Computing Confidence Intervals Research Articles

Advanced Statistics: Bootstrapping Confidence Intervals for Statistics with “Difficult” Distributions

The use of confidence intervals in reporting results of research has increased dramatically and is now required or highly recommended by editors of many scientific journals. Many resources describe methods for computing confidence intervals for statistics with mathematically simple distributions. Computing confidence intervals for descriptive statistics with distributions that are difficult to represent mathematically is more challenging. The bootstrap is a computationally intensive statistical technique that allows the researcher to make inferences from data without making strong distributional assumptions about the data or the statistic being calculated. This allows the researcher to estimate confidence intervals for statistics that do not have simple sampling distributions (e.g., the median). The purposes of this article are to describe the concept of bootstrapping, to demonstrate how to estimate confidence intervals for the median and the Spearman rank correlation coefficient for non-normally-distributed data from a recent clinical study using two commonly used statistical software packages (SAS and Stata), and to discuss specific limitations of the bootstrap.

Estimating Parliamentary Composition Through Electoral Polls

SummaryAll electoral systems have an electoral formula that converts proportions of votes into Parliamentary seats. Pre-electoral polls usually focus on estimating proportions of votes and then apply the electoral formula to give a forecast of Parliamentary composition. We describe the problems that arise from this approach: there will typically be a bias in the forecast. We study the origin of the bias and some methods for evaluating and reducing it. We propose a bootstrap algorithm for computing confidence intervals for the allocation of seats. We show, by Monte Carlo simulation, the performance of the proposed methods using data from Spanish elections in previous years. We also propose graphical methods for visualizing how electoral formulae and Parliamentary forecasts work (or fail).

분포함수를 기초로 일반화가중선형모형

이 논문에서는 일반화가중선형모형이라는 새로운 형태의 선형모형을 제시한다. 일반화가중선형모형은 설명변수와 반응변수의 관계를 설명분포함수의 선형결합이 반응변수의 평균에 대한 연결분포함수를 통해 모형화 되는 형태를 가지는 것으로 가정한다. 이모형은 일반화선형 모형에서 연결함수를 선택할 때 발생할 수 있는 모수공간과 선형 예측값의 공간이 일치하지 않을 수 있다는 문제가 발생하지 않고 모수에 대한 해석이 용이하다는 장점이 있다. 이 논문에서는 설명분포함수와 연결분포함수를 선택하는데 있어 발생할 수 있는 문제와 해결책에 대해 알아본다. 또한 모형에 포함되어 있는 모수를 추정하는데 고려해야 할 주의 사항과 이 사항들을 고려한 최대가능도추정법과 재표집 방법을 이용한 구간추정과 가설검정에 대해 알아본다. In this paper, a new form of linear models referred to as generalized weighted linear models is proposed. The proposed models assume that the relationship between the response variable and explanatory variables can be modelled by a distribution function of the response mean and a weighted linear combination of distribution functions of covariates. This form addresses a structural problem of the link function in the generalized linear models in which the parameter space may not be consistent with the space derived from linear predictors. The maximum likelihood estimation with Lagrange's undetermined multipliers is used to estimate the parameters and resampling method is applied to compute confidence intervals and to test hypotheses.

Factors Affecting Relative Weights: The Influence of Sampling and Measurement Error

Relative weights are used to interpret a regression equation when the researcher is interested in the relative importance of the predictor variables. Like other statistics, relative weights are influenced by sampling and measurement error. To assess the influence of sampling error, a bootstrapping approach to computing confidence intervals around relative weights is described. To assess the influence of measurement error, a Monte Carlo study was conducted in which relative weights were computed before and after correcting correlation matrices for unreliability. The influence of measurement error was generally small but can be substantial in some cases. Correcting for unreliability tended to make the biggest difference when criterion correlations were small, predictor reliabilities were low, and the number of predictors was large.

