A Characterization of Most(More) Powerful Test Statistics with Simple Nonparametric Applications.

Data-driven most powerful tests are statistical hypothesis decision-making tools that deliver the greatest power against a fixed null hypothesis among all corresponding data-based tests of a given size. When the underlying data distributions are known, the likelihood ratio principle can be applied to conduct most powerful tests. Reversing this notion, we consider the following questions. (a) Assuming a test statistic, say T, is given, how can we transform T to improve the power of the test? (b) Can T be used to generate the most powerful test? (c) How does one compare test statistics with respect to an attribute of the desired most powerful decision-making procedure? To examine these questions, we propose one-to-one mapping of the term "most powerful" to the distribution properties of a given test statistic via matching characterization. This form of characterization has practical applicability and aligns well with the general principle of sufficiency. Findings indicate that to improve a given test, we can employ relevant ancillary statistics that do not have changes in their distributions with respect to tested hypotheses. As an example, the present method is illustrated by modifying the usual t-test under nonparametric settings. Numerical studies based on generated data and a real-data set confirm that the proposed approach can be useful in practice.

Influence function methods to assess the effectiveness of influenza vaccine with survey data.

OBJECTIVE: To examine a robust relative risk (RR) estimation for survey data analysis with ideal inferential properties under various model assumptions. DATA SOURCES: We employed secondary data from the Household Component of the 2000-2016 US Medical Expenditure Panel Survey (MEPS). STUDY DESIGN: We investigate a broad range of data-balancing techniques by implementing influence function (IF) methods, which allows us to easily estimate the variability for the RR estimates in the complex survey setting. We conduct a simulation study of seasonal influenza vaccine effectiveness to evaluate these approaches and discuss techniques that show robust inferential performance across model assumptions. DATA COLLECTION/EXTRACTION METHODS: Demographic information, vaccine status, and self-administered questionnaire surveys were obtained from the longitudinal data files. We linked this information with medical condition files and medical event to extract the disease type and associated expenditures for each medical visit. We excluded individuals who were 18 years or younger at the beginning of each panel. PRINCIPAL FINDINGS: Under various model assumptions, the IF methods show robust inferential performance when the data-balancing procedures are incorporated. Once IF methods and data-balancing techniques are implemented, contingency table-based RR estimation yields a comparable result to the generalized linear model approach. We demonstrate the applicability of the proposed methods for complex survey data using 2000-2016 MEPS data. When employing these methods, we find a significant, negative association between vaccine effectiveness (VE) estimates and influenza-incurred expenditures. CONCLUSIONS: We describe and demonstrate a robust method for RR estimation and relevant inferences for influenza vaccine effectiveness using MEPS data. The proposed method is flexible and can be extended to weighted data for survey data analysis. Hence, these methods have great potential for health services research, especially when data are nonexperimental and imbalanced.

The empirical likelihood prior applied to bias reduction of general estimating equations.

The practice of employing empirical likelihood (EL) components in place of parametric likelihood functions in the construction of Bayesian-type procedures has been well-addressed in the modern statistical literature. The EL prior, a Jeffreys-type prior, which asymptotically maximizes the Shannon mutual information between data and the parameters of interest, is rigorously derived. The focus of the proposed approach is on an integrated Kullback-Leibler distance between the EL-based posterior and prior density functions. The EL prior density is the density function for which the corresponding posterior form is asymptotically negligibly different from the EL. The proposed result can be used to develop a methodology for reducing the asymptotic bias of solutions of general estimating equations and M-estimation schemes by removing the first-order term. This technique is developed in a similar manner to methods employed to reduce the asymptotic bias of maximum likelihood estimates via penalizing the underlying parametric likelihoods by their Jeffreys invariant priors. A real data example related to a study of myocardial infarction illustrates the attractiveness of the proposed technique in practical aspects.

A sequential density-based empirical likelihood ratio test for treatment effects.

In health-related experiments, treatment effects can be identified using paired data that consist of pre- and posttreatment measurements. In this framework, sequential testing strategies are widely accepted statistical tools in practice. Since performances of parametric sequential testing procedures vitally depend on the validity of the parametric assumptions regarding underlying data distributions, we focus on distribution-free mechanisms for sequentially evaluating treatment effects. In fixed sample size designs, the density-based empirical likelihood (DBEL) methods provide powerful nonparametric approximations to optimal Neyman-Pearson-type statistics. In this article, we extend the DBEL methodology to develop a novel sequential DBEL testing procedure for detecting treatment effects based on paired data. The asymptotic consistency of the proposed test is shown. An extensive Monte Carlo study confirms that the proposed test outperforms the conventional sequential Wilcoxon signed-rank test across a variety of alternatives. The excellent applicability of the proposed method is exemplified using the ventilator-associated pneumonia study that evaluates the effect of Chlorhexidine Gluconate treatment in reducing oral colonization by pathogens in ventilated patients.

