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## Coverage Probability (coverage + probability)
Kinds of Coverage Probability
## Selected Abstracts## UPPER BOUNDS ON THE MINIMUM COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER MODEL SELECTION AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2009Paul KabailaSummary We consider a linear regression model, with the parameter of interest a specified linear combination of the components of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing the Akaike information criterion , AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest, based on the assumption that the selected model had been given to us,a priori. This assumption is false, and it can lead to a confidence interval with poor coverage properties. We provide an easily computed finite-sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum Bayesian information criterion (BIC), maximum adjusted,R2, minimum Mallows' CP and,t -tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite-sample analogue of an earlier large-sample upper bound due to Kabaila and Leeb. [source] ## A Measure of Representativeness of a Sample for Inferential Purposes INTERNATIONAL STATISTICAL REVIEW, Issue 2 2006Salvatore BertinoSummary After defining the concept of representativeness of a random sample, the author proposes a measure of how much the observed sample represents its parent distribution. This measure is called Representativeness Index. The same measure, seen as a function of a sample and of a distribution, will be called Representativeness Function. For a given sample it provides the value of the index for the different distributions under examination, and for a given distribution it provides a measure of the representativeness of its possible samples. Such Representativeness Function can be used in an inferential framework just as the likelihood function, since it gives to any distribution the "experimental support" provided by the observed sample. This measure is distribution-free and it is shown to be a transformation of the wellknown Cramér,von Mises statistic. By using the properties of that statistic, criteria for providing set estimators and tests of hypotheses are introduced. The utilization of the representativeness function in many standard statistical problems is outlined through examples. The quality of the inferential decisions can be assessed with the usual techniques (MSE, power function, coverage probabilities). The most interesting examples turn out to be those of situations that are "non-regular", as for instance the estimation of parameters involved in the support of the parent distribution, or less explored (model choice). Résumé Après avoir défini le concept de répresentativité d'un échantillon aléatoire, l'auteur propose une mesure de combien l'échantillon observé réprésente la distribution parente. Cette mesure est dite Fonction de Répresentativité. Pour un échantillon donné la fonction donne les valeurs de l'indice pour toutes le distributions de la famille consideée, tandis que, pour une distribution donnée, elle donne la mesure de la répresentativité de chaque possible échantillon. La Fonction de Répresentativité peut être employée dans les problèmes d'inference statistique justement comme la fonction de vraisemblance, puisque elle donne à chaque distribution le "support expérimental" produit par l'échantillon observé. La measure est à distribution libre et on demontre que elle est une tranformation de la bien connue statistique de Cramér,von Mises. En utilisant le propriétés de la dite statistique, on introduit des crières pour obtenir estimateur ensemblistes et test d'hypothèse. L'utilization de la fonction de répresentativité dans plusieurs problèmes statistique est montrée par des examples. La qualité des decisions inférentielles peut être evaluée par les techniques usuelles (MSE, fonction de puissance, probabiliés de couverture). Les examples les plus interessant sont ceux qui concerne les situations "non regulères", par exemple l'estimation de paramètres qui figurent dans le support de la population parente, ou situations moins exploées (choix du modèle). [source] ## Default Bayesian Priors for Regression Models with First-Order Autoregressive Residuals JOURNAL OF TIME SERIES ANALYSIS, Issue 3 2003Malay GhoshAbstract. The objective of this paper is to develop default priors when the parameter of interest is the autocorrelation coefficient in normal regression models with first-order autoregressive residuals. Jeffreys' prior as well as reference priors are found. These priors are compared in the light of how accurately the coverage probabilities of Bayesian credible intervals match the corresponding frequentist coverage probabilities. It is found that the reference priors have a definite edge over Jeffreys' prior in this respect. Also, the credible intervals based on these reference priors seem superior to similar intervals based on certain divergence measures. [source] ## Likelihood and bayesian approaches to inference for the stationary point of a quadratic response surface THE CANADIAN JOURNAL OF STATISTICS, Issue 2 2008Valeria SambuciniAbstract In response surface analysis, a second order polynomial model is often used for inference on the stationary point of the response function. The standard confidence regions for the stationary point are due to Box & Hunter (1954). The authors propose an alternative parametrization, in which the stationary point is the parameter of