Profile Likelihood Interval Research Articles

Practical and Optimal Likelihood Intervals and Regions for Weibull and Gumbel Distributions

Profile likelihood intervals with good coverage probabilities and convenient simple expressions are presented to estimate separately and efficiently each of the parameters and quantiles of Weibull and Gumbel distributions, widely used in reliability and environmental applications. For the first time, likelihood-confidence intervals for these distributions, that were previously unavailable, can now be easily and quickly calculated with the provided R code for sample sizes n ≥ 10 . For simultaneous estimation of both parameters, likelihood-confidence regions are also provided. The proposal takes advantage of the transformation relationship between the Weibull and Gumbel distributions, of the fact that this latter distribution belongs to the Location and Scale family of distributions, and of likelihood symmetrizing parametrizations, all leading to simpler inferences about the parameters. The plot of the relative profile likelihood of a quantile of interest and its corresponding profile likelihood intervals provide decision makers with useful information in many practical applications, particularly when comparing two treatments. The proposed intervals for the Weibull shape parameter facilitate testing if an exponential distribution is reasonable for the data. The proposed methods are illustrated with practical examples.

Wald Intervals via Profile Likelihood for the Mean of the Inverse Gaussian Distribution

The inverse Gaussian distribution, known for its flexible shape, is widely used across various applications. Existing confidence intervals for the mean parameter, such as profile likelihood, reparametrized profile likelihood, and Wald-type reparametrized profile likelihood with observed Fisher information intervals, are generally effective. However, our simulation study identifies scenarios where the coverage probability falls below the nominal confidence level. Wald-type intervals are widely used in statistics and have a symmetry property. We mathematically derive the Wald-type profile likelihood (WPL) interval and the Wald-type reparametrized profile likelihood with expected Fisher information (WRPLE) interval and compare their performance to existing methods. Our results indicate that the WRPLE interval outperforms others in terms of coverage probability, while the WPL typically yields the shortest interval. Additionally, we apply these proposed intervals to a real dataset, demonstrating their potential applicability to other datasets that follow the IG distribution.

Interval estimation for concentration in the ELISA setting

Enzyme-linked immunosorbent assays (ELISAs) are often used to quantify the concentration of biological substances. In a typical analysis only a point estimate of the concentration will be presented as interval estimation continues to present challenges for non-linear dose-response models. In this setting, interval estimates calculated using a Wald approach can suffer from poor coverage and have limits that fall outside parameter boundaries. Here we compare profile likelihood interval estimation procedures to Wald type intervals for the interval estimation of a concentration in the ELISA setting. Through a comprehensive simulation study, it is shown that profile likelihood methods result in interval estimates with superior coverage and that are more robust to differences in assay design when compared to Wald based approaches.

Comparison of some interval estimation methods for the parameters of the gamma distribution

Several methods of finding interval estimators of the parameters of the gamma distribution are considered in the literature. In this work we compare the following methods: Wald confidence intervals; profile likelihood intervals; Bayesian intervals using the Jeffreys prior, the reference prior when α is the parameter of interest and β the nuisance parameter, the reference prior when β is the parameter of interest and α the nuisance parameter; and three fiducial methods. The comparison is done using Montecarlo simulations, in terms of the coverage probabilities and the expected lengths of the intervals, considering small, medium and large sample sizes. As an important result of the simulations we found that the fiducial methods are the best when the sample size is very small, and as the sample size increases all the methods, except the Wald confidence intervals, have a similar behavior. An example of application is shown considering earthquake data of Mexico.

Interval estimation of the overall treatment effect in random-effects meta-analyses: Recommendations from a simulation study comparing frequentist, Bayesian, and bootstrap methods.

