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PubMed2006

An expectation-maximization algorithm for probabilistic reconstructions of full-length isoforms from splice graphs.

Reconstructing full-length transcript isoforms from sequence fragments (such as ESTs) is a major interest and challenge for bioinformatic analysis of pre-mRNA alternative splicing. This problem has been formulated as finding traversals across the splice graph, which is a directed acyclic graph (DAG) representation of gene structure and alternative splicing. In this manuscript we introduce a probabilistic formulation of the isoform reconstruction problem, and provide an expectation-maximization (EM) algorithm for its maximum likelihood solution. Using a series of simulated data and expressed sequences from real human genes, we demonstrate that our EM algorithm can correctly handle various situations of fragmentation and coupling in the input data. Our work establishes a general probabilistic framework for splice graph-based reconstructions of full-length isoforms.

Algorithms↗
PubMed2003

Analysis of a probabilistic record linkage technique without human review.

We previously developed a deterministic record linkage algorithm demonstrating sensitivities approaching 90% while maintaining 100% specificity. Substantially better performance has been reported using probabilistic linkage techniques; however, such methods often incorporate human review into the process. To avoid human review, we employed an estimator function using the Expectation Maximization (EM) algorithm to establish a single true-link threshold. We compared the unsupervised probabilistic results against the manually reviewed gold-standard for two hospital registries, as well against our previous deterministic results. At an estimated specificity of 99.95%, actual specificities were 99.43% and 99.42% for registries A and B, respectively. At an estimated sensitivity of 99.95%, actual sensitivities were 99.19% and 98.99% for registries A and B, respectively. The EM algorithm estimated linkage parameters with acceptable accuracy, and was an improvement over the deterministic algorithm. Such a methodology may be used where record linkage is required, but human intervention is not possible or practical.

Algorithms↗
PubMed2005

Tissue classification of noisy MR brain images using constrained GMM.

We present an automated algorithm for tissue segmentation of noisy, low contrast magnetic resonance (MR) images of the brain. We use a mixture model composed of a large number of Gaussians, with each brain tissue represented by a large number of the Gaussian components in order to capture the complex tissue spatial layout. The intensity of a tissue is considered a global feature and is incorporated into the model through parameter tying of all the related Gaussians. The EM algorithm is utilized to learn the parameter-tied Gaussian mixture model. A new initialization method is applied to guarantee the convergence of the EM algorithm to the global maximum likelihood. Segmentation of the brain image is achieved by the affiliation of each voxel to a selected tissue class. The presented algorithm is used to segment 3D, T1-weighted, simulated and real MR images of the brain into three different tissues, under varying noise conditions. Quantitative results are presented and compared with state-of-the-art results reported in the literature.

Algorithms↗
PubMed1999

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.

Algorithms↗
PubMed1995

Estimation and reliability of molecular sequence alignments.

The problem of estimating the relatedness of a pair of biological sequences is addressed. A stochastic model of sequence evolution is described that allows insertion and deletion as well as replacement of amino acid residues (or substitution of nucleotides) over time. An expectation-maximization (EM) algorithm that obtains maximum likelihood estimates of the model parameters is introduced. The method assumes that the sequences are related by descent from a common ancestor but the alignment (i.e., the precise evolutionary correspondence between residues in each sequence) is unknown. Results from the E-step of the EM algorithm are used to assess the likelihood that any two residues are related by direct descent from a common ancestor.

Algorithms↗
PubMed1997

Regression analysis of interval-censored survival data with covariates using log-linear models.

We considered the regression analysis of the event time data with left-, right-, or interval-censored observations. We extended life-table techniques for censored survival data using log-linear models to incorporate interval-censored failures. The EM algorithm was used to calculate maximum likelihood estimates for the parameters. We assumed that the hazard function was a stepwise function over disjoint intervals of time; thus, the nonparametric model, the parametric exponential model, and the semiparametric Cox proportional hazard model were easily implemented as special cases. We adapted the restricted EM algorithm to test hypotheses and to construct confidence intervals for the parameters. These methods were applied in an analysis of the recurrence time for treated melanoma patients.

