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PubMed1996

Modeling conception as an aggregated Bernoulli outcome with latent variables via the EM algorithm.

We describe a general method for analyzing aggregated Bernoulli outcomes. The research is motivated by an epidemiological reproductive study where the outcome was whether pregnancy was detected in a woman's particular menstrual cycle and the Bernoulli "trials" corresponded to days with intercourse during the cycle. Each cycle is either "viable" or not, i.e., is or is not susceptible to conception. We develop an EM algorithm approach to maximizing the observed-data pseudo-likelihood, based on a set of unobservable latent outcomes linked to the specific days with intercourse. This method is flexible in that it allows one to model effect of covariates on the susceptibility factor and on the latent outcomes. Application of the method to fertility studies enables one to investigate covariates with a long-term or transient effect on the daily conception probability. A complication is that most couples contribute more than one cycle in a prospective study. A generalized estimating equation approach adjusts for the dependency among outcomes within individual couples. The method can be applied in any setting where dependency among Bernoulli trials is induced by a susceptibility factor and the trial outcomes are only observable in the aggregate.

Algorithms↗
PubMed1990

An expectation maximization (EM) algorithm for the identification and characterization of common sites in unaligned biopolymer sequences.

Statistical methodology for the identification and characterization of protein binding sites in a set of unaligned DNA fragments is presented. Each sequence must contain at least one common site. No alignment of the sites is required. Instead, the uncertainty in the location of the sites is handled by employing the missing information principle to develop an "expectation maximization" (EM) algorithm. This approach allows for the simultaneous identification of the sites and characterization of the binding motifs. The reliability of the algorithm increases with the number of fragments, but the computations increase only linearly. The method is illustrated with an example, using known cyclic adenosine monophosphate receptor protein (CRP) binding sites. The final motif is utilized in a search for undiscovered CRP binding sites.

Algorithms↗
PubMed2002

On the inappropriateness of an EM algorithm based procedure for blinded sample size re-estimation.

When planning a clinical trial the sample size calculation is commonly based on an a priori estimate of the variance of the outcome variable. Misspecification of the variance can have substantial impact on the power of the trial. It is therefore attractive to update the planning assumptions during the ongoing trial using an internal estimate of the variance. For this purpose, an EM algorithm based procedure for blinded variance estimation was proposed for normally distributed data. Various simulation studies suggest a number of appealing properties of this procedure. In contrast, we show that (i) the estimates provided by this procedure depend on the initialization, (ii) the stopping rule used is inadequate to guarantee that the algorithm converges against the maximum likelihood estimator, and (iii) the procedure corresponds to the special case of simple randomization which, however, in clinical trials is rarely applied. Further, we show that maximum likelihood estimation leads to no reasonable results for blinded sample size re-estimation due to bias and high variability. The problem is illustrated by a clinical trial in asthma.

Administration, Inhalation↗
PubMed1996

Direct speech feature estimation using an iterative EM algorithm for vocal fold pathology detection.

The focus of this study is to formulate a speech parameter estimation algorithm for analysis/detection of vocal fold pathology. The speech processing algorithm proposed estimates features necessary to formulate a stochastic model to characterize healthy and pathology conditions from speech recordings. The general idea is to separate speech components under healthy and assumed pathology conditions. This problem is addressed using an iterative maximum-likelihood (ML) estimation procedure, based on the estimation-maximization (EM) algorithm. A new feature for characterizing pathology, termed enhanced-spectral-pathology component (ESPC), is estimated and shown to vary consistently between healthy and pathology conditions. It is also shown that the mean-area-peak-value (MAPV) and the weighted-slope (WSLOPE) indexes, which are obtained from the ESPC estimate, are meaningful measures of speech pathology conditions. For classification purposes, a five-state hidden-Markov-model (HMM) recognizer was formulated, based on the MAPV, WSLOPE, and ESPC spectral features. A set of log Mel-frequency filter bank coefficients were used to parameterize the ESPC feature. An evaluation of the HMM-based classifier was performed using speech recordings from healthy and vocal fold cancer patients of sustained vowel sounds. It is shown that while both MAPV and WSLOPE are useful features for vocal fold pathology detection, superior performance was achieved using a finer spectral representation of ESPC (e.g., a detection rate of 88.7% for pathology and 92.8% for healthy condition). One main advantage of the proposed method is that it does not require direct estimation of the glottal flow waveform. Therefore, the limitation of the inability to characterize vocal fold pathology, due to incomplete glottal closure, is no longer an issue. The results suggest that general analysis of the ESPC feature can provide a quantitative, noninvasive approach for analysis, detection, and characterization of speech production under vocal fold pathology.

