Exact Interval Estimation Research Articles

Inference for Projection-Based Wasserstein Distances on Finite Spaces

The Wasserstein distance is a distance between two probability distributions and has recently gained increasing popularity in statistics and machine learning, owing to its attractive properties. One important approach to extending this distance is using low-dimensional projections of distributions to avoid a high computational cost and the curse of dimensionality in empirical estimation, such as the sliced Wasserstein or max-sliced Wasserstein distances. Despite their practical success in machine learning tasks, the availability of statistical inferences for projection-based Wasserstein distances is limited owing to the lack of distributional limit results. In this paper, we consider distances defined by integrating or maximizing Wasserstein distances between low-dimensional projections of two probability distributions. Then we derive limit distributions regarding these distances when the two distributions are supported on finite points. We also propose a bootstrap procedure to estimate quantiles of limit distributions from data. This facilitates asymptotically exact interval estimation and hypothesis testing for these distances. Our theoretical results are based on the arguments of Sommerfeld and Munk (2018) for deriving distributional limits regarding the original Wasserstein distance on finite spaces and the theory of sensitivity analysis in nonlinear programming. Finally, we conduct numerical experiments to illustrate the theoretical results and demonstrate the applicability of our inferential methods to real data analysis.

The person-time ratio distribution for the exact monitoring of adverse events: Historical vs surveillance Poisson data.

In the postmarket drug and vaccine safety surveillance, when the number of adverse events follows a Poisson distribution, the ratio between the exposed and the unexposed person-time information is the random variable that governs the decision rule about the safety of the drug or vaccine. The probability distribution function of such a ratio is derived in this paper. Exact point and interval estimators for the relative risk are discussed as well as statistical hypothesis testing. To the best of our knowledge, this is the first paper that provides an unbiased estimator for the relative risk based on the person-time ratio. The applicability of this new distribution is illustrated through a real data analysis aimed to detect increased risk of occurrence of Myocarditis/Pericarditis following mRNA COVID-19 vaccination in Manitoba, Canada.

Assessing standardized contrast effects in ANCOVA: Confidence intervals, precision evaluations, and sample size requirements.

Standardized effect sizes and confidence intervals are useful statistical assessments for comparing results across different studies when measurement units are not directly comparable. This paper aims to describe and compare confidence interval estimation methods for the standardized contrasts of treatment effects in ANCOVA designs. Sample size procedures are also presented to assure that the resulting confidence intervals yield informative estimation with adequate precision. Exact interval estimation approach has theoretical and empirical advantages in coverage probability and interval width over the approximate interval procedures. Numerical investigations of the existing method reveal that the omission of covariate variables has a negative impact on sample size calculations for precise interval estimation, especially when there is disparity in influential covariate variables. The proposed approaches and developed computer programs fully utilize covariate properties in interval estimation and provide accurate sample size determinations under the precision considerations of the expected interval width and the assurance probability of interval width.

Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring

Two estimation problems are studied based on the general progressively censored samples, and the distributions from the inverted scale family (ISF) are considered as prospective life distributions. One is the exact interval estimation for the unknown parameter θ , which is achieved by constructing the pivotal quantity. Through Monte Carlo simulations, the average 90 % and 95 % confidence intervals are obtained, and the validity of the above interval estimation is illustrated with a numerical example. The other is the estimation of R = P ( Y < X ) in the case of ISF. The maximum likelihood estimator (MLE) as well as approximate maximum likelihood estimator (AMLE) is obtained, together with the corresponding R-symmetric asymptotic confidence intervals. With Bootstrap methods, we also propose two R-asymmetric confidence intervals, which have a good performance for small samples. Furthermore, assuming the scale parameters follow independent gamma priors, the Bayesian estimator as well as the HPD credible interval of R is thus acquired. Finally, we make an evaluation on the effectiveness of the proposed estimations through Monte Carlo simulations and provide an illustrative example of two real datasets.

Exact Confidence Limits on Some New Measures of Concordance and Discordance in Binary Outcomes.

Using a measure of agreement that does not distinguish the "positive" outcome from the "negative" outcome can be sometimes misleading in assessing resemblance. To alleviate this concern, some new indices, including the "positive" and "negative" conditional synchrony measures (CSM) (or the conditional discordant measures [CDM]), as well as their related measures, have been recently proposed elsewhere. We show that one can easily derive exact confidence limits for these new indices. Using Monte Carlo simulation, we find that the asymptotic interval estimator derived from the score test and these exact interval estimators can all perform well in a variety of situations, while the asymptotic interval estimator based on Wald's statistic can lose accuracy. We use the data taken from a cross-sectional validation study assessing the diagnostic performance of the Whooley questions for major depression disorder (MDD) among older adults to illustrate the use of these interval estimators developed here.

