Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

Evaluating the Performance of Simultaneous Stepwise Confidence Intervals for the Difference between two Poisson Rates

We consider the problem of simultaneously estimating Poisson rate differences via applications of the Hsu and Berger stepwise confidence interval method (termed HBM), where comparisons to a common reference group are performed. We discuss continuity-corrected confidence intervals (CIs) and investigate the HBM performance with a moment-based CI, and uncorrected and corrected for continuity Wald and Pooled confidence intervals (CIs). Using simulations, we compare nine individual CIs in terms of coverage probability and the HBM with nine intervals in terms of family-wise error rate (FWER) and overall and local power. The simulations show that these statistical properties depend highly on parameter settings.     

Approximate and Fiducial Confidence Intervals for the Difference Between Two Binomial Proportions

The problem of estimating the difference between two binomial proportions is considered. Closed-form approximate confidence intervals (CIs) and a fiducial CI for the difference between proportions are proposed. The approximate CIs are simple to compute, and they perform better than the classical Wald CI in terms of coverage probabilities and precision. Numerical studies indicate that these approximate CIs can be used safely for practical applications under a simple condition. The fiducial CI is more accurate than the approximate CIs in terms of coverage probabilities. The fiducial CIs, the Newcombe CIs, and the Miettinen–Nurminen CIs are comparable in terms of coverage probabilities and precision. The interval estimation procedures are illustrated using two examples.     

Confidence intervals for product of powers of the generalized variances of k multivariate normal populations

In this paper, some confidence intervals (CIs) for the product of powers of the generalized variances of k multivariate normal populations with possibly different dimensions are proposed. The performance of these CIs in terms of the coverage probabilities and average lengths were evaluated via a Monte Carlo simulation study. The results were found to be satisfactory. To demonstrate utility of the proposed CIs, applications on three real data sets were provided.     

Confidence intervals for the difference of marginal probabilities in clustered matched-pair binary data

Although there are several available test statistics to assess the difference of marginal probabilities in clustered matched‐pair binary data, associated confidence intervals (CIs) are not readily available. Herein, the construction of corresponding CIs is proposed, and the performance of each CI is investigated. The results from Monte Carlo simulation study indicate that the proposed CIs perform well in maintaining the nominal coverage probability: for small to medium numbers of clusters, the intracluster correlation coefficient‐adjusted McNemar statistic and its associated Wald or Score CIs are preferred; however, this statistic becomes conservative when the number of clusters is larger so that alternative statistics and their associated CIs are preferred. In practice, a combination of the intracluster correlation coefficient‐adjusted McNemar statistic with an alternative statistic is recommended. To illustrate the practical application, a real clustered matched‐pair collection of data is used to illustrate testing the difference of marginal probabilities and constructing the associated CIs. Copyright © 2012 John Wiley & Sons, Ltd.     

Assessing Statistical Results: Magnitude,Precision, and Model Uncertainty

ABSTRACT

Evaluating the importance and the strength of empirical evidence requires asking three questions: First, what are the practical implications of the findings? Second, how precise are the estimates? Confidence intervals provide an intuitive way to communicate precision. Although nontechnical audiences often misinterpret confidence intervals (CIs), I argue that the result is less dangerous than the misunderstandings that arise from hypothesis tests. Third, is the model correctly specified? The validity of point estimates and CIs depends on the soundness of the underlying model.     

Interval Estimation for the Sortino Ratio and the Omega Ratio

In this article, asymptotic confidence intervals (CIs) for the Sortino and Omega ratios are proposed and analyzed. First, the CIs are derived under the assumption of temporal independence and identical distribution of returns. Later they are obtained assuming that the returns process is strictly stationary and α-mixing of a certain size. In order to evaluate the minimum sample size for a good coverage accuracy of the asymptotic CIs, a simulation study is performed. It is obtained that the minimum sample sizes are very high, especially under the more realistic assumption of not-iid returns.     

Divergence-based confidence intervals in false-positive misclassification model

In this article, we introduce minimum divergence estimators of parameters of a binary response model when data are subject to false-positive misclassification and obtained using a double-sampling plan. Under this set up, the problem of goodness-of-fit is considered and divergence-based confidence intervals (CIs) for a population proportion parameter are derived. A simulation experiment is carried out to compare the coverage probabilities of the new CIs. An application to real data is also given.     