Algorithms↗
PubMed1999

Finite mixture modeling with mixture outcomes using the EM algorithm.

This paper discusses the analysis of an extended finite mixture model where the latent classes corresponding to the mixture components for one set of observed variables influence a second set of observed variables. The research is motivated by a repeated measurement study using a random coefficient model to assess the influence of latent growth trajectory class membership on the probability of a binary disease outcome. More generally, this model can be seen as a combination of latent class modeling and conventional mixture modeling. The EM algorithm is used for estimation. As an illustration, a random-coefficient growth model for the prediction of alcohol dependence from three latent classes of heavy alcohol use trajectories among young adults is analyzed.

Adolescent↗
PubMed1977

Counting methods (EM algorithm) in human pedigree analysis: linkage and segregation analysis.

The likelihood of human pedigree data can be written in such a form as to allow the computation of derivatives. This is done for various parameters in linkage and segregation analysis. The equations for the maximum likelihood estimates are represented in a particularly appealing form which allows iterative solutions. This process is an extension to pedigress of Smith's (1957) counting methods. All these procedures belong to a general class of MLE methods for incomplete data called EM algorithms (Dempster et al. 1976).

Gene Frequency↗
PubMed1992

Semiparametric estimation of random effects using the Cox model based on the EM algorithm.

Consider a survival experiment where individuals within a certain subset of the population share a common, unobservable, random frailty. Such a frailty could be an unobservable genetic or early environmental effect if individuals were in sibling groups or an environmental effect if individuals were grouped by households. Suppose that if the frailty, omega, is known, the Cox proportional hazards model for the observable covariates is valid with the consequence of the random effect being a multiplicative factor on the hazard rate. Assuming tht the random frailties follow a gamma distribution, estimates of the fixed and random effects are obtained by using an EM algorithm based on a profile likelihood construction. The method developed is applied to the Framingham Heart Study to examine the risks of smoking and cholesterol levels, adjusting for potential random effects.

Adult↗
PubMedDate not supplied

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

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

Algorithms↗
PubMed1995

Maximum-likelihood estimation in linkage heterogeneity models including additional information via the EM algorithm.

In linkage analysis, estimated recombination fractions between a disease gene and several markers are used to assign the disease gene to a particular chromosome region. For rare diseases, locus heterogeneity leads to different recombination fractions in different families, and a set of pedigrees can be regarded as a mixture. Information which can help to classify the different families may be available and can be included in the model. This is demonstrated for a data set of X-linked retinitis pigmentosa families. The joint distribution of the position of the disease gene and the additional information on the penetrance of tapetal reflex among obligate female carriers is studied. A model including the additional information is fitted to the data using the EM algorithm. The algorithm uses for every pedigree the lod score curve which can be obtained from a standard (multipoint) linkage analysis.

Algorithms↗
PubMed1989

Statistical evaluation of cell kinetic data from DNA flow cytometry (FCM) by the EM algorithm.

Flow cytometric DNA measurements yield the amount of DNA for each of a large number of cells. A DNA histogram normally consists of a mixture of one or more constellations of G0/G1-, S-, G2/M-phase cells, together with internal standards, debris, background noise, and one or more populations of clumped cells. We have modelled typical DNA histograms as a mixed distribution with Gaussian densities for the G0/G1 and G2/M phases, an S-phase density, assumed to be uniform between the G0/G1 and G2/M peaks, observed with a Gaussian error, and with Gaussian densities for standards of chicken and trout red blood cells. The debris is modelled as a truncated exponential distribution, and we also have included a uniform background noise distribution over the whole observation interval. We have explored a new approach for maximum-likelihood analyses of complex DNA histograms by the application of the EM algorithm. This algorithm was used for four observed DNA histograms of varying complexity. Our results show that the algorithm works very well, and it converges to reasonable values for all parameters. In simulations from the estimated models, we have investigated bias, variance, and correlations of the estimates.

Algorithms↗
PubMed2007

Analysis of interval-censored data from circular migrant and non-migrant sexual partnerships using the EM algorithm.

