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PubMed2002

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.

Animals↗
PubMed2006

Estimate haplotype frequencies in pedigrees.

BACKGROUND: Haplotype analysis has gained increasing attention in the context of association studies of disease genes and drug responsivities over the last years. The potential use of haplotypes has led to the initiation of the HapMap project which is to investigate haplotype patterns in the human genome in different populations. Haplotype inference and frequency estimation are essential components of this endeavour. RESULTS: We present a two-stage method to estimate haplotype frequencies in pedigrees, which includes haplotyping stage and estimation stage. In the haplotyping stage, we propose a linear time algorithm to determine all zero-recombinant haplotype configurations for each pedigree. In the estimation stage, we use the expectation-maximization (EM) algorithm to estimate haplotype frequencies based on these haplotype configurations. The experiments demonstrate that our method runs much faster and gives more credible estimates than other popular haplotype analysis software that discards the pedigree information. CONCLUSION: Our method suggests that pedigree information is of great importance in haplotype analysis. It can be used to speedup estimation process, and to improve estimation accuracy as well. The result also demonstrates that the whole haplotype configuration space can be substituted by the space of zero-recombinant haplotype configurations in haplotype frequency estimation, especially when the considered haplotype block is relatively short.

Base Sequence↗
PubMed2004

Maximum likelihood estimation for Cox's regression model under nested case-control sampling.

Nested case-control sampling is designed to reduce the costs of large cohort studies. It is important to estimate the parameters of interest as efficiently as possible. We present a new maximum likelihood estimator (MLE) for nested case-control sampling in the context of Cox's proportional hazards model. The MLE is computed by the EM-algorithm, which is easy to implement in the proportional hazards setting. Standard errors are estimated by a numerical profile likelihood approach based on EM aided differentiation. The work was motivated by a nested case-control study that hypothesized that insulin-like growth factor I was associated with ischemic heart disease. The study was based on a population of 3784 Danes and 231 cases of ischemic heart disease where controls were matched on age and gender. We illustrate the use of the MLE for these data and show how the maximum likelihood framework can be used to obtain information additional to the relative risk estimates of covariates.

Adult↗
PubMed1998

Dual matrix ordered subsets reconstruction for accelerated 3D scatter compensation in single-photon emission tomography.

Three-dimensional (3D) iterative maximum likelihood expectation maximization (ML-EM) algorithms for single-photon emission tomography (SPET) are capable of correcting image-degrading effects of non-uniform attenuation, distance-dependent camera response and patient shape-dependent scatter. However, the resulting improvements in quantitation, resolution and signal-to-noise ratio (SNR) are obtained at the cost of a huge computational burden. This paper presents a new acceleration method for ML-EM: dual matrix ordered subsets (DM-OS). DM-OS combines two acceleration methods: (a) different matrices for projection and back-projection and (b) ordered subsets of projections. DM-OS was compared with ML-EM on simulated data and on physical thorax phantom data, for both 180 degrees and 360 degrees orbits. Contrast, normalized standard deviation and mean squared error were calculated for the digital phantom experiment. DM-OS resulted in similar image quality to ML-EM, even for speed-up factors of 200 compared to ML-EM in the case of 120 projections. The thorax phantom data could be reconstructed 50 times faster (60 projections) using DM-OS with preservation of image quality. ML-EM and DM-OS with scatter compensation showed significant improvement of SNR compared to ML-EM without scatter compensation. Furthermore, inclusion of complex image formation models in the computer code is simplified in the case of DM-OS. It is thus shown that DM-OS is a fast and relatively simple algorithm for 3D iterative scatter compensation, with similar results to conventional ML-EM, for both 180 degrees and 360 degrees acquired data.

Algorithms↗
PubMed1992

Estimating polygenic models for multivariate data on large pedigrees.

