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Results for “Likelihood Functions”

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PubMed1991

Bayesian image processing in magnetic resonance imaging.

In the past several years, image processing techniques based on Bayesian models have received considerable attention. In our earlier work, we developed a novel Bayesian approach which was primarily aimed at the processing and reconstruction of images in positron emission tomography. In this paper, we describe how the technique has been adopted to process magnetic resonance images in order to reduce noise and artifacts, thereby improving image quality. In this framework, the image is assumed to be a statistical variable whose posterior probability density conditional on the observed image is modeled by the product of the likelihood function of the observed data with a prior density based our prior knowledge. A Gibbs random field incorporating local continuity information and with edge-detection capability is used as the prior model. Based on the formalism of the posterior density, we can compute an estimate of the image using an iterative technique. We have implemented this technique and applied it to phantom and clinical images. Our results indicate that the approach works reasonably well for reducing noise, enhancing edges, and removing ringing artifact.

Algorithms↗

PubMed1997

Analyses on the temporal patterns of spikes of auditory neurons of the macaque monkey by means of an artificial neural network and tree-based models.

The time scale over which information in the primary auditory cortex is processed was estimated. An artificial neural network was used to learn the temporal patterns of spikes. After learning, test patterns were input to the network. Comparison of the accuracy of the network with that of the maximum likelihood function computed from the spike count reveals that the temporal patterns of spikes are closely related to stimulus discrimination. Next, a tree-based model from a subset of the spike trains with a fixed time resolution was constructed and validated the model with another. By repeating this for different bin widths, it was found that there are no simple models for the time bin width larger than 50 ms. This indicates that the time scale in the auditory cortex is not larger than 50 ms.

Animals↗

PubMed1976

Point and interval estimation in the combination of bioassay results.

A procedure for combining evidence from different biological assays is shown to be equivalent both to generalized least-squares and to maximum-likelihood estimation. By appropriate nesting of hypotheses, the likelihood function can be used to test the agreement between the assays and to obtain probability limits for the combined estimate of potency. The properties of these limits are examined, with particular reference to the situation, unusual but not impossible in practice, in which the values of relative potency that they define consist of several disjoint segments instead of a single interval. The connection with general theory of estimating linear functional relations is pointed out.

Biological Assay↗

PubMed1990

A Bayesian method for synthesizing evidence. The Confidence Profile Method.

This article describes a collection of meta-analysis techniques based on Bayesian statistics for interpreting, adjusting, and combining evidence to estimate parameters and outcomes important to the assessment of health technologies. The result of an analysis by the Confidence Profile Method is a joint posterior probability distribution for the parameters of interest, from which marginal distributions for any particular parameter can be calculated. The method can be used to analyze problems involving a variety of types of outcomes, a variety of measures of effect, and a variety of experimental designs. This article presents the elements necessary for analysis, including prior distributions, likelihood functions, and specific models for experimental designs that include adjustment for biases.

Bayes Theorem↗

PubMed2001

Introduction to Bayesian reasoning.

Interest in Bayesian analyses has increased recently, in part as a response to policy makers wanting sound scientific bases for health technology assessments, and associated healthcare funding decisions. This paper provides a brief and simplified description of Bayesian reasoning. Bayes is illustrated in a clinical setting of an expert helping a woman understand the potential risk of passing on an inheritable disease (hemophilia) to her next child, based on disease occurrence in two living children. The illustration describes fundamental concepts and derivations, such as Bayes theorem, likelihood functions, prior probability, and posterior probability. A second illustration shows the use of Bayes for interpreting clinical trial results. The uncertainty in the clinical effect before and after the trial analyses has been completed is characterized by the Bayes prior and posterior probabilities, respectively. Techniques are also shown for estimating the potential loss (e.g., in lives lost) for making the wrong decision with and without knowledge of the trial results, an estimation that cannot be carried out using techniques of hypotheses testing associated with the frequentist school of statistics. Information from Bayes analysis then may be used to help policy makers decide, or justify, whether the analyses provides a sufficient basis for making a treatment recommendation, or whether there remains a need to request more information. Subsequent papers in this volume offer additional examples and clarification of the use of Bayes in clinical practice and in interpretation of clinical studies.

Bayes Theorem↗

PubMed1998

A study of interval censoring in parametric regression models.

Parametric models for interval censored data can now easily be fitted with minimal programming in certain standard statistical software packages. Regression equations can be introduced, both for the location and for the dispersion parameters. Finite mixture models can also be fitted, with a point mass on right (or left) censored observations, to allow for individuals who cannot have the event (or already have it). This mixing probability can also be allowed to follow a regression equation. Here, models based on nine different distributions are compared for three examples of heavily censored data as well as a set of simulated data. We find that, for parametric models, interval censoring can often be ignored and that the density, at centres of intervals, can be used instead in the likelihood function, although the approximation is not always reliable. In the context of heavily interval censored data, the conclusions from parametric models are remarkably robust with changing distributional assumptions and generally more informative than the corresponding non-parametric models.