Exact interval estimators for some commonly used measures of binary agreement.

We develop exact interval estimators for some commonly used classical measures of agreement in binary responses. We apply Monte Carlo simulation to evaluate the performance of these estimators. When the measure of agreement is homogeneous, we note that extending the results presented here to accommodate stratified analysis is straightforward. We use the data taken from a survey studying the agreement of religious identifications and the data taken from a study assessing the diagnostic performance of Whooley questions for major depression disorder to illustrate the use of these interval estimators.

Notes on Interval Estimation Under the AB/BA Design in Multicenter Trials With Binary Responses

A multicenter trial is often applied to facilitate recruiting patients into a trial and enhance the generalizability of results to a population. Under a distribution-free random effects logistic regression model, we derive three asymptotic interval estimators and one exact interval estimator for the treatment effect when assuming no treatment-by-center interactions. We note that one may apply interval estimators published elsewhere to estimate the treatment effect based on the data from each center separately when there are treatment-by-center interactions. We employ Monte Carlo simulation to evaluate the performance of these interval estimators in a variety of situations. We have found that all interval estimators developed here can actually perform well with the coverage probability larger than or equal to the desired confidence level. We use the data taken from a multicenter trial comparing an active drug with a placebo on the electrocardiogram to illustrate the use of the interval estimators developed here.

Exact Confidence Limits on Some New Measures of Concordance and Discordance in Binary Outcomes

Using a measure of agreement that does not distinguish the "positive" outcome from the "negative" outcome can be sometimes misleading in assessing resemblance. To alleviate this concern, some new indices, including the "positive" and "negative" conditional synchrony measures (CSM) (or the conditional discordant measures [CDM]), as well as their related measures, have been recently proposed elsewhere. We show that one can easily derive exact confidence limits for these new indices. Using Monte Carlo simulation, we find that the asymptotic interval estimator derived from the score test and these exact interval estimators can all perform well in a variety of situations, while the asymptotic interval estimator based on Wald's statistic can lose accuracy. We use the data taken from a cross-sectional validation study assessing the diagnostic performance of the Whooley questions for major depression disorder (MDD) among older adults to illustrate the use of these interval estimators developed here.

Asymptotic and exact interval estimators of the common odds ratio under the sequential parallel comparison design.

When studying treatments for psychiatric or mental diseases in a placebo-controlled trial, we may consider use of the sequential parallel comparison design to reduce the number of patients needed through the reduction of the high placebo response rate. Under the assumption that the odds ratio of responses is constant between phases in the sequential parallel comparison design, we derive the conditional maximum likelihood estimator for the odds ratio. On the basis of the conditional likelihood, we further derive three asymptotic interval and an exact interval estimators for the odds ratio of responses. We employ Monte Carlo simulation to evaluate the performance of these interval estimators in a variety of situations. We find that the asymptotic interval and exact interval estimators developed here can all perform well. We use the double-blind, placebo-controlled study assessing the efficacy of a low dose of aripiprazole adjunctive to antidepressant therapy for treating patients with major depressive disorder to illustrate the use of these estimators.

Use of interval estimations in design and evaluation of multiregional clinical trials with continuous outcomes.

Multiregional clinical trials have been accepted in recent years as a useful means of accelerating the development of new drugs and abridging their approval time. The statistical properties of multiregional clinical trials are being widely discussed. In practice, variance of a continuous response may be different from region to region, but it leads to the assessment of the efficacy response falling into a Behrens-Fisher problem-there is no exact testing or interval estimator for mean difference with unequal variances. As a solution, this study applies interval estimations of the efficacy response based on Howe's, Cochran-Cox's, and Satterthwaite's approximations, which have been shown to have well-controlled type I error rates. However, the traditional sample size determination cannot be applied to the interval estimators. The sample size determination to achieve a desired power based on these interval estimators is then presented. Moreover, the consistency criteria suggested by the Japanese Ministry of Health, Labour and Welfare guidance to decide whether the overall results from the multiregional clinical trial obtained via the proposed interval estimation were also applied. A real example is used to illustrate the proposed method. The results of simulation studies indicate that the proposed method can correctly determine the required sample size and evaluate the assurance probability of the consistency criteria.

