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PubMed2006

Haplotype association analysis for late onset diseases using nuclear family data.

In haplotype-based association studies for late onset diseases, one attractive design is to use available unaffected spouses as controls (Valle et al. [1998] Diab. Care 21:949-958). Given cases and spouses only, the standard expectation-maximization (EM) algorithm (Dempster et al. [1977] J. R. Stat. Soc. B 39:1-38) for case-control data can be used to estimate haplotype frequencies. But often we will have offspring for at least some of the spouse pairs, and offspring genotypes provide additional information about the haplotypes of the parents. Existing methods may either ignore the offspring information, or reconstruct haplotypes for the subjects using offspring information and discard data from those whose haplotypes cannot be reconstructed with high confidence. Neither of these approaches is efficient, and the latter approach may also be biased. For case-control data with some subjects forming spouse pairs and offspring genotypes available for some spouse pairs or individuals, we propose a unified, likelihood-based method of haplotype inference. The method makes use of available offspring genotype information to apportion ambiguous haplotypes for the subjects. For subjects without offspring genotype information, haplotypes are apportioned as in the standard EM algorithm for case-control data. Our method enables efficient haplotype frequency estimation using an EM algorithm and supports probabilistic haplotype reconstruction with the probability calculated based on the whole sample. We describe likelihood ratio and permutation tests to test for disease-haplotype association, and describe three test statistics that are potentially useful for detecting such an association.

Age of Onset↗

PubMed2006

Study of a fast discriminative training algorithm for pattern recognition.

Discriminative training refers to an approach to pattern recognition based on direct minimization of a cost function commensurate with the performance of the recognition system. This is in contrast to the procedure of probability distribution estimation as conventionally required in Bayes' formulation of the statistical pattern recognition problem. Currently, most discriminative training algorithms for nonlinear classifier designs are based on gradient-descent (GD) methods for cost minimization. These algorithms are easy to derive and effective in practice, but are slow in training speed and have difficulty selecting the learning rates. To address the problem, we present our study on a fast discriminative training algorithm. The algorithm initializes the parameters by the expectation-maximization (EM) algorithm, and then uses a set of closed-form formulas derived in this paper to further optimize a proposed objective of minimizing error rate. Experiments in speech applications show that the algorithm provides better recognition accuracy in a fewer iterations than the EM algorithm and a neural network trained by hundreds of GD iterations. Although some convergent properties need further research, the proposed objective and derived formulas can benefit further study of the problem.

Algorithms↗

PubMed1994

Implementation of an accelerated iterative algorithm for cone-beam SPECT.

In this paper we describe the implementation of an accelerated iterative reconstruction algorithm (AIRA) for cone-beam (CB) projections using a single circular orbit in single-photon-emission computed tomography (SPECT). This algorithm is a modified maximum-likelihood-expectation-maximization (ML-EM) algorithm and several approaches have been used to accelerate the reconstruction process. These approaches include: (i) the use of ordered subsets; (ii) the use of active areas and volumes; and (iii) the storing in memory of the transition vector for a given ray (during the forward projection step). This algorithm, which compensates for collimator geometric sensitivity variation as a function of position and makes uniform attenuation corrections has been evaluated using experimentally acquired phantom data. The results demonstrate a two-orders-of-magnitude decrease of the computational time of this algorithm over the conventional ML-EM algorithm with similar convergence properties.

Algorithms↗

PubMed2001

Statistical estimation and pedigree analysis of CCR2-CCR5 haplotypes.

As more SNP marker data becomes available, researchers have used haplotypes of markers, rather than individual polymorphisms, for association analysis of candidate genes. In order to perform haplotype analysis in a population-based case-control study, haplotypes must be determined by estimation in the absence of family information or laboratory methods for establishing phase. Here, we test the accuracy of the Expectation-Maximization (EM) algorithm for estimating haplotype state and frequency in the CCR2-CCR5 gene region by comparison with haplotype state and frequency determined by pedigree analysis. To do this, we have characterized haplotypes comprising alleles at seven biallelic loci in the CCR2-CCR5 chemokine receptor gene region, a span of 20 kb on chromosome 3p21. Three-generation CEPH families (n=40), totaling 489 individuals, were genotyped by the 5'nuclease assay (TaqMan). Haplotype states and frequencies were compared in 103 grandparents who were assumed to have mated at random. Both pedigree analysis and the EM algorithm yielded the same small number of haplotypes for which linkage disequilibrium was nearly maximal. The haplotype frequencies generated by the two methods were nearly identical. These results suggest that the EM algorithm estimation of haplotype states, frequency, and linkage disequilibrium analysis will be an effective strategy in the CCR2-CCR5 gene region. For genetic epidemiology studies, CCR2-CCR5 allele and haplotype frequencies were determined in African-American (n=30), Hispanic (n=24) and European-American (n=34) populations.

