The Coverage Probability of Confidence Intervals in One‐Way Analysis of Covariance after Two F Tests

Volume 3 of Analysis of Messy Data by Milliken & Johnson (2002) provides detailed recommendations about sequential model development for the analysis of covariance. In his review of this volume, Koehler (2002) asks whether users should be concerned about the effect of this sequential model development on the coverage probabilities of confidence intervals for comparing treatments. We present a general methodology for the examination of these coverage probabilities in the context of the two‐stage model selection procedure that uses two F tests and is proposed in Chapter 2 of Milliken & Johnson (2002). We apply this methodology to an illustrative example from this volume and show that these coverage probabilities are typically very far below nominal. Our conclusion is that users should be very concerned about the coverage probabilities of confidence intervals for comparing treatments constructed after this two‐stage model selection procedure.     

Confidence Intervals on Regression Models with Censored Data

Stute (1993, Consistent estimation under random censorship when covariables are present. Journal of Multivariate Analysis 45, 89–103) proposed a new method to estimate regression models with a censored response variable using least squares and showed the consistency and asymptotic normality for his estimator. This article proposes a new bootstrap-based methodology that improves the performance of the asymptotic interval estimation for the small sample size case. Therefore, we compare the behavior of Stute's asymptotic confidence interval with that of several confidence intervals that are based on resampling bootstrap techniques. In order to build these confidence intervals, we propose a new bootstrap resampling method that has been adapted for the case of censored regression models. We use simulations to study the improvement the performance of the proposed bootstrap-based confidence intervals show when compared to the asymptotic proposal. Simulation results indicate that, for the new proposals, coverage percentages are closer to the nominal values and, in addition, intervals are narrower.     

On consistent testing for serial correlation in seasonal time series models

The author considers serial correlation testing in seasonal time series models. He proposes a test statistic based on a spectral approach. Many tests of this type rely on kernel-based spectral density estimators that assign larger weights to low order lags than to high ones. Under seasonality, however, large autocorrelations may occur at seasonal lags that classical kernel estimators cannot take into account. The author thus proposes a test statistic that relies on the spectral density estimator of Shin (2004), whose weighting scheme is more adapted to this context. The distribution of his test statistic is derived under the null hypothesis and he studies its behaviour under fixed and local alternatives. He establishes the consistency of the test under a general fixed alternative. He also makes recommendations for the choice of the smoothing parameters. His simulation results suggest that his test is more powerful against seasonality than alternative procedures based on classical weighting schemes. He illustrates his procedure with monthly statistics on employment among young Americans.     

Constructing narrower confidence intervals by inverting adaptive tests

We begin by describing how to find the limits of confidence intervals by using a few permutation tests of significance. Next, we demonstrate how the adaptive permutation test, which maintains its level of significance, produces confidence intervals that maintain their coverage probabilities. By inverting adaptive tests, adaptive confidence intervals can be found for any single parameter in a multiple regression model. These adaptive confidence intervals are often narrower than the traditional confidence intervals when the error distributions are long‐tailed or skewed. We show how much reduction in width can be achieved for the slopes in several multiple regression models and for the interaction effect in a two‐way design. An R function that can compute these adaptive confidence intervals is described and instructions are provided for its use with real data.     

Confidence Intervals for Reliability of Stress-Strength Models in the Normal Case

In this article, we propose some procedures to get confidence intervals for the reliability in stress-strength models. The confidence intervals are obtained either through a parametric bootstrap procedure or using asymptotic results, and are applied to the particular context of two independent normal random variables. The performance of these estimators and other known approximate estimators are empirically checked through a simulation study which considers several scenarios.     

Confidence Intervals for the Hyperparameters in Structural Models

This article deals with the bootstrap as an alternative method to construct confidence intervals for the hyperparameters of structural models. The bootstrap procedure considered is the classical nonparametric bootstrap in the residuals of the fitted model using a well-known approach. The performance of this procedure is empirically obtained through Monte Carlo simulations implemented in Ox. Asymptotic and percentile bootstrap confidence intervals for the hyperparameters are built and compared by means of the coverage percentages. The results are similar but the bootstrap procedure is better for small sample sizes. The methods are applied to a real time series and confidence intervals are built for the hyperparameters.     

