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PubMed1998

Likelihood ratio tests for detecting positive selection and application to primate lysozyme evolution.

An excess of nonsynonymous substitutions over synonymous ones is an important indicator of positive selection at the molecular level. A lineage that underwent Darwinian selection may have a nonsynonymous/synonymous rate ratio (dN/dS) that is different from those of other lineages or greater than one. In this paper, several codon-based likelihood models that allow for variable dN/dS ratios among lineages were developed. They were then used to construct likelihood ratio tests to examine whether the dN/dS ratio is variable among evolutionary lineages, whether the ratio for a few lineages of interest is different from the background ratio for other lineages in the phylogeny, and whether the dN/dS ratio for the lineages of interest is greater than one. The tests were applied to the lysozyme genes of 24 primate species. The dN/dS ratios were found to differ significantly among lineages, indicating that the evolution of primate lysozymes is episodic, which is incompatible with the neutral theory. Maximum-likelihood estimates of parameters suggested that about nine nonsynonymous and zero synonymous nucleotide substitutions occurred in the lineage leading to hominoids, and the dN/dS ratio for that lineage is significantly greater than one. The corresponding estimates for the lineage ancestral to colobine monkeys were nine and one, and the dN/dS ratio for the lineage is not significantly greater than one, although it is significantly higher than the background ratio. The likelihood analysis thus confirmed most, but not all, conclusions Messier and Stewart reached using reconstructed ancestral sequences to estimate synonymous and nonsynonymous rates for different lineages.

Animals↗

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↗

PubMed1991

Methods for inferring phylogenies from nucleic acid sequence data by using maximum likelihood and linear invariants.

Likelihood methods and methods using invariants are procedures for inferring the evolutionary relationships among species through statistical analysis of nucleic acid sequences. A likelihood-ratio test may be used to determine the feasibility of any tree for which the maximum likelihood can be computed. The method of linear invariants described by Cavender, which includes Lake's method of evolutionary parsimony as a special case, is essentially a form of the likelihood-ratio method. In the case of a small number of species (four or five), these methods may be used to find a confidence set for the correct tree. An exact version of Lake's asymptotic chi 2 test has been mentioned by Holmquist et al. Under very general assumptions, a one-sided exact test is appropriate, which greatly increases power.

Animals↗

PubMed2000

Ab initio low-resolution phasing in crystallography of macromolecules by maximization of likelihood.

Statistical likelihood criteria were tested to select the true (or closest to true) structure-factor phases from an ensemble of phase sets. To define the criterion value for a given trial phase set, the trial 'molecular region' is defined as a region consisting of the points with the highest values in the Fourier synthesis calculated with the observed magnitudes and the trial set of phases. The structure studied is considered as composed of atoms randomly placed inside the trial molecular region. The figure of merit is defined as the likelihood corresponding to this hypothesis, i.e. the probability that the structure-factor magnitudes calculated (from the positions of atoms randomly placed into the trial region) are equal to the observed magnitudes. The concept of generalized likelihood is introduced to make the calculations more straightforward. The tests performed for known structures with the use of experimentally observed magnitudes show that in general it is impossible to unambiguously determine the best phases among a 'population' of trial phase sets. Nevertheless, the random generation of a great number of phase sets and the selection of phase sets with high likelihood values give a collection of variants with a higher concentration of 'good' phase sets than those found in the original population. Averaging the selected phase sets gives a starting solution of the low-resolution phase problem.

Crystallins↗

PubMed2004

Likelihood-enhanced fast rotation functions.

Experiences with the molecular-replacement program Beast have shown that maximum-likelihood rotation targets are more sensitive to the correct orientation than traditional targets. However, this comes at a high computational cost: brute-force rotation searches can take hours or even days of computation time on current desktop computers. Series approximations to the full likelihood target have been developed that can be computed by fast Fourier transforms in minutes. These likelihood-enhanced targets are more sensitive to the correct orientation than the Crowther fast rotation function and they take advantage of information from partial solutions. The likelihood-enhanced rotation targets have been implemented in the program Phaser.

Algorithms↗

PubMed2004

Association between maternal age and the likelihood of a cesarean section: a population-based multivariate logistic regression analysis.

