Search PubMed⌕ Search

SEARCH · Search PubMed

Results for “Likelihood Functions”

Search indexed PubMed citations on genomics, clinical trials, systematic reviews and public health. Explore titles, authors and supplied subject terms, then open the PubMed record.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2Linked to original sources
PubMed2001

An adjustment for a post-randomization variable in the comparison of two treatments for survival.

A method is proposed to infer the randomized treatment effect on survival after an adjustment for a post-randomization variable. The post-randomization variable is made independent of the treatment assignment and is considered a surrogate for baseline prognostic factors. The relationship between the post-randomization surrogate and baseline prognostic factors for survival is considered smooth, but otherwise unknown. The effect of these factors on survival is modelled using the Cox proportional hazards model with a semi-parametric relative risk function. Likelihood based estimates and tests for the treatment effect are demonstrated. An example from a randomized clinical trial in cancer illustrates the methodology.

Antibiotics, Antineoplastic↗
PubMed2004

Does resting two-dimensional echocardiography identify patients with ischemic cardiomyopathy and low likelihood of functional recovery after coronary revascularization?

OBJECTIVE: To evaluate the potential of a simple and widely available technique as two-dimensional (2D) echocardiography to identify patients with ischemic cardiomyopathy and low likelihood of functional recovery after coronary revascularization. METHODS: Two-dimensional echocardiography and radionuclide ventriculography (RNV) were performed before coronary revascularization in 94 patients with ischemic cardiomyopathy. Left ventricular ejection fraction (LVEF) was measured by RNV. Regional wall motion abnormalities, wall motion score index, end-diastolic wall thickness (EDWT), left ventricular (LV) volumes and LV sphericity index were assessed in the echocardiographic images. RNV was repeated 9-12 months after revascularization to assess LVEF change; an improvement >or=5% was considered clinically significant. RESULTS: Nine hundred and ninety-nine segments were severely dysfunctional; 149 out of 999 (15%) had an EDWT or=100 ml/ml) and of the end-systolic volume index (>or=80 ml) was present in 32 (34%) and 21 (22%) patients, respectively. A spherical shape of the LV was observed in 35 (37%) patients. LVEF after revascularization increased in 30 out of 94 patients (32%) from 30+/-8% to 39+/-9% (P<0.0001). On multivariate analysis, the EDVI was the only predictor of no recovery in LVEF [odds ratio, 1.06, confidence interval (CI), 1.04-1.1, P<0.0001]. The cut-off value of EDVI >or=90 ml/ml accurately identified patients that virtually never recover. Post-operatively, LVEF increased in three out of 42 (7%, 95% CI 0-15%) patients with EDVI >or=90 ml/ml as compared to 27 out of 52 (52%) patients with EDVI<90 ml/ml (P<0.0001). CONCLUSIONS: In patients with ischemic cardiomyopathy and severe LV enlargement, improvement of LVEF after revascularization is unlikely to occur. Conversely, in patients with relatively preserved LV size, a higher likelihood of functional recovery may be anticipated.

Aged↗
PubMed2005

A general class of Bayesian survival models with zero and nonzero cure fractions.

We propose a new class of survival models which naturally links a family of proper and improper population survival functions. The models resulting in improper survival functions are often referred to as cure rate models. This class of regression models is formulated through the Box-Cox transformation on the population hazard function and a proper density function. By adding an extra transformation parameter into the cure rate model, we are able to generate models with a zero cure rate, thus leading to a proper population survival function. A graphical illustration of the behavior and the influence of the transformation parameter on the regression model is provided. We consider a Bayesian approach which is motivated by the complexity of the model. Prior specification needs to accommodate parameter constraints due to the non-negativity of the survival function. Moreover, the likelihood function involves a complicated integral on the survival function, which may not have an analytical closed form, and thus makes the implementation of Gibbs sampling more difficult. We propose an efficient Markov chain Monte Carlo computational scheme based on Gaussian quadrature. The proposed method is illustrated with an example involving a melanoma clinical trial.

Adult↗
PubMed1987

A Monte Carlo evaluation of three statistical methods used in path analysis.

