基于EM算法的遗传重组率估计方法

EM算法是在不完全信息资料下实现参数极大似然估计的一种通用方法．本文导出了双位点不同标记类型,包括共显性-共显性,共显性-显性和显性-显性3种模式下,估计遗传重组率的EM算法,以及获得重组率抽样方差的Bootstrap方法;并将之推广到部分个体缺失标记基因型(未检测到电泳谱带)下的重组率估计．通过大量Monte Carlo模拟研究发现: (1)连锁紧密时,样本容量对重组率的估计影响不大;连锁松散时,需要较大样本容量才可检测到连锁以及实现重组率的较精确估计．(2)用包含缺失标记的所有个体估计重组率比仅用其中的非缺失标记个体估计更准确,且可显著提高连锁检测的统计功效．     

Maximum likelihood methods for cure rate models with missing covariates

We propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction, in which the covariates are allowed to be missing. We allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one-dimensional conditional distributions. We propose a novel EM algorithm for maximum likelihood estimation and derive standard errors by using Louis's formula (Louis, 1982, Journal of the Royal Statistical Society, Series B 44, 226-233). Computational techniques using the Monte Carlo EM algorithm are discussed and implemented. A real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology.     

A Probit Latent Class Model with General Correlation Structures for Evaluating Accuracy of Diagnostic Tests

Summary Traditional latent class modeling has been widely applied to assess the accuracy of dichotomous diagnostic tests. These models, however, assume that the tests are independent conditional on the true disease status, which is rarely valid in practice. Alternative models using probit analysis have been proposed to incorporate dependence among tests, but these models consider restricted correlation structures. In this article, we propose a probit latent class model that allows a general correlation structure. When combined with some helpful diagnostics, this model provides a more flexible framework from which to evaluate the correlation structure and model fit. Our model encompasses several other PLC models but uses a parameter‐expanded Monte Carlo EM algorithm to obtain the maximum‐likelihood estimates. The parameter‐expanded EM algorithm was designed to accelerate the convergence rate of the EM algorithm by expanding the complete‐data model to include a larger set of parameters and it ensures a simple solution in fitting the PLC model. We demonstrate our estimation and model selection methods using a simulation study and two published medical studies.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

Latent class models for joint analysis of disease prevalence and high-dimensional semicontinuous biomarker data

High-dimensional biomarker data are often collected in epidemiological studies when assessing the association between biomarkers and human disease is of interest. We develop a latent class modeling approach for joint analysis of high-dimensional semicontinuous biomarker data and a binary disease outcome. To model the relationship between complex biomarker expression patterns and disease risk, we use latent risk classes to link the 2 modeling components. We characterize complex biomarker-specific differences through biomarker-specific random effects, so that different biomarkers can have different baseline (low-risk) values as well as different between-class differences. The proposed approach also accommodates data features that are common in environmental toxicology and other biomarker exposure data, including a large number of biomarkers, numerous zero values, and complex mean-variance relationship in the biomarkers levels. A Monte Carlo EM (MCEM) algorithm is proposed for parameter estimation. Both the MCEM algorithm and model selection procedures are shown to work well in simulations and applications. In applying the proposed approach to an epidemiological study that examined the relationship between environmental polychlorinated biphenyl (PCB) exposure and the risk of endometriosis, we identified a highly significant overall effect of PCB concentrations on the risk of endometriosis.     

Joint mean–covariance random effect model for longitudinal data

In this paper, we consider the inherent association between mean and covariance in the joint mean–covariance modeling and propose a joint mean–covariance random effect model based on the modified Cholesky decomposition for longitudinal data. Meanwhile, we apply M-H algorithm to simulate the posterior distributions of model parameters. Besides, a computationally efficient Monte Carlo expectation maximization (MCEM) algorithm is developed for carrying out maximum likelihood estimation. Simulation studies show that the model taking into account the inherent association between mean and covariance has smaller standard deviations of the estimators of parameters, which makes the statistical inferences much more reliable. In the real data analysis, the estimation of parameters in the mean and covariance structure is highly efficient.     

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.     

A Monte Carlo EM Algorithm for De Novo Motif Discovery in Biomolecular Sequences

Motif discovery methods play pivotal roles in deciphering the genetic regulatory codes (i.e., motifs) in genomes as well as in locating conserved domains in protein sequences. The Expectation Maximization (EM) algorithm is one of the most popular methods used in de novo motif discovery. Based on the position weight matrix (PWM) updating technique, this paper presents a Monte Carlo version of the EM motif-finding algorithm that carries out stochastic sampling in local alignment space to overcome the conventional EM's main drawback of being trapped in a local optimum. The newly implemented algorithm is named as Monte Carlo EM Motif Discovery Algorithm (MCEMDA). MCEMDA starts from an initial model, and then it iteratively performs Monte Carlo simulation and parameter update until convergence. A log-likelihood profiling technique together with the top-k strategy is introduced to cope with the phase shifts and multiple modal issues in motif discovery problem. A novel grouping motif alignment (GMA) algorithm is designed to select motifs by clustering a population of candidate local alignments and successfully applied to subtle motif discovery. MCEMDA compares favorably to other popular PWM-based and word enumerative motif algorithms tested using simulated (l, d)-motif cases, documented prokaryotic, and eukaryotic DNA motif sequences. Finally, MCEMDA is applied to detect large blocks of conserved domains using protein benchmarks and exhibits its excellent capacity while compared with other multiple sequence alignment methods.     

