Confidence Interval Procedures Research Articles

On Two-Stage Confidence Interval Procedures and Their Comparisons for Estimating the Difference of Normal Means

We consider two independent N(μ1, σ2) and N(μ2, a 2σ2) populations where μ1, μ2, σ2 are unknown parameters with −∞ < μ1, μ2 < ∞, 0 < σ < ∞. We assume that a (>0) is known! The problem is one of estimating Δ = μ1 − μ2 by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient at least 1 − α. Here, d (>0) and 0 < α < 1 are both preassigned numbers. First, a two-stage procedure 𝒫1 is designed in the spirit of Stein (1945, Annals of Mathematical Statistics 16: 243–258). Then, another two-stage procedure 𝒫2 is tried in the spirit of Chapman (1950, Annals of Mathematical Statistics 21: 601–606). We report that the Stein-type two-stage procedure 𝒫1 performs better than 𝒫2. Next, a variant of a two-stage procedure due to Aoshima et al. (1996, Sequential Analysis 15: 61–70) is included as our third procedure, 𝒫3. In a variety of situations, we find that 𝒫1 also comes out ahead of 𝒫3. A new alternative sampling design 𝒫4 is then introduced by incorporating a two-stage sampling technique from one population alone followed by a single-stage sampling strategy from the other population. We observe that 𝒫4 compares favorably with 𝒫1 with regard to the achieved confidence level whereas the margin of oversampling in the case of 𝒫4 is justifiably smaller than that associated with 𝒫1. Additionally, 𝒫4 has a significant operational edge over 𝒫1 and hence we suggest implementing 𝒫4 in practice. In the end, we illustrate superiority of the new procedure 𝒫4 with an example using a real data set from horticulture (Mukhopadhyay et al., 2004, Journal of Agricultural, Biological and Environmental Statistics 38: 1384–1391).

Sequential Fixed Width Confidence Intervals for the Offset Between Two Network Clocks

Estimation of the offset between two network clocks has received a lot of attention in the literature, with the motivating force being data networking applications that require synchronous communication protocols. Statistical modeling techniques have been used to develop improved estimation algorithms, with a recent development being the construction of a confidence interval based on a fixed sample size. Lacking in the fixed sample size confidence interval procedures is a useable relationship between sample size and the width of the resulting confidence interval. Were that available, an optimum sample size could be determined to achieve a specified level of precision in the estimator and thereby improve the efficiency of the estimation procedure by reducing unnecessary overhead in the network that is associated with collecting the data used by the estimation schemes. A fixed sample size confidence interval that has a prescribed width is not available for this problem. However, in this paper we develop and compare alternative sequential intervals with fixed width and demonstrate that an effective solution is available.

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.

Approximate Confidence Intervals for p When Double Sampling

This article reviews literature on confidence intervals for p, assuming (Bernoulli) single, double, and multistage sampling. Methods used to produce approximate single-sampling confidence intervals extend to double sampling as well. A new Wilson approximate confidence interval procedure for double sampling is compared with existing procedures in terms of general interval properties, such as interval invariance, and performance (in terms of actual coverage probability and expected length) for double-sampling plans from the literature.

An exact multinomial test for equivalence

AbstractExisting equivalence tests for multinomial data are valid asymptotically, but the $\alpha$ level is not properly controlled for small and moderate sample sizes. We resolve this difficulty by developing an exact multinomial test for equivalence and an associated confidence interval procedure. We also derive a conservative version of the test that is easy to implement even for very large sample sizes. Both tests use a notion of equivalence that is based on the cumulative distribution function, with two probability vectors being considered equivalent if their partial sums never differ by more than some specified constant. We illustrate the methods by applying them to Weldon's dice data, to data on the digits of $\pi$, and to data collected by Mendel. The Canadian Journal of Statistics 37: 47–59; © 2009 Statistical Society of Canada

Data Depth Trimming Counterpart of the Classicalt(orT2) Procedure

The classicalt(orT2in high dimensions) inference procedure for unknown meanμ:X¯±tα(n−1)Sn/n (or {μ:n(x¯−μ)′S−1(x¯−μ)≤χ(1−α)2(p)}) is so fundamental in statistics and so prevailing in practices; it is regarded as an optimal procedure in the mind of many practitioners. It this manuscript we present a new procedure based on data depth trimming and bootstrapping that can outperform the classicalt(orT2in high dimensions) confidence interval (or region) procedure.

