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PubMed1992

A likelihood approach to HLA serology.

A likelihood approach to HLA serology has been developed in which the aim is not to define a recognition set for a serum but to describe the serum's ability to react with each and every antigen in the test cells, this ability being quantified in terms of the probability of a positive reaction. For a given set of probabilities, one for each antigen, it is possible to derive the probability of the observed set of reactions (the likelihood of the set of probabilities). The maximum possible value of the likelihood for any possible combination of the probability set can then be sought, but this requires a maximization of likelihood with respect to 60-100 independent parameters. Theoretical considerations of the shape of the likelihood surface prove that, in this particular case, this is a feasible proposition. This approach allows the recognition of three groups of antigens: those for which there is considerable evidence of a specificity, those for which there is either no specificity or a very weak specificity, and those for which there is insufficient evidence on which to base a conclusion. The existence of a specificity can be tested using a log likelihood ratio as a statistic, but the usual assumption of a chi 2 distribution of this statistic cannot automatically be made in this situation. Therefore, the distribution is estimated by simulation. A serologist using this approach would receive considerably more information as to the serum's reaction patterns and valid statistics for the existence, or not, of a specificity.

HLA Antigens↗

PubMed2002

Multilevel likelihood ratios for identifying exudative pleural effusions(*).

STUDY OBJECTIVES: To determine multilevel likelihood ratios for pleural fluid tests that are commonly used to discriminate between exudative and transudative pleural effusions. DESIGN: Meta-analysis of patient-level data. PATIENT DATA: Selected studies included patients with diagnoses of exudative or transudative pleural effusions who underwent thoracentesis and laboratory analysis of their pleural fluid. MEASUREMENTS AND METHODS: Studies were identified by searching MEDLINE and related bibliographies. Data were obtained for 1,448 patients from seven primary investigators or extracted from dot plots in published reports. Likelihood ratios were calculated from extracted data stratified across ranges of test result values. RESULTS: Sufficient data were available to calculate multilevel likelihood ratios for the elements of Light's criteria (pleural fluid lactate dehydrogenase [LDH], ratio of pleural fluid to serum LDH, and ratio of pleural fluid to serum protein), pleural fluid protein, ratio of pleural fluid to serum cholesterol, pleural fluid cholesterol, and gradient of pleural fluid to serum albumin. Each of these tests provided levels of likelihood ratios through the most clinically relevant range (0 to 10). CONCLUSION: Multilevel likelihood ratios combined with a clinician's estimation of the pretest probability of an exudative effusion improve the diagnostic accuracy of discriminating between exudative and transudative pleural effusions. Likelihood ratios avoid the use of confusing terms, such as "pseudoexudates," that derive from the use of single cutoff points for pleural fluid tests.

Diagnosis, Differential↗

PubMed2006

Analysis of neighborhood dynamics of forest ecosystems using likelihood methods and modeling.

Advances in computing power in the past 20 years have led to a proliferation of spatially explicit, individual-based models of population and ecosystem dynamics. In forest ecosystems, the individual-based models encapsulate an emerging theory of "neighborhood" dynamics, in which fine-scale spatial interactions regulate the demography of component tree species. The spatial distribution of component species, in turn, regulates spatial variation in a whole host of community and ecosystem properties, with subsequent feedbacks on component species. The development of these models has been facilitated by development of new methods of analysis of field data, in which critical demographic rates and ecosystem processes are analyzed in terms of the spatial distributions of neighboring trees and physical environmental factors. The analyses are based on likelihood methods and information theory, and they allow a tight linkage between the models and explicit parameterization of the models from field data. Maximum likelihood methods have a long history of use for point and interval estimation in statistics. In contrast, likelihood principles have only more gradually emerged in ecology as the foundation for an alternative to traditional hypothesis testing. The alternative framework stresses the process of identifying and selecting among competing models, or in the simplest case, among competing point estimates of a parameter of a model. There are four general steps involved in a likelihood analysis: (1) model specification, (2) parameter estimation using maximum likelihood methods, (3) model comparison, and (4) model evaluation. Our goal in this paper is to review recent developments in the use of likelihood methods and modeling for the analysis of neighborhood processes in forest ecosystems. We will focus on a single class of processes, seed dispersal and seedling dispersion, because recent papers provide compelling evidence of the potential power of the approach, and illustrate some of the statistical challenges in applying the methods.

