Exact confidence intervals following a group sequential test

A numerical method is used to compute confidence intervals, which have exact coverage probabilities, for the mean of a normal distribution following a group sequential test. This method, which uses an ordering of the sample space similar to that employed by Siegmund (1978, Biometrika 65, 341-349), is contrasted with the usual confidence interval for the mean.     

Validation of a Demographic Model for Woodland Caribou

Abstract: Wildlife population models are potentially valuable for conservation planning. Validation is necessary to ensure that models are sufficiently robust for predicting management outcomes consistent with conservation objectives. Sorensen et al. (2008) produced a model of woodland caribou (Rangifer tarandus) population growth rate that was recently modified and used as a predictive tool at several scales. We computed confidence intervals and evaluated the performance of this model using novel data. Confidence intervals were wide, and results suggested that the model may have a positive bias, resulting in over-estimation of population growth rates, as well as low predictive power. Wide confidence intervals mean that current understanding of factors governing woodland caribou herd dynamics is not sufficient for wildlife managers to make reliable projections of responses to management.     

Confidence intervals for the mean of diagnostic test charge data containing zeros

In this paper, we consider the problem of interval estimation for the mean of diagnostic test charges. Diagnostic test charge data may contain zero values, and the nonzero values can often be modeled by a log-normal distribution. Under such a model, we propose three different interval estimation procedures: a percentile-t bootstrap interval based on sufficient statistics and two likelihood-based confidence intervals. For theoretical properties, we show that the two likelihood-based one-sided confidence intervals are only first-order accurate and that the bootstrap-based one-sided confidence interval is second-order accurate. For two-sided confidence intervals, all three proposed methods are second-order accurate. A simulation study in finite-sample sizes suggests all three proposed intervals outperform a widely used minimum variance unbiased estimator (MVUE)-based interval except for the case of one-sided lower end-point intervals when the skewness is very small. Among the proposed one-sided intervals, the bootstrap interval has the best coverage accuracy. For the two-sided intervals, when the sample size is small, the bootstrap method still yields the best coverage accuracy unless the skewness is very small, in which case the bias-corrected ML method has the best accuracy. When the sample size is large, all three proposed intervals have similar coverage accuracy. Finally, we analyze with the proposed methods one real example assessing diagnostic test charges among older adults with depression.     

Test‐compatible confidence intervals for adaptive two‐stage single‐arm designs with binary endpoint

Inference after two‐stage single‐arm designs with binary endpoint is challenging due to the nonunique ordering of the sampling space in multistage designs. We illustrate the problem of specifying test‐compatible confidence intervals for designs with nonconstant second‐stage sample size and present two approaches that guarantee confidence intervals consistent with the test decision. Firstly, we extend the well‐known Clopper–Pearson approach of inverting a family of two‐sided hypothesis tests from the group‐sequential case to designs with fully adaptive sample size. Test compatibility is achieved by using a sample space ordering that is derived from a test‐compatible estimator. The resulting confidence intervals tend to be conservative but assure the nominal coverage probability. In order to assess the possibility of further improving these confidence intervals, we pursue a direct optimization approach minimizing the mean width of the confidence intervals. While the latter approach produces more stable coverage probabilities, it is also slightly anti‐conservative and yields only negligible improvements in mean width. We conclude that the Clopper–Pearson‐type confidence intervals based on a test‐compatible estimator are the best choice if the nominal coverage probability is not to be undershot and compatibility of test decision and confidence interval is to be preserved.     

Confidence intervals for the similarity between algal communities

While researchers commonly use similarity measures to compare algal communities, very few researchers have considered the variability of these estimated measures. This paper discusses a recent method for estimating the variance of and confidence intervals for similarity measures proposed by Johnson & Millie (1982, Hydrobiologia 89: 3–8). Applications of this method to data have produced confidence intervals that are too narrow. Two alternative methods, the jackknife method and the bootstrap, are shown to provide superior estimates of the variability.     

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.     