BACKGROUND: A majority of studies examining the relationship between advancing maternal age and the likelihood of cesarean section (CS) use data from regional samples or from a limited number of medical institutions. This study uses population-based data from Taiwan to explore the relationship between maternal age and the likelihood of a CS. METHODS: The National Health Insurance Research Database (NHIRD) on registries of medical facilities and board-certified physicians and monthly claim summaries for inpatients were used. In total, 502 524 singleton deliveries were included in the study. Multivariate logistic regressions were performed with the presence of CS as the dependent variable and maternal age (<20, 20-29, 30-34 and >34 years) as the independent variable. The study controlled for maternal indications, institution characteristics, maternal requests and attending physician characteristics. RESULTS: CS rates for the age groups <20, 20-29, 30-34 and >34 years were 17.7, 27.4, 37.4 and 47.5%, respectively. The regression analyses consistently showed that the likelihood of a CS significantly increased with advancing maternal age within each category of complication after adjusting for medical institution characteristics and characteristics of the attending physician. CONCLUSIONS: This study found that, after adjusting for maternal indications, and healthcare institution and physician characteristics, there was a significant relationship between advancing maternal age and an increased likelihood of a CS. This finding, together with the high CS rate of 32.1% in Taiwan, one of the highest reported in the world today, highlights an imperative need to devise interventions to reduce the frequency of CSs.

Adult↗

PubMed2001

Likelihood ratio tests for a mixture of two von Mises distributions.

It is proposed that the orientation of elongate objects, such as bones, may be used to identify the flow direction of ancient river deposits. If true, elongate objects could be of great value when ancient bedforms such as ripples and dunes are not visible. Two sandstone quarries were investigated wherein the paleoflow direction was determined from both bedforms and elongate dinosaur bones. A mixture of two von Mises distributions captures the observation that elongate bones transported under unidirectional flow conditions will align both parallel and perpendicular to the flow direction. Likelihood ratio tests for a mixture of two von Mises distributions are given. The power of these tests is investigated by simulation since the direction of dinosaur bones agrees with the primary bedforms if the hypothesis test comparing the dominant mean direction of the bones to the paleoflow direction fails to reject. The likelihood ratio test on the dominant mean direction has reasonable power. If the two mean directions in the mixture distribution are pi apart, a more powerful likelihood ratio test can be used. The likelihood ratio test on the hypothesis that the two mean directions are exactly pi apart is useful in determining if the assumptions of the more powerful test are satisfied.

Animals↗

PubMed2002

Likelihood analysis for the ratio of means of two independent log-normal distributions.

Existing methods for comparing the means of two independent skewed log-normal distributions do not perform well in a range of small-sample settings such as a small-sample bioavailability study. In this article, we propose two likelihood-based approaches-the signed log-likelihood ratio statistic and modified signed log-likelihood ratio statistic-for inference about the ratio of means of two independent log-normal distributions. More specifically, we focus on obtaining p-values for testing the equality of means and also constructing confidence intervals for the ratio of means. The performance of the proposed methods is assessed through simulation studies that show that the modified signed log-likelihood ratio statistic is nearly an exact approach even for very small samples. The methods are also applied to two real-life examples.

Biological Availability↗

PubMed2004

A likelihood-based method for testing for nonstochastic variation of diversification rates in phylogenies.

Observed variations in rates of taxonomic diversification have been attributed to a range of factors including biological innovations, ecosystem restructuring, and environmental changes. Before inferring causality of any particular factor, however, it is critical to demonstrate that the observed variation in diversity is significantly greater than that expected from natural stochastic processes. Relative tests that assess whether observed asymmetry in species richness between sister taxa in monophyletic pairs is greater than would be expected under a symmetric model have been used widely in studies of rate heterogeneity and are particularly useful for groups in which paleontological data are problematic. Although one such test introduced by Slowinski and Guyer a decade ago has been applied to a wide range of clades and evolutionary questions, the statistical behavior of the test has not been examined extensively, particularly when used with Fisher's procedure for combining probabilities to analyze data from multiple independent taxon pairs. Here, certain pragmatic difficulties with the Slowinski-Guyer test are described, further details of the development of a recently introduced likelihood-based relative rates test are presented, and standard simulation procedures are used to assess the behavior of the two tests in a range of situations to determine: (1) the accuracy of the tests' nominal Type I error rate; (2) the statistical power of the tests; (3) the sensitivity of the tests to inclusion of taxon pairs with few species; (4) the behavior of the tests with datasets comprised of few taxon pairs; and (5) the sensitivity of the tests to certain violations of the null model assumptions. Our results indicate that in most biologically plausible scenarios, the likelihood-based test has superior statistical properties in terms of both Type I error rate and power, and we found no scenario in which the Slowinski-Guyer test was distinctly superior, although the degree of the discrepancy varies among the different scenarios. The Slowinski-Guyer test tends to be much more conservative (i.e., very disinclined to reject the null hypothesis) in datasets with many small pairs. In most situations, the performance of both the likelihood-based test and particularly the Slowinski-Guyer test improve when pairs with few species are excluded from the computation, although this is balanced against a decline in the tests' power and accuracy as fewer pairs are included in the dataset. The performance of both tests is quite poor when they are applied to datasets in which the taxon sizes do not conform to the distribution implied by the usual null model. Thus, results of analyses of taxonomic rate heterogeneity using the Slowinski-Guyer test can be misleading because the test's ability to reject the null hypothesis (equal rates) when true is often inaccurate and its ability to reject the null hypothesis when the alternative (unequal rates) is true is poor, particularly when small taxon pairs are included. Although not always perfect, the likelihood-based test provides a more accurate and powerful alternative as a relative rates test.