Results of a Monte Carlo study to investigate the properties of three statistical methods used extensively in path analysis of family data are presented. All three methods are based on the maximum likelihood principle and involve the assumptions of multivariate normality and large sample (asymptotic) statistical properties. The methods differ, however, in the specification of the likelihood function. Given a set of correlation estimates, method 1 maximizes the likelihood function under the stipulation that the estimates are independent. Method 2 differs from the former by allowing for covariances among the correlation estimators. Method 3 involves (direct) maximization of the likelihood function for the individual family observations assuming multivariate normality for the vector of family observations. The Monte Carlo study investigated validity of the test statistics and confidence intervals and evaluated the relative efficiency and bias of the parameter estimates based on 1,000 replications of each of several simulation conditions. The effects of violating the two basic assumptions, multivariate normality and asymptotic theory, were investigated by comparing results for non-normally vs normally distributed family data and for small vs large sample sizes. It is shown that method 3 provides valid statistical inferences under multivariate normality and that it is generally robust against minor departures from normality. Method 2 is also robust against minor deviations from normality, but it is sensitive to small sample sizes. Method 1 yields highly conservative test statistics under all conditions studied.

Genetics, Medical↗
PubMed1991

Derivative-free restricted maximum likelihood estimation in animal models with a sparse matrix solver.

Estimation of (co)variance components by derivative-free REML requires repeated evaluation of the log-likelihood function of the data. Gaussian elimination of the augmented mixed model coefficient matrix is often used to evaluate the likelihood function, but it can be costly for animal models with large coefficient matrices. This study investigated the use of a direct sparse matrix solver to obtain the log-likelihood function. The sparse matrix package SPARSPAK was used to reorder the mixed model equations once and then repeatedly to solve the equations by Cholesky factorization to generate the terms required to calculate the likelihood. The animal model used for comparison contained 19 fixed levels, 470 maternal permanent environmental effects, and 1586 direct and 1586 maternal genetic effects, resulting in a coefficient matrix of order 3661 with .3% nonzero elements after including numerator relationships. Compared with estimation via Gaussian elimination of the unordered system, utilization of SPARSPAK required 605 and 240 times less central processing unit time on mainframes and personal computers, respectively. The SPARSPAK package also required less memory and provided solutions for all effects in the model.

Algorithms↗
PubMed1996

Analysis of mortality rates via marginal extended quasi-likelihood.

We use a mixed Poisson regression model with extra variation to analyse mortality data cross-classified by age and geographic region. We use estimates of dispersion parameter and fixed effects parameters, obtained by maximizing a marginal quasi-likelihood function, to estimate mortality rates in an empirical Bayes manner. This is a modification of an earlier method by Tsutakawa that used the likelihood function and is computationally impractical for routine use. In order to apply the extended quasi-likelihood function, a linear variance function (of the mean) is used in place of a quadratic one used previously. We illustrate the method with male lung cancer deaths in the state of Missouri.

Age Distribution↗
PubMed1999

Reciprocal-space solvent flattening.

Solvent flattening is a powerful tool for improving crystallographic phases for macromolecular structures obtained at moderate resolution, but uncertainties in the optimal weighting of experimental phases and modified phases make it difficult to extract all the phase information possible. Solvent flattening is essentially an iterative method for maximizing a likelihood function which consists of (i) experimental phase information and (ii) information on the likelihood of various arrangements of electron density in a map, but the likelihood function is generally not explicitly defined. In this work, a procedure is described for reciprocal-space maximization of a likelihood function based on experimental phases and characteristics of the electron-density map. The procedure can readily be applied to phase improvement based on solvent flattening and can potentially incorporate information on a wide variety of other characteristics of the electron-density map.

Bacterial Proteins↗
PubMed1998

Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions.

We continue the theme of previous papers [J. Opt. Soc. Am. A 7, 1266 (1990); 12, 834 (1995)] on objective (task-based) assessment of image quality. We concentrate on signal-detection tasks and figures of merit related to the ROC (receiver operating characteristic) curve. Many different expressions for the area under an ROC curve (AUC) are derived for an arbitrary discriminant function, with different assumptions on what information about the discriminant function is available. In particular, it is shown that AUC can be expressed by a principal-value integral that involves the characteristic functions of the discriminant. Then the discussion is specialized to the ideal observer, defined as one who uses the likelihood ratio (or some monotonic transformation of it, such as its logarithm) as the discriminant function. The properties of the ideal observer are examined from first principles. Several strong constraints on the moments of the likelihood ratio or the log likelihood are derived, and it is shown that the probability density functions for these test statistics are intimately related. In particular, some surprising results are presented for the case in which the log likelihood is normally distributed under one hypothesis. To unify these considerations, a new quantity called the likelihood-generating function is defined. It is shown that all moments of both the likelihood and the log likelihood under both hypotheses can be derived from this one function. Moreover, the AUC can be expressed, to an excellent approximation, in terms of the likelihood-generating function evaluated at the origin. This expression is the leading term in an asymptotic expansion of the AUC; it is exact whenever the likelihood-generating function behaves linearly near the origin. It is also shown that the likelihood-generating function at the origin sets a lower bound on the AUC in all cases.