Confidence Intervals for fMRI Activation Maps

Neuroimaging activation maps typically color voxels to indicate whether the blood oxygen level-dependent (BOLD) signals measured among two or more experimental conditions differ significantly at that location. This data presentation, however, omits information critical for interpretation of experimental results. First, no information is represented about trends at voxels that do not pass the statistical test. Second, no information is given about the range of probable effect sizes at voxels that do pass the statistical test. This leads to a fundamental error in interpreting activation maps by naïve viewers, where it is assumed that colored, “active” voxels are reliably different from uncolored “inactive” voxels. In other domains, confidence intervals have been added to data graphics to reduce such errors. Here, we first document the prevalence of the fundamental error of interpretation, and then present a method for solving it by depicting confidence intervals in fMRI activation maps. Presenting images where the bounds of confidence intervals at each voxel are coded as color allows readers to visually test for differences between “active” and “inactive” voxels, and permits for more proper interpretation of neuroimaging data. Our specific graphical methods are intended as initial proposals to spur broader discussion of how to present confidence intervals for fMRI data.     

Test-based exact confidence intervals for the difference of two binomial proportions

Confidence intervals are often provided to estimate a treatment difference. When the sample size is small, as is typical in early phases of clinical trials, confidence intervals based on large sample approximations may not be reliable. In this report, we propose test-based methods of constructing exact confidence intervals for the difference in two binomial proportions. These exact confidence intervals are obtained from the unconditional distribution of two binomial responses, and they guarantee the level of coverage. We compare the performance of these confidence intervals to ones based on the observed difference alone. We show that a large improvement can be achieved by using the standardized Z test with a constrained maximum likelihood estimate of the variance.     

Comparison of family-based association tests in chromosome regions selected by linkage-based confidence intervals

We use the Genetic Analysis Workshop 14 simulated data to explore the effectiveness of a two-stage strategy for mapping complex disease loci consisting of an initial genome scan with confidence interval construction for gene location, followed by fine mapping with family-based tests of association on a dense set of single-nucleotide polymorphisms. We considered four types of intervals: the 1-LOD interval, a basic percentile bootstrap confidence interval based on the position of the maximum Zlr score, and asymptotic and bootstrap confidence intervals based on a generalized estimating equations method. For fine mapping we considered two family-based tests of association: a test based on a likelihood ratio statistic and a transmission-disequilibrium-type test implemented in the software FBAT. In two of the simulation replicates, we found that the bootstrap confidence intervals based on the peak Zlr and the 1-LOD support interval always contained the true disease loci and that the likelihood ratio test provided further strong confirmatory evidence of the presence of disease loci in these regions.     

The estimation and applicability of confidence intervals for Stander's Similarity Index (SIMI) in algal assemblage comparisons

Stander's Similarity Index (SIMI) has become a popular measure for comparing algal assemblages. The interpretation of the value produced by this index, however, has been highly variable between studies. Using replicate sampling of natural algal assemblages and computer simulation techniques, a method by which confidence intervals could be applied to SIMI was developed. This paper presents that method and gives an example using hypothetical species counts of two algal assemblages. Since a more realistic range of the actual similarity between two assemblages is produced, the estimation of confidence intervals when using SIMI is recommended.     

Confidence Intervals in Qtl Mapping by Bootstrapping

The determination of empirical confidence intervals for the location of quantitative trait loci (QTLs) was investigated using simulation. Empirical confidence intervals were calculated using a bootstrap resampling method for a backcross population derived from inbred lines. Sample sizes were either 200 or 500 individuals, and the QTL explained 1, 5, or 10% of the phenotypic variance. The method worked well in that the proportion of empirical confidence intervals that contained the simulated QTL was close to expectation. In general, the confidence intervals were slightly conservatively biased. Correlations between the test statistic and the width of the confidence interval were strongly negative, so that the stronger the evidence for a QTL segregating, the smaller the empirical confidence interval for its location. The size of the average confidence interval depended heavily on the population size and the effect of the QTL. Marker spacing had only a small effect on the average empirical confidence interval. The LOD drop-off method to calculate empirical support intervals gave confidence intervals that generally were too small, in particular if confidence intervals were calculated only for samples above a certain significance threshold. The bootstrap method is easy to implement and is useful in the analysis of experimental data.     