Alleles↗

PubMed2006

Imputation methods to improve inference in SNP association studies.

Missing single nucleotide polymorphisms (SNPs) are quite common in genetic association studies. Subjects with missing SNPs are often discarded in analyses, which may seriously undermine the inference of SNP-disease association. In this article, we develop two haplotype-based imputation approaches and one tree-based imputation approach for association studies. The emphasis is to evaluate the impact of imputation on parameter estimation, compared to the standard practice of ignoring missing data. Haplotype-based approaches build on haplotype reconstruction by the expectation-maximization (EM) algorithm or a weighted EM (WEM) algorithm, depending on whether case-control status is taken into account. The tree-based approach uses a Gibbs sampler to iteratively sample from a full conditional distribution, which is obtained from the classification and regression tree (CART) algorithm. We employ a standard multiple imputation procedure to account for the uncertainty of imputation. We apply the methods to simulated data as well as a case-control study on developmental dyslexia. Our results suggest that imputation generally improves efficiency over the standard practice of ignoring missing data. The tree-based approach performs comparably well as haplotype-based approaches, but the former has a computational advantage. The WEM approach yields the smallest bias at a price of increased variance.

Algorithms↗

PubMed2002

SMEM algorithm is not fully compatible with maximum-likelihood framework.

The expectation-maximization (EM) algorithm with split-and-merge operations (SMEM algorithm) proposed by Ueda, Nakano, Ghahramani, and Hinton (2000) is a nonlocal searching method, applicable to mixture models, for relaxing the local optimum property of the EM algorithm. In this article, we point out that the SMEM algorithm uses the acceptance-rejection evaluation method, which may pick up a distribution with smaller likelihood, and demonstrate that an increase in likelihood can then be guaranteed only by comparing log likelihoods.

Journal Article↗

PubMed2002

Adjusting the outputs of a classifier to new a priori probabilities: a simple procedure.

It sometimes happens (for instance in case control studies) that a classifier is trained on a data set that does not reflect the true a priori probabilities of the target classes on real-world data. This may have a negative effect on the classification accuracy obtained on the real-world data set, especially when the classifier's decisions are based on the a posteriori probabilities of class membership. Indeed, in this case, the trained classifier provides estimates of the a posteriori probabilities that are not valid for this real-world data set (they rely on the a priori probabilities of the training set). Applying the classifier as is (without correcting its outputs with respect to these new conditions) on this new data set may thus be suboptimal. In this note, we present a simple iterative procedure for adjusting the outputs of the trained classifier with respect to these new a priori probabilities without having to refit the model, even when these probabilities are not known in advance. As a by-product, estimates of the new a priori probabilities are also obtained. This iterative algorithm is a straightforward instance of the expectation-maximization (EM) algorithm and is shown to maximize the likelihood of the new data. Thereafter, we discuss a statistical test that can be applied to decide if the a priori class probabilities have changed from the training set to the real-world data. The procedure is illustrated on different classification problems involving a multilayer neural network, and comparisons with a standard procedure for a priori probability estimation are provided. Our original method, based on the EM algorithm, is shown to be superior to the standard one for a priori probability estimation. Experimental results also indicate that the classifier with adjusted outputs always performs better than the original one in terms of classification accuracy, when the a priori probability conditions differ from the training set to the real-world data. The gain in classification accuracy can be significant.

Classification↗

PubMed2005

Using evolutionary Expectation Maximization to estimate indel rates.

