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PubMed2002

Likelihood-based refinement. I. Irremovable model errors.

In conventional structure refinement, the discrepancy between the calculated magnitudes and those observed in X-ray experiments is attributed to errors inherent in preliminary assigned values of the model parameters. However, the chosen set of model parameters may not be adequate to describe the structure factors precisely. For example, if some atoms are not included in the current model, then the structure factors calculated from such a partial model contain 'irremovable errors'. These errors cannot be eliminated by any choice of the parameters of the partial structure. Probabilistic modelling suggests a way to take irremovable errors into account. Every trial set of values of the model parameters is now associated with the joint probability distribution of the calculated magnitudes, rather than with a particular set of magnitudes. The new goal of the refinement is formulated as the search for the distribution that is the most consistent with the observed data. The statistical likelihood is a possible measure of the consistency. The suggested quadratic approximation of the likelihood function allows the likelihood-based refinement to be considered as a kind of least-squares refinement that uses appropriate weights and modified targets for the calculated magnitudes. This in turn enables the analysis of tendencies of the likelihood-based refinement in comparison with the classical least-squares refinement.

Journal Article↗
PubMed1994

Genetic analysis combining path analysis with regressive models: the BETA path model of polygenic and familial environmental transmission.

We have extended the class D regressive model for the purpose of combined path and segregation analyses by incorporating the BETA path model. We have done this by expressing correlations among residuals from major genotype (RMGs) of family members under the class D regressive model as functions of path coefficients under the BETA path model. The likelihood function under the combined model was factorized into a product of conditional densities, which is dominated by bivariate normal densities. Statistical inferences under the combined model are analogous to those under the class D regressive model.

Binomial Distribution↗
PubMed1995

MARKOV: a computer program for multi-state Markov models with covariables.

This paper discusses a computer program, called MARKOV, designed to fit a multi-state Markov model with covariables, with a particular emphasis on the analysis of survival data. The Markov model consists of k-1 transient disease states and one absorbing state. The exact transition times are not observed, except in situations such as death. Baseline transition intensities and regression coefficients are estimated via the method of maximum likelihood using a quasi-Newton optimization algorithm. The program's output includes the parameter estimates, the standard error of the estimates, the matrix of the correlation of the estimates and minus two times the log-likelihood function, evaluated at the initial values and at the maximum likelihood estimates. Optionally, survival curves can be generated from each transient state, for one or more combination of covariable values and simple tests about the parameters. The program is illustrated by using a four-state model to determine factors influencing diabetic retinopathy in young subjects with insulin-dependent diabetes mellitus.

Algorithms↗
PubMed2000

Hidden Markov modeling for single channel kinetics with filtering and correlated noise.

Hidden Markov modeling (HMM) can be applied to extract single channel kinetics at signal-to-noise ratios that are too low for conventional analysis. There are two general HMM approaches: traditional Baum's reestimation and direct optimization. The optimization approach has the advantage that it optimizes the rate constants directly. This allows setting constraints on the rate constants, fitting multiple data sets across different experimental conditions, and handling nonstationary channels where the starting probability of the channel depends on the unknown kinetics. We present here an extension of this approach that addresses the additional issues of low-pass filtering and correlated noise. The filtering is modeled using a finite impulse response (FIR) filter applied to the underlying signal, and the noise correlation is accounted for using an autoregressive (AR) process. In addition to correlated background noise, the algorithm allows for excess open channel noise that can be white or correlated. To maximize the efficiency of the algorithm, we derive the analytical derivatives of the likelihood function with respect to all unknown model parameters. The search of the likelihood space is performed using a variable metric method. Extension of the algorithm to data containing multiple channels is described. Examples are presented that demonstrate the applicability and effectiveness of the algorithm. Practical issues such as the selection of appropriate noise AR orders are also discussed through examples.

Algorithms↗
PubMed2005

Modelling bivariate count series with excess zeros.