Simple estimators of the intensity of seasonal occurrence

BackgroundEdwards's method is a widely used approach for fitting a sine curve to a time-series of monthly frequencies. From this fitted curve, estimates of the seasonal intensity of occurrence (i.e., peak-to-low ratio of the fitted curve) can be generated.MethodsWe discuss various approaches to the estimation of seasonal intensity assuming Edwards's periodic model, including maximum likelihood estimation (MLE), least squares, weighted least squares, and a new closed-form estimator based on a second-order moment statistic and non-transformed data. Through an extensive Monte Carlo simulation study, we compare the finite sample performance characteristics of the estimators discussed in this paper. Finally, all estimators and confidence interval procedures discussed are compared in a re-analysis of data on the seasonality of monocytic leukemia.ResultsWe find that Edwards's estimator is substantially biased, particularly for small numbers of events and very large or small amounts of seasonality. For the common setting of rare events and moderate seasonality, the new estimator proposed in this paper yields less finite sample bias and better mean squared error than either the MLE or weighted least squares. For large studies and strong seasonality, MLE or weighted least squares appears to be the optimal analytic method among those considered.ConclusionEdwards's estimator of the seasonal relative risk can exhibit substantial finite sample bias. The alternative estimators considered in this paper should be preferred.

Fiducial Intervals for Variance Components in an Unbalanced Two-Component Normal Mixed Linear Model

In this article we propose a new method for constructing confidence intervals for σα2,σϵ2, and the intraclass correlation ρ==σα2(σα2++σε2) in a two-component mixed-effects linear model. This method is based on an extension of R. A. Fisher's fiducial argument. We conducted a simulation study to compare the resulting interval estimates with other competing confidence interval procedures from the literature. Our results demonstrate that the proposed fiducial intervals have satisfactory performance in terms of coverage probability, as well as shorter average confidence interval lengths overall. We also prove that these fiducial intervals have asymptotically exact frequentist coverage probability. The computations for the proposed procedures are illustrated using real data examples.

On the Estimation of Additive Interaction by Use of the Four-by-two Table and Beyond

A four-by-two table with its four rows representing the presence and absence of gene and environmental factors has been suggested as the fundamental unit in the assessment of gene-environment interaction. For such a table to be more meaningful from a public health perspective, it is important to estimate additive interaction. A confidence interval procedure proposed by Hosmer and Lemeshow has become widespread. This article first reveals that the Hosmer-Lemeshow procedure makes an assumption that confidence intervals for risk ratios are symmetric and then presents an alternative that uses the conventional asymmetric intervals for risk ratios to set confidence limits for measures of additive interaction. For the four-by-two table, the calculation involved requires no statistical programs but only elementary calculations. Simulation results demonstrate that this new approach can perform almost as well as the bootstrap. Corresponding calculations in more complicated situations can be simplified by use of routine output from multiple regression programs. The approach is illustrated with three examples. A Microsoft Excel spreadsheet and SAS codes for the calculations are available from the author and the Journal's website, respectively.

Inference on the Minimum Effective Dose Using Binary Data

This article proposes a confidence interval procedure for an open question posted by Tamhane and Dunnett regarding the inference on the minimum effective dose of a drug for binary data. We use a partitioning approach in conjunction with a confidence interval procedure to provide a solution to this problem. Binary data frequently arise in medical investigations in connection with dichotomous outcomes such as the development of a disease or the efficacy of a drug. The proposed procedure not only detects the minimum effective dose of the drug, but also provides estimation information on the treatment effect of the closest ineffective dose. Such information benefits follow-up investigations in clinical trials. We prove that, when the confidence interval for the pairwise comparison has (or asymptotically controls) confidence level 1 − α, the stepwise procedure strongly controls (or asymptotically controls) the familywise error rate at level α, which is a key criterion in dose finding. The new method is compared with other procedures in terms of power performance and coverage probability using simulations. The simulated results shed new light on the discernible features of the new procedure. An example on the investigation of acetaminophen is included.