In epidemiological studies where subjects are seen periodically on follow-up visits, interval-censored data occur naturally. The exact time the change of state (such as HIV seroconversion) occurs is not known exactly, only that it occurred sometime within a specific time interval. Methods of estimation for interval-censored data are readily available when data are independent. However, methods for correlated interval-censored data are not well developed. This paper considers an approach for estimating the parameters when data are interval-censored and correlated within sexual partnerships. We consider the exact event times for interval-censored observations as unobserved data, only known to be between two time points. Dependency induced by sexual partnerships is modelled as frailties assuming a gamma distribution for frailties and an exponential distribution on the time to infection. This formulation facilitates application of the expectation-maximization (EM) algorithm. Maximization process maximizes the standard survival frailty model. Results show high degree of heterogeneity between sexual partnerships. Intervention strategies aimed at combating the spread of HIV and other sexually transmitted infections (STI)s should treat sexual partnerships as social units and fully incorporate the effects of migration in their strategies.

Adolescent↗
PubMedDate not supplied

Use of a fast EM algorithm for 3D image reconstruction with the YAP-PET tomograph.

OBJECTIVE: We would like to improve the image reconstructions for both signal-to-noise ratio (SNR) and spatial resolution characteristics for the small animal positron emission tomograph YAP-PET, built at the Department of Physics of Ferrara University. The three-dimensional (3D) filtered backprojection (FBP) algorithm, usually used for image reconstruction, has a limited angle restriction due to the tomograph geometry, which causes a serious loss in sensitivity. METHODS: We implemented a 3D iterative reconstruction program using the symmetry and sparse properties of the 'probability matrix', which correlates the emission from each voxel to the detector within a coincidence tube. A fraction only of matrix elements are calculated before the reconstruction and stored on disk: this allows us to avoid on-line computation. A depth dependent function differentiates the voxels in a coincidence tube. Three experimental phantoms with no background were reconstructed by using the program, in comparison with traditionally used FBP. RESULTS: The adopted method allowed us to reduce the computation time significantly. Furthermore, the simple depth dependent function improved the spatial resolution. With 64 x 64 x 20 voxels of 0.625 x 0.625 x 2.0 mm(3) in the field of view, the computation time was less than 4 min per iteration on a Sparc Ultra 450 Workstation, and less than 6 min per iteration on a Mac-PPC G3 300 MHz: the spatial resolution measured with a 0.8 mm diameter 18F-FDG filled capillary reconstructed in this way was 2.0 mm FWHM. By decreasing the voxel size to 0.3125 x 0.3125 x 2.0 mm(3) per voxel the transaxial FWHM was 1.7 mm with a computation time of 15 min per iteration on a Sparc Ultra 450. By using all the acquired data, the SNR improves from 1.3 to 6.0 in the worst measured case, a pair of 0.8mm diameter 18F-FDG filled capillaries, which are 2.5 mm apart each other. CONCLUSION: The adoption of iterative reconstruction allowed us to overcome the loss in sensitivity of previously used FBP: this improved the SNR. The studies of symmetry and sparse properties avoided a severe increase of the reconstruction time and of storing space on disk. This fast EM Algorithm is now routinely used for the image reconstruction with the YAP-PET tomograph.

Algorithms↗
PubMed2003

Optimal step length EM algorithm (OSLEM) for the estimation of haplotype frequency and its application in lipoprotein lipase genotyping.

BACKGROUND: Haplotype based linkage disequilibrium (LD) mapping has become a powerful and cost-effective method for performing genetic association studies, particularly in the search for genetic markers in linkage disequilibrium with complex disease loci. Various methods (e.g. Monte-Carlo (Gibbs sampling); EM (expectation maximization); and Clark's method) have been used to estimate haplotype frequencies from routine genotyping data. RESULTS: These algorithms can be very slow for large number of SNPs. In order to speed them up, we have developed a new algorithm using numerical analysis technology, a so-called optimal step length EM (OSLEM) that accelerates the calculation. By optimizing approximately the step length of the EM algorithm, OSLEM can run at about twice the speed of EM. This algorithm has been used for lipoprotein lipase (LPL) genotyping analysis. CONCLUSIONS: This new optimal step length EM (OSLEM) algorithm can accelerate the calculation for haplotype frequency estimation for genotyping data without pedigree information. An OSLEM on-line server is available, as well as a free downloadable version.

Algorithms↗
PubMed1998

A Markov chain Monte Carlo EM algorithm for analyzing interval-censored data under the Cox proportional hazards model.

This paper proposes a Monte Carlo EM (MCEM) algorithm for fitting the proportional hazards model for interval-censored failure-time data. The algorithm generates orderings of the failures from their probability distribution under the model. We maximize the average of the log-likelihoods from these completed data sets to obtain updated parameter estimates. As with the standard Cox model, this algorithm does not require the estimation of the baseline hazard function. The performance of the algorithm is evaluated using simulations, and the method is applied to data from AIDS and cancer studies. Our results indicate that our method produced more precise and unbiased estimates than methods of right and midpoint imputation.