Bivariate time series of counts with excess zeros relative to the Poisson process are common in many bioscience applications. Failure to account for the extra zeros in the analysis may result in biased parameter estimates and misleading inferences. A class of bivariate zero-inflated Poisson autoregression models is presented to accommodate the zero-inflation and the inherent serial dependency between successive observations. An autoregressive correlation structure is assumed in the random component of the compound regression model. Parameter estimation is achieved via an EM algorithm, by maximizing an appropriate log-likelihood function to obtain residual maximum likelihood estimates. The proposed method is applied to analyze a bivariate series from an occupational health study, in which the zero-inflated injury count events are classified as either musculoskeletal or non-musculoskeletal in nature. The approach enables the evaluation of the effectiveness of a participatory ergonomics intervention at the population level, in terms of reducing the overall incidence of lost-time injury and a simultaneous decline in the two mean injury rates.

Accidents, Occupational↗
PubMed2003

Absence of multiple local minima effects in intensity modulated optimization with dose-volume constraints.

This paper reports on the analysis of intensity modulated radiation treatment optimization problems in the presence of non-convex feasible parameter spaces caused by the specification of dose-volume constraints for the organs-at-risk (OARs). The main aim was to determine whether the presence of those non-convex spaces affects the optimization of clinical cases in any significant way. This was done in two phases: (1) Using a carefully designed two-dimensional mathematical phantom that exhibits two controllable minima and with randomly initialized beamlet weights, we developed a methodology for exploring the nature of the convergence characteristics of quadratic cost function optimizations (deterministic or stochastic). The methodology is based on observing the statistical behaviour of the residual cost at the end of optimizations in which the stopping criterion is progressively more demanding and carrying out those optimizations to very small error changes per iteration. (2) Seven clinical cases were then analysed with dose-volume constraints that are stronger than originally used in the clinic. The clinical cases are two prostate cases differently posed, a meningioma case, two head-and-neck cases, a spleen case and a spine case. Of the 14 different sets of optimizations (with and without the specification of maximum doses allowed for the OARs), 12 fail to show any effect due to the existence of non-convex feasible spaces. The remaining two sets of optimizations show evidence of multiple minima in the solutions, but those minima are very close to each other in cost and the resulting treatment plans are practically identical, as measured by the quality of the dose-volume histograms (DVHs). We discuss the differences between fluence maps resulting from those similar treatment plans. We provide a possible reason for the observed results and conclude that, although the study is necessarily limited, the annealing characteristics of a simulated annealing method may not be justified in clinical optimization in the presence of dose-volume constraints. The results of optimizations by the Newton gradient (NG) method with a quadratic cost function are reported in detail. An adaptive simulated annealing method, optimizing the same function, and the dynamically penalized likelihood method, optimizing a log likelihood function, have also been used in the study. The results of the latter two methods have only been discussed briefly, as they yielded the same conclusions as the NG method.

Algorithms↗
PubMed1999

Conditional inference for predictive agreement.

We introduce the concept, and a measure of predictive agreement, tau, for two raters classifying items into q categories. The measure is based on the linear combination of log odds ratios from 2 x 2 subtables of a q x q cross-classification table. We show that analysis procedures for this measure, and transforms of it, can be based on conditional likelihood procedures. These procedures are exact, which is particularly helpful because of the small tabular cell frequencies which can typically arise with agreement data. To illustrate the advantages of the methodology, examples of typical agreement data arising from medical studies are considered. We demonstrate that the conditional likelihood function portrays the available sample information about tau, often more appropriately than the maximum likelihood estimate and an associated standard error. We highlight the value of combining information via likelihoods in an example involving 24 2 x 2 tables. An example involving three categories is used to illustrate that the methodology for the overall agreement measure can be adapted to examine relative agreement between pairs of categories.

Alcohol Drinking↗
PubMed2006

Comment on "Neutral ecological theory reveals isolation and rapid speciation in a biodiversity hot spot".

Latimer et al. (Reports, 9 September 2005, p. 1722) used an approximate likelihood function to estimate parameters of Hubbell's neutral model of biodiversity. Reanalysis with the exact likelihood not only yields different estimates but also shows that two similar likelihood maxima for very different parameter combinations can occur. This reveals a limitation of using species abundance data to gain insight into speciation and dispersal.

Animals↗
PubMed2005

Maximum likelihood estimation of haplotype effects and haplotype-environment interactions in association studies.