Avoiding Problems With Normal Approximation Confidence Intervals for Probabilities

Although it is well known that modern methods of computing confidence intervals (CIs) based on likelihood or simulation have important advantages, normal approximation confidence interval procedures (NACPs) are still widely used, especially in the analysis of censored data. This is because CIs from NACPs are easy to compute and easy to explain. But when the sample size is not large or when there is heavy censoring, the performance of NACPs can be poor. A transformation can be applied to keep CI endpoints from falling outside the parameter space and improve performance, but the degree of improvement (if any) depends on the chosen function. To obtain CIs for distribution probabilities, some seemingly useful transformation functions will cause the estimated variance to blow up in the tails of the distribution and can lead to nonsensical confidence intervals. This article compares different NACPs for distribution probabilities. Our results suggest that an NACP based on a studentized statistic, which we call the z procedure, has desirable properties compared with alternative NACPs.

Modelling patterns of agreement for nominal scales

The measurement of agreement of repeat rating is the usual method of assessing the reliability of categorical scales. Measurement of agreement is also important in genetic twin studies based on categorical scales. One of the most commonly used methods of analysis for both types of study is the kappa coefficient. For scales with more than two categories, one approach is to use a single summary kappa coefficient. While this may be sufficient for many studies, in some instances investigation of heterogeneity in the pattern of agreement may give additional insights as there may be greater agreement for some pairs of categories than for others. In this paper, kappa-type coefficients are used to model heterogeneity in the pattern of agreement. Constraints are added to the heterogeneous model to obtain simplified models. Procedures for estimation, confidence intervals, and inference for these coefficients are described for the case of two ratings per subject for a single sample and for the comparison of two independent samples. Formulae for sample size and power calculation are derived using the non-central chi-squared distribution. Two simulation studies are carried out to check the empirical test size and power. Methods are illustrated by two examples involving nominal scales with three categories.

Under‐Smoothed Kernel Confidence Intervals for the Hazard Ratio Based on Censored Data

In cancer clinical trials, it is often of interest in estimating the ratios of hazard rates at some specific time points during the study from two independent populations. In this paper, we consider nonparametric confidence interval procedures for the hazard ratio based on kernel estimates for the hazard rates with under-smoothing bandwidths. Two methods are used to derive the confidence intervals: one based on the asymptotic normality of the ratio of the kernel estimates for the hazard rates in two populations and another through Fieller's Theorem. The performances of the proposed confidence intervals are evaluated through Monte-Carlo simulations and applied to the analysis of data from a clinical trial on early breast cancer.

Empirical likelihood confidence intervals for the mean of a long‐range dependent process

Abstract. This paper considers blockwise empirical likelihood for real‐valued linear time processes which may exhibit either short‐ or long‐range dependence. Empirical likelihood approaches intended for weakly dependent time series can fail in the presence of strong dependence. However, a modified blockwise method is proposed for confidence interval estimation of the process mean, which is valid for various dependence structures including long‐range dependence. The finite‐sample performance of the method is evaluated through a simulation study and compared with other confidence interval procedures involving subsampling or normal approximations.

Generalized confidence intervals for process capability indices in the one-way random model

The method of generalized confidence intervals is proposed as an alternative method for constructing confidence intervals for process capability indices under the one-way random model for balanced as well as unbalanced data. The generalized lower confidence limits and the coverage probabilities for three commonly used capability indices were studied via simulation, separately for balanced and unbalanced cases. Simulation results showed that the generalized confidence interval procedure is quite satisfactory both in the balanced and unbalanced cases. Examples are provided to illustrate the results.