Area Under Curve↗
PubMed2005

Extending the resolution and phase-quality limits in automated model building with iterative refinement.

Previously, the direct use of prior phase information from a single-wavelength anomalous diffraction (SAD) experiment with a multivariate likelihood function applied to automated model building with iterative refinement has been proposed [Skubák et al. (2004), Acta Cryst. D60, 2196-2201]. In this approach, the anomalous information from the experimental data is used in refinement to derive phase information in a maximum-likelihood formalism and provided a more theoretically valid way of incorporating prior phase information compared with current approaches. In the present work, the SAD multivariate likelihood function that directly uses prior phase information is tested against currently used functions on many different SAD data sets which exhibit a wide range of resolution limits and anomalous signal. The results clearly show the importance of the more theoretically valid utilization of prior phase information: the SAD function extends the resolution and phase-quality limits needed for successful automated model building with iterative refinement. Indeed, the multivariate likelihood function reduces the overfitting in the refinement procedure and performs consistently better than the current refinement targets in terms of the quality of the models obtained and the number of residues built.

Computational Biology↗
PubMed2001

Statistical analysis of nuclear genome size of plants with flow cytometer data.

BACKGROUND: There is a mismatch between the sophistication of the cytometer and the resulting data, made possible by the computing power of today, and the traditional statistical methods based on the computing power of the early 1930s. The purpose here is to apply modern statistical techniques that similarly take advantage of this computer power. METHODS: Likelihood functions and their graphs are introduced as direct measures of plausibility of the parameters of interest. These methods are valid for samples of any size. They are exemplified on an experimental plant flow cytometer data set with n = 2 replications. RESULTS: The likelihood functions revealed important features of the data that would have been missed by the traditional methods, and in fact would invalidate them. CONCLUSIONS: The likelihood function produced highly informative graphs that allow quantitative comparisons of different aspects of 2C DNA nuclear contents among different groups or varieties of plants.

Analysis of Variance↗
PubMed1986

Maximum likelihood estimation for stationary point processes.

In this paper we derive the log likelihood function for point processes in terms of their stochastic intensities by using the martingale approach. For practical purposes we work with an approximate log likelihood function that is shown to possess the usual asymptotic properties of a log likelihood function. The resulting estimates are strongly consistent and asymptotically normal (under some regularity conditions). As a by-product, a strong law of large numbers and a central limit theorem for martingales in continuous times are derived.

Journal Article↗
PubMed2000

Estimation problems associated with stochastic modeling of proliferation and differentiation of O-2A progenitor cells in vitro.

Our previous research effort has resulted in a stochastic model that provides an excellent fit to our experimental data on proliferation and differentiation of oligodendrocyte type-2 astrocyte progenitor cells at the clonal level. However, methods for estimation of model parameters and their statistical properties still remain far away from complete exploration. The main technical difficulty is that no explicit analytic expression for the joint distribution of the number of progenitor cells and oligodendrocytes, and consequently for the corresponding likelihood function, is available. In the present paper, we overcome this difficulty by using computer-intensive simulation techniques for estimation of the likelihood function. Since the output of our simulation model is essentially random, stochastic optimization methods are necessary to maximize the estimated likelihood function. We use the Kiefer-Wolfowitz procedure for this purpose. Given sufficient computing resources, the proposed estimation techniques significantly extend the spectrum of problems that become approachable. In particular, these techniques can be applied to more complex branching models of multi-type cell systems with dependent evolutions of different types of cells.

Animals↗
PubMed2006

A statistical framework for genome-wide scanning and testing of imprinted quantitative trait loci.

Non-equivalent expression of alleles at a locus results in genomic imprinting. In this article, a statistical framework for genome-wide scanning and testing of imprinted quantitative trait loci (iQTL) underlying complex traits is developed based on experimental crosses of inbred line species in backcross populations. The joint likelihood function is composed of four component likelihood functions with each of them derived from one of four backcross families. The proposed approach models genomic imprinting effect as a probability measure with which one can test the degree of imprinting. Simulation results show that the model is robust for identifying iQTL with various degree of imprinting ranging from no imprinting, partial imprinting to complete imprinting. Under various simulation scenarios, the proposed model shows consistent parameter estimation with reasonable precision and high power in testing iQTL. When a QTL shows Mendelian effect, the proposed model also outperforms traditional Mendelian model. Extension to incorporate maternal effect is also given. The developed model, built within the maximum likelihood framework and implemented with the EM algorithm, provides a quantitative framework for testing and estimating iQTL involved in the genetic control of complex traits.

