Better approximate confidence intervals for a binomial parameter

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x² distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x². Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

Mid-P confidence intervals based on the likelihood ratio for proportions estimated by group testing

Group testing has been used in many fields of study to estimate proportions. When groups are of different size, the derivation of exact confidence intervals is complicated by the lack of a unique ordering of the event space. An exact interval estimation method is described here, in which outcomes are ordered according to a likelihood ratio statistic. The method is compared with another exact method, in which outcomes are ordered by their associated MLE. Plots of the P‐value against the proportion are useful in examining the properties of the methods. Coverage provided by the intervals is assessed using several realistic grouptesting procedures. The method based on the likelihood ratio, with a mid‐P correction, is shown to give very good coverage in terms of closeness to the nominal level, and is recommended for this type of problem.     

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.     

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.     

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.     

Confidence intervals for the closed population size under inverse sampling without replacement

Inverse sampling is an appropriate design for the second phase of capture-recapture experiments which provides an exactly unbiased estimator of the population size. However, the sampling distribution of the resulting estimator tends to be highly right skewed for small recapture samples, so, the traditional Wald-type confidence intervals appear to be inappropriate. The objective of this paper is to study the performance of interval estimators for the population size under inverse recapture sampling without replacement. To this aim, we consider the Wald-type, the logarithmic transformation-based, the Wilson score, the likelihood ratio and the exact methods. Also, we propose some bootstrap confidence intervals for the population size, including the with-replacement bootstrap (BWR), the without replacement bootstrap (BWO), and the Rao–Wu’s rescaling method. A Monte Carlo simulation is employed to evaluate the performance of suggested methods in terms of the coverage probability, error rates and standardized average length. Our results show that the likelihood ratio and exact confidence intervals are preferred to other competitors, having the coverage probabilities close to the desired nominal level for any sample size, with more balanced error rate for exact method and shorter length for likelihood ratio method. It is notable that the BWO and Rao–Wu’s rescaling methods also may provide good intervals for some situations, however, those coverage probabilities are not invariant with respect to the population arguments, so one must be careful to use them.     

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.     

The Maximum Probability 2 × c Contingency Tables and the Maximum Probability Points of the Multivariate Hypergeometric Distribution

Abstract

The problem of obtaining the maximum probability 2 × c contingency table with fixed marginal sums, R  = (R ₁, R ₂) and C  = (C ₁, … , C _c ), and row and column independence is equivalent to the problem of obtaining the maximum probability points (mode) of the multivariate hypergeometric distribution MH(R ₁; C ₁, … , C _c ). The most simple and general method for these problems is Joe's (Joe, H. (1988 Joe, H. 1988. Extreme probabilities for contingency tables under row and column independence with application to Fisher's exact test. Commun. Statist. Theory Meth., 17(11): 3677–3685. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Extreme probabilities for contingency tables under row and column independence with application to Fisher's exact test. Commun. Statist. Theory Meth. 17(11):3677–3685.) In this article we study a family of MH's in which a connection relationship is defined between its elements. Based on this family and on a characterization of the mode described in Requena and Martín (Requena, F., Martín, N. (2000 Requena, F. and Martín, N. 2000. Characterization of maximum probability points in the multivariate hypergeometric distribution. Statist. Probab. Lett., 50: 39–47.  [Google Scholar]). Characterization of maximum probability points in the multivariate hypergeometric distribution. Statist. Probab. Lett. 50:39–47.), we develop a new method for the above problems, which is completely general, non recursive, very simple in practice and more efficient than the Joe's method. Also, under weak conditions (which almost always hold), the proposed method provides a simple explicit solution to these problems. In addition, the well-known expression for the mode of a hypergeometric distribution is just a particular case of the method in this article.     

Comparison of exact tests for association in unordered contingency tables using standard,mid-p,and randomized test versions

Pearson’s chi-square (Pe), likelihood ratio (LR), and Fisher (Fi)–Freeman–Halton test statistics are commonly used to test the association of an unordered r×c contingency table. Asymptotically, these test statistics follow a chi-square distribution. For small sample cases, the asymptotic chi-square approximations are unreliable. Therefore, the exact p-value is frequently computed conditional on the row- and column-sums. One drawback of the exact p-value is that it is conservative. Different adjustments have been suggested, such as Lancaster’s mid-p version and randomized tests. In this paper, we have considered 3×2, 2×3, and 3×3 tables and compared the exact power and significance level of these test’s standard, mid-p, and randomized versions. The mid-p and randomized test versions have approximately the same power and higher power than that of the standard test versions. The mid-p type-I error probability seldom exceeds the nominal level. For a given set of parameters, the power of Pe, LR, and Fi differs approximately the same way for standard, mid-p, and randomized test versions. Although there is no general ranking of these tests, in some situations, especially when averaged over the parameter space, Pe and Fi have the same power and slightly higher power than LR. When the sample sizes (i.e., the row sums) are equal, the differences are small, otherwise the observed differences can be 10% or more. In some cases, perhaps characterized by poorly balanced designs, LR has the highest power.     