Computer Simulation↗

PubMed2006

Sampling variability and estimates of density dependence: a composite-likelihood approach.

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.

Algorithms↗

PubMed2004

Quantitative trait loci mapping for dairy cattle production traits using a maximum likelihood method.

A maximum likelihood method was developed for QTL mapping in half-sib designs and compared to the regression method in analyses of both field and simulated data. The field data consisted of milk production evaluations of 433 progeny tested sons of 6 sires and 64 microsatellite markers distributed over 12 chromosomes. Based on permutation tests, 5 significant QTL were detected in the field data by the regression method compared with 10 by the maximum likelihood method (P < 0.05). In field data analysis, the maximum likelihood method detected more significant QTL and had a smaller residual variance than the regression method. The simulation included 9 scenarios differing in number of families, family size, QTL variance, and marker density, each replicated 100 times. The simulation results suggested that, as for the regression method, the precision of estimating QTL from the maximum likelihood method improves with increasing number of sons per sire, increasing the ratio of QTL to phenotypic variance, and decreasing marker interval. The maximum likelihood method had a smaller dispersion of estimated QTL positions than the regression method in 6 of 9 scenarios simulated. Overall, the maximum likelihood method shows potential advantage in QTL detection over the regression method, especially in the situations with less favorable conditions for QTL detection.

Algorithms↗

PubMed1988

Application of the maximum likelihood principle to separate exponential terms in T2 relaxation of nuclear magnetic resonance.

The method of maximum likelihood has been implemented for the estimation of multiple exponential components of T2 decay curves in spin echo NMR measurements on biologic tissues. Each Each component contributes an exponential term described by two parameters (initial amplitude and T2) to the T2 decay curve. The maximum likelihood method estimates the parameters and their standard errors for all terms simultaneously, avoiding the subjectivity inherent in methods such as graphical peeling. In the model used, it was assumed that water protons are compartmentalized and that the measured spin echo signals from the protons undergoing relaxation obey the Poisson distribution. A system of non-linear equations was derived and solved iteratively for the values of the exponential parameters which maximize the likelihood of obtaining the observed data under these assumptions. The approach was implemented for bi- and tri-exponential models on a MicroVAX II computer (Digital Equipment Corporation, Maynard, MA). Simulations of bi- and tri-exponential data, with and without system noise, were analyzed to assess the accuracy and reproducibility of the method. A subset of the simulations was repeated with non-linear least squares techniques and was compared to the results obtained with maximum likelihood. Rabbit muscle and gerbil brain samples were measured and analyzed with the maximum likelihood method. The simulations showed that within specific limits on relative sizes and relaxation rates of components, these parameters can be estimated with errors less than 5%. The comparison to non-linear least squares analysis showed that the maximum likelihood method is generally superior in estimating the parameters in difficult cases. The results from tissue measurements demonstrate that the method is effective even in cases where graphical peeling would clearly not yield reliable results.

Algorithms↗

PubMed2004

EEG synchronization likelihood in mild cognitive impairment and Alzheimer's disease during a working memory task.

OBJECTIVE: Synchronization likelihood analysis of resting state EEG has shown that cognitive dysfunction in Alzheimer's disease (AD) and its precursor mild cognitive impairment (MCI) are associated with a loss of functional connectivity in high (upper alpha and beta) frequency bands. Working memory tasks are known to change functional connectivity, but it is unknown whether this increases the differences between AD, MCI and healthy controls. Our objective was to investigate the behavior of synchronization likelihood of multichannel EEG in AD, MCI and cognitively healthy controls, both at rest and during a working memory task. METHODS: EEGs (200 Hz sample frequency, 21 channels, average reference) were recorded at rest as well as during a visual working memory task in 14 patients with AD according to the NINCDS-ADRDA criteria (mean age 76.4; SD 13.6), 11 patients with MCI according to the criteria of Petersen (mean age 78.4; SD 6.4) and 14 with subjective memory complaints but no demonstrable memory disturbance (mean age 61.6; SD 26.6). The synchronization likelihood was computed over 19 channels, comparing each channel with all the other channels for the 0.5-4, 4-8, 8-10, 10-12, 12-30, 30-50 Hz frequency bands. RESULTS: The synchronization likelihood was significantly decreased in the upper alpha (10-12) and beta (12-30) bands in AD compared to persons with subjective memory complaints. The working memory task scores strongly correlated with Mini-Mental State Examination scores. During the working memory task the synchronization likelihood was significantly higher in MCI compared to the control subjects in the lower alpha band (8-10 Hz). CONCLUSIONS: Decrease of beta band synchronization occurs in mild AD, both in a resting condition and during a working memory task. SIGNIFICANCE: Decrease of beta band synchronization in mild AD is a robust finding. The present study confirms our findings in a different cohort of patients, using alternative frequency bands. The diagnostic value of the synchronization likelihood in AD and MCI needs to be further established.