There exists a variety of interval estimators for the overall treatment effect in a random-effects meta-analysis. A recent literature review summarizing existing methods suggested that in most situations, the Hartung-Knapp/Sidik-Jonkman (HKSJ) method was preferable. However, a quantitative comparison of those methods in a common simulation study is still lacking. Thus, we conduct such a simulation study for continuous and binary outcomes, focusing on the medical field for application. Based on the literature review and some new theoretical considerations, a practicable number of interval estimators is selected for this comparison: the classical normal-approximation interval using the DerSimonian-Laird heterogeneity estimator, the HKSJ interval using either the Paule-Mandel or the Sidik-Jonkman heterogeneity estimator, the Skovgaard higher-order profile likelihood interval, a parametric bootstrap interval, and a Bayesian interval using different priors. We evaluate the performance measures (coverage and interval length) at specific points in the parameter space, that is, not averaging over a prior distribution. In this sense, our study is conducted from a frequentist point of view. We confirm the main finding of the literature review, the general recommendation of the HKSJ method (here with the Sidik-Jonkman heterogeneity estimator). For meta-analyses including only two studies, the high length of the HKSJ interval limits its practical usage. In this case, the Bayesian interval using a weakly informative prior for the heterogeneity may help. Our recommendations are illustrated using a real-world meta-analysis dealing with the efficacy of an intramyocardial bone marrow stem cell transplantation during coronary artery bypass grafting.

Evaluation of heterogeneity and heterogeneity interval estimators in random-effects meta-analysis of the standardized mean difference in education and psychology.

Meta-analyses are conducted to synthesize the quantitative results of related studies. The random-effects meta-analysis model is based on the assumption that a distribution of true effects exists in the population. This distribution is often assumed to be normal with a mean and variance. The population variance, also called heterogeneity, can be estimated numerous ways. Research exists comparing subsets of heterogeneity estimators over limited conditions. Additionally, heterogeneity is a parameter estimated with uncertainty. Various methods exist for heterogeneity interval estimation, and similar to heterogeneity estimators, these evaluations are limited. The current simulation study examined the performance of Bayesian (with 5 prior specifications) and non-Bayesian estimators over conditions found after a review of meta-analyses of the standardized mean difference in education and psychology research. Three simulation conditions were varied: (a) number of effect sizes per meta-analysis, (b) true heterogeneity, and (c) sample size per effect size within each meta-analysis. Estimators were evaluated over average bias and means square error. Methods of interval estimation were then evaluated with the estimators found to operate optimally. Interval estimators were evaluated based on coverage probability, interval width, and coverage of the estimated value. Overall, the Paule and Mandel estimator, in conjunction with the Jackson method of interval estimation, is recommended if no knowledge exists with regards to the expected value of heterogeneity when synthesizing the standardized mean difference effect size. If heterogeneity is expected to be small (e.g., < .075), then REML with the profile likelihood interval estimator is recommended. Sensitivity analysis evaluating differences in substantive conclusions with a suite of heterogeneity estimators, such as Paule and Mandel, REML, and Hedges and Olkin, is recommended. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Interval Estimation of Diagnostic Odds Ratio in Meta-analysis by Means of Profile Likelihoods

The objectives of this paper are to (1) derive the profile maximum likelihood estimator (PMLE) for a true diagnostic odds ratio over across k studies in meta-analysis, (2) build the confidence intervals by replacing PMLE into the variance of logarithm of each diagnostic odds ratio, leading to two profile likelihood intervals (WPLF, WPLR), (3) create bootstrapping confidence interval (BOOT) from PMLE distribution by using the percentile, (4) compare the interval performance between all profile intervals with the conventional intervals, such as Mantel-Haenszel method (MH) and Weighted least squares method (WLS) in terms of the coverage probability and the width of interval. The results under a simulation plan indicated that for moderated study size (k = 8, 16) and small sample size ▪, there were only three proposed interval estimates (WPLF, WPLR, and BOOT) that could be calibrated the coverage probability at 95% and the interval widths of WPLF and WPLR are narrower than the BOOT. Hence, we recommend to use WPLF and WPLR rather than the conventional intervals in such situations.

Constructing confidence intervals of extreme rainfall quantiles using Bayesian, bootstrap, and profile likelihood approaches

