Expected Interval Width Research Articles

A comparison of Bayesian and score methods for interval estimates of positive/negative likelihood ratios in support of diagnostic device performance evaluation

ABSTRACT Background Positive and negative likelihood ratios (PLR and NLR) are important metrics of accuracy for diagnostic devices with a binary output. However, the properties of Bayesian and frequentist interval estimators of PLR/NLR have not been extensively studied and compared. In this study, we explore the potential use of the Bayesian method for interval estimation of PLR/NLR, and, more broadly, for interval estimation of the ratio of two independent proportions. Methods We develop a Bayesian-based approach for interval estimation of PLR/NLR for use as a part of a diagnostic device performance evaluation. Our approach is applicable to a broader setting for interval estimation of any ratio of two independent proportions. We compare score and Bayesian interval estimators for the ratio of two proportions in terms of the coverage probability (CP) and expected interval width (EW) via extensive experiments and applications to two case studies. A supplementary experiment was also conducted to assess the performance of the proposed exact Bayesian method under different priors. Results Our experimental results show that the overall mean CP for Bayesian interval estimation is consistent with that for the score method (0.950 vs. 0.952), and the overall mean EW for Bayesian is shorter than that for score method (15.929 vs. 19.724). Application to two case studies showed that the intervals estimated using the Bayesian and frequentist approaches are very similar. Discussion Our numerical results indicate that the proposed Bayesian approach has a comparable CP performance with the score method while yielding higher precision (i.e. a shorter EW).

Assessing standardized contrast effects in ANCOVA: Confidence intervals, precision evaluations, and sample size requirements.

Standardized effect sizes and confidence intervals are useful statistical assessments for comparing results across different studies when measurement units are not directly comparable. This paper aims to describe and compare confidence interval estimation methods for the standardized contrasts of treatment effects in ANCOVA designs. Sample size procedures are also presented to assure that the resulting confidence intervals yield informative estimation with adequate precision. Exact interval estimation approach has theoretical and empirical advantages in coverage probability and interval width over the approximate interval procedures. Numerical investigations of the existing method reveal that the omission of covariate variables has a negative impact on sample size calculations for precise interval estimation, especially when there is disparity in influential covariate variables. The proposed approaches and developed computer programs fully utilize covariate properties in interval estimation and provide accurate sample size determinations under the precision considerations of the expected interval width and the assurance probability of interval width.

Chance Constrained Extreme Learning Machine for Nonparametric Prediction Intervals of Wind Power Generation

Confronted with considerable intermittence and variability of wind power, prediction intervals (PIs) serve as a crucial tool to assist power system decision-making under uncertainties. Conventional PIs rely on predetermining the lower and upper quantile proportions and therefore suffer from conservative interval width. This paper innovatively develops a chance constrained extreme learning machine (CCELM) model to generate quality nonparametric proportion-free PIs of wind power generation, which minimizes the expected interval width subject to the PI coverage probability constraint. Due to the independency on the preset PI bounds proportions, the proposed CCELM model merits high adaptivity and taps the latent potentialities for PI shortening. The convexity of extreme learning machine renders the sample average approximation counterpart of stochastic CCELM model equivalent to a parameter searching task in parametric optimization problem with polyhedral feasible region. A novel difference of convex functions optimization based bisection search (DCBS) algorithm is proposed to efficiently construct the CCELM model, which successfully realizes machine learning by means of solving linear programming problems sequentially. Comprehensive numerical experiments based on actual wind farm data demonstrate the significant effectiveness and efficiency of the developed CCELM model and DCBS algorithm.

Optimal confidence interval for the largest exponential location parameter

A single sampling procedure is proposed for obtaining an optimal confidence interval for the largest location parameter of several k independent exponential populations in the cases of known and unknown common scale parameter, where “optimal” is defined in the sense of minimal expected interval width and best allocation at a fixed confidence. The optimal result can be applied to type II censored samples. Tables of percentage points are provided for practitioners.

Exact adaptive confidence intervals for linear regression coefficients

We propose an adaptive confidence interval procedure (CIP) for the coefficients in the normal linear regression model. This procedure has a frequentist coverage rate that is constant as a function of the model parameters, yet provides smaller intervals than the usual interval procedure, on average across regression coefficients. The proposed procedure is obtained by defining a class of CIPs that all have exact $1-\alpha $ frequentist coverage, and then selecting from this class the procedure that minimizes a prior expected interval width. We describe an adaptive approach for estimating the prior distribution from the data, so that the potential risk of a poorly specified prior is reduced. The resulting adaptive confidence intervals maintain exact non-asymptotic $1-\alpha $ coverage if two conditions are met - that the design matrix is full rank (which will be known) and that the errors are normally distributed (which can be checked empirically). No assumptions on the unknown parameters are necessary to maintain exact coverage. Additionally, in a “$p$ growing with $n$” asymptotic scenario, this adaptive FAB procedure is asymptotically Bayes-optimal among $1-\alpha $ frequentist CIPs.

