Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases

A Comparison of Statistical Methods to Construct Confidence Intervals and Fiducial Intervals for Measures of Health Disparities

Health disparities are differences in health status across different socioeconomic groups. Classical methods, e.g., the Delta method, have been used to estimate the standard errors of estimated measures of health disparities and to construct confidence intervals for these measures. However, the confidence intervals constructed using the classical methods do not have good coverage properties for situations involving sparse data. In this article, we introduce three new methods to construct fiducial intervals for measures of health disparities based on approximate fiducial quantities. Through a comprehensive simulation study, We compare the empirical coverage properties of the proposed fiducial intervals against two Monte Carlo simulation-based methods-utilizing either a truncated Normal distribution or the Gamma distribution-as well as the classical method. The findings of the simulation study advocate for the adoption of the Monte Carlo simulation-based method with the Gamma distribution when a unified approach is sought for all health disparity measures.

A fiducial-based confidence interval for the linear combination of multinomial probabilities

Across a broad set of applications, system outcomes may be summarized as probabilities in confusion or contingency tables. In settings with more than two outcomes (e.g., stages of cancer), these outcomes represent multinomial experiments. Measures to summarize system performance have been presented as linear combinations of the resulting multinomial probabilities. Statistical inference on the linear combination of multinomial probabilities has been focused on large-sample and parametric settings and not small-sample settings. Such inference is valuable, however, especially in settings such as those resulting from pilot or low-cost studies. To address this gap, we leverage the fiducial approach to derive confidence intervals around the linear combination of multinomial parameters with desirable frequentist properties. One of the original arguments against the fiducial approach was its inability to extend to multiparameter settings. Therefore, the great novelty of this work is both the derived interval and the logical framework for applying the fiducial approach in multiparameter settings. Through simulation, we demonstrate that the proposed method maintains a minimum coverage of1 - α $1 - \alpha$ , unlike the bootstrap and large-sample methods, at comparable interval lengths. Finally, we illustrate its use in a medical problem of selecting classifiers for diagnosing chronic allograph nephropathy in postkidney transplant patients.

A fiducial approach for testing the non-inferiority of proportion difference in matched-pairs design

The non-inferiority of one treatment/drug to another is a common and important issue in medical and pharmaceutical fields. This study explored a fiducial approach for testing the non-inferiority of proportion difference in matched-pairs design. Approximate tests constructed using fiducial quantities with a combination of different parameters were proposed. Four simulation studies were employed to compare the performance of fiducial tests by comparing their type I errors and powers. The results showed that fiducial quantities with parameter 0.6 ≤ w 1 ≤ 0.8 $$ 0.6\le {w}_1\le 0.8 $$ performed satisfactorily from small to large samples. Therefore, the fiducial tests could be recommended for practical applications. The recommended fiducial tests might be a competitive alternative to other available tests. Three real data sets were analyzed to illustrate the proposed methods were competitive or even better than other tests.

Generalized fiducial methods for testing the homogeneity of a three-sample problem with a mixture structure

Recently, the likelihood ratio (LR) test was proposed to test the homogeneity of a three-sample model with a mixture structure. Because of the presence of the mixture structure, the null limiting distribution of the LR test has a complicated supremum form, which leads to challenges in determining p-values. In addition, the LR test cannot control type-I errors well under small to moderate sample size. In this paper, we propose seven generalized fiducial methods to test the homogeneity of the three-sample model. Via simulation studies, we find that our methods perform significantly better than the LR test method in controlling the type-I errors under small to moderate sample size, while they have comparable powers in most cases. A halibut data example is used to illustrate the proposed methods.