Acquired Immunodeficiency Syndrome↗
PubMed2004

Cox proportional hazards survival regression in haplotype-based association analysis using the Stochastic-EM algorithm.

It is now widely recognized that haplotype information inferred from genotypes can be of great interest to better characterize the role of a candidate gene in the etiology of a complex trait in the context of association studies. Several works have recently advocated the simultaneous estimation of haplotype frequencies and haplotype effects in order to get a better efficiency in parameter estimation. Most of the available models can deal with a binary or a quantitative phenotype, but none has yet discussed the application of haplotype-based association analysis to a survival outcome. We describe how the recently proposed Stochastic-EM (SEM) algorithm can be applied to estimate haplotype effects in censored data analysis using a standard Cox proportional hazards formulation. This model has been implemented in the THESIAS software freely available at http://www.genecanvas.org.

Algorithms↗
PubMed2005

An EM algorithm for shape classification based on level sets.

In this paper, we propose an expectation-maximization (EM) approach to separate a shape database into different shape classes, while simultaneously estimating the shape contours that best exemplify each of the different shape classes. We begin our formulation by employing the level set function as the shape descriptor. Next, for each shape class we assume that there exists an unknown underlying level set function whose zero level set describes the contour that best represents the shapes within that shape class. The level set function for each example shape in the database is modeled as a noisy measurement of the appropriate shape class's unknown underlying level set function. Based on this measurement model and the judicious introduction of the class labels as the hidden data, our EM formulation calculates the labels for shape classification and estimates the shape contours that best typify the different shape classes. This resulting iterative algorithm is computationally efficient, simple, and accurate. We demonstrate the utility and performance of this algorithm by applying it to two medical applications.

Algorithms↗
PubMed1996

Using the EM-algorithm for survival data with incomplete categorical covariates.

Incomplete covariate data is a common occurrence in many studies in which the outcome is survival time. With generalized linear models, when the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM by the method of weights proposed in Ibrahim (1990). In this article, we extend the EM by the method of weights to survival outcomes whose distributions may not fall in the class of generalized linear models. This method requires the estimation of the parameters of the distribution of the covariates. We present a clinical trials example with five covariates, four of which have some missing values.

Algorithms↗
PubMed1998

Soft vector quantization and the EM algorithm.

The relation between hard c-means (HCM), fuzzy c-means (FCM), fuzzy learning vector quantization (FLVQ), soft competition scheme (SCS) of [Yair et al. (1992)] and probabilistic Gaussian mixtures (GM) have been pointed out recently by [Bezdek and Pal (1995)]. We extend this relation to their training, showing that learning rules by these models to estimate the cluster centers can be seen as approximations to the expectation-maximization (EM) method as applied to Gaussian mixtures. HCM and unsupervised, LVQ use 1-of-c type competition. In FCM and FLVQ, membership is the -2/(m-1)th power of the distance. In SCS and GM, Gaussian function is used. If the Gaussian membership function is used, the weighted within-groups sum of squared errors used as the fuzzy objective function corresponds to the maximum likelihood estimate in Gaussian mixtures with equal priors and covariances. The fuzzy clustering method named fuzzy c-means alternating optimization procedure (FCM-AO) proposed to optimize the former is then equivalent to batch EM and SCS's update rule is a variant of the online version of EM. The advantages of the probabilistic framework are: (i) we no longer have spurious spread parameters that needs fine tuning as m in fuzzy vector quantization or beta in SCS; instead we have a variance term that has a sound interpretation and that can be estimated from the sample; (ii) EM guarantees that the likelihood does not decrease, thus it converges to the nearest local optimum; (iii) EM also allows us to estimate the underlying distance norm and the cluster priors which we could not with the other approaches. We compare Gaussian mixtures trained with EM with LVQ (HCM), SCS and FLVQ on the IRIS dataset and see that it is more accurate due to its being able to take into account the covariance information. We finally note that vector quantization is generally an intermediate step before finding a final output for which supervision may be possible. Thus, instead of an uncoupled approach where an unsupervised method is used first to find the cluster parameters followed by supervised training of the mapping based on the memberships, we advocate a coupled approach where the cluster parameters and mapping are trained supervised in a coupled way. The uncoupled approach ignores the error at the outputs which may not be ideal.

Journal Article↗