Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.     

Confidence and Likelihood*

Confidence intervals for a single parameter are spanned by quantiles of a confidence distribution, and one‐sided p‐values are cumulative confidences. Confidence distributions are thus a unifying format for representing frequentist inference for a single parameter. The confidence distribution, which depends on data, is exact (unbiased) when its cumulative distribution function evaluated at the true parameter is uniformly distributed over the unit interval. A new version of the Neyman–Pearson lemma is given, showing that the confidence distribution based on the natural statistic in exponential models with continuous data is less dispersed than all other confidence distributions, regardless of how dispersion is measured. Approximations are necessary for discrete data, and also in many models with nuisance parameters. Approximate pivots might then be useful. A pivot based on a scalar statistic determines a likelihood in the parameter of interest along with a confidence distribution. This proper likelihood is reduced of all nuisance parameters, and is appropriate for meta‐analysis and updating of information. The reduced likelihood is generally different from the confidence density. Confidence distributions and reduced likelihoods are rooted in Fisher–Neyman statistics. This frequentist methodology has many of the Bayesian attractions, and the two approaches are briefly compared. Concepts, methods and techniques of this brand of Fisher–Neyman statistics are presented. Asymptotics and bootstrapping are used to find pivots and their distributions, and hence reduced likelihoods and confidence distributions. A simple form of inverting bootstrap distributions to approximate pivots of the abc type is proposed. Our material is illustrated in a number of examples and in an application to multiple capture data for bowhead whales.     

Confidence Intervals for Variance Components in Measurement System Capability Studies

In Measurement System Analysis a relevant issue is how to find confidence intervals for the parameters used to evaluate the capability of a gauge. In literature approximate solutions are available but they produce so wide intervals that they are often not effective in the decision process. In this article we introduce a new approach and, with particular reference to the parameter γ_R, i.e., the ratio of the variance due to the process and the variance due to the instrument, we show that, under quite realistic assumptions, we obtain confidence intervals narrower than other methods. An application to a real microelectronic case study is reported.     

Mean-Minimum Exact Confidence Intervals

This article introduces mean-minimum (MM) exact confidence intervals for a binomial probability. These intervals guarantee that both the mean and the minimum frequentist coverage never drop below specified values. For example, an MM 95[93]% interval has mean coverage at least 95% and minimum coverage at least 93%. In the conventional sense, such an interval can be viewed as an exact 93% interval that has mean coverage at least 95% or it can be viewed as an approximate 95% interval that has minimum coverage at least 93%. Graphical and numerical summaries of coverage and expected length suggest that the Blaker-based MM exact interval is an attractive alternative to, even an improvement over, commonly recommended approximate and exact intervals, including the Agresti–Coull approximate interval, the Clopper–Pearson (CP) exact interval, and the more recently recommended CP-, Blaker-, and Sterne-based mean-coverage-adjusted approximate intervals.     

Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance

A confidence interval for the between group variance is proposed which is deduced from Wald'sexact confidence interval for the rtio of the two variance components in the one-way random effects model and the exact confidence interval for the error variance resp.an unbiased estimator of the error variance. In a simulation study the confidence coeffecients for these two intervals are compared with the confidence coefficients of two other commonly used confidence intervals. There the confidence interval derived here yields confidence coefficiends which are always greater than the prescriped level.     

Exact Confidence Intervals for Proportions Estimated by Group Testing with Different Group Sizes

Group testing is the process of combining individual samples and testing them as a group for the presence of an attribute. The use of such testing to estimate proportions is an important statistical tool in many applications. When samples are collected and tested in groups of different size, complications arise in the construction of exact confidence intervals. In this case, the numbers of positive groups has a multivariate distribution, and the difficulty stems from a lack of a natural ordering of the sample points. Exact two‐sided intervals such as the equal‐tail method based on maximum likelihood estimation, and those based on joint probability or likelihood ratio statistics, have been previously considered. In this paper several new estimators are developed and assessed. We show that the combined tails (or Blaker) method based on a suitable ordering statistic, is the best choice in this setting. The methods are illustrated using a study involving the infection prevalence of Myxobolus cerebralis among free‐ranging fish.     