Adult↗
PubMed2006

The Parkinson's Disease Questionnaire (PDQ-39): evidence for a method of imputing missing data.

BACKGROUND: The Parkinson's Disease Questionnaire (PDQ-39) is the most widely used Parkinson's specific measure of health status. It is increasingly used in treatment trials, sometimes as a primary end-point, where any missing data can potentially cause difficulties in analyses. OBJECTIVES: The purpose of this article is to evaluate the Expectation Maximisation (EM) algorithm for the imputation of missing dimension scores on the 39-item PDQ-39. METHODS: A postal survey of patients diagnosed with Parkinson's disease (PD). A total of 1,372 patients were surveyed and 839 (61.15%) questionnaires returned completed or partially completed. Of these, complete PDQ data were available in 715 (85.22%) cases. Data were deleted from this complete dataset and a sub-set of 200 respondents from this dataset and then imputed using the EM algorithm; results were then compared to the dataset before data deletion. RESULTS: Results gained from imputation of data closely mirrored that of the complete dataset in each case. Descriptive statistics, mean scores and spread of scores were almost identical between original and imputed datasets. Furthermore, original and imputed datasets were highly correlated [intra-class correlation coefficient (ICC) = 0.93 or greater], and mean differences were small (+/-1.00). CONCLUSIONS: The results suggest that the use of EM for the PDQ-39 provides data that closely mirrors the original when this has been deliberately removed. Consequently, EM is likely to be appropriate for trials using the PDQ that contains missing data points.

Adult↗
PubMed2002

A comparison of imputation techniques for handling missing data.

Researchers are commonly faced with the problem of missing data. This article presents theoretical and empirical information for the selection and application of approaches for handling missing data on a single variable. An actual data set of 492 cases with no missing values was used to create a simulated yet realistic data set with missing at random (MAR) data. The authors compare and contrast five approaches (listwise deletion, mean substitution, simple regression, regression with an error term, and the expectation maximization [EM] algorithm) for dealing with missing data, and compare the effects of each method on descriptive statistics and correlation coefficients for the imputed data (n = 96) and the entire sample (n = 492) when imputed data are inculded. All methods had limitations, although our findings suggest that mean substitution was the least effective and that regression with an error term and the EM algorithm produced estimates closest to those of the original variables.

Algorithms↗
PubMed2026

Genomic wastewater surveillance of human and animal influenza A viruses in California during the 2024-2025 flu season.

BACKGROUND: Wastewater genomic surveillance provides an opportunity to detect human and animal influenza A virus (IAV). We aimed to implement an IAV genomic surveillance framework agnostic to subtype, which enables recovery of IAV from multiple hosts and estimation of proportions across subtypes. METHODS: We conducted IAV genomic surveillance in wastewater during the 2024-2025 flu season at multiple sites in California and compared these data with available human clinical IAV sequences and test positivity. We applied a custom whole-genome, multi-host IAV probe enrichment panel and adapted our custom expectation-maximization (EM) algorithm to deconvolute IAV mixtures in wastewater and infer subtype relative abundances. Absolute IAV concentrations were quantified using RT-PCR-based assays. H5N1 wastewater and clinical sequences were further characterized by constructing a whole-genome maximum-likelihood phylogenetic tree. Finally, we performed variant analysis to examine amino acid substitutions detected in wastewater. FINDINGS: Our IAV probe enrichment method and EM algorithm successfully enriched all eight segments of three circulating IAV subtypes and accurately estimated subclade relative abundances for mixed IAV samples. Seasonal human H1N1pdm09 and H3N2 were detected throughout the study period from both wastewater and clinical sequencing data, with H1N1 subclades 6B.1A.5a.2a.1 and 6B.1A.5a.2a co-circulating, and H3N2 dominated by subclade 3C.2a1b.2a.2a.3a.1. Wastewater surveillance consistently detected H5N1 clade 2.3.4.4b across three monitored wastewater sites, while clinical H5N1 detections, from anywhere in CA, were sporadic and rare. Whole-genome phylogenetic analysis revealed that wastewater H5N1 sequences clustered with reference sequences associated with dairy cow and avian infections, while all human clinical H5N1 sequences clustered exclusively with reference sequences associated with dairy cow infections. Amino acid substitutions were identified across viral segments, and no mutations associated with mammalian adaptation were observed from wastewater samples. INTERPRETATION: When IAV concentrations were dominated by seasonal human subtypes rather than H5N1, subtype patterns aligned between wastewater and clinical data. While sequencing IAV in wastewater was unable to distinguish if H5N1 detections were due to human or animal infections, it was able to provide clade-level information about H5N1 found in wastewater that could be useful in the future. Wastewater genomic surveillance can complement clinical surveillance, increasing ability to detect all circulating IAV subtypes and enhancing public health preparedness from a One Health perspective.

