Confidence limits for the population prevalence rate based on the negative binomial distribution

This paper shows that the extension of the simple procedure of George and Elston in calculation of confidence limits for the underlying prevalence rate to accommodate any finite number of cases in inverse sampling is straightforward. To appreciate the fact that the length of the confidence interval calculated on the basis of the first single case may be too wide for general utility, I include a quantitative discussion on the effect due to an increase in the number of cases requested in the sample on the expected length of confidence intervals. To facilitate further the application of the results presented in this paper, I present a table that summarizes in a variety of situations the minimum required number of cases for the ratio of the expected length of a confidence interval relative to the underlying prevalence rate to be less than or equal to a given value. I also include a discussion on the relation between Cleman'S confidence limits on the expected number of trials before the failure of a given device and those presented here.     

Ensemble confidence intervals for binomial proportions

We propose two measures of performance for a confidence interval for a binomial proportion p: the root mean squared error and the mean absolute deviation. We also devise a confidence interval for p based on the actual coverage function that combines several existing approximate confidence intervals. This “Ensemble” confidence interval has improved statistical properties over the constituent confidence intervals. Software in an R package, which can be used in devising and assessing these confidence intervals, is available on CRAN.     

间接法总体标准化率区间估计的可信因子法

The standardized rate of indirect method is widely used, but no method of interval estimation for its population rate has been reported. The authors have discussed the standard error of standardized rate of indirect method and suggested that a method using this standard error should be used to determine the confidence limits for the population rate. In this paper, the authors put forward another method (Confidence Factors-Method) which can be easily applied to determine the confidence limits. It gives approximately the same result with the method mentioned above.     

Recommended tests and confidence intervals for paired binomial proportions

We describe, evaluate, and recommend statistical methods for the analysis of paired binomial proportions. A total of 24 methods are considered. The best tests for association include the asymptotic McNemar test and the McNemar mid‐ p test. For the difference between proportions, we recommend two simple confidence intervals with closed‐form expressions and the asymptotic score interval. The asymptotic score interval is also recommended for the ratio of proportions, as is an interval with closed‐form expression based on combining two Wilson score intervals for the single proportion. For the odds ratio, we recommend a transformation of the Wilson score interval and a transformation of the Clopper–Pearson mid‐ p interval. We illustrate the practical application of the methods using data from a recently published study of airway reactivity in children before and after stem cell transplantation and a matched case–control study of the association between floppy eyelid syndrome and obstructive sleep apnea‐hypopnea syndrome. Copyright © 2014 John Wiley & Sons, Ltd.     

Accurate confidence limits for stratified clinical trials

For stratified 2 × 2 tables, standard approximate confidence limits can perform poorly from a strict frequentist perspective, even for moderate‐sized samples, yet they are routinely used. In this paper, I show how to use importance sampling to compute highly accurate limits in reasonable time. The methodology is very general and simple to implement, and orders of magnitude are faster than existing alternatives. Copyright © 2013 John Wiley & Sons, Ltd.     

Improved confidence intervals for a binomial parameter following a group sequential phase II clinical trial

Group sequential procedures are widely employed in phase II clinical trials. It is often desirable to provide an interval estimate of the true response rate upon termination of a group sequential phase II clinical trial. The confidence intervals proposed by Jennison and Turnbull (Technometrics 1983; 25: 49-58) are conservative. Utilization of auxiliary statistics based on the overall disease status is proposed to reduce the discreteness of the underlying distribution. Confidence intervals generated by the proposed methods have confidence coefficients closer to the nominal level and have shorter average lengths than JT.     

Improved confidence intervals for the difference between binomial proportions based on paired data

Existing methods for setting confidence intervals for the difference θ between binomial proportions based on paired data perform inadequately. The asymptotic method can produce limits outside the range of validity. The ‘exact’ conditional method can yield an interval which is effectively only one-sided. Both these methods also have poor coverage properties. Better methods are described, based on the profile likelihood obtained by conditionally maximizing the proportion of discordant pairs. A refinement (methods 5 and 6) which aligns 1−α with an aggregate of tail areas produces appropriate coverage properties. A computationally simpler method based on the score interval for the single proportion also performs well (method 10). © 1998 John Wiley & Sons, Ltd.     