We have developed algorithms for the likelihood estimation of additive genetic models for quantitative traits on large pedigrees. The approach uses the expectation L-maximization (EM) algorithm, but avoids intensive computation. In this paper, we focus on extensions of previous work to the case of multivariate data. We exemplify the approach by analyses of bivariate data on a four-generation, 949-member pedigree of the snail Lymnaea elodes, and on a three-generation pedigree of the guppy Poecilia reticulata containing about 400 individuals.

Algorithms↗
PubMed2004

Relation between weight matrix and substitution matrix: motif search by similarity.

MOTIVATION: The discovery of patterns shared by several sequences that differ greatly is a basic task in sequence analysis, and still a challenge. Several methods have been developed for detecting patterns. Methods commonly used for motif search include the Gibbs sampler, Expectation-Maximization (EM) algorithm and some intuitive greedy approaches. One cannot guarantee the optimality of the result produced by the Gibbs sampler in a single run. The deterministic EM methods tend to get trapped by local optima. Solutions found by greedy approaches are rarely sufficiently good. RESULTS: A simple model describing a motif or a portion of local multiple sequence alignment is the weight matrix model, in which a motif is characterized with position-specific probabilities. Two substitution matrices are proposed to relate the sequence similarity with the weight matrix. Combining the substitution matrix and weight matrix, we examine three typical sets of protein sequences with increasing complexity. At a low score threshold for pair similarity, sliding windows are compared with a seed window to find the score sum, which provides a measure of statistical significance for multiple sequence comparison. Such a similarity analysis reveals many aspects of motifs. Blocks determined by similarity can be used to deduce a primary weight matrix or an improved substitution matrix. The algorithm successfully obtains the optimal solution for the test sets by just greedy iteration.

Algorithms↗
PubMed1997

Weighted least-squares reconstruction methods for positron emission tomography.

We present unpenalized and penalized weighted least-squares (WLS) reconstruction methods for positron emission tomography (PET), where the weights are based on the covariance of a model error and depend on the unknown parameters. The penalty function for the latter method is chosen so that certain a priori information is incorporated. The algorithms used to minimize the WLS objective functions guarantee nonnegative estimates and, experimentally, they converged faster than the maximum likelihood expectation-maximization (ML-EM) algorithm and produced images that had significantly better resolution and contrast. Although simulations suggest that the proposed algorithms are globally convergent, a proof of convergence has not yet been found. Nevertheless, we are able to show that the unpenalized method produces estimates that decrease the objective function monotonically with increasing iterations.

Algorithms↗
PubMed2007

Classification of spatiotemporal hemodynamics from brain perfusion MR images using expectation-maximization estimation with finite mixture of multivariate gaussian distributions.

The ability to cluster different perfusion compartments in the brain is critical for analyzing brain perfusion. This study presents a method based on a mixture of multivariate Gaussians (MoMG) and the expectation-maximization (EM) algorithm to dissect various perfusion compartments from dynamic susceptibility contrast (DSC) MR images so that each compartment comprises pixels of similar signal-time curves. This EM-based method provides an objective way to 1) delineate an area to serve as the in-plane arterial input function (AIF) of the feeding artery for adjacent tissues to better quantify the relative cerebral blood volume (rCBV), relative cerebral blood flow (rCBF), and mean transit time (MTT); 2) demarcate regions with abnormal perfusion derangement to facilitate diagnosis; and 3) obtain parametric maps with supplementary information, such as temporal scenarios and recirculation of contrast agent. Results from normal subjects show that perfusion cascade manifests (in order of appearance) the arteries, gray matter (GM), white matter (WM), veins and sinuses, and choroid plexus mixed with cerebrospinal fluid (CSF). The averaged rCBV, rCBF, and MTT ratios between GM and WM are in good agreement with those in the literature. Results from a patient with cerebral arteriovenous malformation (CAVM) showed distinct spatiotemporal characteristics between perfusion patterns, which allowed differentiation between pathological and nonpathological areas.