interest; likelihood techniques and Bayesian analysis are then easier to perform. The authors also suggest an approximate method to get highest posterior density regions for the maximum point (not simply for the stationary point). Furthermore, they study the coverage probabilities of these Bayesian regions through simulations. Approches vraisemblantiste et bayésienne pour I'inference portant sur le point stationnaire d'une surface de réponse quadratique Résumé: Dans l'analyse des surfaces de réponse, un polyn,me du second degré est souvent utilisé pour l'inférence portant sur le point stationnaire de la fonction de réponse. Les régions de confiance standards pour le point stationnaire sont dues à Box & Hunter (1954). Les auteurs proposent une paramétrisation différente dans laquelle le point stationnaire est le paramètre d'intér,t; ceci facilite l'emploi des techniques de vraisemblance et l'analyse bayésienne. Les auteurs suggèrent aussi une façon d'approximer les régions de plus haute densité a posteriori pour le point maximum (et non seulement pour le point stationnaire). De plus, ils étudient les propriétés de couverture des régions bayésiennespar voie de simulation. [source] ## Empirical likelihood for linear regression models under imputation for missing responses THE CANADIAN JOURNAL OF STATISTICS, Issue 4 2001Qihua WangAbstract The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log-likelihood ratio is asymptotically a weighted sum of chi-square variables with unknown weights. They obtain an adjusted empirical log-likelihood ratio which is asymptotically standard chi-square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log-likelihood ratio and use its distribution to approximate that of the empirical log-likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method. [source] ## A Bayesian Spatial Multimarker Genetic Random-Effect Model for Fine-Scale Mapping ANNALS OF HUMAN GENETICS, Issue 5 2008M.-Y. TsaiSummary Multiple markers in linkage disequilibrium (LD) are usually used to localize the disease gene location. These markers may contribute to the disease etiology simultaneously. In contrast to the single-locus tests, we propose a genetic random effects model that accounts for the dependence between loci via their spatial structures. In this model, the locus-specific random effects measure not only the genetic disease risk, but also the correlations between markers. In other words, the model incorporates this relation in both mean and covariance structures, and the variance components play important roles. We consider two different settings for the spatial relations. The first is our proposal, relative distance function (RDF), which is intuitive in the sense that markers nearby are likely to correlate with each other. The second setting is a common exponential decay function (EDF). Under each setting, the inference of the genetic parameters is fully Bayesian with Markov chain Monte Carlo (MCMC) sampling. We demonstrate the validity and the utility of the proposed approach with two real datasets and simulation studies. The analyses show that the proposed model with either one of two spatial correlations performs better as compared with the single locus analysis. In addition, under the RDF model, a more precise estimate for the disease locus can be obtained even when the candidate markers are fairly dense. In all simulations, the inference under the true model provides unbiased estimates of the genetic parameters, and the model with the spatial correlation structure does lead to greater confidence interval coverage probabilities. [source] ## Confidence Intervals for Relative Risks in Disease Mapping BIOMETRICAL JOURNAL, Issue 4 2003M.D. UgarteAbstract Several analysis of the geographic variation of mortality rates in space have been proposed in the literature. Poisson models allowing the incorporation of random effects to model extra-variability are widely used. The typical modelling approach uses normal random effects to accommodate local spatial autocorrelation. When spatial autocorrelation is absent but overdispersion persists, a discrete mixture model is an alternative approach. However, a technique for identifying regions which have significant high or low risk in any given area has not been developed yet when using the discrete mixture model. Taking into account the importance that this information provides to the epidemiologists to formulate hypothesis related to the potential risk factors affecting the population, different procedures for obtaining confidence intervals for relative risks are derived in this paper. These methods are the standard information-based method and other four, all based on bootstrap techniques, namely the asymptotic-bootstrap, the percentile-bootstrap, the BC-bootstrap and the modified information-based method. All of them are compared empirically by their application to mortality data due to cardiovascular diseases in women from Navarra, Spain, during the period 1988,1994. In the small area example considered here, we find that the information-based method is sensible at estimating standard errors of the component means in the discrete mixture model but it is not appropriate for providing standard errors of the estimated relative risks and hence, for constructing confidence intervals for the relative risk associated to each region. Therefore, the bootstrap-based methods are recommended for this matter. More specifically, the BC method seems to provide better coverage