Score confidence intervals and sample sizes for stratified comparisons of binomial proportions

In a series of articles, Gart and Nam construct the efficient score tests and confidence intervals with or without skewness correction for stratified comparisons of binomial proportions on the risk difference, relative risk, and odds ratio effect metrics. However, the stratified score methods and their properties are not well understood. We rederive the efficient score tests, which reveals their theoretical relationship with the contrast-based score tests, and provides a basis for adapting the method by using other weighting schemes. The inverse variance weight is optimal for a common treatment effect in large samples. We explore the behavior of the score approach in the presence of extreme outcomes when either no or all subjects in some strata are responders, and provide guidance on the choice of weights in the analysis of rare events. The score method is recommended for studies with a small number of moderate or large sized strata. A general framework is proposed to calculate the asymptotic power and sample size for the score test in superiority, noninferiority and equivalence clinical trials, or case-control studies. We also describe a nearly exact procedure that underestimates the exact power, but the degree of underestimation can be controlled to a negligible level. The proposed methods are illustrated by numerical examples.     

Exact one-sided confidence limits for the difference between two correlated proportions

We construct exact and optimal one-sided upper and lower confidence bounds for the difference between two probabilities based on matched binary pairs using well-established optimality theory of Buehler. Starting with five different approximate lower and upper limits, we adjust them to have coverage probability exactly equal to the desired nominal level and then compare the resulting exact limits by their mean size. Exact limits based on the signed root likelihood ratio statistic are preferred and recommended for practical use.     

Construction of confidence limits about effect measures: a general approach

It is widely accepted that confidence interval construction has important advantages over significance testing for the presentation of research results, as now facilitated by readily available software. However, for a number of effect measures, procedures are either not available or not satisfactory in samples of small to moderate size. In this paper, we describe a general approach for estimating a difference between effect measures, which can also be used to obtain confidence limits for a risk ratio and a lognormal mean. Numerical evaluation shows that this closed-form procedure outperforms existing methods, including the bootstrap.     

Goodness-of-fit based confidence intervals for estimates of the size of a closed population

Methods for estimating the size of a closed population often consist of fitting some model (e.g. a log-linear model) to data with a missing cell corresponding to the members of the population missed by all reporting sources. Although the use of the asymptotic standard error is the usual method for forming confidence intervals for the population total, the sample sizes are not always large enough to produce valid confidence intervals. We propose a method for forming confidence intervals based upon changes in a goodness-of-fit statistic associated with changes in trial values of the population total.     

Computation of profile likelihood-based confidence intervals for reference limits with covariates

Many biochemical quantities depend on age or some other covariate. Reference limits that allow for these dependencies help physicians to interpret the results of biochemical tests. Because reference limits must be estimated, it is important to assess their precision with, for example, confidence intervals. This paper relies on the assumption that data can be modeled by a generalized linear model and presents a method for calculating approximate profile likelihood-based confidence intervals for reference limits. The calculation of confidence intervals is based on a new method that draws on profile likelihood-based confidence intervals in general statistical models. The asset of this new method is that only two constrained optimization problems have to be solved instead of several in the standard method. We motivate our confidence interval calculation method with two applications. The first is for data on immunoglobulin concentration in the context of a generalized linear model with gamma distribution. This model is compared with the often used lognormal model. The second application handles data on serum alpha-fetoprotein and is presented in a linear regression situation. In the latter application the widths of the calculated profile confidence intervals are compared with exact and approximate regression-based intervals and the actual confidence levels are determined by simulation. Copyright (c) 2007 John Wiley & Sons, Ltd.     