MOTIVATION: The Expectation Maximization (EM) algorithm, in the form of the Baum-Welch algorithm (for hidden Markov models) or the Inside-Outside algorithm (for stochastic context-free grammars), is a powerful way to estimate the parameters of stochastic grammars for biological sequence analysis. To use this algorithm for multiple-sequence evolutionary modelling, it would be useful to apply the EM algorithm to estimate not only the probability parameters of the stochastic grammar, but also the instantaneous mutation rates of the underlying evolutionary model (to facilitate the development of stochastic grammars based on phylogenetic trees, also known as Statistical Alignment). Recently, we showed how to do this for the point substitution component of the evolutionary process; here, we extend these results to the indel process. RESULTS: We present an algorithm for maximum-likelihood estimation of insertion and deletion rates from multiple sequence alignments, using EM, under the single-residue indel model owing to Thorne, Kishino and Felsenstein (the 'TKF91' model). The algorithm converges extremely rapidly, gives accurate results on simulated data that are an improvement over parsimonious estimates (which are shown to underestimate the true indel rate), and gives plausible results on experimental data (coronavirus envelope domains). Owing to the algorithm's close similarity to the Baum-Welch algorithm for training hidden Markov models, it can be used in an 'unsupervised' fashion to estimate rates for unaligned sequences, or estimate several sets of rates for sequences with heterogenous rates. AVAILABILITY: Software implementing the algorithm and the benchmark is available under GPL from http://www.biowiki.org/

Algorithms↗

PubMed2000

Maximum likelihood analysis of quantitative trait loci under selective genotyping.

Selective genotyping is a cost-saving strategy in mapping quantitative trait loci (QTLs). When the proportion of individuals selected for genotyping is low, the majority of the individuals are not genotyped, but their phenotypic values, if available, are still included in the data analysis to correct the bias in parameter estimation. These ungenotyped individuals do not contribute much information about linkage analysis and their inclusion can substantially increase the computational burden. For multiple trait analysis, ungenotyped individuals may not have a full array of phenotypic measurements. In this case, unbiased estimation of QTL effects using current methods seems to be impossible. In this study, we develop a maximum likelihood method of QTL mapping under selective genotyping using only the phenotypic values of genotyped individuals. Compared with the full data analysis (using all phenotypic values), the proposed method performs well. We derive an expectation-maximization (EM) algorithm that appears to be a simple modification of the existing EM algorithm for standard interval mapping. The new method can be readily incorporated into a standard QTL mapping software, e.g. MAPMAKER. A general recommendation is that whenever full data analysis is possible, the full maximum likelihood analysis should be performed. If it is impossible to analyse the full data, e.g. sample sizes are too large, phenotypic values of ungenotyped individuals are missing or composite interval mapping is to be performed, the proposed method can be applied.

Algorithms↗

PubMed2006

Accurate haplotype inference for multiple linked single-nucleotide polymorphisms using sibship data.

Sibships are commonly used in genetic dissection of complex diseases, particularly for late-onset diseases. Haplotype-based association studies have been advocated as powerful tools for fine mapping and positional cloning of complex disease genes. Existing methods for haplotype inference using data from relatives were originally developed for pedigree data. In this study, we proposed a new statistical method for haplotype inference for multiple tightly linked single-nucleotide polymorphisms (SNPs), which is tailored for extensively accumulated sibship data. This new method was implemented via an expectation-maximization (EM) algorithm without the usual assumption of linkage equilibrium among markers. Our EM algorithm does not incur extra computational burden for haplotype inference using sibship data when compared with using unrelated parental data. Furthermore, its computational efficiency is not affected by increasing sibship size. We examined the robustness and statistical performance of our new method in simulated data created from an empirical haplotype data set of human growth hormone gene 1. The utility of our method was illustrated with an application to the analyses of haplotypes of three candidate genes for osteoporosis.

Algorithms↗

PubMed2004

Method for using complete and incomplete trios to identify genes related to a quantitative trait.