Journal Article↗
PubMed2005

Estimating haplotype frequencies in pooled DNA samples when there is genotyping error.

BACKGROUND: Maximum likelihood estimates of haplotype frequencies can be obtained from pooled DNA using the expectation maximization (EM) algorithm. Through simulation, we investigate the effect of genotyping error on the accuracy of haplotype frequency estimates obtained using this algorithm. We explore model parameters including allele frequency, inter-marker linkage disequilibrium (LD), genotyping error rate, and pool size. RESULTS: Pool sizes of 2, 5, and 10 individuals achieved comparable levels of accuracy in the estimation procedure. Common marker allele frequencies and no inter-marker LD result in less accurate estimates. This pattern is observed regardless of the amount of genotyping error simulated. CONCLUSION: Genotyping error slightly decreases the accuracy of haplotype frequency estimates. However, the EM algorithm performs well even in the presence of genotyping error. Overall, pools of 2, 5, and 10 individuals yield similar accuracy of the haplotype frequency estimates, while reducing costs due to genotyping.

DNA↗
PubMed2001

Maximum-likelihood expectation-maximization reconstruction of sinograms with arbitrary noise distribution using NEC-transformations.

The maximum-likelihood (ML) expectation-maximization (EM) [ML-EM] algorithm is being widely used for image reconstruction in positron emission tomography. The algorithm is strictly valid if the data are Poisson distributed. However, it is also often applied to processed sinograms that do not meet this requirement. This may sometimes lead to suboptimal results: streak artifacts appear and the algorithm converges toward a lower likelihood value. As a remedy, we propose two simple pixel-by-pixel methods [noise equivalent counts (NEC)-scaling and NEC-shifting] in order to transform arbitrary sinogram noise into noise which is approximately Poisson distributed (the first and second moments of the distribution match those of the Poisson distribution). The convergence speed associated with both transformation methods is compared, and the NEC-scaling method is validated with both simulations and clinical data. These new methods extend the ML-EM algorithm to a general purpose nonnegative reconstruction algorithm.

Algorithms↗
PubMed2005

Continuous covariates in genetic association studies of case-parent triads: gene and gene-environment interaction effects, population stratification, and power analysis.

We propose a multinomial logistic regression method which permits estimation and likelihood ratio tests for allele effects, their interactions with continuous covariates, and assessment of the degree of population stratification in genetic association studies of case-parent triads. Our approach overcomes the constraint imposed by the categorical nature of explanatory variables in the log-linear model. We also demonstrate that the multinomial logistic method can yield efficient inference in the presence of missing parental genotype data via the use of the Expectation-Maximization (EM) algorithm. We performed simulations to compare the multinomial logistic model with the case-pseudosibling conditional logistic model approach, both of which permit the incorporation of continuous covariates. Simulation results indicate that the multinomial logistic model and the conditional logistic model lead to similar estimates in large samples. A simulation-based method of sample size estimation is also used to show that the two models are approximately equivalent in sample size requirements. When parental genotype data are missing, either completely at random or dependent on covariates, the use of the EM algorithm gives multinomial logistic model greater power. Since the multinomial logistic model offers the possibility of assessing the degree of population stratification in the sample and can also provide efficient inference in the presence of missing parental genotypes, the proposed model has an important application in epidemiological family-based association studies.