Animals↗

PubMed2002

Segregation analysis of prostate cancer in 1,719 white, African-American and Asian-American families in the United States and Canada.

UNLABELLED: Some data suggest that brothers of prostate cancer patients have higher disease risk than their fathers, supporting an X-linked or recessive mode of inheritance. However, higher observed frequencies in brothers than fathers may merely reflect the strong temporal changes in US incidence rates. OBJECTIVES: (a) to evaluate the fit of X-linked, recessive, and dominant modes of inheritance to prostate cancer incidence, specific for calendar year, age, and race, in population-based samples of US and Canadian families; and (b) to evaluate a simple multifactorial model for familial aggregation of prostate cancer due to shared low-penetrance variants of many genes or shared lifestyle factors. METHODS: The data consist of reported prostate cancer incidence in first-degree relatives of 1,719 white, African-American, and Asian-American men with and without prostate cancer at ages <70 years. Model parameters were estimated by maximizing a pseudo-likelihood function of the data, and goodness of model fit was assessed by evaluating discrepancies between observed and expected numbers of pairs of relatives with prostate cancer. RESULTS: After adjusting for temporal trends in prostate cancer incidence rates we found that the X-linked model fit poorly. underpredicting the observed number of affected father-son pairs. This also was true of the recessive model, although the evidence for poor fit did not achieve statistical significance. In contrast, the dominant model provided adequate fit to the data. In this model the race-specific penetrance estimates for carriers of deleterious genotypes were similar among African-Americans and whites, but lower among Asian-Americans: risk by age 80 years for carriers born in 1900 was estimated as 75.3% for African-Americans and whites, and 44.4% for Asian-Americans. None of the Mendelian models fit the data better than did the simple multifactorial model. CONCLUSIONS: The good fit of the multifactorial model suggests that multiple genes, each having low penetrance, may be responsible for most inherited prostate cancer susceptibility, and that the contribution of rare highly penetrant mutations is small.

Adult↗

PubMed1998

A use of Monte Carlo integration for population pharmacokinetics with multivariate population distribution.

This paper describes a use of Monte Carlo integration for population pharmacokinetics with multivariate population distribution. In the proposed approach, a multivariate lognormal distribution is assumed for a population distribution of pharmacokinetic (PK) parameters. The maximum likelihood method is employed to estimate the population means, variances, and correlation coefficients of the multivariate lognormal distribution. Instead of a first-order Taylor series approximation to a nonlinear PK model, the proposed approach employs a Monte Carlo integration for the multiple integral in maximizing the log likelihood function. Observations below the lower limit of detection, which are usually included in Phase 1 PK data, are also incorporated into the analysis. Applications are given to a simulated data set and an actual Phase 1 trial to show how the proposed approach works in practice.

Algorithms↗

PubMed1995

Complex segregation analysis of Gerbera flower colour.

The distribution of hue (CIELAB colour notation) classes among flowers of the Davis population of gerbera (Gerbera jamesonii H. Bolus ex Hooker) appears bimodal. This suggests that the genetic control of hue is determined by the segregation of a gene with large effect modified by additional genes with smaller effects. Complex segregation analysis (CSA), routinely employed in human genetic epidemiology, was used to study both qualitative and quantitative variation. CSA applies pedigree analysis through the consideration of transmission probabilities to optimize likelihood functions of various genetic models. Applying this technique to study flower hue of a sample representing generations 14, 15 and 16 of the Davis population, allowed identification of a putative dominant major gene with genotypic values for the dominant homozygote, heterozygote and recessive homozygote of 32, 32 and 71 degrees, respectively. This corresponds to the modes of the hue frequency distribution for the population. The putative major gene represents 0.66 of the total variation. The residual parent-offspring correlation (rho po = 0.2) measures the genetic contribution to the remainder of the variance.

Heterozygote↗

PubMed1996

Number of lethal equivalents in human populations: how good are the previous estimates?

Lee et al. (1996) recently developed a method for interval estimation of the number of lethal equivalents by using a hierarchical structure of likelihood functions. This hierarchical model consists of two multinomial trials: one of the sampling process of the parents from the population of interest, and the other for the survival of the offspring of the families obtained by mating the parents. The method, initially developed for selfing and full-sib mating, is extended here to include more general mating systems as well as mixtures of mating systems. We applied it to human data sets for which confidence intervals were previously not available. Our point estimates were close to previous ones, and the standard deviations were generally quite small. Thus, even if debate over the meaning of the concept of lethal equivalents has not been entirely resolved, our results showed that the previous estimates are at least statistically meaningful.