probabilities in the case studied, according to a small scale simulation study that has been carried out using a scenario as encountered in the analysis of the real data. [source] ## Evaluating Normal Approximation Confidence Intervals for Measures of 2 × 2 Association with Applications to Twin Data BIOMETRICAL JOURNAL, Issue 1 2003M.M. ShoukriAbstract Twin data are of interest to genetic epidemiologists for exploring the underlying genetic basis of disease development. When the outcome is binary, several indices of 2 × 2 association can be used to measure the degree of within twin similarity. All such measures share a common feature, in that they can be expressed as a monotonic increasing function of the within twin correlation. The sampling distributions of their estimates are influenced by the sample size, the correlation and the marginal distribution of the binary response. In this paper we use Monte-Carlo simulations to estimate the empirical coverage probabilities and evaluate the adequacy of the classical normal confidence intervals on the population values of these measures. [source] ## Profile-Likelihood Inference for Highly Accurate Diagnostic Tests BIOMETRICS, Issue 4 2002John V. TsimikasSummary. We consider profile-likelihood inference based on the multinomial distribution for assessing the accuracy of a diagnostic test. The methods apply to ordinal rating data when accuracy is assessed using the area under the receiver operating characteristic (ROC) curve. Simulation results suggest that the derived confidence intervals have acceptable coverage probabilities, even when sample sizes are small and the diagnostic tests have high accuracies. The methods extend to stratified settings and situations in which the ratings are correlated. We illustrate the methods using data from a clinical trial on the detection of ovarian cancer. [source] ## A propensity score approach to correction for bias due to population stratification using genetic and non-genetic factors GENETIC EPIDEMIOLOGY, Issue 8 2009Huaqing ZhaoAbstract Confounding due to population stratification (PS) arises when differences in both allele and disease frequencies exist in a population of mixed racial/ethnic subpopulations. Genomic control, structured association, principal components analysis (PCA), and multidimensional scaling (MDS) approaches have been proposed to address this bias using genetic markers. However, confounding due to PS can also be due to non-genetic factors. Propensity scores are widely used to address confounding in observational studies but have not been adapted to deal with PS in genetic association studies. We propose a genomic propensity score (GPS) approach to correct for bias due to PS that considers both genetic and non-genetic factors. We compare the GPS method with PCA and MDS using simulation studies. Our results show that GPS can adequately adjust and consistently correct for bias due to PS. Under no/mild, moderate, and severe PS, GPS yielded estimated with bias close to 0 (mean=,0.0044, standard error=0.0087). Under moderate or severe PS, the GPS method consistently outperforms the PCA method in terms of bias, coverage probability (CP), and type I error. Under moderate PS, the GPS method consistently outperforms the MDS method in terms of CP. PCA maintains relatively high power compared to both MDS and GPS methods under the simulated situations. GPS and MDS are comparable in terms of statistical properties such as bias, type I error, and power. The GPS method provides a novel and robust tool for obtaining less-biased estimates of genetic associations that can consider both genetic and non-genetic factors. Genet. Epidemiol. 33:679,690, 2009. © 2009 Wiley-Liss, Inc. [source] ## Inferences for Selected Location Quotients with Applications to Health Outcomes GEOGRAPHICAL ANALYSIS, Issue 3 2010Gemechis Dilba DjiraThe location quotient (LQ) is an index frequently used in geography and economics to measure the relative concentration of activities. This quotient is calculated in a variety of ways depending on which group is used as a reference. Here, we focus on a simultaneous inference for the ratios of the individual proportions to the overall proportion based on binomial data. This is a multiple comparison problem and inferences for LQs with adjustments for multiplicity have not been addressed before. The comparisons are negatively correlated. The quotients can be simultaneously tested against unity, and simultaneous confidence intervals can be constructed for the LQs based on existing probability inequalities and by directly using the asymptotic joint distribution of the associated test statistics. The proposed inferences are appropriate for analysis based on sample surveys. Two real data sets are used to demonstrate the application of multiplicity-adjusted LQs. A simulation study is also carried out to assess the performance of the proposed methods to achieve a nominal coverage probability. For the LQs considered, the coverage of the simple Bonferroni-adjusted Fieller intervals for LQs is observed to be almost as good as the coverage of the method that directly takes the correlations into account. El cociente de localización (LQ) es un índice de uso frecuente en las disciplinas de Geografía y Economía para medir la concentración relativa de actividades. El cálculo del cociente se realiza de una variedad de formas, dependiendo del grupo que se utilice como referencia. El presente artículo aborda el problema de realizar