AlphaCI: un programa de cálculo de intervalos de confianza para el coeficiente alfa de Cronbach

El coeficiente alfa de Cronbach es el modo más habitual de estimar la fiabilidad de pruebas basadas en Teoría Clásica de los Test. En dicha estimación, los investigadores usualmente omiten informar intervalos de confianza para el coeficiente, un aspecto no solo recomendado por los especialistas, sino también requerido explícitamente en las normas editoriales de algunas revistas especializadas. Esta situación puede atribuirse a que los métodos de estimación de intervalos de confianza son poco conocidos, además de no estar disponibles al usuario en los programas estadísticos más populares. Así, en este trabajo se presenta un programa desarrollado dentro del sistema estadístico ViSta que permite calcular intervalos de confianza basados en el enfoque clásico y mediante la técnica bootstrap. Se espera promover la inclusión de intervalos de confianza para medidas de fiabilidad, facilitando el acceso a las herramientas necesarias para su aplicación. El programa es gratuito y puede obtenerse enviando un mail de solicitud al autor del trabajo.

On‐line assessment of prediction risk for wind power production forecasts

AbstractThe article introduces a new methodology for assessing on‐line the prediction risk of short‐term wind power forecasts. The first stage of this methodology consists in computing confidence intervals with a confidence level defined by the end‐user. The resampling approach is used, which, in contrast to existing methods for wind forecasting, does not need to make a restrictive hypothesis on the distribution of the errors. To account for the non‐linearity of the power curve and the cut‐off effect, the errors are classified using appropriate fuzzy sets. The confidence intervals are then fine‐tuned to reduce their width in the case of stable weather conditions. For this purpose an appropriate index, called the ‘meteo‐risk index’ (MRI), is defined reflecting the spread of the available numerical weather predictions. A linear relation between that index and the resulting prediction error is shown. The second part of the methodology is to use the MRI itself as a preventive on‐line tool to derive signals for the operator on the meteorological risk, i.e. the probabilities of the occurrence of high prediction errors depending on the weather stability. Evaluation results of this methodology over a 1 year period on the case study of Ireland are given, where the output of several wind farms is predicted using a dynamic fuzzy neural network‐based model. The proposed methodology is generic and can be applied to all kinds of wind power prediction models. Copyright © 2004 John Wiley & Sons, Ltd.

Rym15 from the Japanese cultivar Chikurin Ibaraki 1 is a new barley mild mosaic virus (BaMMV) resistance gene mapped on chromosome 6H.

Breeding for resistant cultivars is the only way to prevent high yield loss in barley caused by the soil-borne barley mild mosaic virus (BaMMV) complex. We have characterized the BaMMV resistance of barley cv. Chikurin Ibaraki 1. Doubled haploid lines were obtained from the F(1) between the susceptible six-rowed winter barley cultivar, Plaisant, and Chikurin Ibaraki 1. Each line was tested for reaction to BaMMV by mechanical inoculation followed by DAS-ELISA. Of 44 microsatellites that covered the genome, 22 polymorphic markers were tested on one susceptible and one resistant bulk, each comprising 30 lines. Differential markers and additional microsatellite markers in the same region were then tested on the whole population. A bootstrap analysis was used to compute confidence intervals of distances and to test the orders of the resistance gene and the closest markers. A segregation of 84 resistant/98 susceptible lines fitted a 1:1 ratio (chi(2)=1.08, P=0.30), which corresponds to a single gene in this DH lines population. The resistance gene was flanked by two markers near the centromeric region of chromosome 6HS-Bmag0173, at 0.6+/-1.2 cM, and EBmac0874, at 5.8 +/- 3.4 cM. We propose to name this new resistance gene rym15. This resistance gene and associated markers will increase the possibilities to breed efficiently for new cultivars resistant to the barley mosaic disease.

Economic assessment of the secondary prevention of ischaemic stroke with dipyridamole plus aspirin (Aggrenox/Asasantin) in France.