Hydrological risk is highly dependent on the occurrence of extreme rainfalls. This fact has led to a wide range of studies on the estimation and uncertainty analysis of the extremes. In most cases, confidence intervals (CIs) are constructed to represent the uncertainty of the estimates. Since the accuracy of CIs depends on the asymptotic normality of the data and is questionable with limited observations in practice, a Bayesian highest posterior density (HPD) interval, bootstrap percentile interval, and profile likelihood (PL) interval have been introduced to analyze the uncertainty that does not depend on the normality assumption. However, comparison studies to investigate their performances in terms of the accuracy and uncertainty of the estimates are scarce. In addition, the strengths, weakness, and conditions necessary for performing each method also must be investigated. Accordingly, in this study, test experiments with simulations from varying parent distributions and different sample sizes were conducted. Then, applications to the annual maximum rainfall (AMR) time series data in South Korea were performed. Five districts with 38-year (1973–2010) AMR observations were fitted by the three aforementioned methods in the application. From both the experimental and application results, the Bayesian method is found to provide the lowest uncertainty of the design level while the PL estimates generally have the highest accuracy but also the largest uncertainty. The bootstrap estimates are usually inferior to the other two methods, but can perform adequately when the distribution model is not heavy-tailed and the sample size is large. The distribution tail behavior and the sample size are clearly found to affect the estimation accuracy and uncertainty. This study presents a comparative result, which can help researchers make decisions in the context of assessing extreme rainfall uncertainties.

Optimality of profile likelihood intervals for quantiles of extreme value distributions: application to environmental disasters

Large quantiles of extreme value distributions are useful to assess the risk of environmental disasters. Profile likelihood intervals of quantiles are shown here to be optimal for samples of sizes of n ≥ 50. However, they are seldom used, notwithstanding their reasonable coverage frequencies. In contrast, asymptotic maximum likelihood confidence intervals are often used for any sample size, despite their poor coverage frequencies for moderate and small samples and their tendency to underestimate the quantile of interest. Using these intervals may have dangerous consequences in environmental applications. Calibrated interpolated bootstrap intervals have also been considered a good option for estimating quantiles of extreme value distributions but have not been compared before with profile likelihood intervals. Coverage frequencies of these three types of intervals for large quantiles are compared here through simulations for small and moderate sample sizes. The restricted likelihood function was used successfully to overcome the alleged maximum likelihood estimation problems cited in literature that arise with some Weibull and generalized extreme value distributions. Two rainfall datasets are discussed where the profile likelihood intervals were valuable tools for assessing the risk of disasters.Editor D. Koutsoyiannis

A simple method of constructing a confidence interval for the mean probability of detection in collaborative studies of binary measurement methods

We deal with collaborative studies where each of k laboratories performs n repeated binary measurements (measurement result x = 0: “not detected”; measurement result x = 1: “detected”), and present a simple method of constructing a confidence interval for the mean probability of detection of the laboratories. This method is based on an approximation of the distribution of the number y of detections among n independent measurements of a randomly chosen laboratory by a binomial distribution. The confidence interval is not only much easier to calculate but also more accurate than the profile likelihood interval presented by Uhlig et al.

Density estimation of sympatric carnivores using spatially explicit capture–recapture methods and standard trapping grid

Population density is an important state variable in ecology and monitoring of wildlife populations. In the study of large carnivores, traditional density estimation methods have relied on camera trapping to collect capture–recapture data using sampling designs suited to a single target species and ad hoc methods to calculate the effective trapping area. We describe an application of spatially explicit capture–recapture analysis to estimate density for four sympatric carnivore species using a standard spatial sampling grid, camera trap sampling, and maximum-likelihood estimation methods. We used camera traps deployed in four adjacent, sequential camera arrays to construct capture histories for leopard, aardwolf, spotted hyena, and striped hyena in an African savanna/scrub ecosystem. We considered two methods of constructing trapping histories: (1) a simultaneous layout in which all cameras are deployed and active for 21 trapping intervals of 24 h each; and (2) an incomplete layout in which cameras are considered deployed for 84 sampling intervals, but with only a fraction of traps active for a given sampling interval. We estimated density and confidence or profile likelihood intervals for each species using a mean maximum distance moved method and maximum-likelihood estimation procedures with model averaging. Maximum-likelihood densities ranged from 4.93/100 km2 for spotted hyena to 12.03/100 km2 for leopard. Estimates did not differ between simultaneous and incomplete trap layouts. Our approach demonstrates the utility and cost effectiveness of using maximum-likelihood density estimation methods in studies of sympatric species that are individually recognizable.