Pairwise comparisons of matched proportions

Consider the problem of comparing the success rates of c treatments, each of which induce a Bernoulli response. The comparison is to be made on the basis of n matched samples. We present a method for deriving confidence intervals for the pair-wise difference in success rates which has the desirable quality of providing uniformly shorter intervals than the procedure recently proposed by Bhapkar and Somes (1976). Comparisons of the lengths of the respective intervals are provided. Some observations regarding the assumptions required for the use of Cochran’s Q-test (1950) are also made.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

Median estimation through a regression transformation

The author considers the problem of constructing confidence intervals for the median of a future observation at certain values of exogenous variables, following a normalizing transformation. He shows that when this transformation is estimated, the usual interval obtained through an inverse transformation needs to be corrected, even when the sample size is large. He then gives a simple analytical solution to this problem and provides simulation results confirming the good small‐sample properties of the corrected interval. He also presents two concrete illustrations.     

Generalized confidence intervals for proportions of total variance in mixed linear models

Exact confidence intervals for a proportion of total variance, based on pivotal quantities, only exist for mixed linear models having two variance components. Generalized confidence intervals (GCIs) introduced by Weerahandi [1993. Generalized confidence intervals (Corr: 94V89 p726). J. Am. Statist. Assoc. 88, 899–905] are based on generalized pivotal quantities (GPQs) and can be constructed for a much wider range of models. In this paper, the author investigates the coverage probabilities, as well as the utility of GCIs, for a proportion of total variance in mixed linear models having more than two variance components. Particular attention is given to the formation of GPQs and GCIs in mixed linear models having three variance components in situations where the data exhibit complete balance, partial balance, and partial imbalance. The GCI procedure is quite general and provides a useful method to construct confidence intervals in a variety of applications.     

Corrected likelihood-ratio tests in logistic regression using small-sample data

Likelihood-ratio tests (LRTs) are often used for inferences on one or more logistic regression coefficients. Conventionally, for given parameters of interest, the nuisance parameters of the likelihood function are replaced by their maximum likelihood estimates. The new function created is called the profile likelihood function, and is used for inference from LRT. In small samples, LRT based on the profile likelihood does not follow χ² distribution. Several corrections have been proposed to improve LRT when used with small-sample data. Additionally, complete or quasi-complete separation is a common geometric feature for small-sample binary data. In this article, for small-sample binary data, we have derived explicitly the correction factors of LRT for models with and without separation, and proposed an algorithm to construct confidence intervals. We have investigated the performances of different LRT corrections, and the corresponding confidence intervals through simulations. Based on the simulation results, we propose an empirical rule of thumb on the use of these methods. Our simulation findings are also supported by real-world data.     

Permutation methods in relative risk regression models

In this paper, we develop a weighted permutation (WP) method to construct confidence intervals for regression parameters in relative risk regression models. The WP method is a generalized permutation approach. It constructs a resampled history which mimics the observed history for individuals under study. Inference procedures are based on studentized score statistics that are insensitive to the forms of the relative risk function. This makes the WP method appealing in the general framework of the relative risk regression model. First-order accuracy of the WP method is established using counting process approach with a partial likelihood filtration. A simulation study indicates that the method typically improves accuracy over asymptotic confidence intervals.     

A GEE approach to estimating accuracy and its confidence intervals for correlated data

In this paper, we provide a method for constructing confidence interval for accuracy in correlated observations, where one sample of patients is being rated by two or more diagnostic tests. Confidence intervals for other measures of diagnostic tests, such as sensitivity, specificity, positive predictive value, and negative predictive value, have already been developed for clustered or correlated observations using the generalized estimating equations (GEE) method. Here, we use the GEE and delta‐method to construct confidence intervals for accuracy, the proportion of patients who are correctly classified. Simulation results verify that the estimated confidence intervals exhibit consistent/appropriate coverage rates.     