Adolescent↗
PubMed2000

Standard errors for EM estimates in generalized linear models with random effects.

A procedure is derived for computing standard errors of EM estimates in generalized linear models with random effects. Quadrature formulas are used to approximate the integrals in the EM algorithm, where two different approaches are pursued, i.e., Gauss-Hermite quadrature in the case of Gaussian random effects and nonparametric maximum likelihood estimation for an unspecified random effect distribution. An approximation of the expected Fisher information matrix is derived from an expansion of the EM estimating equations. This allows for inferential arguments based on EM estimates, as demonstrated by an example and simulations.

Algorithms↗
PubMed2004

Bayesian inference for randomized clinical trials with treatment failures.

During the course of a clinical trial, subjects may experience treatment failure. For ethical reasons, it is necessary to administer emergency or rescue medications for such subjects. However, the rescue medications may bias the set of response measurements. This bias is of particular concern if a subject has been randomized to the control group, and the rescue medications improve the subject's condition. The standard approach to analysing data from a clinical trial is to perform an intent-to-treat (ITT) analysis, wherein the data are analysed according to treatment randomization. Supplementary analyses may be performed in addition to the ITT analysis to account for the effect of treatment failures and rescue medications. A Bayesian, counterfactual approach, which uses the data augmentation (DA) algorithm, is proposed for supplemental analysis. A simulation study is conducted to compare the operating characteristics of this procedure with a likelihood-based, counterfactual approach based on the EM algorithm. An example from the Asthma Clinical Research Network (ACRN) is used to illustrate the Bayesian procedure.

Algorithms↗
PubMed2004

An accelerated convergent ordered subsets algorithm for emission tomography.

We propose an algorithm, E-COSEM (enhanced complete-data ordered subsets expectation-maximization), for fast maximum likelihood (ML) reconstruction in emission tomography. E-COSEM is founded on an incremental EM approach. Unlike the familiar OSEM (ordered subsets EM) algorithm which is not convergent, we show that E-COSEM converges to the ML solution. Alternatives to the OSEM include RAMLA, and for the related maximum a posteriori (MAP) problem, the BSREM and OS-SPS algorithms. These are fast and convergent, but require ajudicious choice of a user-specified relaxation schedule. E-COSEM itself uses a sequence of iteration-dependent parameters (very roughly akin to relaxation parameters) to control a tradeoff between a greedy, fast but non-convergent update and a slower but convergent update. These parameters are computed automatically at each iteration and require no user specification. For the ML case, our simulations show that E-COSEM is nearly as fast as RAMLA.

Algorithms↗
PubMed1989

Stochastic models for heterogeneous DNA sequences.

The composition of naturally occurring DNA sequences is often strikingly heterogeneous. In this paper, the DNA sequence is viewed as a stochastic process with local compositional properties determined by the states of a hidden Markov chain. The model used is a discrete-state, discrete-outcome version of a general model for non-stationary time series proposed by Kitagawa (1987). A smoothing algorithm is described which can be used to reconstruct the hidden process and produce graphic displays of the compositional structure of a sequence. The problem of parameter estimation is approached using likelihood methods and an EM algorithm for approximating the maximum likelihood estimate is derived. The methods are applied to sequences from yeast mitochondrial DNA, human and mouse mitochondrial DNAs, a human X chromosomal fragment and the complete genome of bacteriophage lambda.

Base Sequence↗
PubMed2000

Improvement of brain perfusion SPET using iterative reconstruction with scatter and non-uniform attenuation correction.

