On construction of asymptotically correct confidence intervals

In this paper we discuss constructing confidence intervals based on asymptotic generalized pivotal quantities (AGPQs). An AGPQ associates a distribution with the corresponding parameter, and then an asymptotically correct confidence interval can be derived directly from this distribution like Bayesian or fiducial interval estimates. We provide two general procedures for constructing AGPQs. We also present several examples to show that AGPQs can yield new confidence intervals with better finite-sample behaviors than traditional methods.     

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

Simultaneous confidence sets and confidence intervals for multiple ratios

This paper deals with the problem of simultaneously estimating multiple ratios. In the simplest case of only one ratio parameter, Fieller's theorem (J. Roy. Statist. Soc. Ser. B 16 (1954) 175) provides a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that exactly satisfy a given familywise confidence level. Many of the methods in use are conservative since they are based on probability inequalities. In this paper, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discussed. The methods are applied to ratios of linear combinations of the means in the one-way layout and ratios of parameter combinations in the general linear model. Extensive Monte Carlo simulation is carried out to compare the performance of the various methods with respect to the stability of the estimated critical points and of the coverage probabilities.     

A note on effective sample size for constructing confidence intervals for the difference of two proportions

Proportion differences are often used to estimate and test treatment effects in clinical trials with binary outcomes. In order to adjust for other covariates or intra-subject correlation among repeated measures, logistic regression or longitudinal data analysis models such as generalized estimating equation or generalized linear mixed models may be used for the analyses. However, these analysis models are often based on the logit link which results in parameter estimates and comparisons in the log-odds ratio scale rather than in the proportion difference scale. A two-step method is proposed in the literature to approximate the calculation of confidence intervals for the proportion difference using a concept of effective sample sizes. However, the performance of this two-step method has not been investigated in their paper. On this note, we examine the properties of the two-step method and propose an adjustment to the effective sample size formula based on Bayesian information theory. Simulations are conducted to evaluate the performance and to show that the modified effective sample size improves the coverage property of the confidence intervals.     

Data-randomized confidence intervals for discrete distributions

In practice non-randomized conservative confidence intervals for the parameter of a discrete distribution are used instead of the randomized uniformly most accurate intervals. We suggest in this paper that a part of the data be used as the random mechanism to create “data-randomized” confidence intervals. A thoughtful utilization of the data leads to intervals that are shorter than the usual conservative intervals but avoids the arbitrariness of the randomized uniformly most accurate intervals. Examples are given using the binomial, Poisson, and extended hypergeometric distributions, as well as applications to a metched case-control study and a randomized clinical trial.     

Analytical approximations for iterated bootstrap confidence intervals

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.     

Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals

We construct bootstrap confidence intervals for smoothing spline estimates based on Gaussian data, and penalized likelihood smoothing spline estimates based on data from .exponential families. Several vari- ations of bootstrap confidence intervals are considered and compared. We find that the commonly used ootstrap percentile intervals are inferior to the T intervals and to intervals based on bootstrap estimation of mean squared errors. The best variations of the bootstrap confidence intervals behave similar to the well known Bayesian confidence intervals. These bootstrap confidence intervals have an average coverage probability across the function being estimated, as opposed to a pointwise property.     

Short confidence intervals for variance components

Four approximate methods are proposed to construct confidence intervals for the estimation of variance components in unbalanced mixed models. The first three methods are modifications of the Wald, arithmetic and harmonic mean procedures, see Harville and Fenech (1985), while the fourth is an adaptive approach, combining the arithmetic and harmonic mean procedures. The performances of the proposed methods were assessed by a Monte Carlo simulation study. It was found that the intervals based on Wald's method maintained the nominal confidence levels across all designs and values of the parameters under study. On the other hand, the arithmetic (harmonic) mean method performed well for small (large) values of the variance component, relative to the error variance component. The adaptive procedure performed rather well except for extremely unbalanced designs. Further, compared with equal tails intervals, the intervals which use special tables, e.g., Table 678 of Tate and Klett (1959), provided adequate coverage while having much shorter lengths and are thus recommended for use in practice.     

A program to calculate bootstrap confidence intervals for the process capability index Cpk

Franklin and Wasserman (1991) introduced the use of Bootstrap sampling procedures for deriving nonparametric confidence intervals for the process capability index, C_pk, which are applicable for instances when at least twenty data points are available. This represents a significant reduction in the usually recommended sample requirement of 100 observations (see Gunther 1989). To facilitate and encourage the use of these procedures. a FORTRAN program is provided for computation of confidence intervals for C_pk. Three methods are provided for this calculation including the standard method, the percentile confidence interval, and the biased - corrected percentile confidence interval.     

Nonparametric confidence intervals for Tmax in sequence-stratified crossover studies

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.     

Asymptotic confidence intervals in ridge regression based on the Edgeworth expansion

Ridge Regression techniques have been found useful to reduce mean square errors of parameter estimates when multicollinearity is present. But the usefulness of the method rest not only upon its ability to produce good parameter estimates, with smaller mean squared error than Ordinary Least Squares, but also on having reasonable inferential procedures. The aim of this paper is to develop asymptotic confidence intervals for the model parameters based on Ridge Regression estimates and the Edgeworth expansion. Some simulation experiments are carried out to compare these confidence intervals with those obtained from the application of Ordinary Least Squares. Also, an example will be provided based on the well known data set of Hald.     