A critical issue of using the variance of the total in the linearization method - In the context of unequal probability sampling.

Publicly available national survey data are useful for the evidence-based research to advance our understanding of important questions in the health and biomedical sciences. Appropriate variance estimation is a crucial step to evaluate the strength of evidence in the data analysis. In survey data analysis, the conventional linearization method for estimating the variance of a statistic of interest uses the variance estimator of the total based on linearized variables. We warn that this common practice may result in undesirable consequences such as susceptibility to data shift and severely inflated variance estimates, when unequal weights are incorporated into variance estimation. We propose to use the variance estimator of the mean (mean-approach) instead of the variance estimator of the total (total-approach). We show a superiority of the mean-approach through analytical investigations. A real data example (the National Comorbidity Survey Replication) and simulation-based studies strongly support our conclusion.

Empirical likelihood ratio tests with power one.

In the 1970s, Professor Robbins and his coauthors extended the Vile and Wald inequality in order to derive the fundamental theoretical results regarding likelihood ratio based sequential tests with power one. The law of the iterated logarithm confirms an optimal property of the power one tests. In parallel with Robbins's decision-making procedures, we propose and examine sequential empirical likelihood ratio (ELR) tests with power one. In this setting, we develop the nonparametric one- and two-sided ELR tests. It turns out that the proposed sequential ELR tests significantly outperform the classical nonparametric t-statistic-based counterparts in many scenarios based on different underlying data distributions.

To t-Test or Not to t-Test? A p-Values-Based Point of View in the Receiver Operating Characteristic Curve Framework.

A common statistical doctrine supported by many introductory courses and textbooks is that t-test type procedures based on normally distributed data points are anticipated to provide a standard in decision-making. In order to motivate scholars to examine this convention, we introduce a simple approach based on graphical tools of receiver operating characteristic (ROC) curve analysis, a well-established biostatistical methodology. In this context, we propose employing a p-values-based method, taking into account the stochastic nature of p-values. We focus on the modern statistical literature to address the expected p-value (EPV) as a measure of the performance of decision-making rules. During the course of our study, we extend the EPV concept to be considered in terms of the ROC curve technique. This provides expressive evaluations and visualizations of a wide spectrum of testing mechanisms' properties. We show that the conventional power characterization of tests is a partial aspect of the presented EPV/ROC technique. We desire that this explanation of the EPV/ROC approach convinces researchers of the usefulness of the EPV/ROC approach for depicting different characteristics of decision-making procedures, in light of the growing interest regarding correct p-values-based applications.

A Cautionary Note on Beta Families of Distributions and the Aliases Within.

In this note we examine the four parameter beta family of distributions in the context of the beta-normal and beta-logistic distributions. In the process we highlight the concept of numerical and limiting alias distributions, which in turn relate to numerical instabilities in the numerical maximum likelihood fitting routines for these families of distributions. We conjecture that the numerical issues pertaining to fitting these multiparameter distributions may be more widespread than has originally been reported across several families of distributions.

Expected p-values in light of an ROC curve analysis applied to optimal multiple testing procedures.

Many statistical studies report p-values for inferential purposes. In several scenarios, the stochastic aspect of p-values is neglected, which may contribute to drawing wrong conclusions in real data experiments. The stochastic nature of p-values makes their use to examine the performance of given testing procedures or associations between investigated factors to be difficult. We turn our focus on the modern statistical literature to address the expected p-value (EPV) as a measure of the performance of decision-making rules. During the course of our study, we prove that the EPV can be considered in the context of receiver operating characteristic (ROC) curve analysis, a well-established biostatistical methodology. The ROC-based framework provides a new and efficient methodology for investigating and constructing statistical decision-making procedures, including: (1) evaluation and visualization of properties of the testing mechanisms, considering, e.g. partial EPVs; (2) developing optimal tests via the minimization of EPVs; (3) creation of novel methods for optimally combining multiple test statistics. We demonstrate that the proposed EPV-based approach allows us to maximize the integrated power of testing algorithms with respect to various significance levels. In an application, we use the proposed method to construct the optimal test and analyze a myocardial infarction disease dataset. We outline the usefulness of the "EPV/ROC" technique for evaluating different decision-making procedures, their constructions and properties with an eye towards practical applications.

A density based empirical likelihood approach for testing bivariate normality.

Sample entropy based tests, methods of sieves and Grenander estimation type procedures are known to be very efficient tools for assessing normality of underlying data distributions, in one-dimensional nonparametric settings. Recently, it has been shown that the density based empirical likelihood (EL) concept extends and standardizes these methods, presenting a powerful approach for approximating optimal parametric likelihood ratio test statistics, in a distribution-free manner. In this paper, we discuss difficulties related to constructing density based EL ratio techniques for testing bivariate normality and propose a solution regarding this problem. Toward this end, a novel bivariate sample entropy expression is derived and shown to satisfy the known concept related to bivariate histogram density estimations. Monte Carlo results show that the new density based EL ratio tests for bivariate normality behave very well for finite sample sizes. In order to exemplify the excellent applicability of the proposed approach, we demonstrate a real data example related to a study of biomarkers associated with myocardial infarction.