To assess the cost effectiveness of aspirin 25 mg plus dipyridamole 200 mg twice daily in the secondary prevention of ischaemic stroke, according to the French social security perspective, using efficacy data from the second European Stroke Prevention Study (ESPS-2). The ESPS-2 was a double-blind, placebo-controlled clinical trial which assessed the efficacy of four secondary prevention strategies: (i) placebo; (ii) aspirin (acetylsalicylic acid) 25 mg twice daily; (iii) dipyridamole 200 mg twice daily; and (iv) aspirin 25 mg plus dipyridamole 200 mg twice daily. We performed a cost-effectiveness analysis with Monte Carlo simulations to compute confidence intervals. We combined data from various sources including the Dijon Stroke Registry, Institut National de la Statistique et des Etudes Economiques, Etude du Coût de l'Infarctus Cérébral (Study of the Cost of Cerebral Infarction [ECIC]) study and the ESPS-2 trial. According to our findings, a preventive strategy with aspirin 25 mg plus dipyridamole 200 mg twice daily is associated with net benefits per avoided stroke recurrence amounting to USD 23,932 (95% CI -USD 32,609, USD 35,772) compared with aspirin 25 mg twice daily alone, and USD 31,555 (95% CI USD 4921, USD 74,515) compared with dipyridamole alone (1997 values). Sensitivity analysis demonstrated that dipyridamole plus aspirin was still cost effective when the average cost of adverse effects per episode (ignored in the original estimation of the cost-effectiveness ratios due to a lack of data) was assumed to be USD 8600 (50,000 French francs); this cost is unlikely as most of the adverse effects associated with aspirin plus dipyridamole are only slight to moderate in severity. In the secondary prevention of stroke in France, this study suggests, given its underlying assumptions and data, that aspirin 25 mg plus dipyridamole 200 mg twice daily is likely to be a cost-effective strategy from the social security perspective, when compared with other relevant strategies that were evaluated in the ESPS-2 trial.

Application of geostatistical inverse modeling to contaminant source identification at Dover AFB, Delaware

Analysis of subsurface soil cores from the site of a field-scale groundwater remediation experiment at Dover Air Force Base, Delaware, has revealed that tetrachloroethene and trichloroethene contamination extends into an aquitard underlying a groundwater aquifer. Geostatistical inverse modeling is used to make inferences regarding the historical concentration conditions in the overlying aquifer. Because geostatistical inverse modeling is a stochastic approach, it treats parameters as jointly distributed random fields. Therefore, this approach is used to compute confidence intervals in addition to best estimates. This framework is also used to compute large numbers of conditional realizations, which are equally probable solutions given the data, and which allow for a better understanding of the form of the unknown function. Finally, a Markov Chain Monte Carlo method combined with the application of Lagrange multipliers is used to enforce concentration non-negativity.

Inferences for the Fatigue Life Model Based on the Birnbaum–Saunders Distribution

Time to fracture from cracks in materials under fluctuating stress is often well approximated by the Birnbaum–Saunders (BISA) distribution. Unlike the Weibull and lognormal model, there is not much research on the accuracy of the parameter estimation on the BISA distribution under Type I (time) censoring. This article explores and compares different procedures to compute confidence intervals for parameters and quantiles of the BISA distribution for complete and Type I censored data. The procedures are based on using maximum likelihood estimators and can be classified into three groups as the commonly-used normal-approximation, the likelihood ratio, and the parametric bootstrap procedures. The procedures in all three groups are justified on the basis of large-sample asymptotic theory. We use Monte Carlo simulation to investigate the finite sample properties of these procedures and give suggestions of which procedure to use according to proportion failing and number of failures.

Measuring inequality in the presence of intra-household correlation

This note develops a method for computing confidence intervals for estimates of decomposable inequality indices when the sample is characterized by contemporaneous dependence across members of the same household. This is particularly relevant in the context of measuring inequality among wage earners. An application to the German wage distribution shows that the error made by not taking into account intra-household correlation is probably small in practical applications.

Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE

A standard part of any oceanic pressure gauge or current meter analysis is the separation of tidal from non-tidal components of the signal. The tidal signal can either be discarded, or its characteristics described in some fashion useful for further analysis. Although tidal signals can be removed by standard high or bandpass filtering techniques, their relatively deterministic character and large amplitude make special techniques more effective. In classical harmonic analysis, the tidal signal is modelled as the sum of a finite set of sinusoids at specific frequencies related to astronomical parameters. A set of programs has been written in MATLAB to (a) perform classical harmonic analysis for periods of about 1 year or shorter, (b) account for (some) unresolved constituents using nodal corrections, and (c) compute confidence intervals for the analyzed components.

Comments on Chiang (2001)

We read with great interest the article by Andy K. L. Chiang (2001). The author has written a very clear article explaining the idea of surrogate variables for computing confidence intervals for any function of the variance components in a mixedor random-effects model with balanced data. However, we would like to point out that the idea is not new and has already been introduced under the name generalized confidence interval by Weerahandi (1993). In fact, Hamada and Weerahandi (2000) have actually applied the generalized confidence interval idea to gauge RR the idea was also given in the book by Weerahandi (1995, p. 255), for the one-way random model. Using the notations for the two-way random model given in Professor Chiang's article, suppose that we want a confidence interval for 72 = a0 + a2B + oa. Note that

A statistical method for the estimation of window-period risk of transfusion-transmitted HIV in donor screening under non-steady state.

Human immunodeficiency virus (HIV) can be transmitted by transfusion of blood even if the blood unit is test-negative for HIV. This is largely due to a time period following an infection, called the window period, during which antibodies against HIV are not detectable. Window-period risk refers to the probability for a test-negative blood unit to be infectious because of its donation during the window period. Estimation of window-period risk is important in public health for evaluating the safety of donated blood. The standard method for this estimation problem has been based on so-called incidence/window-period (IWP) models in which blood-donation and HIV-infection processes are assumed to be stochastically stationary and independent. Here we propose a new approach in which we relax this key assumption of the IWP models. We estimate window-period risk for each unit of donated blood using a given distribution of window-period risk. The proposed method utilizes the actual observed donation intervals including those of seroconversions, thereby relaxing the assumption that may not be met in practice. Bootstrap is used to compute confidence intervals without specifying the complex dynamics of the donation and infection processes. A simulation study illustrates the usefulness of the proposed method over the IWP method in scenarios where the IWP assumptions do not hold. A real application of the proposed method is presented using blood bank data from a province of northern Thailand. Advantages and limitations of the proposed method are discussed and compared with the IWP models.

Parameter identification in dynamical models of anaerobic waste water treatment

Biochemical reactions can often be formulated mathematically as ordinary differential equations. In the process of modeling, the main questions that arise are concerned with structural identifiability, parameter estimation and practical identifiability. To clarify these questions and the methods how to solve them, we analyze two different second order models for anaerobic waste water treatment processes using two data sets obtained from different experimental setups. In both experiments only biogas production rate was measured which complicates the analysis considerably. We show that proving structural identifiability of the mathematical models with currently used methods fails. Therefore, we introduce a new, general method based on the asymptotic behavior of the maximum likelihood estimator to show local structural identifiability. For parameter estimation we use the multiple shooting approach which is described. Additionally we show that the Hessian matrix approach to compute confidence intervals fails in our examples while a method based on Monte Carlo Simulation works well.

Computing Correct Confidence Intervals for Anova Fixed-and Random-Effects Effect Sizes

Most textbooks explain how to compute confidence intervals for means, correlation coefficients, and other statistics using “central” test distributions (e.g., t, F) that are appropriate for such statistics. However, few textbooks explain how to use “noncentral” test distributions (e.g., noncentral t, noncentral F) to evaluate power or to compute confidence intervals for effect sizes. This article illustrates the computation of confidence intervals for effect sizes for some ANOVA applications; the use of intervals invoking noncentral distributions is made practical by newer software. Greater emphasis on both effect sizes and confidence intervals was recommended by the APA Task Force on Statistical Inference and is consistent with the editorial policies of the 17 journals that now explicitly require effect size reporting.