Comparison of populations as a function of confidence intervals of gene probability

A new method for population comparison has been developed in order to determine the most discriminating systems in relation to diverse populations. The method compares pairs of populations on the basis of confidence intervals for allelic frequencies and does not require any previous assumption, such as normality of the population. A data set of 12 populations studied for Polymarker Kit (Roche) genetic systems, 13 biallelic systems (LDLR, GYPA, D7S8) and two systems with three alleles each (HBGG, GC), was considered. The analysis produced a total of 144 confidence intervals (12 populations x 12 alleles). The raw data were then analyzed comparing pairs of populations, allele by allele, through their respective confidence intervals. The method calculates the matrix of similarity between pairs of populations for each allele and eventually for each genetic system. Finally, the 12 populations were classified depending on the number of nondiscriminating alleles. The results of this method are compared to those produced by such usual techniques as Principal Component Analysis.     

Productivity Estimates From Upland Bird Harvests: Estimating Variance and Necessary Sample Sizes

Abstract: Harvest of upland game birds in concert with sampling of age ratios from wings can yield important biological information about populations. Although estimates of productivity are commonly produced, they are often not accompanied by a measure of variance. Thus, we developed standard error estimates for sample productivity ratios, compared 4 methods for creating confidence intervals for population productivity ratios, and developed a test and the corresponding sample size requirements for comparing 2 population productivity ratios. We applied these techniques to greater sage-grouse (Centrocercus urophasianus) wing-data collected in Oregon, USA (1993–2005). Computer simulations indicated that backtransforming the Wilson's score interval on the proportion of immatures in the sample results in the most reliable confidence intervals among the methods considered. We recommend to managers measuring conservation action outcomes with productivity ratios to consider the appropriate sample sizes for the spatial and temporal scale of their monitoring programs.     

A remark on a paper by misra

MISRA (1978) sets confidence intervals for a double linear compound of multivariate normal regression coefficients by using ROY'S maximum root test criterion. The exact test statistic to be used is STUDENT'S t. The t statistic gives narrower confidence bounds than those given by ROY's maximum root statistic. A result given by MORRISON (1975, p. 18, equation 10) for profile analysis is also obtained by using the STUDENT'S t test.     

A Fast Method that Uses Polygenic Scores to Estimate the Variance Explained by Genome-wide Marker Panels and the Proportion of Variants Affecting a Trait

Several methods have been proposed to estimate the variance in disease liability explained by large sets of genetic markers. However, current methods do not scale up well to large sample sizes. Linear mixed models require solving high-dimensional matrix equations, and methods that use polygenic scores are very computationally intensive. Here we propose a fast analytic method that uses polygenic scores, based on the formula for the non-centrality parameter of the association test of the score. We estimate model parameters from the results of multiple polygenic score tests based on markers with p values in different intervals. We estimate parameters by maximum likelihood and use profile likelihood to compute confidence intervals. We compare various options for constructing polygenic scores, based on nested or disjoint intervals of p values, weighted or unweighted effect sizes, and different numbers of intervals, in estimating the variance explained by a set of markers, the proportion of markers with effects, and the genetic covariance between a pair of traits. Our method provides nearly unbiased estimates and confidence intervals with good coverage, although estimation of the variance is less reliable when jointly estimated with the covariance. We find that disjoint p value intervals perform better than nested intervals, but the weighting did not affect our results. A particular advantage of our method is that it can be applied to summary statistics from single markers, and so can be quickly applied to large consortium datasets. Our method, named AVENGEME (Additive Variance Explained and Number of Genetic Effects Method of Estimation), is implemented in R software.     