Biodiversity↗

PubMed1999

The usefulness of the likelihood ratio in the diagnosis of dyspepsia and gastroesophageal reflux disease.

Clinicians are familiar with the concepts of sensitivity and specificity to describe the accuracy of a diagnostic test. These measures do not always express the probability that a patient has a disease with a given test result as this will vary with the prevalence of the disorder in the population. The likelihood ratio is a more clinically relevant method of reporting accuracy, and the probability of having a disease after a positive or negative test can be calculated. The likelihood ratio can be applied to the clinical problem of dyspepsia management. This suggests that Helicobacter pylori (H. pylori) test and treat will detect and treat most peptic ulcers with only 0.5% of H. pylori-negative patients having peptic ulcer disease. Serology is possibly acceptable in populations with an H. pylori prevalence of approximately 50%. The urea breath tests are more appropriate in more extreme prevalence ranges. Once the prevalence of H. pylori falls below 10%, then the urea breath test becomes inaccurate, and screening and treatment may be less appropriate. The absolute probability of having peptic ulcer disease in a largely H. pylori-negative population will be very small, however, and the appropriateness of performing any investigation in these circumstances is debatable. Finally, likelihood ratios indicate that the clinical diagnosis of gastroesophageal reflux disease (GERD) is not straightforward. Traditionally, it is believed that patients with dominant heartburn are likely to have GERD. Likelihood ratios predict, however, that patients with these symptoms have a little over 50% chance of having GERD as defined by 24-h esophageal pH studies.

Dyspepsia↗

PubMed2003

Computerized scheme for determination of the likelihood measure of malignancy for pulmonary nodules on low-dose CT images.

An automated computerized scheme has been developed for determination of the likelihood measure of malignancy of pulmonary nodules on low-dose helical CT (LDCT) images. Our database consisted of 76 primary lung cancers (147 slices) and 413 benign nodules (576 slices). With this automated computerized scheme, the location of a nodule was first indicated by a radiologist. The outline of the nodule was segmented automatically by use of a dynamic programming technique. Various objective features on the nodules were determined by use of outline analysis and image analysis, and the likelihood measure of malignancy was determined by use of linear discriminant analysis (LDA). The effect of many different combinations of features and the performance of LDA in distinguishing benign nodules from malignant ones were evaluated by means of receiver operating characteristic (ROC) analysis. The Az value (area under the ROC curve) obtained by the computerized scheme in distinguishing benign nodules from malignant ones was 0.828 when a single slice was employed for each of the nodules. However, the Az value was improved to 0.846 when multiple slices were used for determination of the likelihood measure of malignancy. The Az values obtained by the computerized scheme on LDCT images were significantly greater than the Az value of 0.70, which was obtained from our previous observer studies by radiologists in distinguishing benign nodules from malignant ones on LDCT images. The automated computerized scheme for determination of the likelihood measure of malignancy would be useful in assisting radiologists to distinguish between benign and malignant pulmonary nodules on LDCT images.

Algorithms↗

PubMed2006

A maximum likelihood approach to density estimation with semidefinite programming.

Density estimation plays an important and fundamental role in pattern recognition, machine learning, and statistics. In this article, we develop a parametric approach to univariate (or low-dimensional) density estimation based on semidefinite programming (SDP). Our density model is expressed as the product of a nonnegative polynomial and a base density such as normal distribution, exponential distribution, and uniform distribution. When the base density is specified, the maximum likelihood estimation of the polynomial is formulated as a variant of SDP that is solved in polynomial time with the interior point methods. Since the base density typically contains just one or two parameters, computation of the maximum likelihood estimate reduces to a one- or two-dimensional easy optimization problem with this use of SDP. Thus, the rigorous maximum likelihood estimate can be computed in our approach. Furthermore, such conditions as symmetry and unimodality of the density function can be easily handled within this framework. AIC is used to choose the best model. Through applications to several instances, we demonstrate flexibility of the model and performance of the proposed procedure. Combination with a mixture approach is also presented. The proposed approach has possible other applications beyond density estimation. This point is clarified through an application to the maximum likelihood estimation of the intensity function of a nonstationary Poisson process.