Precise confidence intervals of regression-based reference limits: Method comparisons and sample size requirements

Covariate-dependent reference limits have been extensively applied in biology and medicine for determining the substantial magnitude and relative importance of quantitative measurements. Confidence interval and sample size procedures are available for studying regression-based reference limits. However, the existing popular methods employ different technical simplifications and are applicable only in certain limited situations. This paper describes exact confidence intervals of regression-based reference limits and compares the exact approach with the approximate methods under a wide range of model configurations. Using the ratio between the widths of confidence interval and reference interval as the relative precision index, optimal sample size procedures are presented for precise interval estimation under expected ratio and tolerance probability considerations. Simulation results show that the approximate interval methods using normal distribution have inaccurate confidence limits. The exact confidence intervals dominate the approximate procedures in one- and two-sided coverage performance. Unlike the current simplifications, the proposed sample size procedures integrate all key factors including covariate features in the optimization process and are suitable for various regression-based reference limit studies with potentially diverse configurations. The exact interval estimation has theoretical and practical advantages over the approximate methods. The corresponding sample size procedures and computing algorithms are also presented to facilitate the data analysis and research design of regression-based reference limits.

Progression-Free Survival after Proton Therapy for Childhood Craniopharyngioma: Early Results From a Prospective Trial

Clinical trials that include proton therapy for children should be designed to compare disease control rates to those observed when using photon therapy. We designed a prospective phase II clinical trial to estimate progression-free for children with craniopharyngioma treated with proton therapy using a 5-mm clinical target volume (CTV) margin (NCT01419067) while monitoring for evidence that the 3-year PFS was unacceptably low compared to that observed for a historical cohort of patients treated with photon therapy (NCT00187226). The study accrued 94 children and young adults with craniopharyngioma who started proton therapy between 9/19/2011 and 02/22/2016 using the passive-scattering method at the University of Florida Health Proton Therapy Institute. The median age and range at the time of proton therapy was 9.4 years (range (2-20.2 years). Proton therapy was planned using an anatomically constrained CTV 5-mm margin surrounding the gross tumor volume (GTV) defined as residual tumor and post-operative tumor (when applicable) and 3-mm planning target volume margin. Patients were monitored once weekly using MR imaging. Cyst expansion required re-planning (n = 8), drainage (n = 4), or both (n = 2) to maintain coverage. The total prescribed dose was 54CGE in 30 fractions and the median elapsed days was 43 (range 40-50). Dated from the start of proton therapy, patients were followed at St. Jude Children’s Research Hospital in Memphis, Tennessee with clinical evaluation and MR imaging at baseline and every 3 months for the first year and then every 6 months through 5 years. Compliance with on-site follow-up was 98% and 869 on-site MRI evaluations contributed to this report. With a median follow-up 2.65 years (range 0.50-5.23 years), the progression-free survival estimated at 36 months was 97.8% ± 2.5%. More than 46 patients completed 3 years of follow-up at the time of this report. The results were compared to a cohort of 101 children with craniopharyngioma treated at St. Jude Children’s Research Hospital with photon-based conformal or intensity-modulated radiation therapy (54 Gy in 30 fractions) between 4/6/1998 and 1/22/2014. The progression-free survival estimate at 36 months was 96.0% ± 1.9% for the photon cohort. Using the targeting guidelines and proton therapy delivery methods described in this clinical trial, the progression-free survival estimates were comparable to those observed when using photon therapy. Prior reports from this series revealed similar risks of severe treatment-related complications. Final analysis will be performed 3 years after the last patient has been treated with proton therapy, with one additional year to calculate 95% exact confidence interval estimates of the binomial probability of completing three years free of failure and Kaplan-Meier estimates of PFS and overall survival.

Interval estimation for a lower-truncated distribution based on the double Type-II censored sample

ABSTRACTThe interval estimation problem is investigated for the parameters of a general lower truncated distribution under double Type-II censoring scheme. The exact, asymptotic and bootstrap interval estimates are derived for the unknown model parameter and the lower truncated threshold bound. One real-life example and a numerical study are presented to illustrate performance of our methods.

Notes on estimation in Poisson frequency data under an incomplete block crossover design

For comparison of two experimental treatments with a placebo under an incomplete block crossover design, we develop the weighted-least-squares estimator (WLSE) and the conditional maximum likelihood estimator (CMLE) of the relative treatment effects in Poisson frequency data. We further develop the interval estimator based on the WLSE, the interval estimator based on the CMLE, the interval estimator based on the conditional-likelihood-ratio test and the interval estimator based on the exact conditional distribution. Using Monte Carlo simulations, we find that all interval estimators developed here can perform well in a variety of situations. The exact interval estimator derived here can be especially of use when both the number of patients and the mean number of event occurrences are small in a trial. We use the data taken as part of a double-blind randomized crossover trial comparing salbutamol and salmeterol with a placebo with respect to the number of exacerbations in asthma patients to illustrate the use of these estimators.

Estimation of the treatment effect under an incomplete block crossover design in binary data - A conditional likelihood approach.