Simultaneous confidence intervals by iteratively adjusted alpha for relative effects in the one-way layout

A bootstrap based method to construct 1−α simultaneous confidence intervals for relative effects in the one-way layout is presented. This procedure takes the stochastic correlation between the test statistics into account and results in narrower simultaneous confidence intervals than the application of the Bonferroni correction. Instead of using the bootstrap distribution of a maximum statistic, the coverage of the confidence intervals for the individual comparisons are adjusted iteratively until the overall confidence level is reached. Empirical coverage and power estimates of the introduced procedure for many-to-one comparisons are presented and compared with asymptotic procedures based on the multivariate normal distribution.     

Estimating kurtosis and confidence intervals for the variance under nonnormality

Exact confidence intervals for variances rely on normal distribution assumptions. Alternatively, large-sample confidence intervals for the variance can be attained if one estimates the kurtosis of the underlying distribution. The method used to estimate the kurtosis has a direct impact on the performance of the interval and thus the quality of statistical inferences. In this paper the author considers a number of kurtosis estimators combined with large-sample theory to construct approximate confidence intervals for the variance. In addition, a nonparametric bootstrap resampling procedure is used to build bootstrap confidence intervals for the variance. Simulated coverage probabilities using different confidence interval methods are computed for a variety of sample sizes and distributions. A modification to a conventional estimator of the kurtosis, in conjunction with adjustments to the mean and variance of the asymptotic distribution of a function of the sample variance, improves the resulting coverage values for leptokurtically distributed populations.     

Assessment of clinical relevance by considering point estimates and associated confidence intervals

For the proof of efficacy of a new drug in a placebo‐controlled clinical trial it is not sufficient merely to demonstrate a statistically significant treatment difference. In recent years, regulatory authorities have strongly recommended assessing additionally whether the observed effect size is also of clinical relevance. This opinion is reflected in various guidelines which are of the utmost importance for the successful approval of a new drug. Clinical relevance can be investigated by responder analyses or by considering the point estimates on the original scale together with the associated confidence intervals. In this paper, we focus on the latter approach and discuss the suitability of different criteria which are commonly applied in medical research. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust generalized confidence intervals

Most interval estimates are derived from computable conditional distributions conditional on the data. In this article, we call the random variables having such conditional distributions confidence distribution variables and define their finite-sample breakdown values. Based on this, the definition of breakdown value of confidence intervals is introduced, which covers the breakdowns in both the coverage probability and interval length. High-breakdown confidence intervals are constructed by the structural method in location-scale families. Simulation results are presented to compare the traditional confidence intervals and their robust analogues.     

Better nonparametric bootstrap confidence intervals for the correlation coefficient

We respond to criticism leveled at bootstrap confidence intervals for the correlation coefficient by recent authors by arguing that in the correlation coefficient case, non–standard methods should be employed. We propose two such methods. The first is a bootstrap coverage coorection algorithm using iterated bootstrap techniques (Hall, 1986; Beran, 1987a; Hall and Martin, 1988) applied to ordinary percentile–method intervals (Efron, 1979), giving intervals with high coverage accuracy and stable lengths and endpoints. The simulation study carried out for this method gives results for sample sizes 8, 10, and 12 in three parent populations. The second technique involves the construction of percentile–t bootstrap confidence intervals for a transformed correlation coefficient, followed by an inversion of the transformation, to obtain “transformed percentile–t” intervals for the correlation coefficient. In particular, Fisher's z–transformation is used, and nonparametric delta method and jackknife variance estimates are used to Studentize the transformed correlation coefficient, with the jackknife–Studentized transformed percentile–t interval yielding the better coverage accuracy, in general. Percentile–t intervals constructed without first using the transformation perform very poorly, having large expected lengths and erratically fluctuating endpoints. The simulation study illustrating this technique gives results for sample sizes 10, 15 and 20 in four parent populations. Our techniques provide confidence intervals for the correlation coefficient which have good coverage accuracy (unlike ordinary percentile intervals), and stable lengths and endpoints (unlike ordinary percentile–t intervals).     

Simultaneous Confidence Intervals for Ratios of Fixed Effect Parameters in Linear Mixed Models

In multiple comparisons of fixed effect parameters in linear mixed models, treatment effects can be reported as relative changes or ratios. Simultaneous confidence intervals for such ratios had been previously proposed based on Bonferroni adjustments or multivariate normal quantiles accounting for the correlation among the multiple contrasts. We propose Fieller-type intervals using multivariate t quantiles and the application of Markov chain Monte Carlo techniques to sample from the joint posterior distribution and construct percentile-based simultaneous intervals. The methods are compared in a simulation study including bioassay problems with random intercepts and slopes, repeated measurements designs, and multicenter clinical trials.     

Wavelet-based confidence intervals for the self-similarity parameter

We propose and compare several methods of constructing wavelet-based confidence intervals for the self-similarity parameter in heavy-tailed observations. We use empirical coverage probabilities to assess the procedures by applying them to Linear Fractional Stable Motion with many choices of parameters. We find that the asymptotic confidence intervals provide empirical coverage often much lower than nominal. We recommend the use of resampling confidence intervals. We also propose a procedure for monitoring the constancy of the self-similarity parameter and apply it to Ethernet data sets.     

Prediction intervals for general balanced linear random models

The main interest of prediction intervals lies in the results of a future sample from a previously sampled population. In this article, we develop procedures for the prediction intervals which contain all of a fixed number of future observations for general balanced linear random models. Two methods based on the concept of a generalized pivotal quantity (GPQ) and one based on ANOVA estimators are presented. A simulation study using the balanced one-way random model is conducted to evaluate the proposed methods. It is shown that one of the two GPQ-based and the ANOVA-based methods are computationally more efficient and they also successfully maintain the simulated coverage probabilities close to the nominal confidence level. Hence, they are recommended for practical use. In addition, one example is given to illustrate the applicability of the recommended methods.