A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution

A popular way to represent clustered binary, count, or other data is via the generalized linear mixed model framework, which accommodates correlation through incorporation of random effects. A standard assumption is that the random effects follow a parametric family such as the normal distribution; however, this may be unrealistic or too restrictive to represent the data. We relax this assumption and require only that the distribution of random effects belong to a class of 'smooth' densities and approximate the density by the seminonparametric (SNP) approach of Gallant and Nychka (1987). This representation allows the density to be skewed, multi-modal, fat- or thin-tailed relative to the normal and includes the normal as a special case. Because an efficient algorithm to sample from an SNP density is available, we propose a Monte Carlo EM algorithm using a rejection sampling scheme to estimate the fixed parameters of the linear predictor, variance components and the SNP density. The approach is illustrated by application to a data set and via simulation.     

A Fast EM Algorithm for BayesA-Like Prediction of Genomic Breeding Values

Prediction accuracies of estimated breeding values for economically important traits are expected to benefit from genomic information. Single nucleotide polymorphism (SNP) panels used in genomic prediction are increasing in density, but the Markov Chain Monte Carlo (MCMC) estimation of SNP effects can be quite time consuming or slow to converge when a large number of SNPs are fitted simultaneously in a linear mixed model. Here we present an EM algorithm (termed “fastBayesA”) without MCMC. This fastBayesA approach treats the variances of SNP effects as missing data and uses a joint posterior mode of effects compared to the commonly used BayesA which bases predictions on posterior means of effects. In each EM iteration, SNP effects are predicted as a linear combination of best linear unbiased predictions of breeding values from a mixed linear animal model that incorporates a weighted marker-based realized relationship matrix. Method fastBayesA converges after a few iterations to a joint posterior mode of SNP effects under the BayesA model. When applied to simulated quantitative traits with a range of genetic architectures, fastBayesA is shown to predict GEBV as accurately as BayesA but with less computing effort per SNP than BayesA. Method fastBayesA can be used as a computationally efficient substitute for BayesA, especially when an increasing number of markers bring unreasonable computational burden or slow convergence to MCMC approaches.     

Stochastic Approximation Boosting for Incomplete Data Problems

Summary Boosting is a powerful approach to fitting regression models. This article describes a boosting algorithm for likelihood‐based estimation with incomplete data. The algorithm combines boosting with a variant of stochastic approximation that uses Markov chain Monte Carlo to deal with the missing data. Applications to fitting generalized linear and additive models with missing covariates are given. The method is applied to the Pima Indians Diabetes Data where over half of the cases contain missing values.     

Generalized character process models: estimating the genetic basis of traits that cannot be observed and that change with age or environmental conditions

The genetic analysis of characters that change as a function of some independent and continuous variable has received increasing attention in the biological and statistical literature. Previous work in this area has focused on the analysis of normally distributed characters that are directly observed. We propose a framework for the development and specification of models for a quantitative genetic analysis of function-valued characters that are not directly observed, such as genetic variation in age-specific mortality rates or complex threshold characters. We employ a hybrid Markov chain Monte Carlo algorithm involving a Monte Carlo EM algorithm coupled with a Markov chain approximation to the likelihood, which is quite robust and provides accurate estimates of the parameters in our models. The methods are investigated using simulated data and are applied to a large data set measuring mortality rates in the fruit fly, Drosophila melanogaster.     

Accelerated maximum likelihood parameter estimation for stochastic biochemical systems

ABSTRACT: BACKGROUND: A prerequisite for the mechanistic simulation of a biochemical system is detailed knowledge of its kinetic parameters. Despite recent experimental advances, the estimation of unknown parameter values from observed data is still a bottleneck for obtaining accurate simulation results. Many methods exist for parameter estimation in deterministic biochemical systems; methods for discrete stochastic systems are less well developed. Given the probabilistic nature of stochastic biochemical models, a natural approach is to choose parameter values that maximize the probability of the observed data with respect to the unknown parameters, a.k.a. the maximum likelihood parameter estimates (MLEs). MLE computation for all but the simplest models requires the simulation of many system trajectories that are consistent with experimental data. For models with unknown parameters, this presents a computational challenge, as the generation of consistent trajectories can be an extremely rare occurrence. RESULTS: We have developed Monte Carlo Expectation-Maximization with Modified Cross-Entropy Method (MCEM2): an accelerated method for calculating MLEs that combines advances in rare event simulation with a computationally efficient version of the Monte Carlo expectation-maximization (MCEM) algorithm. Our method requires no prior knowledge regarding parameter values, and it automatically provides a multivariate parameter uncertainty estimate. We applied the method to five stochastic systems of increasing complexity, progressing from an analytically tractable pure-birth model to a computationally demanding model of yeast-polarization. Our results demonstrate that MCEM2 substantially accelerates MLE computation on all tested models when compared to a stand-alone version of MCEM. Additionally, we show how our method identifies parameter values for certain classes of models more accurately than two recently proposed computationally efficient methods. CONCLUSIONS: This work provides a novel, accelerated version of a likelihood-based parameter estimation method that can be readily applied to stochastic biochemical systems. In addition, our results suggest opportunities for added efficiency improvements that will further enhance our ability to mechanistically simulate biological processes.     