Filtered back-projection (FBP) is generally used as the reconstruction method for single-photon emission tomography although it produces noisy images with apparent streak artefacts. It is possible to improve the image quality by using an algorithm with iterative correction steps. The iterative reconstruction technique also has an additional benefit in that computation of attenuation correction can be included in the process. A commonly used iterative method, maximum-likelihood expectation maximisation (ML-EM), can be accelerated using ordered subsets (OS-EM). We have applied to the OS-EM algorithm a Bayesian one-step late correction method utilising median root prior (MRP). Methodological comparison was performed by means of measurements obtained with a brain perfusion phantom and using patient data. The aim of this work was to quantitate the accuracy of iterative reconstruction with scatter and non-uniform attenuation corrections and post-filtering in SPET brain perfusion imaging. SPET imaging was performed using a triple-head gamma camera with fan-beam collimators. Transmission and emission scans were acquired simultaneously. The brain phantom used was a high-resolution three-dimensional anthropomorphic JB003 phantom. Patient studies were performed in ten chronic pain syndrome patients. The images were reconstructed using conventional FBP and iterative OS-EM and MRP techniques including scatter and nonuniform attenuation corrections. Iterative reconstructions were individually post-filtered. The quantitative results obtained with the brain perfusion phantom were compared with the known actual contrast ratios. The calculated difference from the true values was largest with the FBP method; iteratively reconstructed images proved closer to the reality. Similar findings were obtained in the patient studies. The plain OS-EM method improved the contrast whereas in the case of the MRP technique the improvement in contrast was not so evident with post-filtering.

Brain↗
PubMed2004

Simultaneous estimation of haplotype frequencies and quantitative trait parameters: applications to the test of association between phenotype and diplotype configuration.

The analysis of the haplotype-phenotype relationship has become more and more important. We have developed an algorithm, using individual genotypes at linked loci as well as their quantitative phenotypes, to estimate the parameters of the distribution of the phenotypes for subjects with and without a particular haplotype by an expectation-maximization (EM) algorithm. We assumed that the phenotype for a diplotype configuration follows a normal distribution. The algorithm simultaneously calculates the maximum likelihood (L0max) under the null hypothesis (i.e., nonassociation between the haplotype and phenotype), and the maximum likelihood (Lmax) under the alternative hypothesis (i.e., association between the haplotype and phenotype). Then we tested the association between the haplotype and the phenotype using a test statistic, -2 log(L0max/Lmax). The above algorithm along with some extensions for different modes of inheritance was implemented as a computer program, QTLHAPLO. Simulation studies using single-nucleotide polymorphism (SNP) genotypes have clarified that the estimation was very accurate when the linkage disequilibrium between linked loci was rather high. Empirical power using the simulated data was high enough. We applied QTLHAPLO for the analysis of the real data of the genotypes at the calpain 10 gene obtained from diabetic and control subjects in various laboratories.

Algorithms↗
PubMed1999

A projector/backprojector with slice-to-slice blurring for efficient three-dimensional scatter modeling.

Scatter correction is an important factor in single photon emission computed tomography (SPECT). Many scatter correction techniques, such as multiple-window subtraction and intrinsic modeling with iterative algorithms, have been under study for many years. Previously, we developed an efficient slice-to-slice blurring technique to model attenuation and system geometric response in a projector/backprojector pair, which was used in an ML-EM algorithm to reconstruct SPECT data. This paper proposes a projector/backprojector that models the three-dimensional (3-D) first-order scatter in SPECT, also using an efficient slice-to-slice blurring technique. The scatter response is estimated from a known nonuniform attenuation distribution map. It is assumed that the probability of detection of a first-order scattered photon from a photon that is emitted in a given source voxel and scattered in a given scatter voxel is proportional to the attenuation coefficient value at that voxel. Monte Carlo simulations of point sources and an MCAT torso phantom were used to verify the accuracy of the proposed projector/backprojector model. An experimental Jaszczak torso/cardiac phantom SPECT study was also performed. For a 64 x 64 x 64 image volume, it took 8.7 s to perform each iteration per slice on a Sun ULTRA Enterprise 3000 (167 MHz, 1 Gbyte RAM) computer, when modeling 3-D scatter, attenuation, and system geometric response functions. The main advantage of the proposed method is its easy implementation and the possibility of performing reconstruction in clinically acceptable time.

