Likelihood-based Confidence Intervals Research Articles

Empirical likelihood-based confidence intervals for data with possible zero observations

In statistical applications, we often encounter a situation where a substantial number of observations takes zero value and at the same time the non-zero observations are highly skewed. We propose empirical likelihood-based non-parametric confidence intervals for the mean parameter which have two unique features. One is that the information contained in the zero observations is fully utilized. The other is that the proposed confidence intervals are more reflective to the skewness in the non-zero observations than those based on the asymptotic normality.

Confidence Intervals and Accuracy Estimation for Heavy-Tailed Generalized Pareto Distributions

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.

Empirical likelihood confidence regions for comparison distributions and roc curves

AbstractAbstract: The authors derive empirical likelihood confidence regions for the comparison distribution of two populations whose distributions are to be tested for equality using random samples. Another application they consider is to ROC curves, which are used to compare measurements of a diagnostic test from two populations. The authors investigate the smoothed empirical likelihood method for estimation in this context, and empirical likelihood based confidence intervals are obtained by means of the Wilks theorem. A bootstrap approach allows for the construction of confidence bands. The method is illustrated with data analysis and a simulation study.

Interpreting Statistical Evidence by using Imperfect Models: Robust Adjusted Likelihood Functions

SummaryThe strength of statistical evidence is measured by the likelihood ratio. Two key performance properties of this measure are the probability of observing strong misleading evidence and the probability of observing weak evidence. For the likelihood function associated with a parametric statistical model, these probabilities have a simple large sample structure when the model is correct. Here we examine how that structure changes when the model fails. This leads to criteria for determining whether a given likelihood function is robust (continuing to perform satisfactorily when the model fails), and to a simple technique for adjusting both likelihoods and profile likelihoods to make them robust. We prove that the expected information in the robust adjusted likelihood cannot exceed the expected information in the likelihood function from a true model. We note that the robust adjusted likelihood is asymptotically fully efficient when the working model is correct, and we show that in some important examples this efficiency is retained even when the working model fails. In such cases the Bayes posterior probability distribution based on the adjusted likelihood is robust, remaining correct asymptotically even when the model for the observable random variable does not include the true distribution. Finally we note a link to standard frequentist methodology—in large samples the adjusted likelihood functions provide robust likelihood-based confidence intervals.

Likelihood-based confidence intervals for a log-normal mean.

To construct a confidence interval for the mean of a log-normal distribution in small samples, we propose likelihood-based approaches - the signed log-likelihood ratio and modified signed log-likelihood ratio methods. Extensive Monte Carlo simulation results show the advantages of the modified signed log-likelihood ratio method over the signed log-likelihood ratio method and other methods. In particular, the modified signed log-likelihood ratio method produces a confidence interval with a nearly exact coverage probability and highly accurate and symmetric error probabilities even for extremely small sample sizes. We then apply the methods to two sets of real-life data.

Estimating the size of spotted hyaena (Crocuta crocuta) populations through playback recordings allowing for non-response

Conventional census techniques are inappropriate for estimating numbers of nocturnal animals such as the spotted hyaena (Crocuta crocuta). A technique based on attracting animals by taped sounds to calling stations is described. A simple independent experiment to estimate the probability of response to the sounds, as well as a probability model for estimating the population size in a given habitat based on the response counts, is proposed. The model allows for the possibility that some animals may not respond to sounds. The estimate of the population size has a simple form and the sensitivity of the estimate to changes in the response probability can be assessed. The model allows statistical comparisons to be made between different habitats or different times. Likelihood-based confidence intervals can be found for all estimates. The results of surveys in the Kruger National Park illustrate the technique which can be applied in different areas and possibly to other species.

Gauge Capability for Pass—Fail Inspection

Quantitative measurement is an accepted ideal, but pass–fail inspection remains a fact of life, even in high-technology industries. For pass–fail data, variance components do not separate gauge and material variation. This article focuses on maximum likelihood estimation of conditional misclassification rates, with and without reference evaluations to anchor the analysis. Likelihood-based confidence intervals and testing for reproducibility effects are also discussed.

Confidence intervals for the mean of diagnostic test charge data containing zeros.

In this paper, we consider the problem of interval estimation for the mean of diagnostic test charges. Diagnostic test charge data may contain zero values, and the nonzero values can often be modeled by a log-normal distribution. Under such a model, we propose three different interval estimation procedures: a percentile-t bootstrap interval based on sufficient statistics and two likelihood-based confidence intervals. For theoretical properties, we show that the two likelihood-based one-sided confidence intervals are only first-order accurate and that the bootstrap-based one-sided confidence interval is second-order accurate. For two-sided confidence intervals, all three proposed methods are second-order accurate. A simulation study in finite-sample sizes suggests all three proposed intervals outperform a widely used minimum variance unbiased estimator (MVUE)-based interval except for the case of one-sided lower end-point intervals when the skewness is very small. Among the proposed one-sided intervals, the bootstrap interval has the best coverage accuracy. For the two-sided intervals, when the sample size is small, the bootstrap method still yields the best coverage accuracy unless the skewness is very small, in which case the bias-corrected ML method has the best accuracy. When the sample size is large, all three proposed intervals have similar coverage accuracy. Finally, we analyze with the proposed methods one real example assessing diagnostic test charges among older adults with depression.