Using Trimmed Means to Compare K Measures Corresponding to Two Independent Groups

Consider two independent groups with K measures for each subject. For the jth group and kth measure, let μtjk be the population trimmed mean, j = 1, 2; k = 1, ..., K. This article compares several methods for testing H 0 : u1k = t2k such that the probability of at least one Type I error is, and simultaneous probability coverage is 1 - α when computing confidence intervals for μt1k - μt2k . The emphasis is on K = 4 and α = .05. For zero trimming the problem reduces to comparing means, but it is well known that when comparing means, arbitrarily small departures from normality can result in extremely low power relative to using say 20% trimming. Moreover, when skewed distributions are being compared, conventional methods for comparing means can be biased for reasons reviewed in the article. A consequence is that in some realistic situations, the probability of rejecting can be higher when the null hypothesis is true versus a situation where the means differ by a half standard deviation. Switching to robust measures of location is known to reduce this problem, and combining robust measures of location with some type of bootstrap method reduces the problem even more. Published articles suggest that for the problem at hand, the percentile t bootstrap, combined with a 20% trimmed mean, will perform relatively well, but there are known situations where it does not eliminate all problems. In this article we consider an extension of the percentile bootstrap approach that is found to give better results.

Estimating Lyapunov exponents in biomedical time series.

Among nonlinear dynamical invariants, determination of the largest Lyapunov exponent is well suited to positive identification of chaos in observed time series. When analyzing the dynamics of biomedical series, such as an electro-encephalogram (EEG), model-based methods should be used. Moreover, in the absence of any well founded theoretical model, and because of unexplained variability in the data, candidate models must provide for a stochastic component. Here we use nonlinear autoregressive stochastic modeling to estimate the dominant Lyapunov exponent in an EEG series and compute confidence intervals from surrogate data. The results are found to differ from those of approaches which aim at deleting noise prior to analysis.

Modeling the effects of health on economic growth

This paper investigates the effects of health indicators such as adult survival rates (ASR) on GDP growth rates at 5-year intervals in several countries. Panel data were analyzed on GDP series based on purchasing power adjustments and on exchange rates. First, we developed a framework for modeling the inter-relationships between GDP growth rates and explanatory variables by re-examining the life expectancy–income relationship. Second, models for growth rates were estimated taking into account the interaction between ASR and lagged GDP level; issues of endogeneity and reverse causality were addressed. Lastly, we computed confidence intervals for the effect of ASR on growth rate and applied a test for parameter stability. The results showed positive effects of ASR on GDP growth rates in low-income countries.

Improved Inference of Mutation Rates: I. An Integral Representation for the Luria–Delbrück Distribution

The estimation of mutation rates is ordinarily performed using results based on the Luria–Delbrück distribution. There are certain difficulties associated with the use of this distribution in practice, some of which we address in this paper (others in the companion paper, Oprea and Kepler, Theor. Popul. Biol., 2001). The distribution is difficult to compute exactly, especially for large values of the random variable. To overcome this problem, we derive an integral representation of the Luria–Delbrück distribution that can be computed easily for large culture sizes. In addition, we introduce the usual assumption of very small probability of having a large proportion of mutants only after the generating function has been computed. Thus, we obtain information on the moments for the more general case. We examine the asymptotic behavior of this system. We find a scaling or “standardization” technique that reduces the family of distributions parameterized by three parameters (mutation rate, initial cell number, and final cell number) to a single distribution with no parameters, valid so long as the product of the mutation rate and the final culture is sufficiently large. We provide a pair of techniques for computing confidence intervals for the mutation rate. In the second paper of this series, we use the distribution derived here to find approximate distributions for the case where the cell cycle time is not well-described as an exponential random variable as is implicitly assumed by Luria–Delbrück distribution.