The associations between haplotypes and disease phenotypes offer valuable clues about the genetic determinants of complex diseases. It is highly challenging to make statistical inferences about these associations because of the unknown gametic phase in genotype data. We describe a general likelihood-based approach to inferring haplotype-disease associations in studies of unrelated individuals. We consider all possible phenotypes (including disease indicator, quantitative trait, and potentially censored age at onset of disease) and all commonly used study designs (including cross-sectional, case-control, cohort, nested case-control, and case-cohort). The effects of haplotypes on phenotype are characterized by appropriate regression models, which allow various genetic mechanisms and gene-environment interactions. We present the likelihood functions for all study designs and disease phenotypes under Hardy-Weinberg disequilibrium. The corresponding maximum likelihood estimators are approximately unbiased, normally distributed, and statistically efficient. We provide simple and efficient numerical algorithms to calculate the maximum likelihood estimators and their variances, and implement these algorithms in a freely available computer program. Extensive simulation studies demonstrate that the proposed methods perform well in realistic situations. An application to the Carolina Breast Cancer Study reveals significant haplotype effects and haplotype-smoking interactions in the development of breast cancer.

Algorithms↗
PubMed2002

Maximum likelihood fitting of FROC curves under an initial-detection-and-candidate-analysis model.

We have developed a model for FROC curve fitting that relates the observer's FROC performance not to the ROC performance that would be obtained if the observer's responses were scored on a per image basis, but rather to a hypothesized ROC performance that the observer would obtain in the task of classifying a set of "candidate detections" as positive or negative. We adopt the assumptions of the Bunch FROC model, namely that the observer's detections are all mutually independent, as well as assumptions qualitatively similar to, but different in nature from, those made by Chakraborty in his AFROC scoring methodology. Under the assumptions of our model, we show that the observer's FROC performance is a linearly scaled version of the candidate analysis ROC curve, where the scaling factors are just given by the FROC operating point coordinates for detecting initial candidates. Further, we show that the likelihood function of the model parameters given observational data takes on a simple form, and we develop a maximum likelihood method for fitting a FROC curve to this data. FROC and AFROC curves are produced for computer vision observer datasets and compared with the results of the AFROC scoring method. Although developed primarily with computer vision schemes in mind, we hope that the methodology presented here will prove worthy of further study in other applications as well.

Algorithms↗
PubMed2005

Estimating heterogeneity in random effects models for longitudinal data.

In this paper, we are interested in estimating parameters entering nonlinear mixed effects models using a likelihood maximization approach. As the accuracy of the likelihood approximation is likely to govern the quality of the derived estimates of both the distribution of the random effects and the fixed parameters, we propose a methodological approach based on the adaptive Gauss Hermite quadrature to better approximate the likelihood function. This work presents improvements of this quadrature that render it accurate and computationally efficient in the problem of likelihood approximation with, an application to mixture models, models which allow the description of coexistence of several different homogeneous subpopulations specifying the distribution of random effects as a mixture of Gaussian distributions. These improvements are based on a new choice of the scaling matrix followed by its optimisation. An application to a phase III clinical trial of an anticoagulant molecule is proposed and estimation results are compared to those obtained with the most frequently used method in population pharmacokinetic analysis. Moreover, in order to evaluate the accuracy of the estimations, an analysis of simulated pharmacokinetic data derived from the model and the a priori values of population parameters of the previous study are presented.

Algorithms↗
PubMed2006

Design of multi-centre trials with binary response.

We propose a correlated beta-binomial model for the binary response in multi-centre trials. The likelihood function in this case has a closed-form and we avoid multivariate numerical integrations in determining the maximum likelihood estimator. Based on derived asymptotic variance-covariance matrix of the MLE, we obtain relatively simple formulae that relate the number of centres, the total number of patients and the precision of the parameter estimate.

Biometry↗
PubMed1992

Maximum likelihood estimation and testing of a poisson regression model.

A Poisson regression model is proposed for the analysis of incidence rates presented in a two-way table classified by two categorical variables. It is shown that the likelihood function is the same as that using Glasser's exponential covariate model. An algorithm is given to solve the maximum likelihood estimates of the regression parameters. The model is evaluated via deviance and the method is illustrated with an example. Some extensions of the model are discussed.

Adult↗
PubMed2002

Modeling spatial survival data using semiparametric frailty models.