A note on confidence limits for quartiles with right censored data

Two new methods are proposed for constructing the confidence limits for quartiles. The small sample behaviour of these two methods is compared with the Jennison and Turnbull modified Brookmeyer-Crowley method at three quartiles. The simulation study indicates that one of the new methods, the smoothed modified reflected method, is preferred over the other methods when the censoring rate is less than 40 per cent while the Jennison and Turnbull method is preferred for higher censoring. The results for the upper quartile are similar to the median. For the lower quartile with small sample and high censoring, semi-infinite intervals may often occur. The correct practice is to permit the semi-infinite intervals without modifying them since it is more informative and gives closer to nominal level coverage.     

Unconditional efficient one-sided confidence limits for the odds ratio based on conditional likelihood

We compare various one-sided confidence limits for the odds ratio in a 2 x 2 table. The first group of limits relies on first-order asymptotic approximations and includes limits based on the (signed) likelihood ratio, score and Wald statistics. The second group of limits is based on the conditional tilted hypergeometric distribution, with and without mid-P correction. All these limits have poor unconditional coverage properties and so we apply the general transformation of Buehler (J. Am. Statist. Assoc. 1957; 52:482-493) to obtain limits which are unconditionally exact. The performance of these competing exact limits is assessed across a range of sample sizes and parameter values by looking at their mean size. The results indicate that Buehler limits generated from the conditional likelihood have the best performance, with a slight preference for the mid-P version. This confidence limit has not been proposed before and is recommended for general use, especially when the underlying probabilities are not extreme.     

The effects of model selection on confidence intervals for the size of a closed population

One encounters in the literature estimates of some rates of genetic and congenital disorders based on log-linear methods to model possible interactions among sources. Often the analyst chooses the simplest model consistent with the data for estimation of the size of a closed population and calculates confidence intervals on the assumption that this simple model is correct. However, despite an apparent excellent fit of the data to such a model, we note here that the resulting confidence intervals may well be misleading in that they can fail to provide an adequate coverage probability. We illustrate this with a simulation for a hypothetical population based on data reported in the literature from three sources. The simulated nominal 95 per cent confidence intervals contained the modelled population size only 30 per cent of the time. Only if external considerations justify the assumption of plausible interactions of sources would use of the simpler model's interval be justified.     

A quasi-exact method for the confidence intervals of the difference of two independent binomial proportions in small sample cases

In this paper we propose a quasi-exact alternative to the exact unconditional method by Chan and Zhang (1999) estimating confidence intervals for the difference of two independent binomial proportions in small sample cases. The quasi-exact method is an approximation to a modified version of Chan and Zhang's method, where the two-sided p-value of an observation is defined by adding to the one-sided p-value the sum of all probabilities of more "extreme" events in the unobserved tail. We show that distinctively less conservative interval estimates can be derived following the modified definition of the two-sided p-value. The approximations applied in the quasi-exact method help to simplify the computations greatly, while the resulting infringements to the nominal level are low. Compared with other approximate methods, including the mid-p quasi-exact methods and the Miettinen and Nurminen (M&N) asymptotic method, our quasi-exact method demonstrates much better reliability in small sample cases.     

Fully specified bootstrap confidence intervals for the difference of two independent binomial proportions based on the median unbiased estimator

In studies in which a binary response for each subject is observed, the success probability and functions of this quantity are of interest. The use of confidence intervals has been increasingly encouraged as complementary to, and indeed preferable to, p‐values as the primary expression of the impact of sampling uncertainty on the findings. The asymptotic confidence interval, based on a normal approximation, is often considered, but this interval can have poor statistical properties when the sample size is small and/or when the success probability is near 0 or 1. In this paper, an estimate of the risk difference based on median unbiased estimates (MUEs) of the two group probabilities is proposed. A corresponding confidence interval is derived using a fully specified bootstrap sample space. The proposed method is compared with Chen's quasi‐exact method, Wald intervals and Agresti and Caffo's method with regard to mean square error and coverage probability. For a variety of settings, the MUE‐based estimate of risk difference has mean square error uniformly smaller than maximum likelihood estimate within a certain range of risk difference. The fully specified bootstrap had better coverage probability in the tail area than Chen's quasi‐exact method, Wald intervals and Agresti and Caffo's intervals. Copyright © 2009 John Wiley & Sons, Ltd.