Dependence and independence: Structure and inference.

Evaluations of relationships between pairs of variables, including testing for independence, are increasingly important. Erich Leo Lehmann noted that "the study of the power and efficiency of tests of independence is complicated by the difficulty of defining natural classes of alternatives to the hypothesis of independence." This paper presents a general review, discussion and comparison of classical and novel tests of independence. We investigate a broad spectrum of dependence structures with/without random effects, including those that are well addressed in both the applied and the theoretical scientific literatures as well as scenarios when the classical tests of independence may break down completely. Motivated by practical considerations, the impact of random effects in dependence structures are studied in the additive and multiplicative forms. A novel index of dependence is proposed based on the area under the Kendall plot. In conjunction with the scatterplot and the Kendall plot, the proposed method provides a comprehensive presentation of the data in terms of graphing and conceptualizing the dependence. We also present a graphical methodology based on heat maps to effectively compare the powers of various tests. Practical examples illustrate the use of various tests of independence and the graphical representations of dependence structures.

Bayesian Empirical Likelihood Methods for Quantile Comparisons.

Bayes factors, practical tools of applied statistics, have been dealt with extensively in the literature in the context of hypothesis testing. The Bayes factor based on parametric likelihoods can be considered both as a pure Bayesian approach as well as a standard technique to compute p-values for hypothesis testing. We employ empirical likelihood methodology to modify Bayes factor type procedures for the nonparametric setting. The paper establishes asymptotic approximations to the proposed procedures. These approximations are shown to be similar to those of the classical parametric Bayes factor approach. The proposed approach is applied towards developing testing methods involving quantiles, which are commonly used to characterize distributions. We present and evaluate one and two sample distribution free Bayes factor type methods for testing quantiles based on indicators and smooth kernel functions. An extensive Monte Carlo study and real data examples show that the developed procedures have excellent operating characteristics for one-sample and two-sample data analysis.

Easy and accurate variance estimation of the nonparametric estimator of the partial area under the ROC curve and its application.

The receiver operating characteristic (ROC) curve is a popular technique with applications, for example, investigating an accuracy of a biomarker to delineate between disease and non-disease groups. A common measure of accuracy of a given diagnostic marker is the area under the ROC curve (AUC). In contrast with the AUC, the partial area under the ROC curve (pAUC) looks into the area with certain specificities (i.e., true negative rate) only, and it can be often clinically more relevant than examining the entire ROC curve. The pAUC is commonly estimated based on a U-statistic with the plug-in sample quantile, making the estimator a non-traditional U-statistic. In this article, we propose an accurate and easy method to obtain the variance of the nonparametric pAUC estimator. The proposed method is easy to implement for both one biomarker test and the comparison of two correlated biomarkers because it simply adapts the existing variance estimator of U-statistics. In this article, we show accuracy and other advantages of the proposed variance estimation method by broadly comparing it with previously existing methods. Further, we develop an empirical likelihood inference method based on the proposed variance estimator through a simple implementation. In an application, we demonstrate that, depending on the inferences by either the AUC or pAUC, we can make a different decision on a prognostic ability of a same set of biomarkers. Copyright © 2016 John Wiley & Sons, Ltd.

Measures of biomarker dependence using a copula-based multivariate epsilon-skew-normal family of distributions.

In this note we develop a new multivariate copula model based on epsilon-skew-normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon-skew-normal distribution.

A toolkit for clinical statisticians to fix problems based on biomarker measurements subject to instrumental limitations: from repeated measurement techniques to a hybrid pooled-unpooled design.

The aim of this chapter is to review and examine different methods in order to display correct and efficient statistical techniques based on complete/incomplete data subject to different sorts of measurement error (ME) problems. Instrument inaccuracies, biological variations, and/or errors in questionnaire-based self-report data can lead to significant MEs in various clinical experiments. Ignoring MEs can cause bias or inconsistency of statistical inferences. The biostatistical literature well addresses two categories of MEs: errors related to additive models and errors caused by the limit of detection (LOD). Several statistical approaches have been developed to analyze data affected by MEs, including the parametric/nonparametric likelihood methodologies, Bayesian methods, the single and multiple imputation techniques, and the repeated measurement design of experiment. We present a novel hybrid pooled-unpooled design as one of the strategies to provide correct statistical inferences when data is subject to MEs. This hybrid design and the classical techniques are compared to show the advantages and disadvantages of the considered methods.

A Simple Density-Based Empirical Likelihood Ratio Test for Independence.