We propose a new class of semiparametric frailty models for spatially correlated survival data. Specifically, we extend the ordinary frailty models by allowing random effects accommodating spatial correlations to enter into the baseline hazard function multiplicatively. We prove identifiability of the models and give sufficient regularity conditions. We propose drawing inference based on a marginal rank likelihood. No parametric forms of the baseline hazard need to be assumed in this semiparametric approach. Monte Carlo simulations and the Laplace approach are used to tackle the intractable integral in the likelihood function. Different spatial covariance structures are explored in simulations and the proposed methods are applied to the East Boston Asthma Study to detect prognostic factors leading to childhood asthma.

Asthma↗
PubMed1999

Multiple local maxima for likelihoods of phylogenetic trees: a simulation study.

Steel demonstrated that the maximum-likelihood function for a phylogenetic tree may have multiple local maxima. If this phenomenon were general, it would compromise the applicability of maximum likelihood as an optimality criterion for phylogenetic trees. In several simulation studies reported on in this paper, the true tree, and other trees of very high likelihood, rarely had multiple maxima. Our results thus provide reassurance that the value of maximum likelihood as a tree selection criterion is not compromised by the presence of multiple local maxima--the best estimates of the true tree are not likely to have them. This result holds true even when an incorrect nucleotide substitution model is used for tree selection.

Base Sequence↗
PubMed2001

Map-likelihood phasing.

The recently developed technique of maximum-likelihood density modification [Terwilliger (2000), Acta Cryst. D56, 965-972] allows a calculation of phase probabilities based on the likelihood of the electron-density map to be carried out separately from the calculation of any prior phase probabilities. Here, it is shown that phase-probability distributions calculated from the map-likelihood function alone can be highly accurate and that they show minimal bias towards the phases used to initiate the calculation. Map-likelihood phase probabilities depend upon expected characteristics of the electron-density map, such as a defined solvent region and expected electron-density distributions within the solvent region and the region occupied by a macromolecule. In the simplest case, map-likelihood phase-probability distributions are largely based on the flatness of the solvent region. Though map-likelihood phases can be calculated without prior phase information, they are greatly enhanced by high-quality starting phases. This leads to the technique of prime-and-switch phasing for removing model bias. In prime-and-switch phasing, biased phases such as those from a model are used to prime or initiate map-likelihood phasing, then final phases are obtained from map-likelihood phasing alone. Map-likelihood phasing can be applied in cases with solvent content as low as 30%. Potential applications of map-likelihood phasing include unbiased phase calculation from molecular-replacement models, iterative model building, unbiased electron-density maps for cases where 2F(o) - F(c) or sigma(A)-weighted maps would currently be used, structure validation and ab initio phase determination from solvent masks, non-crystallographic symmetry or other knowledge about expected electron density.

Crystallography, X-Ray↗
PubMed2004

Statistical framework for phylogenomic analysis of gene family expression profiles.

Microarray technology has produced massive expression data that are invaluable for investigating the genome-wide evolutionary pattern of gene expression. To this end, phylogenetic expression analysis is highly desirable. On the basis of the Brownian process, we developed a statistical framework (called the E(0) model), assuming the independent expression of evolution between lineages. Several evolutionary mechanisms are integrated to characterize the pattern of expression diversity after gene duplications, including gradual drift and dramatic shift (punctuated equilibrium). When the phylogeny of a gene family is given, we show that the likelihood function follows a multivariate normal distribution; the variance-covariance matrix is determined by the phylogenetic topology and evolutionary parameters. Maximum-likelihood methods for multiple microarray experiments are developed, and likelihood-ratio tests are designed for testing the evolutionary pattern of gene expression. To reconstruct the evolutionary trace of expression diversity after gene (or genome) duplications, we developed a Bayesian-based method and use the posterior mean as predictors. Potential applications in evolutionary genomics are discussed.

Algorithms↗
PubMed2004

Modeling zero-inflated count series with application to occupational health.

A zero-inflated Poisson mixed autoregression model is presented for analyzing time series of count events with excess zeros. The model is motivated by the evaluation of a participatory ergonomics intervention intended to reduce manual handling workplace injuries over a specified time period. Random effects are introduced into the linear predictor of the model to account for serial correlation between successive observations. Parameter estimation is achieved by maximizing an appropriate log-likelihood function to obtain approximate residual maximum likelihood estimates. The method enables the evaluation of occupational intervention using population level aggregated count data series containing extra zeros.

Occupational Health↗