Adaptive design: estimation and inference with censored data in a semiparametric model

In this article, we provide a method of estimation for the treatment effect in the adaptive design for censored survival data with or without adjusting for risk factors other than the treatment indicator. Within the semiparametric Cox proportional hazards model, we propose a bias-adjusted parameter estimator for the treatment coefficient and its asymptotic confidence interval at the end of the trial. The method for obtaining an asymptotic confidence interval and point estimator is based on a general distribution property of the final test statistic from the weighted linear rank statistics at the interims with or without considering the nuisance covariates. The computation of the estimates is straightforward. Extensive simulation studies show that the asymptotic confidence intervals have reasonable nominal probability of coverage, and the proposed point estimators are nearly unbiased with practical sample sizes.     

Permutation‐based inference for the AUC: A unified approach for continuous and discontinuous data

We investigate rank‐based studentized permutation methods for the nonparametric Behrens–Fisher problem, that is, inference methods for the area under the ROC curve. We hereby prove that the studentized permutation distribution of the Brunner‐Munzel rank statistic is asymptotically standard normal, even under the alternative. Thus, incidentally providing the hitherto missing theoretical foundation for the Neubert and Brunner studentized permutation test. In particular, we do not only show its consistency, but also that confidence intervals for the underlying treatment effects can be computed by inverting this permutation test. In addition, we derive permutation‐based range‐preserving confidence intervals. Extensive simulation studies show that the permutation‐based confidence intervals appear to maintain the preassigned coverage probability quite accurately (even for rather small sample sizes). For a convenient application of the proposed methods, a freely available software package for the statistical software R has been developed. A real data example illustrates the application.     

Improved confidence intervals in quantitative trait loci mapping by permutation bootstrapping

The nonparametric bootstrap approach is known to be suitable for calculating central confidence intervals for the locations of quantitative trait loci (QTL). However, the distribution of the bootstrap QTL position estimates along the chromosome is peaked at the positions of the markers and is not tailed equally. This results in conservativeness and large width of the confidence intervals. In this study three modified methods are proposed to calculate nonparametric bootstrap confidence intervals for QTL locations, which compute noncentral confidence intervals (uncorrected method I), correct for the impact of the markers (weighted method I), or both (weighted method II). Noncentral confidence intervals were computed with an analog of the highest posterior density method. The correction for the markers is based on the distribution of QTL estimates along the chromosome when the QTL is not linked with any marker, and it can be obtained with a permutation approach. In a simulation study the three methods were compared with the original bootstrap method. The results showed that it is useful, first, to compute noncentral confidence intervals and, second, to correct the bootstrap distribution of the QTL estimates for the impact of the markers. The weighted method II, combining these two properties, produced the shortest and less biased confidence intervals in a large number of simulated configurations.     

Comparison of different methods for decision-making in bioequivalence assessment

If the regulatory requirements are symmetrical, the use of symmetrical confidence intervals as a decision rule for bioequivalence assessment leads, as shown by simulations, to better level properties and an inferior power compared to a rule based on shortest confidence intervals. A choice between these two approaches will have to depend on a loss function. For asymmetric regulatory requirements, symmetrical confidence intervals should not be used; however, a decision can still be based on posterior probabilities, pr (theta epsilon [r1, r2]/x), or shortest confidence intervals. For purposes of inference, presentation and interpretation of results, we think that the use of symmetrical confidence intervals alone can be misleading and we therefore recommend that the posterior probabilities and densities, or at least the shortest confidence intervals, be given.     

Confidence Intervals and Tests of Hypotheses on Variance Components in an Unbalanced Two-fold Nested Design

Confidence intervals and tests of hypotheses on variance components are required in studies that employ a random effects design. The unbalanced random two-fold nested design is considered in this paper and confidence intervals are proposed for the variance components σ2/A and σ2/B. Computer simulation is used to show that even in very unbalanced designs, these intervals generally maintain the stated confidence coefficient. The hypothesis test for σ2/A based on the lower bound of the recommended confidence interval is shown to be better than previously proposed approximate tests.