Aged↗

PubMed1982

The determination of the post-test likelihood for coronary disease using Bayes Theorem.

This report describes a method of using Bayes Theorem for determining the post-test likelihood for coronary disease from the exercise test. The pre-test likelihood is determined from the age, sex, and type of chest pain. Discriminant analysis of the amount of S-T depression, maximum pressure rate product and sex of the patient is then used to determine the sensitivity and specificity of the patient's exercise test for coronary artery disease. The post-test likelihood for coronary artery disease can then be calculated using Bayes Theorem. The discriminant function was estimated from a training set of 174 patients. This was then applied to 113 new patients who had both exercise tests and coronary arteriograms. In this new set, 47 patients had a post-test likelihood of 90% or greater for coronary disease. Only one of these patients had normal coronary arteriograms, a predictive accuracy of 98%. Of the 25 patients with a post-test likelihood of 10% or less for coronary disease, four had multivessel disease and four had single vessel disease. The predictive accuracy for the absence of coronary disease was 68%. The predictive accuracy for the exclusion of multivessel disease was 84%. Eight of 10 patients with left main disease had a post-test likelihood for coronary disease of greater than 90%.

Age Factors↗

PubMed2003

Seizure detection in the neonatal EEG with synchronization likelihood.

OBJECTIVE: To investigate whether epileptic seizure activity can be distinguished from non-epileptic background activity in the neonatal electroenceplalogram (EEG), using synchronization likelihood as a measure of synchronization between EEG channels. METHODS: Forty-two 21s EEG epochs and two complete EEGs from 21 different neonatal patients in a 12-channel bipolar recording were studied (AD-conversion 16bit; sample frequency 200Hz; filter setting 0.5-30Hz). For EEG of each patient, we selected one epoch with epileptic discharges and one without. Synchronization was calculated in all epochs. In two complete EEGs, synchronization was calculated and correlated with a visual scoring of the EEG. RESULTS: Synchronization likelihood was higher in all the epochs with epileptic seizures as compared to the epochs without epileptic activity (P<0.01). When synchronization likelihood exceeded 0.11, the sensitivity for the presence of epileptic activity was 0.85 (95% confidence limits [CL(95)]=0.69-1) and the specificity was 0.75 (CL(95)=0.56-0.94).Analysis of EEG score and synchronization likelihood of two complete EEGs revealed a high correlation between the occurrence of epileptic seizures and elevated synchronization likelihood (Spearman r=0.707, P<0.001). CONCLUSIONS: The results of this study demonstrate that synchronization likelihood is a potential tool in the automatic monitoring of high-risk infants for epileptic activity on neonatal wards.

Cortical Synchronization↗

PubMed2004

The dud-alternative effect in likelihood judgment.

The judged likelihood of a focal outcome should generally decrease as the list of alternative possibilities increases. For example, the likelihood that a runner will win a race goes down when 2 new entries are added to the field. However, 6 experiments demonstrate that the presence of implausible alternatives (duds) often increases the judged likelihood of a focal outcome. This dud-alternative effect was detected for judgments involving uncertainty about trivia facts and stochastic events. Nonnumeric likelihood measures and betting measures reliably detected the effect, but numeric likelihood measures did not. Time pressure increased the magnitude of the effect. The results were consistent with a contrast-effect account: The inclusion of duds increases the perceived strength of the evidence for the focal outcome, thereby affecting its judged likelihood.