A number of tests for linkage and association with qualitative traits have been developed, with the most well-known being the transmission/disequilibrium test (TDT). For quantitative traits, varying extensions of the TDT have been suggested. The quantitative trait approach we propose is based on extending the log-linear model for case-parent trio data (Weinberg et al. [1998] Am. J. Hum. Genet. 62:969-978). Like the log-linear approach for qualitative traits, our proposed polytomous logistic approach for quantitative traits allows for population admixture by conditioning on parental genotypes. Compared to other methods, simulations demonstrate good power and robustness of the proposed test under various scenarios of the genotype effect, distribution of the quantitative trait, and population stratification. In addition, missing parental genotype data can be accommodated through an expectation-maximization (EM) algorithm approach. The EM approach allows recovery of most of the lost power due to incomplete trios.

Computer Simulation↗

PubMed2006

Precise segmentation of multimodal images.

We propose new techniques for unsupervised segmentation of multimodal grayscale images such that each region-of-interest relates to a single dominant mode of the empirical marginal probability distribution of grey levels. We follow the most conventional approaches in that initial images and desired maps of regions are described by a joint Markov-Gibbs random field (MGRF) model of independent image signals and interdependent region labels. However, our focus is on more accurate model identification. To better specify region borders, each empirical distribution of image signals is precisely approximated by a linear combination of Gaussians (LCG) with positive and negative components. We modify an expectation-maximization (EM) algorithm to deal with the LCGs and also propose a novel EM-based sequential technique to get a close initial LCG approximation with which the modified EM algorithm should start. The proposed technique identifies individual LCG models in a mixed empirical distribution, including the number of positive and negative Gaussians. Initial segmentation based on the LCG models is then iteratively refined by using the MGRF with analytically estimated potentials. The convergence of the overall segmentation algorithm at each stage is discussed. Experiments show that the developed techniques segment different types of complex multimodal medical images more accurately than other known algorithms.

Algorithms↗

PubMed2002

A thematic analysis of the AIDS literature.

Faced with the need for human comprehension of any large collection of objects, a time honored approach has been to cluster the objects into groups of closely related objects. Individual groups are then summarized in some convenient manner to provide a more manageable view of the data. Such methods have been applied to document collections with mixed results. If a hard clustering of the data into mutually exclusive clusters is performed then documents are frequently forced into one cluster when they may contain important information that would also appropriately make them candidates for other clusters. If a soft clustering is used there still remains the problem of how to provide a useful summary of the data in a cluster. Here we introduce a new algorithm to produce a soft clustering of document collections that is based on the concept of a theme. A theme is conceptually a subject area that is discussed by multiple documents in the database. A theme has two potential representations that may be viewed as dual to each other. First it is represented by the set of documents that discuss the subject or theme and second it is also represented by the set of key terms that are typically used to discuss the theme. Our algorithm is an EM algorithm in which the term representation and the document representation are explicit components and each is used to refine the other in an alternating fashion. Upon convergence the term representation provides a natural summary of the document representation (the cluster). We describe how to optimize the themes produced by this process and give the results of applying the method to a database of over fifty thousand PubMed documents dealing with the subject of AIDS. How themes may improve access to a document collection is also discussed.

Acquired Immunodeficiency Syndrome↗

PubMed2001

Receiver operating characteristic (ROC) analysis of images reconstructed with iterative expectation maximization algorithms.

PURPOSE: The quality of images reconstructed by means of the maximum likelihood-expectation maximization (ML-EM) and ordered subset (OS)-EM algorithms, was examined with parameters such as the number of iterations and subsets, then compared with the quality of images reconstructed by the filtered back projection method. METHODS: Phantoms showing signals inside signals, which mimicked single-photon emission computed tomography (SPECT) images of cerebral blood flow and myocardial perfusion, and phantoms showing signals around the signals obtained by SPECT of bone and tumor were used for experiments. To determine signals for recognition, SPECT images in which the signals could be appropriately recognized with a combination of fewer iterations and subsets of different sizes and densities were evaluated by receiver operating characteristic (ROC) analysis. The results of ROC analysis were applied to myocardial phantom experiments and scintigraphy of myocardial perfusion. RESULTS: Taking the image processing time into consideration, good SPECT images were obtained by OS-EM at iteration No. 10 and subset 5. CONCLULSION: This study will be helpful for selection of parameters such as the number of iterations and subsets when using the ML-EM or OS-EM algorithms.

Algorithms↗

PubMed2002

Haplotype frequency estimation in patient populations: the effect of departures from Hardy-Weinberg proportions and collapsing over a locus in the HLA region.