Algorithms↗
PubMed1998

Incorporation of prior phase information strengthens maximum-likelihood structure refinement.

The application of a maximum-likelihood analysis to the problem of structure refinement has led to striking improvements over the traditional least-squares methods. Since the method of maximum likelihood allows for a rational incorporation of other sources of information, we have derived a likelihood function that incorporates experimentally determined phase information. In a number of different test cases, this target function performs better than either a least-squares target or a maximum-likelihood function lacking prior phases. Furthermore, this target gives significantly better results compared with other functions incorporating phase information. When combined with a procedure to mask 'unexplained' density, the phased likelihood target also makes it possible to refine very incomplete models.

Animals↗
PubMed2005

Segmentation of fetal ultrasound images.

This paper describes a new method for segmentation of fetal anatomic structures from echographic images. More specifically, we estimate and measure the contours of the femur and of cranial cross-sections of fetal bodies, which can thus be automatically measured. Contour estimation is formulated as a statistical estimation problem, where both the contour and the observation model parameters are unknown. The observation model (or likelihood function) relates, in probabilistic terms, the observed image with the underlying contour. This likelihood function is derived from a region-based statistical image model. The contour and the observation model parameters are estimated according to the maximum likelihood (ML) criterion, via deterministic iterative algorithms. Experiments reported in the paper, using synthetic and real images, testify for the adequacy and good performance of the proposed approach.

Algorithms↗
PubMed2000

Analysis of single gene multitrait effects in livestock by the use of Gibbs sampling.

The paper presents a method of multivariate data analysis described by a model which involves fixed effects, additive polygenic individual effects and the effects of a major gene. To find the estimates of model parameters, the maximization of likelihood function method is applied. The maximum of likelihood function is computed by the use of the Gibbs sampling approach. In this approach, following the conditional posterior distributions, values of all unknown parameters are generated. On the basis of the obtained samples the marginal posterior densities as well as the estimates of fixed effects, gene frequency, genotypic values, major gene, polygenic and error (co)variances are calculated. A numerical example, supplemented to theoretical considerations, deals with data simulated according to the considered model.

Journal Article↗
PubMed1974

A model for analysis of population structure.

Arguments have been presented for the appropriateness of a multinomial Dirichlet distribution for describing single-locus genotypic frequencies in a subdivided population. This distribution is defined as a function of allele frequency, the average (over the entire population) inbreeding coefficient and the correlation between genotypes within a subdivision. Alternative parameterizations and their genetic interpretations are given.-We then show how information from a sample drawn from this subdivided population, in the absence of pedigrees, can be combined with the multinomial Dirichlet model to form a likelihood function. This likelihood function is then used as the basis for estimation and testing hypotheses concerning the genetic parameters of the model. Comparisons of this approach to the alternative procedure of Cockerham (1969) and (1973) are made using human data obtained from Tecumseh, Michigan and Monte Carlo simulations.-Finally, implications of these results to statistical inference and to mutation rates are presented.

Alleles↗
PubMed2004

Ascertainment-adjusted maximum likelihood estimation for the additive genetic gamma frailty model.

The additive genetic gamma frailty model has been proposed for genetic linkage analysis for complex diseases to account for variable age of onset and possible covariates effects. To avoid ascertainment biases in parameter estimates, retrospective likelihood ratio tests are often used, which may result in loss of efficiency due to conditioning. This paper considers when the sibships are ascertained by having at least two affected sibs with the disease before a given age and provides two approaches for estimating the parameters in the additive gamma frailty model. One approach is based on the likelihood function conditioning on the ascertainment event, the other is based on maximizing a full ascertainment-adjusted likelihood. Explicit forms for these likelihood functions are derived. Simulation studies indicate that when the baseline hazard function can be correctly pre-specified, both approaches give accurate estimates of the model parameters. However, when the baseline hazard function has to be estimated simultaneously, only the ascertainment-adjusted likelihood method gives an unbiased estimate of the parameters. These results imply that the ascertainment-adjusted likelihood ratio test in the context of the additive genetic gamma frailty may be used for genetic linkage analysis.

Age of Onset↗