Humans↗

PubMed2000

The likelihood function in fiber diffraction

The likelihood function is an appropriate target function for refinement of molecular structures using fiber diffraction data. However, its practical application to fiber diffraction faces two significant obstacles: (i) the intensities of layer lines in a fiber diffraction pattern usually arise from the superposition of several terms, each equivalent to a crystallographic structure factor, thereby making the calculation significantly more complex than for the crystallographic case; (ii) to describe a molecular structure at the atomic level based on fiber diffraction data, the radial and phase parts of the atomic coordinates must be treated separately owing to the uniaxial symmetry of the structure. These issues are addressed here in order to derive equations of likelihood functions for fiber diffraction. The special case of a single term on a layer line is treated first followed by extension of the method to the multiterm case. A practical difficulty in implementation of likelihood for the multiterm case is that each term has a different variance. An analytical technique is described that allows the conversion of the unequal-variance case to an equal-variance case. This makes it possible to express the likelihood by an explicit formula, allowing a direct implementation of the likelihood calculation. A cylindrically symmetric model is proposed for error distribution of the atomic coordinates in a helical structure. Variances and offset coefficients of the contributing terms in the likelihood functions are expressed in terms of the variance of the atomic coordinates in the cylindrical reference system.

Journal Article↗

PubMed2001

Maximum-likelihood density modification using pattern recognition of structural motifs.

The likelihood-based approach to density modification [Terwilliger (2000), Acta Cryst. D56, 965-972] is extended to include the recognition of patterns of electron density. Once a region of electron density in a map is recognized as corresponding to a known structural element, the likelihood of the map is reformulated to include a term that reflects how closely the map agrees with the expected density for that structural element. This likelihood is combined with other aspects of the likelihood of the map, including the presence of a flat solvent region and the electron-density distribution in the protein region. This likelihood-based pattern-recognition approach was tested using the recognition of helical segments in a largely helical protein. The pattern-recognition method yields a substantial phase improvement over both conventional and likelihood-based solvent-flattening and histogram-matching methods. The method can potentially be used to recognize any common structural motif and incorporate prior knowledge about that motif into density modification.

Amino Acid Motifs↗

PubMed2006

The consistency of emergency physicians' and cardiologists' ECG interpretation and likelihood classification of chest pain patients.

Patients presenting to the emergency department with chest pain are evaluated by emergency physicians in hospitals without cardiology cover 24 h a day. The purpose of this study is to determine the consistency of electrocardiography (ECG) interpretation and chest pain likelihood classification between emergency physicians and cardiologists. This randomised prospective cross-sectional study was performed in a tertiary care university hospital emergency department. The study form included ECG interpretation and chest pain likelihood classification according to American College of Cardiology (ACC)/American Heart Association (AHA) guideline which were recorded by emergency physicians and cardiologists separately in a blinded fashion. All chest pain patients who consulted with a cardiologist were enrolled into the study during the study period. The consistency between the two groups and the kappa value were calculated. Recorded study forms of 133 patients with cardiology consultations were evaluated. The consistency in the interpretation of ECG between the emergency physicians and cardiologists was found to be 94.6% (kappa = 0.85) for ST segment elevation, 78.6% (kappa = 0.57) for ischaemic ECG findings and 79.3% (kappa = 0.36) for dynamic ECG changes. The consistency for the likelihood classification between two groups for predicting the pain as angina or non-cardiac was 90.8% (kappa = 0.30), for classifying as acute coronary syndrome or stable angina pectoris (SAP) was 95.6% (kappa = 0.26) and for classifying patients as low likelihood or intermediate-high likelihood was 86.3% (kappa = 0.61). A strong consistency was shown between the emergency physicians' and cardiologists' ECG interpretation especially in determining the ST segment elevation. And also, there is a strong concordance in the likelihood classification of chest pain patients.

Angina Pectoris↗

PubMed2002

Alcohol's direct and indirect effects on men's self-reported sexual aggression likelihood.