inferencias simultáneas con tasas que describen proporciones individuales en relación a proporciones globales, para el caso de datos en escala binomial. Este problema puede ser caracterizado como uno de tipo de comparaciones múltiples (multiple comparison problem). Salvo el estudio presente, no existen precedentes de métodos diseñados para realizar inferencias de LQ que estén ajustados para abordar comparaciones múltiples. Las comparaciones están correlacionadas negativamente. Los cocientes pueden ser evaluados simultáneamente para verificar la propiedad de unidad (unity), y se pueden construir intervalos de confianza simultáneos para un LQ basado en la desigualdad de probabilidades existentes y por medio del uso directo de la distribución asintótica conjunta (asymtotic joint distribution) de los test o pruebas estadísticas asociadas. El tipo de inferencias propuestas por los autores son las adecuadas para el análisis de encuestas por muestreo. Para demostrar la aplicación del LQ desarrollado por el estudio, se utilizan dos conjuntos de datos del mundo real. Asimismo se lleva a cabo un estudio de simulación para evaluar el desempeño de los métodos propuestos con el fin de alcanzar una probabilidad de cobertura nominal (nominal coverage). Para los LQs seleccionados, la cobertura de los intervalos de confianza simples Fieller-Bonferroni ajustados para LQ, producen resultados casi tan buenos como la cobertura de métodos que toma en cuenta las correlaciones directamente. [source] ## Evaluating predictive performance of value-at-risk models in emerging markets: a reality check JOURNAL OF FORECASTING, Issue 2 2006Yong BaoAbstract We investigate the predictive performance of various classes of value-at-risk (VaR) models in several dimensions,unfiltered versus filtered VaR models, parametric versus nonparametric distributions, conventional versus extreme value distributions, and quantile regression versus inverting the conditional distribution function. By using the reality check test of White (2000), we compare the predictive power of alternative VaR models in terms of the empirical coverage probability and the predictive quantile loss for the stock markets of five Asian economies that suffered from the 1997,1998 financial crisis. The results based on these two criteria are largely compatible and indicate some empirical regularities of risk forecasts. The Riskmetrics model behaves reasonably well in tranquil periods, while some extreme value theory (EVT)-based models do better in the crisis period. Filtering often appears to be useful for some models, particularly for the EVT models, though it could be harmful for some other models. The CaViaR quantile regression models of Engle and Manganelli (2004) have shown some success in predicting the VaR risk measure for various periods, generally more stable than those that invert a distribution function. Overall, the forecasting performance of the VaR models considered varies over the three periods before, during and after the crisis. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Non-parametric confidence bands in deconvolution density estimation JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 3 2007Nicolai BissantzSummary., Uniform confidence bands for densities f via non-parametric kernel estimates were first constructed by Bickel and Rosenblatt. In this paper this is extended to confidence bands in the deconvolution problem g=f*, for an ordinary smooth error density ,. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L, -distance) between a deconvolution kernel estimator and f. Further consistency of the simple non-parametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data-driven bandwidth selector that is based on heuristic arguments, which aims at minimizing the L, -distance between and f. Although not constructed explicitly to undersmooth the estimator, a simulation study reveals that the bandwidth selector suggested performs well in finite samples, in terms of both area and coverage probability of the resulting confidence bands. Finally the methodology is applied to measurements of the metallicity of local F and G dwarf stars. Our results confirm the ,G dwarf problem', i.e. the lack of metal poor G dwarfs relative to predictions from ,closed box models' of stellar formation. [source] ## Small confidence sets for the mean of a spherically symmetric distribution JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 3 2005Richard SamworthSummary., Suppose that X has a k -variate spherically symmetric distribution with mean vector , and identity covariance matrix. We present two spherical confidence sets for ,, both centred at a positive part Stein estimator . In the first, we obtain the radius by approximating the upper , -point of the sampling distribution of by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed. [source] ## The adjustment of prediction intervals to account for errors in parameter estimation JOURNAL OF TIME SERIES ANALYSIS, Issue 3 2004Paul KabailaAbstract., Standard approximate 1 , , prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1 , ,. Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1 , ,, conditional on an appropriate statistic T. For an autoregressive process of order p, it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1 , ,. In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1 , ,. We illustrate the application of this new criterion to a Gaussian zero-mean autoregressive process of order 1 and one-step-ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1 , , conditional on the last observation is the better of the two adjustments. [source] ## Nonparametric confidence intervals for Tmax in sequence-stratified crossover studies PHARMACEUTICAL STATISTICS: THE JOURNAL OF APPLIED STATISTICS IN THE PHARMACEUTICAL INDUSTRY, Issue 1 2008Susan A. WillavizeAbstract Tmax is the time associated with the maximum serum or plasma drug concentration achieved following a dose. While Tmax is continuous in theory, it is usually discrete in practice because it is equated to a nominal sampling time in the noncompartmental pharmacokinetics approach. For a 2-treatment crossover design, a Hodges,Lehmann method exists for a confidence interval on treatment differences. For appropriately designed crossover studies with more than two treatments, a new median-scaling method is proposed to obtain estimates and confidence intervals for treatment effects. A simulation study was done comparing this new method with two previously described rank-based nonparametric methods, a stratified ranks method and a signed ranks method due to Ohrvik. The Normal theory, a nonparametric confidence interval approach without adjustment for periods, and a nonparametric bootstrap method were also compared. Results show that less dense sampling and period effects cause increases in confidence interval length. The Normal theory method can be liberal (i.e. less than nominal coverage) if there is a true treatment effect. The nonparametric methods tend to be conservative with regard to coverage probability and among them the median-scaling method is least conservative and has shortest confidence intervals. The stratified ranks method was the most conservative and had very long confidence intervals. The bootstrap method was generally less conservative than the median-scaling method, but it tended to have longer confidence intervals. Overall, the median-scaling method had the best combination of coverage and confidence interval length. All methods performed adequately with respect to bias. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On the bootstrap in cube root asymptotics THE CANADIAN JOURNAL OF STATISTICS, Issue 1 2006Christian LégerAbstract The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoff's estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff's estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1! A propos du bootstrap pour des estimateurs convergeant a la vitesse racine cubique Les auteurs étudient l'application du bootstrap à une classe d'estimateurs qui convergent à une vitesse et vers une loi non standard. Ils présentent un cadre théorique pour l'étude de son comportement asymptotique. Une simulation démontre que dans le cas d'un estimateur du mode de Chernoff, la probabilité de couverture de l'intervalle de confiance bootstrap de base est grandement inférieure au niveau prescrit, alors que celle des intervalles de type percentile dépasse le niveau prescrit. C'est un rare cas où les intervalles de confiance de base et percentile ont un comportement si différent malgré des longueurs identiques. Dans le cas de l'estimateur de Chernoff, si la distribution est symétrique, il est possible d'appliquer le bootstrap à partir d'un estimateur lisse et symétrique de la distribution qui mènera à des intervalles bootstrap de base dont la probabilité de couverture asymptotique sera la bonne, alors que celle de l'intervalle percentile convergera vers 1! [source] ## The behrens-fisher problem revisited: A bayes-frequentist synthesis THE CANADIAN JOURNAL OF STATISTICS, Issue 1 2001Malay GhoshAbstract The Behrens-Fisher problem concerns the inference for the difference between the means of two normal populations whose ratio of variances is unknown. In this situation, Fisher's fiducial interval differs markedly from the Neyman-Pearson confidence interval. A prior proposed by Jeffreys leads to a credible interval that is equivalent to Fisher's solution but it carries a different interpretation. The authors propose an alternative prior leading to a credible interval whose asymptotic coverage probability matches the frequentist coverage probability more accurately than the interval of Jeffreys. Their simulation results indicate excellent matching even in small samples. Le problème de Bahrens-Fisher concerne l'inférence pour la différence entre les moyennes de deux populations normales dont le repport des variances est inconnu. Dans cette situation, l'intervalle de confiance fiduciaire de Fisher est fort different de celui de Neyman-Pearson. Une loi a priori proposée par Jeffreys conduit à un intervalle de crédibilité équivalent à la solution de Fisher, mais qui n'a pas la même interprétation. Les auteurs proposent une nouvelle loi a priori menant à un intervalle de crédibilité dont le taux de couverture asymptotique est plus près du taux fréquentiste que celui de l'intervalle de Jeffreys. Leurs simulations montrent que cette observation reste valable dans de petits échantillons. [source] ## UPPER BOUNDS ON THE MINIMUM COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER MODEL SELECTION AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2009Paul KabailaSummary We consider a linear regression model, with the parameter of interest a specified linear combination of the components of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing the Akaike information criterion , AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest, based on the assumption