Consistent testing for non‐correlation of two cointegrated ARMA time series

A consistent approach to the problem of testing non‐correlation between two univariate infinite‐order autoregressive models was proposed by Hong (1996). His test is based on a weighted sum of squares of residual cross‐correlations, with weights depending on a kernel function. In this paper, the author follows Hong's approach to test non‐correlation of two cointegrated (or partially non‐stationary) ARMA time series. The test of Pham, Roy & Cédras (2003) may be seen as a special case of his approach, as it corresponds to the choice of a truncated uniform kernel. The proposed procedure remains valid for testing non‐correlation between two stationary invertible multivariate ARMA time series. The author derives the asymptotic distribution of his test statistics under the null hypothesis and proves that his procedures are consistent. He also studies the level and power of his proposed tests in finite samples through simulation. Finally, he presents an illustration based on real data.     

A comparative study of confidence intervals for negative binomial proportion

In this paper, we investigate four existing and three new confidence interval estimators for the negative binomial proportion (i.e., proportion under inverse/negative binomial sampling). An extensive and systematic comparative study among these confidence interval estimators through Monte Carlo simulations is presented. The performance of these confidence intervals are evaluated in terms of their coverage probabilities and expected interval widths. Our simulation studies suggest that the confidence interval estimator based on saddlepoint approximation is more appealing for large coverage levels (e.g., nominal level≤1% ) whereas the score confidence interval estimator is more desirable for those commonly used coverage levels (e.g., nominal level>1% ). We illustrate these confidence interval construction methods with a real data set from a maternal congenital heart disease study.     

Interval estimators for reliability: the bivariate normal case

This paper proposes procedures to provide confidence intervals (CIs) for reliability in stress–strength models, considering the particular case of a bivariate normal set-up. The suggested CIs are obtained by employing either asymptotic variances of maximum-likelihood estimators or a bootstrap procedure. The coverage and the accuracy of these intervals are empirically checked through a simulation study and compared with those of another proposal in the literature. An application to real data is provided.     

Distribution‐free statistical intervals via ranked‐set sampling

The author shows how to construct distribution‐free statistical intervals from ranked‐set sampling. He considers the cases of confidence, tolerance, and prediction intervals. He shows how to compute their coverage probabilities and he compares their performance to that of intervals based on simple random sampling. He finds that the ranked‐set sampling‐based intervals are at least as good as their classical counterpart in most settings of interest, although the nature of the advantage depends on the type of interval considered.     

Confidence Intervals for the Weighted Sum of Two Independent Binomial Proportions

Confidence intervals for the difference of two binomial proportions are well known, however, confidence intervals for the weighted sum of two binomial proportions are less studied. We develop and compare seven methods for constructing confidence intervals for the weighted sum of two independent binomial proportions. The interval estimates are constructed by inverting the Wald test, the score test and the Likelihood ratio test. The weights can be negative, so our results generalize those for the difference between two independent proportions. We provide a numerical study that shows that these confidence intervals based on large‐sample approximations perform very well, even when a relatively small amount of data is available. The intervals based on the inversion of the score test showed the best performance. Finally, we show that as for the difference of two binomial proportions, adding four pseudo‐outcomes to the Wald interval for the weighted sum of two binomial proportions improves its coverage significantly, and we provide a justification for this correction.     

Markov chain Monte Carlo exact inference for binomial and multinomial logistic regression models

We develop Metropolis-Hastings algorithms for exact conditional inference, including goodness-of-fit tests, confidence intervals and residual analysis, for binomial and multinomial logistic regression models. We present examples where the exact results, obtained by enumeration, are available for comparison. We also present examples where Monte Carlo methods provide the only feasible approach for exact inference.     

Asymptotic results in partially non-regular log-exponential distributions

We propose approximations to the moments, different possibilities for the limiting distributions and approximate confidence intervals for the maximum-likelihood estimator of a given parametric function when sampling from partially non-regular log-exponential models. Our results are applicable to the two-parameter exponential, power-function and Pareto distribution. Asymptotic confidence intervals for quartiles in several Pareto models have been simulated. These are compared to asymptotic intervals based on sample quartiles. Our intervals are superior since we get shorter intervals with similar coverage probability. This superiority is even assessed probabilistically. Applications to real data are included.