Bootstrap methods for bias correction and confidence interval estimation for nonlinear quantile regression of longitudinal data

This paper examines the use of bootstrapping for bias correction and calculation of confidence intervals (CIs) for a weighted nonlinear quantile regression estimator adjusted to the case of longitudinal data. Different weights and types of CIs are used and compared by computer simulation using a logistic growth function and error terms following an AR(1) model. The results indicate that bias correction reduces the bias of a point estimator but fails for CI calculations. A bootstrap percentile method and a normal approximation method perform well for two weights when used without bias correction. Taking both coverage and lengths of CIs into consideration, a non-bias-corrected percentile method with an unweighted estimator performs best.     

A Gaussian alternative to using improper confidence intervals

The problem posed by exact confidence intervals (CIs) which can be either all-inclusive or empty for a nonnegligible set of sample points is known to have no solution within CI theory. Confidence belts causing improper CIs can be modified by using margins of error from the renewed theory of errors initiated by J. W. Tukey—briefly described in the article—for which an extended Fraser's frequency interpretation is given. This approach is consistent with Kolmogorov's axiomatization of probability, in which a probability and an error measure obey the same axioms, although the connotation of the two words is different. An algorithm capable of producing a margin of error for any parameter derived from the five parameters of the bivariate normal distribution is provided. Margins of error correcting Fieller's CIs for a ratio of means are obtained, as are margins of error replacing Jolicoeur's CIs for the slope of the major axis. Margins of error using Dempster's conditioning that can correct optimal, but improper, CIs for the noncentrality parameter of a noncentral chi-square distribution are also given.     

Confidence intervals for assessing equivalence of two treatments with combined unilateral and bilateral data

Responses from the paired organs are generally highly correlated in bilateral studies, statistical procedures ignoring the correlation could lead to incorrect results. Note the intraclass correlation in the study of combined unilateral and bilateral outcomes; 11 confidence intervals (CIs) including 7 asymptotic CIs and 4 Bootstrap-resampling CIs for assessing the equivalence of 2 treatments are derived under Rosner''s correlated binary data model. Performance is evaluated with respect to the empirical coverage probability (ECP), the empirical coverage width (ECW) and the ratio of the mesial non-coverage probability to the non-coverage probability (RMNCP) via simulation studies. Simulation results show that (i) all CIs except for the Wald CI and the bias-corrected Bootstrap percentile CI generally produce satisfactory ECPs and hence are recommended; (ii) all CIs except for the bias-corrected Bootstrap percentile CI provide preferred RMNCPs and are more symmetrical; (iii) as the measurement of the dependence increases, the ECWs of all CIs except for the score CI and the profile likelihood CI show increasing patterns that look like linear, while there is no obvious pattern on the ECPs of all CIs except for the profile likelihood CI. A data set from an otolaryngologic study is used to illustrate the proposed methods.     

Multiplicity: discussion points from the Statisticians in the Pharmaceutical Industry multiplicity expert group

In May 2012, the Committee of Health and Medicinal Products issued a concept paper on the need to review the points to consider document on multiplicity issues in clinical trials. In preparation for the release of the updated guidance document, Statisticians in the Pharmaceutical Industry held a one‐day expert group meeting in January 2013. Topics debated included multiplicity and the drug development process, the usefulness and limitations of newly developed strategies to deal with multiplicity, multiplicity issues arising from interim decisions and multiregional development, and the need for simultaneous confidence intervals (CIs) corresponding to multiple test procedures. A clear message from the meeting was that multiplicity adjustments need to be considered when the intention is to make a formal statement about efficacy or safety based on hypothesis tests. Statisticians have a key role when designing studies to assess what adjustment really means in the context of the research being conducted. More thought during the planning phase needs to be given to multiplicity adjustments for secondary endpoints given these are increasing in importance in differentiating products in the market place. No consensus was reached on the role of simultaneous CIs in the context of superiority trials. It was argued that unadjusted intervals should be employed as the primary purpose of the intervals is estimation, while the purpose of hypothesis testing is to formally establish an effect. The opposing view was that CIs should correspond to the test decision whenever possible. Copyright © 2013 John Wiley & Sons, Ltd.     

Interval estimation for misclassification rate parameters in a complementary Poisson model

We investigate three interval estimators for binomial misclassification rates in a complementary Poisson model where the data are possibly misclassified: a Wald-based interval, a score-based interval, and an interval based on the profile log-likelihood statistic. We investigate the coverage and average width properties of these intervals via a simulation study. For small Poisson counts and small misclassification rates, the intervals can perform poorly in terms of coverage. The profile log-likelihood confidence interval (CI) is often proved to outperform the other intervals with good coverage and width properties. Lastly, we apply the CIs to a real data set involving traffic accident data that contain misclassified counts.     

