Corrected profile likelihood confidence interval for binomial paired incomplete data

Clinical trials often use paired binomial data as their clinical endpoint. The confidence interval is frequently used to estimate the treatment performance. Tang et al. (2009) have proposed exact and approximate unconditional methods for constructing a confidence interval in the presence of incomplete paired binary data. The approach proposed by Tang et al. can be overly conservative with large expected confidence interval width (ECIW) in some situations. We propose a profile likelihood‐based method with a Jeffreys' prior correction to construct the confidence interval. This approach generates confidence interval with a much better coverage probability and shorter ECIWs. The performances of the method along with the corrections are demonstrated through extensive simulation. Finally, three real world data sets are analyzed by all the methods. Statistical Analysis System (SAS) codes to execute the profile likelihood‐based methods are also presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Exact likelihood ratio and score confidence intervals for the binomial proportion

Many methods are available for computing a confidence interval for the binomial parameter, and these methods differ in their operating characteristics. It has been suggested in the literature that the use of the exact likelihood ratio (LR) confidence interval for the binomial proportion should be considered. This paper provides an evaluation of the operating characteristics of the two‐sided exact LR and exact score confidence intervals for the binomial proportion and compares these results to those for three other methods that also strictly maintain nominal coverage: Clopper‐Pearson, Blaker, and Casella. In addition, the operating characteristics of the two‐sided exact LR method and exact score method are compared with those of the corresponding asymptotic methods to investigate the adequacy of the asymptotic approximation. Copyright © 2013 John Wiley & Sons, Ltd.     

Confidence Intervals for Difference Between Two Poisson Rates

In this article, we develop four explicit asymptotic two-sided confidence intervals for the difference between two Poisson rates via a hybrid method. The basic idea of the proposed method is to estimate or recover the variances of the two Poisson rate estimates, which are required for constructing the confidence interval for the rate difference, from the confidence limits for the two individual Poisson rates. The basic building blocks of the approach are reliable confidence limits for the two individual Poisson rates. Four confidence interval estimators that have explicit solutions and good coverage levels are employed: the first normal with continuity correction, Rao score, Freeman and Tukey, and Jeffreys confidence intervals. Using simulation studies, we examine the performance of the four hybrid confidence intervals and compare them with three existing confidence intervals: the non-informative prior Bayes confidence interval, the t confidence interval based on Satterthwait's degrees of freedom, and the Bayes confidence interval based on Student's t confidence coefficient. Simulation results show that the proposed hybrid Freeman and Tukey, and the hybrid Jeffreys confidence intervals can be highly recommended because they outperform the others in terms of coverage probabilities and widths. The other methods tend to be too conservative and produce wider confidence intervals. The application of these confidence intervals are illustrated with three real data sets.     

Coverage‐adjusted Confidence Intervals for a Binomial Proportion

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap‐type plots for comparing confidence intervals, we show that the coverage‐adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.     

Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance

A confidence interval for the between group variance is proposed which is deduced from Wald'sexact confidence interval for the rtio of the two variance components in the one-way random effects model and the exact confidence interval for the error variance resp.an unbiased estimator of the error variance. In a simulation study the confidence coeffecients for these two intervals are compared with the confidence coefficients of two other commonly used confidence intervals. There the confidence interval derived here yields confidence coefficiends which are always greater than the prescriped level.     

Confidence Intervals for the Youden Index and Corresponding Optimal Cut-Point

A global measure of biomarker effectiveness is the Youden index, the maximum difference between sensitivity, the probability of correctly classifying diseased individuals, and 1-specificity, the probability of incorrectly classifying healthy individuals. The cut-point leading to the index is the optimal cut-point when equal weight is given to sensitivity and specificity. Using the delta method, we present approaches for estimating confidence intervals for the Youden index and corresponding optimal cut-point for normally distributed biomarkers and also those following gamma distributions. We also provide confidence intervals using various bootstrapping methods. A comparison of interval width and coverage probability is conducted through simulation over a variety of parametric situations. Confidence intervals via delta method are shown to have both closer to nominal coverage and shorter interval widths than confidence intervals from the bootstrapping methods.     