Alleles↗

PubMed2002

Morphological and mechanical information of coronary arteries obtained with intravascular elastography; feasibility study in vivo.

AIMS: Plaque composition is a major determinant of coronary related clinical syndromes. In vitro experiments on human coronary and femoral arteries have demonstrated that different plaque types were detectable with intravascular ultrasound elastography. The aim of this study was to investigate the feasibility of applying intravascular elastography during interventional catheterization procedures. METHODS AND RESULTS: Data were acquired in patients (n=12) during PTCA procedures with an EndoSonics InVision echoapparatus equipped with radiofrequency output. The systemic pressure was used to strain the tissue, and the strain was determined using cross-correlation analysis of sequential frames. A likelihood function was determined to obtain the frames with minimal motion of the catheter in the lumen, since motion of the catheter prevents reliable strain estimation. Minimal motion was observed near end-diastole. Reproducible strain estimates were obtained within one pressure cycle and over several pressure cycles. Validation of the results was limited to the information provided by the echogram. Strain in calcified material (0.20%+/-0.07) was lower (P<0.001) than in non-calcified tissue (0.51%+/-0.20). CONCLUSION: In vivo intravascular elastography is feasible. Significantly higher strain values were found in non-calcified plaques than in calcified plaques.

Adult↗

PubMed2003

Segmentation-free statistical image reconstruction for polyenergetic x-ray computed tomography with experimental validation.

This paper describes a statistical image reconstruction method for x-ray CT that is based on a physical model that accounts for the polyenergetic x-ray source spectrum and the measurement nonlinearities caused by energy-dependent attenuation. Unlike our earlier work, the proposed algorithm does not require pre-segmentation of the object into the various tissue classes (e.g., bone and soft tissue) and allows mixed pixels. The attenuation coefficient of each voxel is modelled as the product of its unknown density and a weighted sum of energy-dependent mass attenuation coefficients. We formulate a penalized-likelihood function for this polyenergetic model and develop an iterative algorithm for estimating the unknown density of each voxel. Applying this method to simulated x-ray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artefacts relative to conventional beam hardening correction methods. We also apply the method to real data acquired from a phantom containing various concentrations of potassium phosphate solution. The algorithm reconstructs an image with accurate density values for the different concentrations, demonstrating its potential for quantitative CT applications.

Absorptiometry, Photon↗

PubMed1992

AIDS in Ireland: the reporting delay distribution and the implementation of integral equation models.

This paper deals with two basic aspects concerning the modelling of AIDS incidence in the context of Irish data. We describe initially the adjustment of the number of AIDS cases (Xij) to allow for reporting delays, where a simple form of the likelihood function for the Xij is supported by GLIM. Subsequently, we consider the accessibility of numerical solution (through a NAG routine) of the integral equation models generated by the back-projection method for the adjusted AIDS cases. Results for the Irish data are summarized for various choices of the incidence distribution.

Acquired Immunodeficiency Syndrome↗

PubMed1996

Estimating the age of the common ancestor of a DNA sample using the number of segregating sites.

The number of segregating sites in a sample of DNA sequences and the age of the most recent common ancestor (MRCA) of the sequences in the sample are positively correlated. The value of the former can be used to estimate the value of the latter. Using the coalescent approach, we derive in this paper the joint probability distribution of the number of segregating sites and the age of the MRCA of a sample under the neutral Wright-Fisher model. From this distribution, we are able to compute the likelihood function of the number of segregating sites and the posterior probability of the age of the MRCA of a sample. Three point estimators and one interval estimator of the age of the MRCA are developed; their relationships and properties are investigated. The estimation of the age of the MRCA of human Y chromosomes from a sample of no variation is discussed.

DNA↗

PubMed1997

The sampling distribution of disease-associated alleles.

A theory is developed that provides the sampling distribution of low frequency alleles at a single locus under the assumption that each allele is the result of a unique mutation. The numbers of copies of each allele is assumed to follow a linear birth-death process with sampling. If the population is of constant size, standard results from theory of birth-death processes show that the distribution of numbers of copies of each allele is logarithmic and that the joint distribution of numbers of copies of k alleles found in a sample of size n follows the Ewens sampling distribution. If the population from which the sample was obtained was increasing in size, if there are different selective classes of alleles, or if there are differences in penetrance among alleles, the Ewens distribution no longer applies. Likelihood functions for a given set of observations are obtained under different alternative hypotheses. These results are applied to published data from the BRCA1 locus (associated with early onset breast cancer) and the factor VIII locus (associated with hemophilia A) in humans. In both cases, the sampling distribution of alleles allows rejection of the null hypothesis, but relatively small deviations from the null model can account for the data. In particular, roughly the same population growth rate appears consistent with both data sets.