Approximate Confidence Limits for a Proportion of the Pólya Distribution

The problem of constructing approximate confidence limits for a proportion parameter of the Pólya distribution is discussed. Three different methods for determining approximate one-sided and two-sided confidence limits for that parameter of the Pólya distribution have been proposed and compared. Particular cases of those confidence bounds are confidence intervals for the parameter of the binomial and the hypergeometric distributions.     

Confidence intervals for the regression coefficient in a simple regression model with a balanced two-fold nested error structure

ABSTRACT

In applications using a simple regression model with a balanced two-fold nested error structure, interest focuses on inferences concerning the regression coefficient. This article derives exact and approximate confidence intervals on the regression coefficient in the simple regression model with a balanced two-fold nested error structure. Eleven methods are considered for constructing the confidence intervals on the regression coefficient. Computer simulation is performed to compare the proposed confidence intervals. Recommendations are suggested for selecting an appropriate method.     

Percentile-t Bootstrap Confidence Intervals for Period Using Periodogram

The magnitude of light intensity of many stars varies over time in a periodic way. Therefore, estimation of period and making inference about this parameter are of great interest in astronomy. The periodogram can be used to estimate period, properly. Bootstrap confidence intervals for period suggested here, are based on using the periodogram and constructed by percentile-t methods. We prove that the equal-tailed percentile-t bootstrap confidence intervals for period have an error of order n ^?1. We also show that the symmetric percentile-t bootstrap confidence intervals reduce the error to order n ^?2, and hence have a better performance. Finally, we assess the theoretical results by conducting a simulation study, compare the results with the coverages of percentile bootstrap confidence intervals for period and then analyze a real data set related to the eclipsing system R Canis Majoris collected by Shiraz Biruni Observatory.     

Performance of Confidence Intervals in Regression Models with Unbalanced One-Fold Nested Error Structures

Abstract

In this article we consider the problem of constructing confidence intervals for a linear regression model with unbalanced nested error structure. A popular approach is the likelihood-based method employed by PROC MIXED of SAS. In this article, we examine the ability of MIXED to produce confidence intervals that maintain the stated confidence coefficient. Our results suggest that intervals for the regression coefficients work well, but intervals for the variance component associated with the primary level cannot be recommended. Accordingly, we propose alternative methods for constructing confidence intervals on the primary level variance component. Computer simulation is used to compare the proposed methods. A numerical example and SAS code are provided to demonstrate the methods.     

Model‐Averaged Confidence Intervals

We develop an approach to evaluating frequentist model averaging procedures by considering them in a simple situation in which there are two‐nested linear regression models over which we average. We introduce a general class of model averaged confidence intervals, obtain exact expressions for the coverage and the scaled expected length of the intervals, and use these to compute these quantities for the model averaged profile likelihood (MPI) and model‐averaged tail area confidence intervals proposed by D. Fletcher and D. Turek. We show that the MPI confidence intervals can perform more poorly than the standard confidence interval used after model selection but ignoring the model selection process. The model‐averaged tail area confidence intervals perform better than the MPI and postmodel‐selection confidence intervals but, for the examples that we consider, offer little over simply using the standard confidence interval for θ under the full model, with the same nominal coverage.     

Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models

We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n ? k tends to infinity fast enough in relation to the thresholding parameter.     

Construction of Simultaneous Confidence Intervals for Ratios of Means of Lognormal Distributions

For constructing simultaneous confidence intervals for ratios of means for lognormal distributions, two approaches using a two-step method of variance estimates recovery are proposed. The first approach proposes fiducial generalized confidence intervals (FGCIs) in the first step followed by the method of variance estimates recovery (MOVER) in the second step (FGCIs–MOVER). The second approach uses MOVER in the first and second steps (MOVER–MOVER). Performance of proposed approaches is compared with simultaneous fiducial generalized confidence intervals (SFGCIs). Monte Carlo simulation is used to evaluate the performance of these approaches in terms of coverage probability, average interval width, and time consumption.     

Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data

Let ( X , Y ) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a non-parametric likelihood ratio approach. This approach was introduced by Thomas & Grunkemeier (1975 ), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.     

Confidence intervals for variance components in unbalanced one-way random effects model using non-normal distributions

In scenarios where the variance of a response variable can be attributed to two sources of variation, a confidence interval for a ratio of variance components gives information about the relative importance of the two sources. For example, if measurements taken from different laboratories are nine times more variable than the measurements taken from within the laboratories, then 90% of the variance in the responses is due to the variability amongst the laboratories and 10% of the variance in the responses is due to the variability within the laboratories. Assuming normally distributed sources of variation, confidence intervals for variance components are readily available. In this paper, however, simulation studies are conducted to evaluate the performance of confidence intervals under non-normal distribution assumptions. Confidence intervals based on the pivotal quantity method, fiducial inference, and the large-sample properties of the restricted maximum likelihood (REML) estimator are considered. Simulation results and an empirical example suggest that the REML-based confidence interval is favored over the other two procedures in unbalanced one-way random effects model.     

Confidence intervals for variance component ratios in unbalanced linear mixed models

Methods for constructing confidence intervals for variance component ratios in general unbalanced mixed models are developed. The methods are based on inverting the distribution of the signed root of the log-likelihood ratio statistic constructed from either the restricted maximum likelihood or the full likelihood. As this distribution is intractable, the inversion is rather based on using a saddlepoint approximation to its distribution. Apart from Wald's exact method, the resulting intervals are unrivalled in terms of achieving accuracy in overall coverage, underage, and overage. Issues related to the proper “reference set” with which to judge the coverage as well as issues connected to variance ratios being nonnegative with lower bound 0 are addressed. Applications include an unbalanced nested design and an unbalanced crossed design.     

Confidence bounds and selection of the t best populations

Confidence statements about location (or scale) parameters associated with K populations, which may be used in making selection decisions about those populations, are investigated. When a subset of fixed size t is selected from the K populations a lower bound is obtained for the minimum selected parameter as a function of the maximum non-selected parameter. Tables are produced for the normal means case when the variance is common but unknown. It is pointed out that these tables may be used to find confidence intervals discussed by Hsu (1984     

Simulation-Based Confidence Intervals for Functions With Complicated Derivatives

In many scientific problems, the quantity of interest is a function of parameters that index the model, and confidence intervals are constructed by applying the delta method. However, when the function of interest has complicated derivatives, this standard approach is unattractive and alternative algorithms are required. This article discusses a simple simulation-based algorithm for estimating the variance of a transformation, and demonstrates its simplicity and accuracy by applying it to several statistical problems.     

Adjusted Confidence Bands in Nonparametric Regression

Suppose we have {(x _i , y _i )} i = 1, 2,…, n, a sequence of independent observations. We wish to find approximate 1 ? α simultaneous confidence bands for the regression curve. Many previous confidence bands in the literature have practical difficulties. In this article, the local linear smoother is used to estimate the regression curve. The bias of the estimator is considered. Different methods of constructing confidence bands are discussed. Finally, a possible method incorporating logistic regression in an innovative way is proposed to construct the bands for random designs. Simulations are used to study the performance or properties of the methods. The procedure for constructing confidence bands is entirely data-driven. The advantage of the proposed method is that it is simple to use and can be applied to random designs. It can be considered as a practically useful and efficient method.     

基于贝叶斯的Bootstrap置信区间

在许多领域中,Bootstrap成为一种数据处理的有效方法。很多情况下,模型中感兴趣的参数的置信区间难以构建,为了解决这一问题,文章提出了一个新的贝叶斯Bootstrap置信区间的估计量,并做了蒙特卡洛模拟比较,结果比经典区间估计方法和经典Bootstrap方法更优,并进行了实例分析。