OBJECTIVE: This study investigated pathways through which alcohol's direct and indirect expectancy effects and direct physiological effects influenced men's self-reported sexual aggression likelihood after they read a violent pornographic story. The indirect effects of participants' affective responses and cognitive judgments of story characters were also examined. METHOD: Male social drinkers (N = 135), recruited through newspaper ads in a large western city, were randomly assigned to one of three beverage conditions: alcohol, placebo or control. After completing pretest measures, subjects read a violent pornographic story and reported their sexual arousal, affect, cognitive judgments and sexual aggression likelihood. RESULTS: Pre-existing expectancies operated directly and interactively with alcohol consumption on reported sexual aggression likelihood. The influence of expectancies on sexual aggression likelihood also occurred indirectly through positive affect and cognitive judgments of assailant force and victim enjoyment. Situational consumption effects were influenced by cognitive judgments. The expectation of receiving alcohol indirectly affected sexual aggression likelihood through its effect on perception of the assailant's typicality. CONCLUSIONS: Among men who have contact with violent pornography, alcohol can have both direct and indirect effects on reported sexual aggression likelihood. In addition to the presence of situational myopia and expectancy effects, pre-existing expectancies can play a significant role both alone and interactively in affecting this tendency.

Adult↗

PubMed2006

Likelihood ratio of sonographic findings in discriminating hydrosalpinx from other adnexal masses.

OBJECTIVE: The objective of our study was to describe the "waist" sign as a feature of hydrosalpinx and to calculate the likelihood ratio of sonographic findings for predicting that a cystic adnexal mass is a hydrosalpinx. MATERIALS AND METHODS: Consecutive institutional records were reviewed to identify patients with surgically excised cystic adnexal masses and preoperative sonograms that had been used to identify the mass. Two radiologists blinded to outcome reviewed the sonograms and recorded the presence of five specific sonographic findings: incomplete septation, short linear projection, small round projection, tubular shape, and presence of a waist. The likelihood ratios of these signs and combinations of signs for the diagnosis of hydrosalpinx were calculated. RESULTS: There were 55 patients with 67 cystic adnexal masses. Of the 67 cystic adnexal masses, there were 26 hydrosalpinges (39%), 36 cystic ovarian neoplasms (54%), and five paraovarian masses (7%). The likelihood ratios for the sonographic findings were as follows: incomplete septation, 2.1; short linear projection, 3.5; small round projection, 2.7; tubular shape, 10.5; and waist sign, 20.5. The waist sign in combination with tubular shape was found in 12 hydrosalpinges and no other masses (likelihood ratio of between 18.9 and infinity). Small round projection combined with tubular shape was found in 14 hydrosalpinges and one other mass (likelihood ratio of 22.1). CONCLUSION: Hydrosalpinx can be diagnosed with the highest likelihood when a tubular mass with the waist sign or a tubular mass with small round projections is encountered. Incomplete septations and short linear projections are less discriminating findings of hydrosalpinx.

Adnexal Diseases↗

PubMed1999

A perspective on standardizing the predictive power of noninvasive cardiovascular tests by likelihood ratio computation: 1. Mathematical principles.

The current practice of reporting positive and negative predictive value (PV), sensitivity (Se), and specificity (Sp) as measures of the power of noninvasive cardiovascular tests has significant limitations. A test result's PV and its comparison with other test results are highly dependent on the pretest disease prevalence at which it is determined; the citation of sensitivity and specificity provides no succinct or explicit quantitation of the rule-in and rule-out power of a test. This article presents a rationale for the use of an alternative standard for expressing predictive power in the form of positive and negative likelihood ratios, (+)LR and (-)LR. The likelihood ratios are composite expressions of test power, which incorporate the Se and Sp and their respective complements [(1 - Se) and (1 - Sp)], thus yielding single unambiguous measures of positive and negative predictive power. The likelihood ratios are calculated as follows: (+)LR = Se/(1 - Sp) and (-)LR = Sp/(1 - Se). On analysis of the predictive value equations, the likelihood ratios equal the quotients of the posttest predictive value odds to the pretest prevalence odds for disease and no disease, respectively, as follows: (+)LR = (+)PVOd/POD and (-)LR = (-)PVOn/PON, where (+)PVOd is positive predictive value odds for disease, POD is prevalence odds for disease, (-)PVOn is negative predictive value odds for no disease, and PON is prevalence odds for no disease. Thus, the likelihood ratios are measures of the odds advantage in posttest probability of disease or no disease relative to pretest probability, independent of disease prevalence in the tested population. The quotients of the (+)LR or the (-)LR among test results studied in a common population are direct expressions of their relative predictive power in that population. The likelihood ratio principle is applicable to the evaluation of the predictive power of multiple tests performed in a common population and to estimating predictive power at multiple test thresholds.