Data Interpretation, Statistical↗

PubMed1994

Likelihood ratios for a sleep apnea clinical prediction rule.

Nocturnal polysomnography, the standard diagnostic test for sleep apnea, is an expensive and limited resource. In order to help identify the urgency of need for treatment, we determined which clinical features were most useful for establishing an accurate estimate of the probability that a patient had sleep apnea. Of 263 physician-referred patients, 200 were eligible for the study and 180 (90%) completed it. All patients had their histories recorded with a standard questionnaire, and underwent anthropomorphic measurements and nocturnal polysomnography. Sleep apnea was defined as more than 10 episodes of apnea or hypopnea per hour of sleep. Multiple linear and logistic regression models predictive of sleep apnea were compared with physicians' subjective impressions and previously reported models. Likelihood ratios were calculated for several levels of a sleep apnea clinical score produced by one of the linear models. Predictors of sleep apnea in the final model (R2 = 0.34) included neck circumference, hypertension, habitual snoring, and bed partner reports of nocturnal gasping/choking respirations. This model was superior to physician impression, slightly inferior to more detailed linear and logistic models, and comparable to previously reported models. A sleep apnea clinical score of less than 5 had a likelihood ratio of 0.25 (95% CI: 0.15 to 0.42) and a corresponding posttest probability of 17%, while a score of greater than 15 had a likelihood ratio of 5.17 (95% CI: 2.54 to 10.51) and posttest probability of 81%. These likelihood ratios can simply and accurately determined the probability of whether a patient has sleep apnea.

Adult↗

PubMedDate not supplied

A comparison of Bayesian and maximum likelihood methods to determine the performance of a point of care test for Helicobacter pylori in the office setting.

OBJECTIVE: Evaluations of point of care tests (PCT) are often hampered by a lack of appropriate gold standards. This study aimed to compare the results of a Bayesian statistical analysis and a maximum likelihood method to evaluate the performance of a PCT for Helicobacter pylori in primary care. METHODS: The Helisal Rapid Blood Test (Cortecs Diagnostics) was performed in 311 patients from 6 primary care centers, and a concurrent venous sample was taken for 2 enzyme-linked immunosorbent assays (ELISA) performed at the laboratory, blind to the PCT result. The Bayesian analysis was conducted using Markov Chain Monte Carlo methods (WinBUGS). The performance characteristics of the PCT and the 2 ELISA tests were estimated together with 95% credible intervals (95% CIs). RESULTS: The estimate of prevalence of H. pylori in this population was 64% (95% CI, 59% to 70%), the sensitivity and specificity of the PCT were 89% (84% to 94%) and 84% (77% to 91%), respectively (likelihood ratios positive 5.6, negative 0.13). The equivalent maximum likelihood results were prevalence, 65%; sensitivity, 90%; and specificity, 83%. CONCLUSIONS: The Helisal Rapid Blood Test performed as well as laboratory-based ELISA tests in this cohort of patients. The Bayesian analysis and the maximum likelihood method gave similar results, the Bayesian method also simultaneously estimating 95% CIs.

Aged↗

PubMed2004

The effect of correlation structure on treatment contrasts estimated from incomplete clinical trial data with likelihood-based repeated measures compared with last observation carried forward ANOVA.