We develop a novel nonparametric likelihood ratio test for independence between two random variables using a technique that is free of the common constraints of defining a given set of specific dependence structures. Our methodology revolves around an exact density-based empirical likelihood ratio test statistic that approximates in a distribution-free fashion the corresponding most powerful parametric likelihood ratio test. We demonstrate that the proposed test is very powerful in detecting general structures of dependence between two random variables, including non-linear and/or random-effect dependence structures. An extensive Monte Carlo study confirms that the proposed test is superior to the classical nonparametric procedures across a variety of settings. The real-world applicability of the proposed test is illustrated using data from a study of biomarkers associated with myocardial infarction.

Evaluations and comparisons of treatment effects based on best combinations of biomarkers with applications to biomedical studies.

Many clinical and biomedical studies evaluate treatment effects based on multiple biomarkers that commonly consist of pre- and post-treatment measurements. Some biomarkers can show significant positive treatment effects, while other biomarkers can reflect no effects or even negative effects of the treatments, giving rise to a necessity to develop methodologies that may correctly and efficiently evaluate the treatment effects based on multiple biomarkers as a whole. In the setting of pre- and post-treatment measurements of multiple biomarkers, we propose to apply a receiver operating characteristic (ROC) curve methodology based on the best combination of biomarkers maximizing the area under the receiver operating characteristic curve (AUC)-type criterion among all possible linear combinations. In the particular case with independent pre- and post-treatment measurements, we show that the proposed method represents the well-known Su and Liu's (1993) result. Further, proceeding from derived best combinations of biomarkers' measurements, we propose an efficient technique via likelihood ratio tests to compare treatment effects. We show an extensive Monte Carlo study that confirms the superiority of the proposed test in comparison with treatment effects based on multiple biomarkers in a paired data setting. For practical applications, the proposed method is illustrated with a randomized trial of chlorhexidine gluconate on oral bacterial pathogens in mechanically ventilated patients as well as a treatment study for children with attention deficit-hyperactivity disorder and severe mood dysregulation.

Empirical Likelihood Approaches to Two-Group Comparisons of Upper Quantiles Applied to Biomedical Data.

In many biomedical studies, a difference in upper quantiles is of specific interest since the upper quantile represents the upper range of biomarkers and/or is used as the cut-off value for a disease classification. In this article, we investigate two-group comparisons of an upper quantile based on the empirical likelihood methodology. Two approaches, the classical empirical likelihood and 'plug-in' empirical likelihood are used to construct the test statistics and their properties are theoretically investigated. Although the plug-in method is developed by the frame work of the empirical likelihood, the test statistic is not based on the maximization of the empirical likelihood, and is simplified by using indicator function in its construction, making it a unique test to investigate. Extensive simulation results demonstrate that the 'plug-in' empirical likelihood approach performs better to compare upper quantiles across various underlying distributions and sample sizes. For the actual application, we employ the developed methods to test the differences in upper quantiles in two different studies, the oral colonization of pneumonia pathogens for intensive care unit patients treated by two different oral treatments, and the biomarker expressions of normal and abnormal bronchial epithelial cells.

Computing Critical Values of Exact Tests by Incorporating Monte Carlo Simulations Combined with Statistical Tables.

Various exact tests for statistical inference are available for powerful and accurate decision rules provided that corresponding critical values are tabulated or evaluated via Monte Carlo methods. This article introduces a novel hybrid method for computing p-values of exact tests by combining Monte Carlo simulations and statistical tables generated a priori. To use the data from Monte Carlo generations and tabulated critical values jointly, we employ kernel density estimation within Bayesian-type procedures. The p-values are linked to the posterior means of quantiles. In this framework, we present relevant information from the Monte Carlo experiments via likelihood-type functions, whereas tabulated critical values are used to reflect prior distributions. The local maximum likelihood technique is employed to compute functional forms of prior distributions from statistical tables. Empirical likelihood functions are proposed to replace parametric likelihood functions within the structure of the posterior mean calculations to provide a Bayesian-type procedure with a distribution-free set of assumptions. We derive the asymptotic properties of the proposed nonparametric posterior means of quantiles process. Using the theoretical propositions, we calculate the minimum number of needed Monte Carlo resamples for desired level of accuracy on the basis of distances between actual data characteristics (e.g. sample sizes) and characteristics of data used to present corresponding critical values in a table. The proposed approach makes practical applications of exact tests simple and rapid. Implementations of the proposed technique are easily carried out via the recently developed STATA and R statistical packages.

Exact Inference for the Dispersion Matrix.

We develop a new and novel exact permutation test for prespecified correlation structures such as compound symmetry or spherical structures under standard assumptions. The key feature of the work contained in this note is the distribution free aspect of our procedures that frees us from the standard and sometimes unrealistic multivariate normality constraint commonly needed for other methods.