Confidence intervals construction for difference of two means with incomplete correlated data.

BackgroundIncomplete data often arise in various clinical trials such as crossover trials, equivalence trials, and pre and post-test comparative studies. Various methods have been developed to construct confidence interval (CI) of risk difference or risk ratio for incomplete paired binary data. But, there is little works done on incomplete continuous correlated data. To this end, this manuscript aims to develop several approaches to construct CI of the difference of two means for incomplete continuous correlated data.MethodsLarge sample method, hybrid method, simple Bootstrap-resampling method based on the maximum likelihood estimates (B1) and Ekbohm’s unbiased estimator (B2), and percentile Bootstrap-resampling method based on the maximum likelihood estimates (B3) and Ekbohm’s unbiased estimator (B4) are presented to construct CI of the difference of two means for incomplete continuous correlated data. Simulation studies are conducted to evaluate the performance of the proposed CIs in terms of empirical coverage probability, expected interval width, and mesial and distal non-coverage probabilities.ResultsEmpirical results show that the Bootstrap-resampling-based CIs B1, B2, B4 behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and the ratio of their mesial non-coverage probabilities to the non-coverage probabilities could be well controlled in the interval [0.4, 0.6].ConclusionsIf one would like a CI with the shortest interval width, the Bootstrap-resampling-based CIs B1 is the optimal choice.

Confidence interval construction for the difference between two correlated proportions with missing observations

Under the assumption of missing at random, eight confidence intervals (CIs) for the difference between two correlated proportions in the presence of incomplete paired binary data are constructed on the basis of the likelihood ratio statistic, the score statistic, the Wald-type statistic, the hybrid method incorporated with the Wilson score and Agresti–Coull (AC) intervals, and the Bootstrap-resampling method. Extensive simulation studies are conducted to evaluate the performance of the presented CIs in terms of coverage probability and expected interval width. Our empirical results evidence that the Wilson-score-based hybrid CI and the Wald-type CI together with the constrained maximum likelihood estimates perform well for small-to-moderate sample sizes in the sense that (i) their empirical coverage probabilities are quite close to the prespecified confidence level, (ii) their expected interval widths are shorter, and (iii) their ratios of the mesial non-coverage to non-coverage probabilities lie in interval [0.4, 0.6]. An example from a neurological study is used to illustrate the proposed methodologies.

Estimating the Parameters of Rayleigh Cumulative Exposure Model in Simple Step-Stress Testing. Natasha Beretvas is an

Assumes the life distribution of a test unit for any stress follows a Rayleigh distribution with scale parameterθ , and that Ln(θ ) is a linear function of the stress level. Maximum likelihood estimators of the parameters under a cumulative exposure model are obtained. The approximate variance estimates obtained from the asymptotic normal distribution of the maximum likelihood estimators are used to construct confidence intervals for the model parameters. A simulation study was conducted to study the performance of the estimators. Simulation results showed that in terms of bias, mean squared error, attainment of the nominal confidence level, symmetry of lower and upper error rates and the expected interval width, the estimators are very accurate and have a high level of precision.

Robust Confidence Intervals for the Bernoulli Parameter

Despite the simplicity of the Bernoulli process, developing good confidence interval procedures for its parameter—the probability of success p—is deceptively difficult. The binary data yield a discrete number of successes from a discrete number of trials, n. This discreteness results in actual coverage probabilities that oscillate with the n for fixed values of p (and with p for fixed n). Moreover, this oscillation necessitates a large sample size to guarantee a good coverage probability when p is close to 0 or 1. It is well known that the Wilson procedure is superior to many existing procedures because it is less sensitive to p than any other procedures, therefore it is less costly. The procedures proposed in this article work as well as the Wilson procedure when 0.1 ≤p ≤ 0.9, and are even less sensitive (i.e., more robust) than the Wilson procedure when p is close to 0 or 1. Specifically, when the nominal coverage probability is 0.95, the Wilson procedure requires a sample size 1, 021 to guarantee that the coverage probabilities stay above 0.92 for any 0.001 ≤ min {p, 1 −p} <0.01. By contrast, our procedures guarantee the same coverage probabilities but only need a sample size 177 without increasing either the expected interval width or the standard deviation of the interval width.