Computer Simulation↗
PubMed1999

An adaptive segmentation algorithm for time-of-flight MRA data.

A three-dimensional (3-D) representation of cerebral vessel morphology is essential for neuroradiologists treating cerebral aneurysms. However, current imaging techniques cannot provide such a representation. Slices of MR angiography (MRA) data can only give two-dimensional (2-D) descriptions and ambiguities of aneurysm position and size arising in X-ray projection images can often be intractable. To overcome these problems, we have established a new automatic statistically based algorithm for extracting the 3-D vessel information from time-of-flight (TOF) MRA data. We introduce distributions for the data, motivated by a physical model of blood flow, that are used in a modified version of the expectation maximization (EM) algorithm. The estimated model parameters are then used to classify statistically the voxels into vessel or other brain tissue classes. The algorithm is adaptive because the model fitting is performed recursively so that classifications are made on local subvolumes of data. We present results from applying our algorithm to several real data sets that contain both artery and aneurysm structures of various sizes.

Algorithms↗
PubMed2004

Evaluations of maximization procedures for estimating linkage parameters under heterogeneity.

Locus heterogeneity is a major problem plaguing the mapping of disease genes responsible for complex genetic traits via linkage analysis. A common feature of several available methods to account for heterogeneity is that they involve maximizing a multidimensional likelihood to obtain maximum likelihood estimates. The high dimensionality of the likelihood surface may be due to multiple heterogeneity (mixing) parameters, linkage parameters, and/or regression coefficients corresponding to multiple covariates. Here, we focus on this nontrivial computational aspect of incorporating heterogeneity by considering several likelihood maximization procedures, including the expectation maximization (EM) algorithm and the stochastic expectation maximization (SEM) algorithm. The wide applicability of these procedures is demonstrated first through a general formulation of accounting for heterogeneity, and then by applying them to two specific formulations. Furthermore, our simulation studies as well as an application to the Genetic Analysis Workshop 12 asthma datasets show that, among other observations, SEM performs better than EM. As an aside, we illustrate a limitation of the popular admixture approach for incorporating heterogeneity, proved elsewhere. We also show how to obtain standard errors (SEs) for EM and SEM estimates, using methods available in the literature. These SEs can then be combined with the corresponding estimates to provide confidence intervals of the parameters.

Algorithms↗
PubMed2003

A comparison of two methods for the estimation of precision with incomplete longitudinal data, jointly modelled with a time-to-event outcome.

Several methods for the estimation and comparison of rates of change in longitudinal studies with staggered entry and informative drop-outs have been recently proposed. For multivariate normal linear models, REML estimation is used. There are various approaches to maximizing the corresponding log-likelihood; in this paper we use a restricted iterative generalized least squares method (RIGLS) combined with a nested EM algorithm. An important statistical problem in such approaches is the estimation of the standard errors adjusted for the missing data (observed data information matrix). Louis has provided a general technique for computing the observed data information in terms of completed data quantities within the EM framework. The multiple imputation (MI) method for obtaining variances can be regarded as an alternative to this. The aim of this paper is to develop, apply and compare the Louis and a modified MI method in the setting of longitudinal studies where the source of missing data is either death or disease progression (informative) or end of the study (assumed non-informative). Longitudinal data are simultaneously modelled with the missingness process. The methods are illustrated by modelling CD4 count data from an HIV-1 clinical trial and evaluated through simulation studies. Both methods, Louis and MI, are used with Monte Carlo simulations of the missing data using the appropriate conditional distributions, the former with 100 simulations, the latter with 5 and 10. It is seen that naive SEs based on the completed data likelihood can be seriously biased. This bias was largely corrected by Louis and modified MI methods, which gave broadly similar estimates. Given the relative simplicity of the modified MI method, it may be preferable.

Algorithms↗