鉴定多基因存在的方法,分布参数的估计采用ＩＥＣＭ算法,以水稻株高性状为例说明该方法的应用。     

Multilist population estimation with incomplete and partial stratification

Multilist capture-recapture methods are commonly used to estimate the size of elusive populations. In many situations, lists are stratified by distinguishing features, such as age or sex. Stratification has often been used to reduce biases caused by heterogeneity in the probability of list membership among members of the population; however, it is increasingly common to find lists that are structurally not active in all strata. We develop a general method to deal with cases when not all lists are active in all strata using an expectation maximization (EM) algorithm. We use a flexible log-linear modeling framework that allows for list dependencies and differential probabilities of ascertainment in each list. Finally, we apply our method of estimating population size to two examples.     

Fitting mixture distributions to phenylthiocarbamide (PTC) sensitivity.

A technique for fitting mixture distributions to phenylthiocarbamide (PTC) sensitivity is described. Under the assumptions of Hardy-Weinberg equilibrium, a mixture of three normal components is postulated for the observed distribution, with the mixing parameters corresponding to the proportions of the three genotypes associated with two alleles A and a acting at a single locus. The corresponding genotypes AA, Aa, and aa are then considered to have separate means and variances. This paper is concerned with estimating the parameters of the model, and their standard errors, by using an application of the EM algorithm. This technique also caters for the fact that the sensitivity measurements are only known to lie between the endpoints of certain intervals and that the exact measurement of the attribute is not possible.     

High Resolution of Quantitative Traits into Multiple Loci via Interval Mapping

A very general method is described for multiple linear regression of a quantitative phenotype on genotype [putative quantitative trait loci (QTLs) and markers] in segregating generations obtained from line crosses. The method exploits two features, (a) the use of additional parental and F(1) data, which fixes the joint QTL effects and the environmental error, and (b) the use of markers as cofactors, which reduces the genetic background noise. As a result, a significant increase of QTL detection power is achieved in comparison with conventional QTL mapping. The core of the method is the completion of any missing genotypic (QTL and marker) observations, which is embedded in a general and simple expectation maximization (EM) algorithm to obtain maximum likelihood estimates of the model parameters. The method is described in detail for the analysis of an F(2) generation. Because of the generality of the approach, it is easily applicable to other generations, such as backcross progenies and recombinant inbred lines. An example is presented in which multiple QTLs for plant height in tomato are mapped in an F(2) progeny, using additional data from the parents and their F(1) progeny.     

Application of Random Coefficient Regression Model to Myopia Data: A Case Study

A one-year birth cohort from Northern Finland has been followed up since 1966. As a part of this study, we are in this paper concerned with analysing the progression of myopia (nearsightness) up to the age of 20 years. The random coefficient regression model was chosen for the analysis because of the large individual variation in the development of myopia. Maximum likelihood estimates for the parameters in the model were obtained via the expectation maximization (EM) algorithm. It is shown how the estimated model can be used to predict future observations for an individual using the previously recorded refractive error measurements as well as other relevant data on the patient in question.     

Use of the EM algorithm to detect QTL affecting multiple-traits in an across half-sib family analysis

QTL detection experiments in livestock species commonly use the half-sib design. Each male is mated to a number of females, each female producing a limited number of progeny. Analysis consists of attempting to detect associations between phenotype and genotype measured on the progeny. When family sizes are limiting experimenters may wish to incorporate as much information as possible into a single analysis. However, combining information across sires is problematic because of incomplete linkage disequilibrium between the markers and the QTL in the population. This study describes formulæ for obtaining MLEs via the expectation maximization (EM) algorithm for use in a multiple-trait, multiple-family analysis. A model specifying a QTL with only two alleles, and a common within sire error variance is assumed. Compared to single-family analyses, power can be improved up to fourfold with multi-family analyses. The accuracy and precision of QTL location estimates are also substantially improved. With small family sizes, the multi-family, multi-trait analyses reduce substantially, but not totally remove, biases in QTL effect estimates. In situations where multiple QTL alleles are segregating the multi-family analysis will average out the effects of the different QTL alleles.     

The full EM algorithm for the MLEs of QTL effects and positions and their estimated variances in multiple-interval mapping

The advent of complete genetic linkage maps of DNA markers has made systematic studies of mapping quantitative trait loci (QTL) in experimental organisms feasible. The method of multiple-interval mapping provides an appropriate way for mapping QTL using genetic markers. However, efficient algorithms for the computation involved remain to be developed. In this article, a full EM algorithm for the simultaneous computation of the MLEs of QTL effects and positions is developed. EM-based formulas are derived for computing the observed Fisher information matrix. The full EM algorithm is compared with an ECM algorithm developed by Kao and Zeng (1997, Biometrics 53, 653-665). The validity of the inverted observed Fisher information matrix as an estimate of the variance matrix of the MLEs is demonstrated by a simulation study.