Journal Article↗
PubMed1994

Massively parallel computers for 3D single-photon-emission computed tomography.

Since the introduction of the expectation-maximization (EM) algorithm for generating maximum-likelihood (ML) and maximum a posteriori (MAP) estimates in emission tomography, there have been many investigators applying the ML method. However, almost all of the previous work has been restricted to two-dimensional (2D) reconstructions. The major focus and contribution of this paper is to demonstrate a fully three-dimensional (3D) implementation of the MAP method for single-photon-emission computed tomography (SPECT). The 3D reconstruction exhibits an improvement in resolution when compared to the generation of the series of separate 2D slice reconstructions. As has been noted, the iterative EM algorithm for 2D reconstruction is highly computational; the 3D algorithm is far worse. To accommodate the computational complexity, we have extended our previous work in the 2D arena and demonstrate an implementation on the class of massively parallel processors of the 3D algorithm. Using a 16000 processor MasPar machine, the algorithm is demonstrated to execute at 1.24 s/EM iteration for the entire 64 x 64 x 64 cube of 64 planar measurements obtained from the Siemens Orbiter rotating camera operating in the high-resolution mode.

Algorithms↗
PubMed1994

A simulation of emission and transmission noise propagation in cardiac SPECT imaging with nonuniform attenuation correction.

Transmission computed tomography provides information needed for nonuniform attenuation correction of cardiac single photon emission computed tomography (SPECT). Nonuniform attenuation correction is accomplished using an iterative ML-EM algorithm and a projection-backprojection operation that incorporates attenuation factors measured from the reconstructed transmission map. The precision and accuracy of the attenuation corrected emission reconstruction is a function of emission and transmission statistics. This paper presents an error propagation analysis that uses a mathematical cardiac chest phantom to simulate various combinations of total emission counts C and transmission flux I0 under ideal imaging conditions (without geometric response distortion and without scatter). The spatial average, spatial variance, and accuracy measures for a 4 x 4 pixel region in the heart are tabulated after 30 iterations of the ML-EM algorithm. The confidence intervals for these measures were determined from 1000 realizations of reconstructions from projections randomly generated with the same transmission and emission statistics. It can be shown empirically from the simulation results that the spatial %rms uncertainty for the simulated cardiac region has a simple expression: %rms2 = K1/C+K2/I0(2)+B2 where K1 and K2 are least-square estimates based on the simulation results, and B is the measured spatial %rms uncertainty for the simulation at infinite statistics. For a transmission incident flux of 1500 events per projection bin of 0.712 cm and typical clinical emission events totaling 1 x 10(5), the spatial %rms uncertainty is approximately 14%. At clinical transmission and emission statistics, the statistical noise in the simulated attenuation-corrected reconstructions are dominated by the emission statistics.

Algorithms↗
PubMedDate not supplied

Assessment of the Gould-Shih procedure for sample size re-estimation.

The power of a clinical trial is partly dependent upon its sample size. With continuous data, the sample size needed to attain a desired power is a function of the within-group standard deviation. An estimate of this standard deviation can be obtained during the trial itself based upon interim data; the estimate is then used to re-estimate the sample size. Gould and Shih proposed a method, based on the EM algorithm, which they claim produces a maximum likelihood estimate of the within-group standard deviation while preserving the blind, and that the estimate is quite satisfactory. However, others have claimed that the method can produce non-unique and/or severe underestimates of the true within-group standard deviation. Here the method is thoroughly examined to resolve the conflicting claims and, via simulation, to assess its validity and the properties of its estimates. The results show that the apparent non-uniqueness of the method's estimate is due to an apparently innocuous alteration that Gould and Shih made to the EM algorithm. When this alteration is removed, the method is valid in that it produces the maximum likelihood estimate of the within-group standard deviation (and also of the within-group means). However, the estimate is negatively biased and has a large standard deviation. The simulations show that with a standardized difference of 1 or less, which is typical in most clinical trials, the standard deviation from the combined samples ignoring the groups is a better estimator, despite its obvious positive bias.