Confidence Intervals for Standardized Effect Sizes: Theory, Application, and Implementation

The behavioral, educational, and social sciences are undergoing a paradigmatic shift in methodology, from disciplines that focus on the dichotomous outcome of null hypothesis significance tests to disciplines that report and interpret effect sizes and their corresponding confidence intervals. Due to the arbitrariness of many measurement instruments used in the behavioral, educational, and social sciences, some of the most widely reported effect sizes are standardized. Although forming confidence intervals for standardized effect sizes can be very beneficial, such confidence interval procedures are generally difficult to implement because they depend on noncentral t, F, and x² distributions. At present, no main-stream statistical package provides exact confidence intervals for standardized effects without the use of specialized programming scripts. Methods for the Behavioral, Educational, and Social Sciences (MBESS) is an R package that has routines for calculating confidence intervals for noncentral t, F, and x² distributions, which are then used in the calculation of exact confidence intervals for standardized effect sizes by using the confidence interval transformation and inversion principles. The present article discusses the way in which confidence intervals are formed for standardized effect sizes and illustrates how such confidence intervals can be easily formed using MBESS in R.

Exact Inference for a Population Proportion Based on a Ranked Set Sample

ABSTRACT This article develops and investigates a confidence interval and hypothesis testing procedure for a population proportion based on a ranked set sample (RSS). The inference is exact, in the sense that it is based on the exact distribution of the total number of successes observed in the RSS. Furthermore, this distribution can be readily computed with the well-known and freely available R statistical software package. A data example that illustrates the methodology is presented. In addition, the properties of the inference procedures are compared with their simple random sample (SRS) counterparts. In regards to expected lengths of confidence intervals and the power of tests, the RSS inference procedures are superior to the SRS methods.

Application of Sequential Interval Estimation to Adaptive Mastery Testing

In this paper, we apply sequential one-sided confidence interval estimation procedures with β-protection to adaptive mastery testing. The procedures of fixed-width and fixed proportional accuracy confidence interval estimation can be viewed as extensions of one-sided confidence interval procedures. It can be shown that the adaptive mastery testing procedure based on a one-sided confidence interval with β-protection is more efficient in terms of test length than a testing procedure based on a two-sided/fixed-width confidence interval. Some simulation studies applying the one-sided confidence interval procedure and its extensions mentioned above to adaptive mastery testing are conducted. For the purpose of comparison, we also have a numerical study of adaptive mastery testing based on Wald's sequential probability ratio test. The comparison of their performances is based on the correct classification probability, averages of test length, as well as the width of the “indifference regions.” From these empirical results, we found that applying the one-sided confidence interval procedure to adaptive mastery testing is very promising.

Second-Order Accurate Inference on Simple, Partial, and Multiple Correlations

This article develops confidence interval procedures for functions of simple, partial, and squared multiple correlation coefficients. It is assumed that the observed multivariate data represent a random sample from a distribution that possesses infinite moments, but there is no requirement that the distribution be normal. The coverage error of conventional one-sided large sample intervals decreases at rate 1√n as n increases, where n is an index of sample size. The coverage error of the proposed intervals decreases at rate 1/n as n increases. The results of a simulation study that evaluates the performance of the proposed intervals is reported and the intervals are illustrated on a real data set.

Quantifying responsiveness of quality of life measures without an external criterion

The responsiveness of a quality of life measure has received considerable attention in the literature. A two time-point (pre-/post-) study design is usually adopted to evaluate this property when a gold standard is not available. Among many indices, Cohen's effect size and the standardized response mean (SRM) are usually computed. To interpret the results, researchers commonly appeal to an arbitrary criterion for both indices even though they are different by definition. In this paper, we demonstrate their close algebraic relationship and conceptual differences, showing that only the SRM is necessary to quantify responsiveness. To facilitate interpretation, we transform the SRM to the 'probability of change' with a value of 0.5 denoting null responsiveness and 1.0 perfect responsiveness. Simple confidence interval procedures are provided and evaluated. We also discuss the possibility of applying the results to the analysis of data from a two independent groups pre-/post- design. Two examples are provided.