Model‐Averaged Confidence Intervals

We develop an approach to evaluating frequentist model averaging procedures by considering them in a simple situation in which there are two‐nested linear regression models over which we average. We introduce a general class of model averaged confidence intervals, obtain exact expressions for the coverage and the scaled expected length of the intervals, and use these to compute these quantities for the model averaged profile likelihood (MPI) and model‐averaged tail area confidence intervals proposed by D. Fletcher and D. Turek. We show that the MPI confidence intervals can perform more poorly than the standard confidence interval used after model selection but ignoring the model selection process. The model‐averaged tail area confidence intervals perform better than the MPI and postmodel‐selection confidence intervals but, for the examples that we consider, offer little over simply using the standard confidence interval for θ under the full model, with the same nominal coverage.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

A confidence region with multiple intervals in multivariate bioassay

The exact confidence region for log relative potency resulting from likelihood score methods (Williams (1988) An exact confidence interval for the relative potency estimated from a multivariate bioassay, Biometrics, 44:861-868) will very likely consist of two disjoint confidence intervals. The two methods proposed by Williams which aim to select just one (the same) confidence interval from the confidence region are nearly – but not completely – consistent. The likelihood score interval and likelihood ratio interval are asymptotically equivalent. Williams's very strong claim concerning the confidence coefficient in the second selection method is still theoretically unproved; yet, simulations show that it is true for a wide range of practical experimental situations.     

Reliability inference for a general lower-truncated family of distributions under records

Based on record values, point and interval estimators are proposed in this paper for the parameters of a general lower-truncated family of distributions. Maximum likelihood and bias-corrected estimators are obtained for unknown model parameters. Based on a sufficient and complete statistic, the bias-corrected estimator is also shown to be uniformly minimum variance unbiased estimator. Different exact confidence intervals and exact confidence regions are constructed for the both model and truncated parameters, and other confidence interval estimates based on asymptotic distribution theory and bootstrap approaches are obtained as well. Finally, two real-life examples and a numerical study are presented to illustrate the performance of our methods.     

A procedure for approximate negative binomial tolerance intervals

In this article, we present a procedure for approximate negative binomial tolerance intervals. We utilize an approach that has been well-studied to approximate tolerance intervals for the binomial and Poisson settings, which is based on the confidence interval for the parameter in the respective distribution. A simulation study is performed to assess the coverage probabilities and expected widths of the tolerance intervals. The simulation study also compares eight different confidence interval approaches for the negative binomial proportions. We recommend using those in practice that perform the best based on our simulation results. The method is also illustrated using two real data examples.     

Bayesian Confidence Interval for the Ratio of Marginal Probabilities in the Matched-Pair Design

This article studies the construction of a Bayesian confidence interval for the ratio of marginal probabilities in matched-pair designs. Under a Dirichlet prior distribution, the exact posterior distribution of the ratio is derived. The tail confidence interval and the highest posterior density (HPD) interval are studied, and their frequentist performances are investigated by simulation in terms of mean coverage probability and mean expected length of the interval. An advantage of Bayesian confidence interval is that it is always well defined for any data structure and has shorter mean expected width. We also find that the Bayesian tail interval at Jeffreys prior performs as well as or better than the frequentist confidence intervals.     

CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN TWO PARTIAL AUCS

As new diagnostic tests are developed and marketed, it is very important to be able to compare the accuracy of a given two continuous‐scale diagnostic tests. An effective method to evaluate the difference between the diagnostic accuracy of two tests is to compare partial areas under the receiver operating characteristic curves (AUCs). In this paper, we review existing parametric methods. Then, we propose a new semiparametric method and a new nonparametric method to investigate the difference between two partial AUCs. For the difference between two partial AUCs under each method, we derive a normal approximation, define an empirical log‐likelihood ratio, and show that the empirical log‐likelihood ratio follows a scaled chi‐square distribution. We construct five confidence intervals for the difference based on normal approximation, bootstrap, and empirical likelihood methods. Finally, extensive simulation studies are conducted to compare the finite‐sample performances of these intervals, and a real example is used as an application of our recommended intervals. The simulation results indicate that the proposed hybrid bootstrap and empirical likelihood intervals outperform other existing intervals in most cases.     