Algorithms↗
PubMed1994

Analysis with missing data in drug prevention research.

Missing data problems have been a thorn in the side of prevention researchers for years. Although some solutions for these problems have been available in the statistical literature, these solutions have not found their way into mainstream prevention research. This chapter is meant to serve as an introduction to the systematic application of the missing data analysis solutions presented recently by Little and Rubin (1987) and others. The chapter does not describe a complete strategy, but it is relevant for (1) missing data analysis with continuous (but not categorical) data, (2) data that are reasonably normally distributed, and (3) solutions for missing data problems for analyses related to the general linear model in particular, analyses that use (or can use) a covariance matrix as input. The examples in the chapter come from drug prevention research. The chapter discusses (1) the problem of wanting to ask respondents more questions than most individuals can answer; (2) the problem of attrition and some solutions; and (3) the problem of special measurement procedures that are too expensive or time consuming to obtain for all subjects. The authors end with several conclusions: Whenever possible, researchers should use the Expectation-Maximization (EM) algorithm (or other maximum likelihood procedure, including the multiple-group structural equation-modeling procedure or, where appropriate, multiple imputation, for analyses involving missing data [the chapter provides concrete examples]); If researchers must use other analyses, they should keep in mind that these others produce biased results and should not be relied upon for final analyses; When data are missing, the appropriate missing data analysis procedures do not generate something out of nothing but do make the most out of the data available; When data are missing, researchers should work hard (especially when planning a study) to find the cause of missingness and include the cause in the analysis models; and Researchers should sample the cases originally missing (whenever possible) and adjust EM algorithm parameter estimates accordingly.

Algorithms↗
PubMed2004

Adjust quality scores from alignment and improve sequencing accuracy.

In shotgun sequencing, statistical reconstruction of a consensus from alignment requires a model of measurement error. Churchill and Waterman proposed one such model and an expectation-maximization (EM) algorithm to estimate sequencing error rates for each assembly matrix. Ewing and Green defined Phred quality scores for base-calling from sequencing traces by training a model on a large amount of data. However, sample preparations and sequencing machines may work under different conditions in practice and therefore quality scores need to be adjusted. Moreover, the information given by quality scores is incomplete in the sense that they do not describe error patterns. We observe that each nucleotide base has its specific error pattern that varies across the range of quality values. We develop models of measurement error for shotgun sequencing by combining the two perspectives above. We propose a logistic model taking quality scores as covariates. The model is trained by a procedure combining an EM algorithm and model selection techniques. The training results in calibration of quality values and leads to a more accurate construction of consensus. Besides Phred scores obtained from ABI sequencers, we apply the same technique to calibrate quality values that come along with Beckman sequencers.

Algorithms↗
PubMed1999

A general maximum likelihood analysis of variance components in generalized linear models.

This paper describes an EM algorithm for nonparametric maximum likelihood (ML) estimation in generalized linear models with variance component structure. The algorithm provides an alternative analysis to approximate MQL and PQL analyses (McGilchrist and Aisbett, 1991, Biometrical Journal 33, 131-141; Breslow and Clayton, 1993; Journal of the American Statistical Association 88, 9-25; McGilchrist, 1994, Journal of the Royal Statistical Society, Series B 56, 61-69; Goldstein, 1995, Multilevel Statistical Models) and to GEE analyses (Liang and Zeger, 1986, Biometrika 73, 13-22). The algorithm, first given by Hinde and Wood (1987, in Longitudinal Data Analysis, 110-126), is a generalization of that for random effect models for overdispersion in generalized linear models, described in Aitkin (1996, Statistics and Computing 6, 251-262). The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully nonparametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters can be sensitive to the specification of a parametric form for the mixing distribution. The nonparametric analysis can be extended straightforwardly to general random parameter models, with full NPML estimation of the joint distribution of the random parameters. This can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters. A simple method is described for obtaining correct standard errors for parameter estimates when using the EM algorithm. Several examples are discussed involving simple variance component and longitudinal models, and small-area estimation.

Adrenergic beta-Antagonists↗