Model-Averaged Profile Likelihood Intervals

Model-averaging is commonly used as a means of allowing for model uncertainty in parameter estimation. In the frequentist framework, a model-averaged estimate of a parameter is the weighted mean of the estimates from each of the candidate models, the weights typically being chosen using an information criterion. Current methods for calculating a model-averaged confidence interval assume approximate normality of the model-averaged estimate, i.e., they are Wald intervals. As in the single-model setting, we might improve the coverage performance of this interval by a one-to-one transformation of the parameter, obtaining a Wald interval, and then back-transforming the endpoints. However, a transformation that works in the single-model setting may not when model-averaging, due to the weighting and the need to estimate the weights. In the single-model setting, a natural alternative is to use a profile likelihood interval, which generally provides better coverage than a Wald interval. We propose a method for model-averaging a set of single-model profile likelihood intervals, making use of the link between profile likelihood intervals and Bayesian credible intervals. We illustrate its use in an example involving negative binomial regression, and perform two simulation studies to compare its coverage properties with the existing Wald intervals.

A coverage study of the CMSSM based on ATLAS sensitivity using fast neural networks techniques

We assess the coverage properties of confidence and credible intervals on the CMSSM parameter space inferred from a Bayesian posterior and the profile likelihood based on an ATLAS sensitivity study. In order to make those calculations feasible, we introduce a new method based on neural networks to approximate the mapping between CMSSM parameters and weak-scale particle masses. Our method reduces the computational effort needed to sample the CMSSM parameter space by a factor of ~ 10^4 with respect to conventional techniques. We find that both the Bayesian posterior and the profile likelihood intervals can significantly over-cover and identify the origin of this effect to physical boundaries in the parameter space. Finally, we point out that the effects intrinsic to the statistical procedure are conflated with simplifications to the likelihood functions from the experiments themselves.

Hierarchical models in ecology: confidence intervals, hypothesis testing, and model selection using data cloning

Hierarchical statistical models are increasingly being used to describe complex ecological processes. The data cloning (DC) method is a new general technique that uses Markov chain Monte Carlo (MCMC) algorithms to compute maximum likelihood (ML) estimates along with their asymptotic variance estimates for hierarchical models. Despite its generality, the method has two inferential limitations. First, it only provides Wald-type confidence intervals, known to be inaccurate in small samples. Second, it only yields ML parameter estimates, but not the maximized likelihood values used for profile likelihood intervals, likelihood ratio hypothesis tests, and information-theoretic model selection. Here we describe how to overcome these inferential limitations with a computationally efficient method for calculating likelihood ratios via data cloning. The ability to calculate likelihood ratios allows one to do hypothesis tests, construct accurate confidence intervals and undertake information-based model selection with hierarchical models in a frequentist context. To demonstrate the use of these tools with complex ecological models, we reanalyze part of Gause's classic Paramecium data with state-space population models containing both environmental noise and sampling error. The analysis results include improved confidence intervals for parameters, a hypothesis test of laboratory replication, and a comparison of the Beverton-Holt and the Ricker growth forms based on a model selection index.

Score and profile likelihood confidence intervals for contingency table parameters

A straightforward approach to computing score and profile likelihood confidence intervals for contingency table parameters is described. The computational approach herein avoids two main limitations of existing methods: (1) Compared with existing methods, this paper's approach is applicable to a much broader class of parameters. (2) Unlike existing methods, this paper's approach is not case-specific and, hence, lends itself to a general, yet very simple, computational algorithm. This paper describes the 'sliding quadratic' computational algorithm and illustrates its use on examples that have not been previously considered in the literature. A small-scale simulation study highlights the advantage of using score and profile likelihood intervals rather than Wald intervals.

Confidence intervals for parameters of two diagnostic tests in the absence of a gold standard

We derive a profile-likelihood confidence interval and a score based confidence interval to estimate the population prevalences, test sensitivities, and test specificities of two conditionally independent diagnostic tests when no gold standard is available. We are motivated by a real-data example on the study of the properties for two fallible diagnostic tests for bovine immunodeficiency virus. We compare the coverage and average width of two new intervals with an interval based on the asymptotic normality of the maximum likelihood estimator and a Bayesian interval estimator via Monte Carlo simulation. We determine that for the parameter configurations considered here, the profile-likelihood, score, and Bayesian intervals all perform adequately in terms of coverage, but overall, the profile-likelihood interval performs best in terms of yielding at least nominal coverage with minimum expected width.