that the selected model had been given to us,a priori. This assumption is false, and it can lead to a confidence interval with poor coverage properties. We provide an easily computed finite-sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum Bayesian information criterion (BIC), maximum adjusted,R2, minimum Mallows' CP and,t -tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite-sample analogue of an earlier large-sample upper bound due to Kabaila and Leeb. [source] ## CONFIDENCE INTERVALS UTILIZING PRIOR INFORMATION IN THE BEHRENS,FISHER PROBLEM AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 4 2008Paul KabailaSummary Consider two independent random samples of size f,+ 1, one from an N (,1, ,21) distribution and the other from an N (,2, ,22) distribution, where ,21/,22, (0, ,). The Welch ,approximate degrees of freedom' (,approximate t -solution') confidence interval for ,1,,2 is commonly used when it cannot be guaranteed that ,21/,22= 1. Kabaila (2005, Comm. Statist. Theory and Methods,34, 291,302) multiplied the half-width of this interval by a positive constant so that the resulting interval, denoted by J0, has minimum coverage probability 1 ,,. Now suppose that we have uncertain prior information that ,21/,22= 1. We consider a broad class of confidence intervals for ,1,,2 with minimum coverage probability 1 ,,. This class includes the interval J0, which we use as the standard against which other members of will be judged. A confidence interval utilizes the prior information substantially better than J0 if (expected length of J)/(expected length of J0) is (a) substantially less than 1 (less than 0.96, say) for ,21/,22= 1, and (b) not too much larger than 1 for all other values of ,21/,22. For a given f, does there exist a confidence interval that satisfies these conditions? We focus on the question of whether condition (a) can be satisfied. For each given f, we compute a lower bound to the minimum over of (expected length of J)/(expected length of J0) when ,21/,22= 1. For 1 ,,= 0.95, this lower bound is not substantially less than 1. Thus, there does not exist any confidence interval belonging to that utilizes the prior information substantially better than J0. [source] ## Complementary Log,Log Regression for the Estimation of Covariate-Adjusted Prevalence Ratios in the Analysis of Data from Cross-Sectional Studies BIOMETRICAL JOURNAL, Issue 3 2009Alan D. PenmanAbstract We assessed complementary log,log (CLL) regression as an alternative statistical model for estimating multivariable-adjusted prevalence ratios (PR) and their confidence intervals. Using the delta method, we derived an expression for approximating the variance of the PR estimated using CLL regression. Then, using simulated data, we examined the performance of CLL regression in terms of the accuracy of the PR estimates, the width of the confidence intervals, and the empirical coverage probability, and compared it with results obtained from log,binomial regression and stratified Mantel,Haenszel analysis. Within the range of values of our simulated data, CLL regression performed well, with only slight bias of point estimates of the PR and good confidence interval coverage. In addition, and importantly, the computational algorithm did not have the convergence problems occasionally exhibited by log,binomial regression. The technique is easy to implement in SAS (SAS Institute, Cary, NC), and it does not have the theoretical and practical issues associated with competing approaches. CLL regression is an alternative method of binomial regression that warrants further assessment. [source] ## Youden Index and Optimal Cut-Point Estimated from Observations Affected by a Lower Limit of Detection BIOMETRICAL JOURNAL, Issue 3 2008Marcus D. RuoppAbstract The receiver operating characteristic (ROC) curve is used to evaluate a biomarker's ability for classifying disease status. The Youden Index (J), the maximum potential effectiveness of a biomarker, is a common summary measure of the ROC curve. In biomarker development, levels may be unquantifiable below a limit of detection (LOD) and missing from the overall dataset. Disregarding these observations may negatively bias the ROC curve and thus J. Several correction methods have been suggested for mean estimation and testing; however, little has been written about the ROC curve or its summary measures. We adapt non-parametric (empirical) and semi-parametric (ROC-GLM [generalized linear model]) methods and propose parametric methods (maximum likelihood (ML)) to estimate J and the optimal cut-point (c *) for a biomarker affected by a LOD. We develop unbiased estimators of J and c * via ML for normally and gamma distributed biomarkers. Alpha level confidence intervals are proposed using delta and bootstrap methods for the ML, semi-parametric, and non-parametric approaches respectively. Simulation studies are conducted over a range of distributional scenarios and sample sizes evaluating estimators' bias, root-mean square error, and coverage probability; the average bias was less than one percent for ML and GLM methods across scenarios and decreases with increased sample size. An example using polychlorinated biphenyl levels to classify women with and without endometriosis illustrates the potential benefits of these methods. We address the limitations and usefulness of each method in order to give researchers guidance in constructing appropriate estimates of biomarkers' true discriminating