A random effects logistic regression model is proposed for an incomplete block crossover trial comparing three treatments when the underlying patient response is dichotomous. On the basis of the conditional distributions, the conditional maximum likelihood estimator for the relative effect between treatments and its estimated asymptotic standard error are derived. Asymptotic interval estimator and exact interval estimator are also developed. Monte Carlo simulation is used to evaluate the performance of these estimators. Both asymptotic and exact interval estimators are found to perform well in a variety of situations. When the number of patients is small, the exact interval estimator with assuring the coverage probability larger than or equal to the desired confidence level can be especially of use. The data taken from a crossover trial comparing the low and high doses of an analgesic with a placebo for the relief of pain in primary dysmenorrhea are used to illustrate the use of estimators and the potential usefulness of the incomplete block crossover design.

Testing equality and interval estimation of the generalized odds ratio in ordinal data under a three-period crossover design.

The crossover design can be of use to save the number of patients or improve power of a parallel groups design in studying treatments to noncurable chronic diseases. We propose using the generalized odds ratio for paired sample data to measure the relative effects in ordinal data between treatments and between periods. We show that one can apply the commonly used asymptotic and exact test procedures for stratified analysis in epidemiology to test non-equality of treatments in ordinal data, as well as obtain asymptotic and exact interval estimators for the generalized odds ratio under a three-period crossover design. We further show that one can apply procedures for testing the homogeneity of the odds ratio under stratified sampling to examine whether there are treatment-by-period interactions. We use the data taken from a three-period crossover trial studying the effects of low and high doses of an analgesic versus a placebo for the relief of pain in primary dysmenorrhea to illustrate the use of these test procedures and estimators proposed here.

Exact likelihood-based point and interval estimation for Laplace distribution based on Type-II right censored samples

In this paper, we first present explicit expressions for the maximum likelihood estimators (MLEs) of the location and scale parameters of the Laplace distribution based on a Type-II right censored sample under different cases. Next, we derive the joint moment generating function of the MLEs of the two parameters and use it to obtain the bias and mean squared error of the MLEs for all the cases. We then derive the exact density functions of the MLEs and utilize them to develop exact confidence intervals for the parameters. Next, a Monte Carlo simulation study is carried out to evaluate the performance of the developed inferential results. Finally, some examples are presented to illustrate the point and interval estimation methods.

Interval estimation for the Gini index of shifted exponential distribution under Type-II double censoring

Given a shifted exponential distribution, the exact sampling distribution of the Gini index has been derived by Moothathu (Ann Inst Stat Math 37: 473–479, 1985). However, direct derivation of confidence interval of the Gini index via inverting the sampling distribution requires highly intensive computational power. In this note, we propose an exact interval estimation procedure through the concept of generalized confidence interval introduced by Weerahandi (J Am Stat Assoc 88: 899–905, 1993). The proposed method is much more computationally efficient and can be readily extended for Type-II doubly censored data. The idea is further developed for the interval estimation of the difference of Gini indexes from two shifted exponential distributions. Numerical studies show that the generalized interval estimator is better than commonly used asymptotic approximation and bootstrap in term of coverage probability, particularly for small to median sample size. The proposed approaches are illustrated via an application to a previously published data set.

Notes on Interval Estimation of Odds Ratio in Matched Pairs under Stratified Sampling

We develop four asymptotic interval estimators and one exact interval estimator for the odds ratio (OR) under stratified random sampling with matched pairs. We apply Monte Carlo simulation to evaluate the performance of these five interval estimators. We note that the conditional score test-based interval estimator with a monotonic transformation and the interval estimator based on the Mantel–Haenszel (MH) type point estimator with the logarithmic transformation are generally preferable to the others considered here. We also note that the conditional exact confidence interval can be of use when the total number of matched pairs with discordant responses is small.

Accounting for uncertainty in willingness to pay for environmental benefits

Previous literature on the distribution of willingness to pay has focused on its heterogeneity distribution without addressing exact interval estimation. In this paper we derive and analyze Bayesian confidence sets for quantifying uncertainty in the determination of willingness to pay for carbon dioxide abatement. We use two empirical case studies: household decisions of energy-efficient heating versus insulation, and purchase decisions of ultra-low-emission vehicles. We first show that deriving credible sets using the posterior distribution of the willingness to pay is straightforward in the case of deterministic consumer heterogeneity. However, when using individual estimates, which is the case for the random parameters of the mixed logit model, it is complex to define the distribution of interest for the interval estimation problem. This latter problem is actually more involved than determining the moments of the heterogeneity distribution of the willingness to pay using frequentist econometrics. A solution that we propose is to derive and then summarize the distribution of point estimates of the individual willingness to pay under different loss functions.