A comparative study of confidence intervals for negative binomial proportion

In this paper, we investigate four existing and three new confidence interval estimators for the negative binomial proportion (i.e., proportion under inverse/negative binomial sampling). An extensive and systematic comparative study among these confidence interval estimators through Monte Carlo simulations is presented. The performance of these confidence intervals are evaluated in terms of their coverage probabilities and expected interval widths. Our simulation studies suggest that the confidence interval estimator based on saddlepoint approximation is more appealing for large coverage levels (e.g., nominal level≤1% ) whereas the score confidence interval estimator is more desirable for those commonly used coverage levels (e.g., nominal level>1% ). We illustrate these confidence interval construction methods with a real data set from a maternal congenital heart disease study.

Confidence intervals for the risk ratio under inverse sampling

In this paper, we investigate various confidence intervals for the risk ratio under inverse sampling (also known as negative binomial sampling). Three existing confidence intervals (namely, the confidence intervals that are based on Fieller's theorem, the delta method and the F-statistic) are reviewed and three new confidence intervals (namely, the score, likelihood ratio and saddlepoint approximation (SA)-based confidence intervals) are developed. Comparative studies among these confidence intervals through Monte Carlo simulations are evaluated in terms of their coverage probabilities and expected interval widths under different settings. Our simulation results suggest that the SA-based confidence interval is generally more appealing. We illustrate these confidence interval construction methods with real data sets from a drug comparison study and a congenital heart disease study.

Interval estimation of binomial proportion in clinical trials with a two‐stage design

Generally, a two-stage design is employed in Phase II clinical trials to avoid giving patients an ineffective drug. If the number of patients with significant improvement, which is a binomial response, is greater than a pre-specified value at the first stage, then another binomial response at the second stage is also observed. This paper considers interval estimation of the response probability when the second stage is allowed to continue. Two asymptotic interval estimators, Wald and score, as well as two exact interval estimators, Clopper-Pearson and Sterne, are constructed according to the two binomial responses from this two-stage design, where the binomial response at the first stage follows a truncated binomial distribution. The mean actual coverage probability and expected interval width are employed to evaluate the performance of these interval estimators. According to the comparison results, the score interval is recommended for both Simon's optimal and minimax designs.

Confidence intervals for a ratio of binomial proportions based on paired data

Four interval estimation methods for the ratio of marginal binomial proportions are compared in terms of expected interval width and exact coverage probability. Two new methods are proposed that are based on combining two Wilson score intervals. The new methods are easy to compute and perform as well or better than the method recently proposed by Nam and Blackwelder. Two sample size formulas are proposed to approximate the sample size required to achieve an interval estimate with desired confidence level and width.

Statistical Inference For Risk Difference in an Incomplete Correlated 2 × 2 Table

AbstractIn some infectious disease studies and 2‐step treatment studies, 2 × 2 table with structural zero could arise in situations where it is theoretically impossible for a particular cell to contain observations or structural void is introduced by design. In this article, we propose a score test of hypotheses pertaining to the marginal and conditional probabilities in a 2 × 2 table with structural zero via the risk/rate difference measure. Score test‐based confidence interval will also be outlined. We evaluate the performance of the score test and the existing likelihood ratio test. Our empirical results evince the similar and satisfactory performance of the two tests (with appropriate adjustments) in terms of coverage probability and expected interval width. Both tests consistently perform well from small‐ to moderate‐sample designs. The score test however has the advantage that it is only undefined in one scenario while the likelihood ratio test can be undefined in many scenarios. We illustrate our method by a real example from a two‐step tuberculosis skin test study.

Interval Estimation of Bivariate Correlations With Missing Data on Both Variables: A Bayesian Approach

The posterior distribution of the bivariate correlation ( ρxy) is analytically derived given a data set consisting N1 cases measured on both x and y, N2 cases measured only on x, and N3 cases measured only on y. The posterior distribution is shown to be a function of the subsample sizes, the sample correlation ( rxy) computed from the N1 complete cases, a set of four statistics which measure the extent to which the missing data are not missing completely at random, and the specified prior distribution for ρxy. A sampling study suggests that in small ( N = 20) and moderate ( N = 50) sized samples, posterior Bayesian interval estimates will dominate maximum likelihood based estimates in terms of coverage probability and expected interval widths when the prior distribution for ρxy is simply uniform on (0, 1). The advantage of the Bayesian method when more informative priors based on beta densities are employed is not as consistent.

Confidence interval scatterplots for evaluating confidence interval performance

Confidence intervals are widely used to determine an interval, estimated from data, that contains a population parameter with a prescribed probability. Kang and Schmeiser (1986) introduce a scatterplot that is useful in comparing the effectiveness of different confidence interval procedures based on performance measures such as the coverage, the expected interval width and the variance of the interval width. This paper examines the usefulness of this confidence interval scatterplot (CSI) for evaluating approximate confidence intervals, as illustrated by the comparison of simultation output analysis techniques for correlated observations.