A Comparison of Large-Sample Confidence Interval Methods for the Difference of Two Binomial Probabilities

A simulation study was done to compare seven confidence interval methods, based on the normal approximation, for the difference of two binomial probabilities. Cases considered included minimum expected cell sizes ranging from 2 to 15 and smallest group sizes (NMIN) ranging from 6 to 100. Our recommendation is to use a continuity correction of 1/(2 NMIN) combined with the use of (N ? 1) rather than N in the estimate of the standard error. For all of the cases considered with minimum expected cell size of at least 3, this method gave coverage probabilities close to or greater than the nominal 90% and 95%. The Yates method is also acceptable, but it is slightly more conservative. At the other extreme, the usual method (with no continuity correction) does not provide adequate coverage even at the larger sample sizes. For the 99% intervals, our recommended method and the Yates correction performed equally well and are reasonable for minimum expected cell sizes of at least 5. None of the methods performed consistently well for a minimum expected cell size of 2.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Simultaneous confidence sets and confidence intervals for multiple ratios

This paper deals with the problem of simultaneously estimating multiple ratios. In the simplest case of only one ratio parameter, Fieller's theorem (J. Roy. Statist. Soc. Ser. B 16 (1954) 175) provides a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that exactly satisfy a given familywise confidence level. Many of the methods in use are conservative since they are based on probability inequalities. In this paper, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discussed. The methods are applied to ratios of linear combinations of the means in the one-way layout and ratios of parameter combinations in the general linear model. Extensive Monte Carlo simulation is carried out to compare the performance of the various methods with respect to the stability of the estimated critical points and of the coverage probabilities.     

A comment on sample size calculations for binomial confidence intervals

In this article we examine sample size calculations for a binomial proportion based on the confidence interval width of the Agresti–Coull, Wald and Wilson Score intervals. We pointed out that the commonly used methods based on known and fixed standard errors cannot guarantee the desired confidence interval width given a hypothesized proportion. Therefore, a new adjusted sample size calculation method was introduced, which is based on the conditional expectation of the width of the confidence interval given the hypothesized proportion. With the reduced sample size, the coverage probability can still maintain at the nominal level and is very competitive to the converge probability for the original sample size.     

The Fate Worse than Death and other Curiosities and Stupidities

The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sterne's procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.     

Confidence intervals for the difference between independent binomial proportions: comparison using a graphical approach and moving averages

This paper uses graphical methods to illustrate and compare the coverage properties of a number of methods for calculating confidence intervals for the difference between two independent binomial proportions. We investigate both small‐sample and large‐sample properties of both two‐sided and one‐sided coverage, with an emphasis on asymptotic methods. In terms of aligning the smoothed coverage probability surface with the nominal confidence level, we find that the score‐based methods on the whole have the best two‐sided coverage, although they have slight deficiencies for confidence levels of 90% or lower. For an easily taught, hand‐calculated method, the Brown‐Li ‘Jeffreys’ method appears to perform reasonably well, and in most situations, it has better one‐sided coverage than the widely recommended alternatives. In general, we find that the one‐sided properties of many of the available methods are surprisingly poor. In fact, almost none of the existing asymptotic methods achieve equal coverage on both sides of the interval, even with large sample sizes, and consequently if used as a non‐inferiority test, the type I error rate (which is equal to the one‐sided non‐coverage probability) can be inflated. The only exception is the Gart‐Nam ‘skewness‐corrected’ method, which we express using modified notation in order to include a bias correction for improved small‐sample performance, and an optional continuity correction for those seeking more conservative coverage. Using a weighted average of two complementary methods, we also define a new hybrid method that almost matches the performance of the Gart‐Nam interval. Copyright © 2014 John Wiley & Sons, Ltd.     

Conservative confidence intervals for a single parameter

The method of constructing confidence intervals from hypothesis tests is studied in the case in which there is a single unknown parameter and is proved to provide confidence intervals with coverage probability that is at least the nominal level. The confidence intervals obtained by the method in several different contexts are seen to compare favorably with confidence intervals obtained by traditional methods. The traditional intervals are seen to have coverage probability less than the nominal level in several instances, This method can be applied to all confidence interval problems and reduces to the traditional method when an exact pivotal statistic is known.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

Saddlepoint <Emphasis Type="Italic">p</Emphasis>-values and confidence intervals for a class of two sample permutation tests for current status and panel count data

The current status and panel count data frequently arise from cancer and tumorigenicity studies when events currently occur. A common and widely used class of two sample tests, for current status and panel count data, is the permutation class. We manipulate the double saddlepoint method to calculate the exact mid-p-values of the underlying permutation distributions of this class of tests. Permutation simulations are replaced by analytical saddlepoint computations which provide extremely accurate mid-p-values that are exact for most practical purposes and almost always more accurate than normal approximations. The method is illustrated using two real tumorigenicity panel count data. To compare the saddlepoint approximation with the normal asymptotic approximation, a simulation study is conducted. The speed and accuracy of the saddlepoint method facilitate the calculation of the confidence interval for the treatment effect. The inversion of the mid-p-values to calculate the confidence interval for the mean rate of development of the recurrent event is discussed.