Two sample comparisons including zero-inflated continuous data: A parametric approach with applications to microarray experiment

Micro-array experiments are important fields in molecular biology where zero values mixed with a continuous outcome are frequently encountered leading to a mixed distribution with a clump at zero. Comparison of two mixed populations, e.g. of a control and a treated group; of two groups with different types of cancer, to name a few, are often encountered in these contexts. Fairly skewed distribution of the continuous part coupled with small sample sizes are issues of main concern to be attended for the quality of inference in such situations while popularly used nonparametric methods rely on asymptotic distribution of the underlying test statistics which are valid only under large sample sizes. We address the aforementioned issues via a newly proposed exact test for location-scale family distributions and Generalized pivot quantity (GPQ) based parametric test procedures for non-location-scale distributions. Simulation based assessment showed their superior performance with respect to size and power in comparison to the popular two-part tests (Wilcoxon rank sum, t-test, Kolmogrov-Smirnov, Ansari-Bradley and Sigel-Tukey) more prominently for small sample sizes.

Generalized Confidence Intervals and Fiducial Intervals for Some Epidemiological Measures

For binary outcome data from epidemiological studies, this article investigates the interval estimation of several measures of interest in the absence or presence of categorical covariates. When covariates are present, the logistic regression model as well as the log-binomial model are investigated. The measures considered include the common odds ratio (OR) from several studies, the number needed to treat (NNT), and the prevalence ratio. For each parameter, confidence intervals are constructed using the concepts of generalized pivotal quantities and fiducial quantities. Numerical results show that the confidence intervals so obtained exhibit satisfactory performance in terms of maintaining the coverage probabilities even when the sample sizes are not large. An appealing feature of the proposed solutions is that they are not based on maximization of the likelihood, and hence are free from convergence issues associated with the numerical calculation of the maximum likelihood estimators, especially in the context of the log-binomial model. The results are illustrated with a number of examples. The overall conclusion is that the proposed methodologies based on generalized pivotal quantities and fiducial quantities provide an accurate and unified approach for the interval estimation of the various epidemiological measures in the context of binary outcome data with or without covariates.

A nonparametric fiducial interval for the Youden index in multi-state diagnostic settings

The Youden index is a commonly employed metric to characterize the performance of a diagnostic test at its optimal point. For tests with three or more outcome classes, the Youden index has been extended; however, there are limited methods to compute a confidence interval (CI) about its value. Often, outcome classes are assumed to be normally distributed, which facilitates computational formulas for the CI bounds; however, many scenarios exist for which these assumptions cannot be made. In addition, many of these existing CI methods do not work well for small sample sizes. We propose a method to compute a nonparametric interval about the Youden index utilizing the fiducial argument. This fiducial interval ensures that CI coverage is met regardless of sample size, underlying distributional assumptions, or use of a complex classifier for diagnosis. Two alternate fiducial intervals are also considered. A simulation was conducted, which demonstrates the coverage and interval length for the proposed methods. Comparisons were made using no distributional assumptions on the outcome classes and for when outcomes were assumed to be normally distributed. In general, coverage probability was consistently met, and interval length was reasonable. The proposed fiducial method was also demonstrated in data examining biomarkers in subjects to predict diagnostic stages ranging from normal kidney function to chronic allograph nephropathy. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.

Inference for Surrogate Endpoint Validation in the Binary Case

Surrogate endpoint validation for a binary surrogate endpoint and a binary true endpoint is investigated using the criteria of proportion explained (PE) and the relative effect (RE). The concepts of generalized confidence intervals and fiducial intervals are used for computing confidence intervals for PE and RE. The numerical results indicate that the proposed confidence intervals are satisfactory in terms of coverage probability, whereas the intervals based on Fieller's theorem and the delta method fall short in this regard. Our methodology can also be applied to interval estimation problems in a causal inference-based approach to surrogate endpoint validation.

Closed-form fiducial confidence intervals for some functions of independent binomial parameters with comparisons

Approximate closed-form confidence intervals (CIs) for estimating the difference, relative risk, odds ratio, and linear combination of proportions are proposed. These CIs are developed using the fiducial approach and the modified normal-based approximation to the percentiles of a linear combination of independent random variables. These confidence intervals are easy to calculate as the computation requires only the percentiles of beta distributions. The proposed confidence intervals are compared with the popular score confidence intervals with respect to coverage probabilities and expected widths. Comparison studies indicate that the proposed confidence intervals are comparable with the corresponding score confidence intervals, and better in some cases, for all the problems considered. The methods are illustrated using several examples.