Computing empirical likelihood from the bootstrap

The close relationship between the bootstrap and empirical likelihood has been noted in the literature. The purpose of this paper is to show how to construct a bootstrap likelihood from a single bootstrap, without any nested bootstrapping nor any smoothing. For a wide class of M-estimators the likelihood agrees with the empirical likelihood up to order O( n −1/2). The resulting likelihood may be used for display purpose, for computing likelihood-based confidence intervals or for future use in combining information.

Likelihood-based experimental design for estimation of ED50.

In selecting the best dosage choice for the estimation of ED50, it is natural to try to minimize the length of the confidence intervals. In this presentation, the dose allocation that minimizes the length of the likelihood-based confidence intervals is presented and compared with alternative allocations that have been proposed based on the length of different types of confidence intervals, such as those based on the asymptotic variance or on Fieller's Theorem. Effective strategies to deal with the parameter dependence of these allocations are explored. A series of experiments to evaluate the effect of small doses per fraction on the radiation tolerance of the rat cervical spinal cord provide the motivation and an illustration for the proposed procedures.

Inference from Grouped Data in Three-Parameter Weibull Models with Applications to Breakdown-Voltage Experiments

An important objective of breakdown-voltage experiments is to estimate either the threshold below which no breakdown occurs or other voltage levels below which breakdown occurs with a small probability. A commonly used statistical technique in these studies is to fit a three-parameter Weibull distribution to the experimental data. Difficulties in applying the method of maximum likelihood to three-parameter Weibull models due to nonregularity have led to a variety of alternative approaches in the literature. In practice, however, voltage breakdown and other failure-time data are usually discretized or rounded off. For grouped data from the Weibull model, there are still difficulties in applying likelihood methods when the sample size is moderate because of the so-called “embedded-model problem.” After a brief review of this problem and related likelihood theory, we show how the difficulties can be resolved and how likelihood-based confidence intervals for the Weibull parameters and quantiles can be constru...

Inference From Grouped Data in Three-Parameter Weibull Models With Applications to Breakdown-Voltage Experiments

An important objective of breakdown-voltage experiments is to estimate either the threshold below which no breakdown occurs or other voltage levels below which breakdown occurs with a small probability. A commonly used statistical technique in these studies is to fit a three-parameter Weibull distribution to the experimental data. Difficulties in applying the method of maximum likelihood to three-parameter Weibull models due to nonregularity have led to a variety of alternative approaches in the literature. In practice, however, voltage breakdown and other failure-time data are usually discretized or rounded off. For grouped data from the Weibull model, there are still difficulties in applying likelihood methods when the sample size is moderate because of the so-called “embedded-model problem.” After a brief review of this problem and related likelihood theory, we show how the difficulties can be resolved and how likelihood-based confidence intervals for the Weibull parameters and quantiles can be constructed from grouped data, under positivity constraints and for moderately sized samples that commonly occur in practice, by using reparameterization and enlarging the parameter space.

The use of likelihood-based confidence intervals in genetic models.

This article describes the computation and relative merits of likelihood-based confidence intervals, compared to other measures of error in parameter estimates. Likelihood-based confidence intervals have the advantage of being asymmetric, which is often the case with structural equation models for genetically informative studies. We show how the package Mx provides confidence intervals for parameters and functions of parameters in the context of a simple additive genetic, common, and specific environment threshold model for binary data. Previously published contingency tables for major depression in adult female twins are used for illustration. The support for the model shows a marked skew as the additive genetic parameter is systematically varied from zero to one. The impact of allowing different prevalence rates in MZ vs. DZ twins is explored by fitting a model with separate threshold parameters and comparing the confidence intervals. Despite the improvement in fit of the different prevalence model, the confidence intervals on all parameters broaden, owing to their covariance.

Measurement of Interrater Agreement with Adjustment for Covariates

The kappa coefficient measures chance-corrected agreement between two observers in the dichotomous classification of subjects. The marginal probability of classification by each rater may depend on one or more confounding variables, however. Failure to account for these confounders may lead to inflated estimates of agreement. A multinomial model is used that assumes both raters have the same marginal probability of classification, but this probability may depend on one or more covariates. The model may be fit using software for conditional logistic regression. Additionally, likelihood-based confidence intervals for the parameter representing agreement may be computed. A simple example is discussed to illustrate model-fitting and application of the technique.

A likelihood approach to meta-analysis with random effects.