Haplotype analyses are an important area in the study of the genetic components of human disease. Associations between markers and disease loci that are not evident with a single marker locus may be identified in multi-locus marker analyses using estimated haplotype frequencies (HFs). Procedures that make use of the expectation-maximization (EM) algorithm to estimate HFs from unphased genotype data are in common use in genetic studies. The EM algorithm uses these unphased genotype frequencies along with the assumption of Hardy-Weinberg proportions (HWP) to converge on HF estimates. In this paper, we assess the accuracy of EM estimates of HFs in patients with type I diabetes for whom the true haplotypes are known, but the data are analyzed ignoring family information to allow comparison between estimated and true frequencies. The data consist of six HLA loci with high levels of polymorphism and a range of departures from HWP and linkage equilibrium. While the overall accuracy of the EM estimates is good, there can be large over- and underestimates of particular HFs, even for common haplotypes, especially when the loci involved deviate significantly from HWP. Estimating HFs for three or more loci and then collapsing over loci so as to generate two locus haplotypes can improve the accuracy of the estimation. The collapsing procedure is most beneficial when one of the loci in the two-locus haplotype of interest deviates significantly from HWP and the locus collapsed over is in linkage disequilibrium with the other loci.

Algorithms↗

PubMed1999

Stochastic algorithms for Markov models estimation with intermittent missing data.

Multistate Markov models are frequently used to characterize disease processes, but their estimation from longitudinal data is often hampered by complex patterns of incompleteness. Two algorithms for estimating Markov chain models in the case of intermittent missing data in longitudinal studies, a stochastic EM algorithm and the Gibbs sampler, are described. The first can be viewed as a random perturbation of the EM algorithm and is appropriate when the M step is straightforward but the E step is computationally burdensome. It leads to a good approximation of the maximum likelihood estimates. The Gibbs sampler is used for a full Bayesian inference. The performances of the two algorithms are illustrated on two simulated data sets. A motivating example concerned with the modelling of the evolution of parasitemia by Plasmodium falciparum (malaria) in a cohort of 105 young children in Cameroon is described and briefly analyzed.

Algorithms↗

PubMed2004

Statistical list-mode image reconstruction for the high resolution research tomograph.

We have investigated statistical list-mode reconstruction applicable to a depth-encoding high resolution research tomograph. An image non-negativity constraint has been employed in the reconstructions and is shown to effectively remove the overestimation bias introduced by the sinogram non-negativity constraint. We have furthermore implemented a convergent subsetized (CS) list-mode reconstruction algorithm, based on previous work (Hsiao et al 2002 Conf. Rec. SPIE Med. Imaging 4684 10-19; Hsiao et al 2002 Conf. Rec. IEEE Int. Symp. Biomed. Imaging 409-12) on convergent histogram OSEM reconstruction. We have demonstrated that the first step of the convergent algorithm is exactly equivalent (unlike the histogram-mode case) to the regular subsetized list-mode EM algorithm, while the second and final step takes the form of additive updates in image space. We have shown that in terms of contrast, noise as well as FWHM width behaviour, the CS algorithm is robust and does not result in limit cycles. A hybrid algorithm based on the ordinary and the convergent algorithms is also proposed, and is shown to combine the advantages of the two algorithms (i.e. it is able to reach a higher image quality in fewer iterations while maintaining the convergent behaviour), making the hybrid approach a good alternative to the ordinary subsetized list-mode EM algorithm.

Algorithms↗

PubMed1989

Probits of mixtures.

The tolerances of individuals (insects, parasites) in a population have a frequency or probability distribution called a tolerance distribution. Many tolerance distributions in bioassay studies can be the result of a rather heterogeneous population of individuals and can often be modelled as a mixture of a number of standard unimodal distributions. A probit analysis can be generalized to the case where the tolerance distribution is a mixture of location and scale parameter distributions. In this article, the existence and determination of the maximum likelihood estimates are investigated. An expectation-maximization (EM) algorithm for probits of mixtures is developed and it is shown that by application of the EM algorithm, the problem of probits of mixtures can be separated into a series of probits of individual component tolerance distributions.

Algorithms↗