Statistical analysis of second harmonic generation experiments: a phenomenological model

We discuss issues arising in fitting theoretically derived nonlinear models with complex coefficients to data from surface Second Harmonic Generation (SHG) experiments conducted at the air/liquid interface. We explore different parametrisations for the complex parameters and show that the Euler (magnitude and phase angle) parametrisation is preferable to the real and imaginary parts parametrisation, both theoretically and empirically. We emphasise the importance and value of diagnostic plots for evaluating the quality of model fit. We derive approximate standard errors for the parameter estimates and discuss issues of making inference about ratios of parameters. We consider approximate confidence intervals (using the approximate standard errors), profile likelihood intervals, Fieller's method and bootstrap intervals. Fieller's method (and the bootstrap intervals) provide useful information on the value of the simpler approximate confidence intervals. We also propose and implement a likelihood ratio test to assess whether a common model can be fitted to several independent data sets. Finally, the methods are applied to data sets obtained from SHG experiments on l-phenylalanine at the air/water interface and toluene.

Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations

Increased population survival rate after an episode of seasonal exploitation is considered a type of compensatory population response. Lack of an increase is interpreted as evidence that exploitation results in added annual mortality in the population. Despite its importance to management of exploited species, there are limited statistical techniques for comparing relative support for these two alternative models. For exploited bird species, the most common technique is to use a fixed effect, deterministic ultrastructure model incorporated into band recovery models to estimate the relationship between harvest and survival rate. We present a new likelihood-based technique within a framework that assumes that survival and harvest are random effects that covary through time. We conducted a Monte Carlo simulation study under this framework to evaluate the performance of these two techniques. The ultrastructure models performed poorly in all simulated scenarios, due mainly to pathological distributional properties. The random effects estimators and their associated estimators of precision had relatively small negative bias under most scenarios, and profile likelihood intervals achieved nominal coverage. We suggest that the random effects estimation method approach has many advantages compared to the ultrastructure models, and that evaluation of robustness and generalization to more complex population structures are topics for additional research.

Interval Estimation for Log-Linear Models with One Variable Subject to Non-Ignorable Non-Response

SummaryLog-linear models for multiway contingency tables where one variable is subject to non-ignorable non-response will often yield boundary solutions, with the probability of non-respondents being classified in some cells of the table estimated as 0. The paper considers the effect of this non-standard behaviour on two methods of interval estimation based on the distribution of the maximum likelihood estimator. The first method relies on the estimator being approximately normally distributed with variance equal to the inverse of the information matrix. It is shown that the information matrix is singular for boundary solutions, but intervals can be calculated after a simple transformation. For the second method, based on the bootstrap, asymptotic results suggest that the coverage properties may be poor for boundary solutions. Both methods are compared with profile likelihood intervals in a simulation study based on data from the British General Election Panel Study. The results of this study indicate that all three methods perform poorly for a parameter of the non-response model, whereas they all perform well for a parameter of the margin model, irrespective of whether or not there is a boundary solution.

Male‐biased sex ratio in a small tuatara population

AimWe estimated population size, survival, longevity and sex ratio of tuatara (Sphenodon guntheri Buller 1877) on North Brother Island and determined whether recruitment was occurring, to aid management and identify potential problems for population viability.LocationThis study was conducted on North Brother Island, Cook Strait, New Zealand. This 4‐ha island supports the sole remnant population of S. guntheri.MethodsDemographic variables were estimated using capture–recapture methods. Tuatara were individually marked and recaptured during twelve trips, spanning a decade. Population size was estimated for selected trips using the CAPTURE package and finite mixture models, and survival was analysed using the Jolly–Seber (J–S) model. Longevity was estimated using tuatara individually marked in 1957.ResultsApproximately 350 adult tuatara profile likelihood interval (PLI) (294–427) inhabit North Brother Island, and the sex ratio is strongly biased towards males (1.7M : 1F). Annual adult survivorship is high (0.95) for both sexes and some tuatara live for at least 61 years.Main conclusionsThe small size and biased sex ratio of this population may make it susceptible to demographic stochasticity, Allee effects, and/or loss of genetic variation. Harvesting for translocation could exacerbate such problems. In addition, tuatara have temperature‐dependent sex determination (TSD), with males produced at higher incubation temperatures. Global warming may therefore skew the sex ratio further unless female nest site choice or adaptive shifts in pivotal temperatures compensate for rising temperatures.