Valid analyses of longitudinal data can be problematic, particularly when subjects dropout prior to completing the trial for reasons related to the outcome. Regulatory agencies often favor the last observation carried forward (LOCF) approach for imputing missing values in the primary analysis of clinical trials. However, recent evidence suggests that likelihood-based analyses developed under the missing at random framework provide viable alternatives. The within-subject error correlation structure is often the means by which such methods account for the bias from missing data. The objective of this study was to extend previous work that used only one correlation structure by including several common correlation structures in order to assess the effect of the correlation structure in the data, and how it is modeled, on Type I error rates and power from a likelihood-based repeated measures analysis (MMRM), using LOCF for comparison. Data from four realistic clinical trial scenarios were simulated using autoregressive, compound symmetric and unstructured correlation structures. When the correct correlation structure was fit, MMRM provided better control of Type I error and power than LOCF. Although misfitting the correlation structure in MMRM inflated Type I error and altered power, misfitting the structure was typically less deleterious than using LOCF. In fact, simply specifying an unstructured matrix for use in MMRM, regardless of the true correlation structure, yielded superior control of Type I error than LOCF in every scenario. The present and previous investigations have shown that the bias in LOCF is influenced by several factors and interactions between them. Hence, it is difficult to precisely anticipate the direction and magnitude of bias from LOCF in practical situations. However, in scenarios where the overall tendency is for patient improvement, LOCF tends to: 1) overestimate a drug's advantage when dropout is higher in the comparator and underestimate the advantage when dropout is lower in the comparator; 2) overestimate a drug's advantage when the advantage is maximum at intermediate time points and underestimate the advantage when the advantage increases over time; and 3) have a greater likelihood of overestimating a drug's advantage when the advantage is small. In scenarios in which the overall tendency is for patient worsening, the above biases are reversed. In the simulation scenarios considered in this study, which were patterned after acute phase neuropsychiatric clinical trials, the likelihood-based repeated measures approach, implemented with standard software, was more robust to the bias from missing data than LOCF, and choice of correlation structure was not an impediment to its implementation.

Analysis of Variance↗

PubMed2003

Computed tomographic angiography for detecting cerebral aneurysms: implications of aneurysm size distribution for the sensitivity, specificity, and likelihood ratios.

OBJECTIVE: This study examined the efficacy of computed tomographic angiography (CTA) for detection of ruptured and unruptured aneurysms after adjustment for their size distributions under various conditions of aneurysm prevalence. METHODS: A systematic review was used to estimate 1) the aneurysm size-specific sensitivity and specificity of CTA, and 2) the size distributions of ruptured and unruptured aneurysms. Probabilistic computer simulation was used to estimate the efficacy of CTA in the detection of aneurysms. RESULTS: The sensitivity of CTA ranged from 53% (95% confidence interval [CI], 44-62%) for 2-mm aneurysms to 95% (95% CI, 92-97%) for 7-mm aneurysms. The overall specificity was 98.9% (95% CI, 91.5-99.99%), but there was between-study heterogeneity. The estimated negative likelihood ratios for ruptured, unruptured, and at least 6-mm unruptured aneurysms were 0.081, 0.18, and 0.012, respectively. The positive likelihood ratio for CTA was sensitive to the pretest probability, the size of the positive finding, and the clinical context. With a small pretest probability, the positive likelihood ratio for unruptured aneurysms ranged from 15 for 2-mm aneurysms to 61 for 5-mm aneurysms. The positive likelihood ratio for ruptured aneurysms with an intermediate pretest probability (50%) ranged from 3.9 to 56 for 2- to 5-mm aneurysms. CONCLUSION: Small aneurysms detected on CTA should be investigated further unless there is a high pretest probability of a ruptured aneurysm. During screening for ruptured aneurysms, a negative CTA should be investigated further. During screening for unruptured aneurysms, a negative CTA results in a very low probability of a clinically important aneurysm.

Aneurysm, Ruptured↗

PubMed1994

Corrected likelihood for proportional hazards measurement error model and its application.

Consider the case where the exact values of covariates in the proportional hazards model may not be observed but instead, only surrogates for them involving measurement errors are available. The maximum likelihood estimate based on the partial likelihood with the true covariate replaced by the observed surrogate is even asymptotically biased and may cause seriously misleading results in covariance analysis based on the partial likelihood. These facts are illustrated by Monte Carlo simulation. A correction to partial likelihood proposed by the first author is studied to gain insight into its merits and limitations in practical applications. The results indicate that when the "effective magnitude of the measurement error" as defined in this article is small, which is indeed the case for most applications, the method will be useful. Some other correction methods for the measurement error in censored survival models are also reviewed and discussed.

Adult↗

PubMed2005

Maximum-likelihood estimation of coalescence times in genealogical trees.

We develop a method for maximum-likelihood estimation of coalescence times in genealogical trees, based on population genetics data. For this purpose, a Viterbi-type algorithm is constructed to maximize the joint likelihood of the coalescence times. Marginal confidence intervals for the coalescence times based on the profile likelihoods are also computed. Our method of finding MLEs and calculating C.I.'s appears to be more accurate than alternative numerical maximization methods, and maximum-likelihood inference appears to be more accurate than other existing model-free approaches to estimating coalescent times. We demonstrate the method on two different data sets: human Y chromosome DNA data and fungus DNA data.

Algorithms↗