capabilities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Estimation and Confidence Intervals after Adjusting the Maximum Information, BIOMETRICAL JOURNAL, Issue 2 2003John LawrenceAbstract In a comparative clinical trial, if the maximum information is adjusted on the basis of unblinded data, the usual test statistic should be avoided due to possible type I error inflation. An adaptive test can be used as an alternative. The usual point estimate of the treatment effect and the usual confidence interval should also be avoided. In this article, we construct a point estimate and a confidence interval that are motivated by an adaptive test statistic. The estimator is consistent for the treatment effect and the confidence interval asymptotically has correct coverage probability. [source] ## Statistical Inference For Risk Difference in an Incomplete Correlated 2 × 2 Table BIOMETRICAL JOURNAL, Issue 1 2003Nian-Sheng TangAbstract In some infectious disease studies and 2-step treatment studies, 2 × 2 table with structural zero could arise in situations where it is theoretically impossible for a particular cell to contain observations or structural void is introduced by design. In this article, we propose a score test of hypotheses pertaining to the marginal and conditional probabilities in a 2 × 2 table with structural zero via the risk/rate difference measure. Score test-based confidence interval will also be outlined. We evaluate the performance of the score test and the existing likelihood ratio test. Our empirical results evince the similar and satisfactory performance of the two tests (with appropriate adjustments) in terms of coverage probability and expected interval width. Both tests consistently perform well from small- to moderate-sample designs. The score test however has the advantage that it is only undefined in one scenario while the likelihood ratio test can be undefined in many scenarios. We illustrate our method by a real example from a two-step tuberculosis skin test study. [source] ## Marginal Analysis of Incomplete Longitudinal Binary Data: A Cautionary Note on LOCF Imputation BIOMETRICS, Issue 3 2004Richard J. CookSummary In recent years there has been considerable research devoted to the development of methods for the analysis of incomplete data in longitudinal studies. Despite these advances, the methods used in practice have changed relatively little, particularly in the reporting of pharmaceutical trials. In this setting, perhaps the most widely adopted strategy for dealing with incomplete longitudinal data is imputation by the "last observation carried forward" (LOCF) approach, in which values for missing responses are imputed using observations from the most recently completed assessment. We examine the asymptotic and empirical bias, the empirical type I error rate, and the empirical coverage probability associated with estimators and tests of treatment effect based on the LOCF imputation strategy. We consider a setting involving longitudinal binary data with longitudinal analyses based on generalized estimating equations, and an analysis based simply on the response at the end of the scheduled follow-up. We find that for both of these approaches, imputation by LOCF can lead to substantial biases in estimators of treatment effects, the type I error rates of associated tests can be greatly inflated, and the coverage probability can be far from the nominal level. Alternative analyses based on all available data lead to estimators with comparatively small bias, and inverse probability weighted analyses yield consistent estimators subject to correct specification of the missing data process. We illustrate the differences between various methods of dealing with drop-outs using data from a study of smoking behavior. [source] ## Sample Size Determination for Establishing Equivalence/Noninferiority via Ratio of Two Proportions in Matched,Pair Design BIOMETRICS, Issue 4 2002Man-Lai TangSummary. In this article, we propose approximate sample size formulas for establishing equivalence or noninferiority of two treatments in match-pairs design. Using the ratio of two proportions as the equivalence measure, we derive sample size formulas based on a score statistic for two types of analyses: hypothesis testing and confidence interval estimation. Depending on the purpose of a study, these formulas can be used to provide a sample size estimate that guarantees a prespecified power of a hypothesis test at a certain significance level or controls the width of a confidence interval with a certain confidence level. Our empirical results confirm that these score methods are reliable in terms of true size, coverage probability, and skewness. A liver scan detection study is used to illustrate the proposed methods. [source] ## On Small-Sample Confidence Intervals for Parameters in Discrete Distributions BIOMETRICS, Issue 3 2001Alan AgrestiSummary. The traditional definition of a confidence interval requires the coverage probability at any value of the parameter to be at least the nominal confidence level. In constructing such intervals for parameters in discrete distributions, less conservative behavior results from inverting a single two-sided test than inverting two separate one-sided tests of half the nominal level each. We illustrate for a variety of discrete problems, including interval estimation of a binomial parameter, the difference and the ratio of two binomial parameters for independent samples, and the odds ratio. [source] |