In a meta-analysis of a set of clinical trials, a crucial but problematic component is providing an estimate and confidence interval for the overall treatment effect theta. Since in the presence of heterogeneity a fixed effect approach yields an artificially narrow confidence interval for theta, the random effects method of DerSimonian and Laird, which incorporates a moment estimator of the between-trial components of variance sigma B2, has been advocated. With the additional distributional assumptions of normality, a confidence interval for theta may be obtained. However, this method does not provide a confidence interval for sigma B2, nor a confidence interval for theta which takes account of the fact that sigma B2 has to be estimated from the data. We show how a likelihood based method can be used to overcome these problems, and use profile likelihoods to construct likelihood based confidence intervals. This approach yields an appropriately widened confidence interval compared with the standard random effects method. Examples of application to a published meta-analysis and a multicentre clinical trial are discussed. It is concluded that likelihood based methods are preferred to the standard method in undertaking random effects meta-analysis when the value of sigma B2 has an important effect on the overall estimated treatment effect.

Improving Markov-Recapture Population Estimation with Multiple Marking

Markov-recapture population size estimates rely on insects (or other animals) marking themselves before arrival in a trap. Assumptions about rates of marking and capture are based upon the sampling scheme, and the estimate is based on the resulting multinomial probability distribution. The original scheme used one marking station per trap, yielding an assumption of equal rates of marking and capture. This paper explores more general schemes that provide better resolution, sometimes with added degrees of freedom, and which provide estimates more quickly than the original scheme (reducing the risk of violating assumptions). The alternative schemes involve varying the ratio of marking stations to traps and multiple marking. The paper continues to describe the maximum likelihood estimates of population size based on these sampling schemes and gives a rough correction for the bias of the population estimate, found by using a jackknife technique. Two standard error confidence limits are explicit expressions obtained using asymptotic maximum likelihood arguments; these work well in conjunction with the corrected estimate if trapping has continued for an extended period. Relative likelihood-based confidence intervals perform better than two standard error intervals over a wide range of parameter values, especially where trapping has not continued for an extended period. The method tends to become biased when used in an open or growing population. Goodness-of-fit tests are possible with the added degrees of freedom but are not very powerful.

MARKOV-RECAPTURE POPULATION ESTIMATES

This paper reviews recent development of a method for estimating insect populations. It is like mark-recapture methods, except that marking is done passively at bait stations by the insects themselves, and capture probabilities are generated using a simple Markov process model. Assumptions about rates of marking and capture are made from the sampling scheme, and the estimate is based upon the resulting multinomial probability distribution and maximum likelihood methods. The paper continues to review the sampling distributions for the population estimate, revealed by simulation, and explores correction of the bias. Relative likelihood based confidence intervals are compared with two standard error intervals, and found to perform better over a wide range of parameter values, especially where the number of recaptures is small. The method tends to become biased when used in an open or growing population. Goodness of fit tests are possible with the added degrees of freedom, but are not very powerful.

Interval estimation of parameters in models for seasonal growth in fishes

Several non-linear models have been proposed in the literature to study seasonal growth in fishes. In this paper, we present interval estimation of parameters in models of Pauly and Gaschutz and Hoenig and Hanumara by inverting the likelihood ratio statistic. These intervals may differ from the Wald intervals that can be obtained from fitting the models with available software packages. It is shown that the differences between the two intervals relate to parameter effects non-linearity using profile t-plots and in those cases likelihood based confidence intervals are to be employed. The likelihood interval is also given for the growth performance index of Pauly and Munro.

Likelihood-based confidence intervals for estimating floods with given return periods

This paper discusses aspects of the calculation of likelihood-based confidence intervals for T-year floods, with particular reference to (1) the two-parameter gamma distribution; (2) the Gumbel distribution; (3) the two-parameter log-normal distribution, and other distributions related to the normal by Box-Cox transformations. Calculation of the confidence limits is straightforward using the Nelder-Mead algorithm with a constraint incorporated, although care is necessary to ensure convergence either of the Nelder-Mead algorithm, or of the Newton-Raphson calculation of maximum-likelihood estimates. Methods are illustrated using records from 18 gauging stations in the basin of the River Itajai-Acu, State of Santa Catarina, southern Brazil. A small and restricted simulation compared likelihood-based confidence limits with those given by use of the central limit theorem; for the same confidence probability, the confidence limits of the simulation were wider than those of the central limit theorem, which failed more frequently to contain the true quantile being estimated. The paper discusses possible applications of likelihood-based confidence intervals in other areas of hydrological analysis.

Statistical Modelling in GLIM

Part 1 Introducing GLIM 3: getting started in GLIM 3. Part 2 Statistical modelling and statistical inference: the Bernoulli distribution for binary data types of variables definition of a statistical model model criticism likelihood-based confidence intervals. Part 3 Normal regression and analysis of variance: the normal distribution and the Box-Cox transformation family link functions and transformations regression models for prediction the use of regression models for calibration fatorial designs midding data. Part 4 Binomial response data: binary responses transformations and link functions contingency table construction from binary data multidimensional contingency tables with a binary response. Part 5: multinomial and Poisson response data. Part 6 Survival data: probability plotting with censored data - the Kaplan-Meier estimator the Weibull distribution the Cox proportional hazards model and the piecewise exponential distribution the logistic and log logistic distributions time-dependent explanatory variables. Appendices: discussion GLIM directives system defined structures in GLIM datasets and macros.