Bayesian Confidence Interval for the Difference of Two Proportions in the Matched-Paired Design

This article studies the construction of Bayesian confidence interval for the difference of two proportions in the matched-pair design, and applies it to the equiva-lence or non inferiority test. Under the Dirichlet prior distribution, the exact posterior distribution of difference of two proportions is derived. The tail confidence interval and the highest posterior density (HPD) interval are studied, and their frequentist performance are investigated by simulation in terms of the mean coverage probability of interval. Our results suggest to use tail interval at Jeffreys prior for testing equivalence or non inferiority in matched-pair design.     

Comparison of Poisson Confidence Intervals

Abstract

The standard method of obtaining a two-sided confidence interval for the Poisson mean produces an interval which is exact but can be shortened without violating the minimum coverage requirement. We classify the intervals proposed as alternatives to the standard method interval. We carry out the classification using two desirable properties of two-sided confidence intervals.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

Distribution‐free statistical intervals via ranked‐set sampling

The author shows how to construct distribution‐free statistical intervals from ranked‐set sampling. He considers the cases of confidence, tolerance, and prediction intervals. He shows how to compute their coverage probabilities and he compares their performance to that of intervals based on simple random sampling. He finds that the ranked‐set sampling‐based intervals are at least as good as their classical counterpart in most settings of interest, although the nature of the advantage depends on the type of interval considered.     

Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring

Epstein (1954) introduced the Type-I hybrid censoring scheme as a mixture of Type-I and Type-II censoring schemes. Childs et al. (2003) introduced the Type-II hybrid censoring scheme as an alternative to Type-I hybrid censoring scheme, and provided the exact distribution of the maximum likelihood estimator of the mean of a one-parameter exponential distribution based on Type-II hybrid censored samples. The associated confidence interval also has been provided. The main aim of this paper is to consider a two-parameter exponential distribution, and to derive the exact distribution of the maximum likelihood estimators of the unknown parameters based on Type-II hybrid censored samples. The marginal distributions and the exact confidence intervals are also provided. The results can be used to derive the exact distribution of the maximum likelihood estimator of the percentile point, and to construct the associated confidence interval. Different methods are compared using extensive simulations and one data analysis has been performed for illustrative purposes.     

Inference on effect size indices from several two-armed experiments

The statistical inference problem on effect size indices is addressed using a series of independent two-armed experiments from k arbitrary populations. The effect size parameter simply quantifies the difference between two groups. It is a meaningful index to be used when data are measured on different scales. In the context of bivariate statistical models, we define estimators of the effect size indices and propose large sample testing procedures to test the homogeneity of these indices. The null and non-null distributions of the proposed testing procedures are derived and their performance is evaluated via Monte Carlo simulation. Further, three types of interval estimation of the proposed indices are considered for both combined and uncombined data. Lower and upper confidence limits for the actual effect size indices are obtained and compared via bootstrapping. It is found that the length of the intervals based on the combined effect size estimator are almost half the length of the intervals based on the uncombined effect size estimators. Finally, we illustrate the proposed procedures for hypothesis testing and interval estimation using a real data set.     

Interval estimation for a binomial proportion: a bootstrap approach

This paper discusses the classic but still current problem of interval estimation of a binomial proportion. Bootstrap methods are presented for constructing such confidence intervals in a routine, automatic way. Three confidence intervals for a binomial proportion are compared and studied by means of a simulation study, namely: the Wald confidence interval, the Agresti–Coull interval and the bootstrap-t interval. A new confidence interval, the Agresti–Coull interval with bootstrap critical values, is also introduced and its good behaviour related to the average coverage probability is established by means of simulations.