Genetic analysis of growth curves using the SAEM algorithm

The analysis of nonlinear function-valued characters is very important in genetic studies, especially for growth traits of agricultural and laboratory species. Inference in nonlinear mixed effects models is, however, quite complex and is usually based on likelihood approximations or Bayesian methods. The aim of this paper was to present an efficient stochastic EM procedure, namely the SAEM algorithm, which is much faster to converge than the classical Monte Carlo EM algorithm and Bayesian estimation procedures, does not require specification of prior distributions and is quite robust to the choice of starting values. The key idea is to recycle the simulated values from one iteration to the next in the EM algorithm, which considerably accelerates the convergence. A simulation study is presented which confirms the advantages of this estimation procedure in the case of a genetic analysis. The SAEM algorithm was applied to real data sets on growth measurements in beef cattle and in chickens. The proposed estimation procedure, as the classical Monte Carlo EM algorithm, provides significance tests on the parameters and likelihood based model comparison criteria to compare the nonlinear models with other longitudinal methods.     

A hidden Markov model for continuous longitudinal data with missing responses and dropout

We propose a hidden Markov model for multivariate continuous longitudinal responses with covariates that accounts for three different types of missing pattern: (I) partially missing outcomes at a given time occasion, (II) completely missing outcomes at a given time occasion (intermittent pattern), and (III) dropout before the end of the period of observation (monotone pattern). The missing-at-random (MAR) assumption is formulated to deal with the first two types of missingness, while to account for the informative dropout, we rely on an extra absorbing state. Estimation of the model parameters is based on the maximum likelihood method that is implemented by an expectation-maximization (EM) algorithm relying on suitable recursions. The proposal is illustrated by a Monte Carlo simulation study and an application based on historical data on primary biliary cholangitis.     

Swift block-updating EM and pseudo-EM procedures for Bayesian shrinkage analysis of quantitative trait loci

Introduction

Virtually all existing expectation-maximization (EM) algorithms for quantitative trait locus (QTL) mapping overlook the covariance structure of genetic effects, even though this information can help enhance the robustness of model-based inferences.

Results

Here, we propose fast EM and pseudo-EM-based procedures for Bayesian shrinkage analysis of QTLs, designed to accommodate the posterior covariance structure of genetic effects through a block-updating scheme. That is, updating all genetic effects simultaneously through many cycles of iterations.

Conclusion

Simulation results based on computer-generated and real-world marker data demonstrated the ability of our method to swiftly produce sensible results regarding the phenotype-to-genotype association. Our new method provides a robust and remarkably fast alternative to full Bayesian estimation in high-dimensional models where the computational burden associated with Markov chain Monte Carlo simulation is often unwieldy. The R code used to fit the model to the data is provided in the online supplementary material.     

Alternative implementations of Monte Carlo EM algorithms for likelihood inferences

Two methods of computing Monte Carlo estimators of variance components using restricted maximum likelihood via the expectation-maximisation algorithm are reviewed. A third approach is suggested and the performance of the methods is compared using simulated data.     

非线性再生散度随机效应模型的极大似然估计及EM算法

非线性再生散度随机效应模型是指数族非线性随机效应模型和非线性再生散度模型的推广和发展．通过视模型中的随机效应为假想的缺失数据和应用Metropolis-Hastings（MH）算法,提出了模型参数极大似然估计的Monte-Carlo EM（MCEM）算法,并用模拟研究和实例分析说明了该算法的可行性．     

High‐Dimensional Variable Selection in Meta‐Analysis for Censored Data

Summary This article considers the problem of selecting predictors of time to an event from a high‐dimensional set of candidate predictors using data from multiple studies. As an alternative to the current multistage testing approaches, we propose to model the study‐to‐study heterogeneity explicitly using a hierarchical model to borrow strength. Our method incorporates censored data through an accelerated failure time model. Using a carefully formulated prior specification, we develop a fast approach to predictor selection and shrinkage estimation for high‐dimensional predictors. For model fitting, we develop a Monte Carlo expectation maximization (MC‐EM) algorithm to accommodate censored data. The proposed approach, which is related to the relevance vector machine (RVM), relies on maximum a posteriori estimation to rapidly obtain a sparse estimate. As for the typical RVM, there is an intrinsic thresholding property in which unimportant predictors tend to have their coefficients shrunk to zero. We compare our method with some commonly used procedures through simulation studies. We also illustrate the method using the gene expression barcode data from three breast cancer studies.