Confidence Intervals↗

PubMed2005

The perceived likelihood of dental hygienists to report abuse before and after a training program.

PURPOSE: The rise of abuse, mandatory reporting, and penalties for not reporting abuse make this study significant for oral health care personnel. The purpose of this research was to determine survey results pertaining to the likelihood of dental hygienists reporting abuse before and after a training program, in order to influence and encourage similar training programs in other locations and to impact dental hygiene curricula. METHODS: Exempt status was obtained from the University of New England Institutional Review Board for the Protection of Human Subjects. A convenience sample was taken of registered dental hygienists who attended a training program and who volunteered to complete a 10-item statement form. A three-category ordinal Likert-type scale was employed. The statement form was filled out before and after a tested training program for the recognition and reporting of abuse (violence) and neglect. The terms family violence, child abuse, and elder abuse were defined as umbrella terms to encompass all abuse, except where explicit. This study explores two research questions: Do dental hygienists perceive the likelihood to make a report if confronted with suspected abuse, and would training make a difference in the perceived likelihood to report? The 10 statements were grouped into three sets for analysis: 1) training and experience in reporting, 2) knowledge of responsibilities, signs, symptoms, and interviewing, and 3) likelihood of making a report. Data were analyzed using descriptive statistics to explain the population's knowledge characteristics and the likelihood of reporting abuse. RESULTS: Twenty-six surveys were administered and 25 surveys were returned for a 96% response rate (n=25). Survey results supported training to increase compliance with mandatory reporting. Of the subgroup having experience with reporting (n=5; 20%), over half (n=3) knew all aspects of abuse. The entire group knew more about child abuse than elder abuse. Prior to training, 40% definitely knew that they would likely report abuse, 40% somewhat knew that they would likely report it, and 20% didn't know or said it would be unlikely that they would report. Only 5% stated that they definitely knew how to make a report before the training. After training, 100% reported that they would be likely to make a report, an overall increase of 60%. In the pre-survey, 60% said they did not know how to make a report, compared to 96% indicating in the post-survey that they knew how to make a report after training. CONCLUSION: Evidence from the dental hygienists attending a continuing education program supports that training increases the self-perceived likelihood to report abuse. This study also acknowledged areas for investigation of curricular augmentation, such as providing more information on elder abuse and presenting a guide for filing a report of abuse to the appropriate agencies. It is imperative for educators to include adequate information in dental and dental hygiene curricula for training in reporting abuse. It is also incumbent upon dental hygiene clinicians to identify their own educational needs and to seek out appropriate continuing education. These identified outcomes are an important reinforcement to providing adequate instruction in dental hygiene curricula.

Curriculum↗

PubMedDate not supplied

On non-parametric maximum likelihood estimation of the bivariate survivor function.

The likelihood function for the bivariate survivor function F, under independent censorship, is maximized to obtain a non-parametric maximum likelihood estimator &Fcirc;. &Fcirc; may or may not be unique depending on the configuration of singly- and doubly-censored pairs. The likelihood function can be maximized by placing all mass on the grid formed by the uncensored failure times, or half lines beyond the failure time grid, or in the upper right quadrant beyond the grid. By accumulating the mass along lines (or regions) where the likelihood is flat, one obtains a partially maximized likelihood as a function of parameters that can be uniquely estimated. The score equations corresponding to these point mass parameters are derived, using a Lagrange multiplier technique to ensure unit total mass, and a modified Newton procedure is used to calculate the parameter estimates in some limited simulation studies. Some considerations for the further development of non-parametric bivariate survivor function estimators are briefly described.

Computer Simulation↗