Alleles↗

PubMed1999

Quantitative evaluation of alternative mechanisms of blood and testes disposition of di(2-ethylhexyl) phthalate and mono(2-ethylhexyl) phthalate in rats.

Di(2-ethylhexyl) phthalate (DEHP), a commercially important plasticizer, induces testicular toxicity in laboratory animals at high doses. After oral exposure, most of the DEHP is rapidly metabolized in the gut to mono(2-ethylhexyl) phthalate (MEHP), which is the active metabolite for induction of testicular toxicity. To quantify the testes dose of MEHP with various routes of exposure and dose levels, we developed a physiologically based pharmacokinetic (PBPK) model for DEHP and MEHP in rats. Tissue:blood partition coefficients for DEHP were estimated from the n-octanol: water partition coefficient, while partition coefficients for MEHP were determined experimentally using a vial equilibration technique. All other parameters were either found in the literature or estimated from blood or tissue levels following oral or intravenous exposure to DEHP or MEHP. A flow-limited model failed to adequately simulate the available data. Alternative plausible mechanisms were explored, including diffusion-limited membrane transport, enterohepatic circulation, and MEHP ionization (pH-trapping model). In the pH-trapping model, only nonionized MEHP is free to become partitioned into the tissues, where it is equilibrated and trapped as ionized MEHP until it is deionized and released. All three alternative models significantly improved predictions of DEHP and MEHP blood concentrations over the flow-limited model predictions. The pH-trapping model gave the best predictions with the largest value of the log likelihood function. Predicted MEHP blood and testes concentrations were compared to measured concentrations in juvenile rats to validate the pH-trapping model. Thus, MEHP ionization may be an important mechanism of MEHP blood and testes disposition in rats.

Animals↗

PubMed2000

Quantitative evaluation of alternative mechanisms of blood disposition of di(n-butyl) phthalate and mono(n-butyl) phthalate in rats.

Phthalate esters are ubiquitous, low-level environmental contaminants that induce testicular toxicity in laboratory animals. The diester is rapidly metabolized in the gut to the monoester, which causes the testicular toxicity. Several physiologically based pharmacokinetic (PBPK) model structures have been evaluated for di(2-ethylhexyl) phthalate (DEHP) and mono(2-ethylhexyl) phthalate (MEHP). The objective of this study was to test these PBPK models for a less lipophilic phthalate diester, di(n-butyl) phthalate (DBP), and monoester, mono(n-butyl) phthalate (MBP). Alternate models describing enterohepatic circulation, diffusion-limitation, tissue pH gradients (pH trapping), and a simpler, flow-limited model were evaluated. A combined diffusion-limited and pH trapping model was also tested. MBP tissue:blood partition coefficients were similar when determined either experimentally by a nonvolatile, vial equilibration technique or algorithmically. All other parameters were obtained from the literature or estimated from MBP blood concentrations following intravenous or oral exposure to DBP or MBP. A flow-limited model was unable to predict MBP blood levels, whereas each alternative model had statistically better predictions. The combined diffusion-limited and pH trapping model was the best overall, having the highest log-likelihood function value. This result is consistent with a previous finding that the pH trapping model was the best model for describing DEHP and MEHP blood dosimetry, though it was necessary to extend the model to include diffusion-limitation. The application of the pH trapping model is a step toward developing a generic model structure for all phthalate esters, though more work is required before a generic structure can be identified with confidence. Development of a PBPK model structure applicable to all phthalate esters would support more realistic assessments of risk to human health from exposure to one or more members of this class of compounds.

Administration, Oral↗

PubMed1997

Level detection in ion channel records via idealization by statistical filtering and likelihood optimization.

A parameter-free method is presented for the level detection in ion channel records via recovery of step wise current changes. No assumptions about ion channel mechanism are made. The primary detection of the transitions is made by statistical filtering the data using the Student's t-test. The event currents are calculated as the average value of the current between two adjacent transitions. An optimal ideal trace is found by maximization of a likelihood function. The distribution of event currents recovered from the raw data is then analysed, again by using the Student's t-test, for their grouping into separate statistical ensembles, defining current levels. The method is subjected to rigorous test using simulated data, and is compared with several other methods. It produces the levels of channel current, their noise amplitudes and distributions of dwell times, the desired information for constructing the channel mechanism.

Ion Channels↗