Wilson Confidence Intervals for the Two-Sample Log-Odds-Ratio in Stratified 2 × 2 Contingency Tables

Large-sample Wilson-type confidence intervals (CIs) are derived for a parameter of interest in many clinical trials situations: the log-odds-ratio, in a two-sample experiment comparing binomial success proportions, say between cases and controls. The methods cover several scenarios: (i) results embedded in a single 2 × 2 contingency table; (ii) a series of K 2 × 2 tables with common parameter; or (iii) K tables, where the parameter may change across tables under the influence of a covariate. The calculations of the Wilson CI require only simple numerical assistance, and for example are easily carried out using Excel. The main competitor, the exact CI, has two disadvantages: It requires burdensome search algorithms for the multi-table case and results in strong over-coverage associated with long confidence intervals. All the application cases are illustrated through a well-known example. A simulation study then investigates how the Wilson CI performs among several competing methods. The Wilson interval is shortest, except for very large odds ratios, while maintaining coverage similar to Wald-type intervals. An alternative to the Wald CI is the Agresti-Coull CI, calculated from the Wilson and Wald CIs, which has same length as the Wald CI but improved coverage.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

Evaluation of alternative confidence intervals to address non-inferiority through the stratified difference between proportions

The confidence interval (CI) for the difference between two proportions has been an important and active research topic, especially in the context of non-inferiority hypothesis testing. Issues concerning the Type 1 error rate, power, coverage rate and aberrations have been extensively studied for non-stratified cases. However, stratified confidence intervals are frequently used in non-inferiority trials and similar settings. In this paper, several methods for stratified confidence intervals for the difference between two proportions, including existing methods and novel extensions from unstratified CIs, are evaluated across different scenarios. When sparsity across the strata is not a concern, adding imputed observations to the stratification analysis can strengthen Type-1 error control without substantial loss of power. When sparseness of data is a concern, most of the evaluated methods fail to control Type-1 error; the modified stratified t-test CI is an exception. We recommend the modified stratified t-test CI as the most useful and flexible method across the respective scenarios; the modified stratified Wald CI may be useful in settings where sparsity is unlikely. These findings substantially contribute to the application of stratified CIs for non-inferiority testing of differences between two proportions.     

Tests and confidence intervals for the shape parameter of a gamma distribution

Inference based on the Central Limit Theorem has only first order accuracy. We give tests and confidence intervals (CIs) of second orderaccuracy for the shape parameter ρ of a gamma distribution for both the unscaled and scaled cases.

Tests and CIs based on moment and cumulant estimates are considered as well as those based on the maximum likelihood estimate (MLE).

For the unscaled case the MLE is the moment estimate of order zero; the most efficient moment estimate of integral order is the sample mean, having asymptotic relative efficiency (ARE) .61 when ρ= 1.

For the scaled case the most efficient moment estimate is a functionof the mean and variance. Its ARE is .39 when ρ = 1.

Our motivation for constructing these tests of ρ = 1 and CIs forρ is to provide a simple and convenient method for testing whether a distribution is exponential in situations such as rainfall models where such an assumption is commonly made.     

Confidence Intervals of Performance Measures for an M/G/1 Queueing System

In this article, we discuss constructing confidence intervals (CIs) of performance measures for an M/G/1 queueing system. Fiducial empirical distribution is applied to estimate the service time distribution. We construct fiducial empirical quantities (FEQs) for the performance measures. The relationship between generalized pivotal quantity and fiducial empirical quantity is illustrated. We also present numerical examples to show that the FEQs can yield new CIs dominate the bootstrap CIs in relative coverage (defined as the ratio of coverage probability to average length of CI) for performance measures of an M/G/1 queueing system in most of the cases.     

Exact inference for competing risks model with generalized type-I hybrid censored exponential data

Recently, exact inference under hybrid censoring scheme has attracted extensive attention in the field of reliability analysis. However, most of the authors neglect the possibility of competing risks model. This paper mainly discusses the exact likelihood inference for the analysis of generalized type-I hybrid censoring data with exponential competing failure model. Based on the maximum likelihood estimates for unknown parameters, we establish the exact conditional distribution of parameters by conditional moment generating function, and then obtain moment properties as well as exact confidence intervals (CIs) for parameters. Furthermore, approximate CIs are constructed by asymptotic distribution and bootstrap method as well. We also compare their performances with exact method through the use of Monte Carlo simulations. And finally, a real data set is analysed to illustrate the validity of all the methods developed here.