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.

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.

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 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.

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.

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.

Density-based empirical likelihood procedures for testing symmetry of data distributions and K-sample comparisons.

In practice, parametric likelihood-ratio techniques are powerful statistical tools. In this article, we propose and examine novel and simple distribution-free test statistics that efficiently approximate parametric likelihood ratios to analyze and compare distributions of K groups of observations. Using the density-based empirical likelihood methodology, we develop a Stata package that applies to a test for symmetry of data distributions and compares K-sample distributions. Recognizing that recent statistical software packages do not sufficiently address K-sample nonparametric comparisons of data distributions, we propose a new Stata command, vxdbel, to execute exact density-based empirical likelihood-ratio tests using K samples. To calculate p-values of the proposed tests, we use the following methods: 1) a classical technique based on Monte Carlo p-value evaluations; 2) an interpolation technique based on tabulated critical values; and 3) a new hybrid technique that combines methods 1 and 2. The third, cutting-edge method is shown to be very efficient in the context of exact-test p-value computations. This Bayesian-type method considers tabulated critical values as prior information and Monte Carlo generations of test statistic values as data used to depict the likelihood function. In this case, a nonparametric Bayesian method is proposed to compute critical values of exact tests.

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.

Nonparametric Bayes Factors Based On Empirical Likelihood Ratios.

Bayes methodology provides posterior distribution functions based on parametric likelihoods adjusted for prior distributions. A distribution-free alternative to the parametric likelihood is use of empirical likelihood (EL) techniques, well known in the context of nonparametric testing of statistical hypotheses. Empirical likelihoods have been shown to exhibit many of the properties of conventional parametric likelihoods. In this article, we propose and examine Bayes factors (BF) methods that are derived via the EL ratio approach. Following Kass & Wasserman [10], we consider Bayes factors type decision rules in the context of standard statistical testing techniques. We show that the asymptotic properties of the proposed procedure are similar to the classical BF's asymptotic operating characteristics. Although we focus on hypothesis testing, the proposed approach also yields confidence interval estimators of unknown parameters. Monte Carlo simulations were conducted to evaluate the theoretical results as well as to demonstrate the power of the proposed test.

An Exact Density-Based Empirical Likelihood Ratio Test for Paired Data.

The Wilcoxon rank-sum test and its variants are historically well-known to be very powerful nonparametric decision rules for testing no location difference between two groups given paired data versus a shift alternative. In this article, we propose a new alternative empirical likelihood (EL) ratio approach for testing the equality of marginal distributions given that sampling is from a continuous bivariate population. We show that in various shift alternative scenarios the proposed exact test is superior to the classic nonparametric procedures, which may break down completely or are frequently inferior to the density-based EL ratio test. This is particularly true in the cases where there is a non-constant shift under the alternative or the data distributions are skewed. An extensive Monte Carlo study shows that the proposed test has excellent operating characteristics. We apply the density-based EL ratio test to analyze real data from two medical studies.

Two-sample density-based empirical likelihood ratio tests based on paired data, with application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation.

It is a common practice to conduct medical trials to compare a new therapy with a standard-of-care based on paired data consisted of pre- and post-treatment measurements. In such cases, a great interest often lies in identifying treatment effects within each therapy group and detecting a between-group difference. In this article, we propose exact nonparametric tests for composite hypotheses related to treatment effects to provide efficient tools that compare study groups utilizing paired data. When correctly specified, parametric likelihood ratios can be applied, in an optimal manner, to detect a difference in distributions of two samples based on paired data. The recent statistical literature introduces density-based empirical likelihood methods to derive efficient nonparametric tests that approximate most powerful Neyman-Pearson decision rules. We adapt and extend these methods to deal with various testing scenarios involved in the two-sample comparisons based on paired data. We show that the proposed procedures outperform classical approaches. An extensive Monte Carlo study confirms that the proposed approach is powerful and can be easily applied to a variety of testing problems in practice. The proposed technique is applied for comparing two therapy strategies to treat children's attention deficit/hyperactivity disorder and severe mood dysregulation.

Estimation and testing based on data subject to measurement errors: from parametric to non-parametric likelihood methods.

Measurement error (ME) problems can cause bias or inconsistency of statistical inferences. When investigators are unable to obtain correct measurements of biological assays, special techniques to quantify MEs need to be applied. Sampling based on repeated measurements is a common strategy to allow for ME. This method has been well addressed in the literature under parametric assumptions. The approach with repeated measures data may not be applicable when the replications are complicated because of cost and/or time concerns. Pooling designs have been proposed as cost-efficient sampling procedures that can assist to provide correct statistical operations based on data subject to ME. We demonstrate that a mixture of both pooled and unpooled data (a hybrid pooled-unpooled design) can support very efficient estimation and testing in the presence of ME. Nonparametric techniques have not been well investigated to analyze repeated measures data or pooled data subject to ME. We propose and examine both the parametric and empirical likelihood methodologies for data subject to ME. We conclude that the likelihood methods based on the hybrid samples are very efficient and powerful. The results of an extensive Monte Carlo study support our conclusions. Real data examples demonstrate the efficiency of the proposed methods 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.

Two-sample density-based empirical likelihood tests for incomplete data in application to a pneumonia study.

In clinical trials examining the incidence of pneumonia it is a common practice to measure infection via both invasive and non-invasive procedures. In the context of a recently completed randomized trial comparing two treatments the invasive procedure was only utilized in certain scenarios due to the added risk involved, and given that the level of the non-invasive procedure surpassed a given threshold. Hence, what was observed was bivariate data with a pattern of missingness in the invasive variable dependent upon the value of the observed non-invasive observation within a given pair. In order to compare two treatments with bivariate observed data exhibiting this pattern of missingness we developed a semi-parametric methodology utilizing the density-based empirical likelihood approach in order to provide a non-parametric approximation to Neyman-Pearson-type test statistics. This novel empirical likelihood approach has both a parametric and non-parametric components. The non-parametric component utilizes the observations for the non-missing cases, while the parametric component is utilized to tackle the case where observations are missing with respect to the invasive variable. The method is illustrated through its application to the actual data obtained in the pneumonia study and is shown to be an efficient and practical method.

ROC curve inference for best linear combination of two biomarkers subject to limits of detection.

The receiver operating characteristic (ROC) curve is a tool commonly used to evaluate biomarker utility in clinical diagnosis of disease. Often, multiple biomarkers are developed to evaluate the discrimination for the same outcome. Levels of multiple biomarkers can be combined via best linear combination (BLC) such that their overall discriminatory ability is greater than any of them individually. Biomarker measurements frequently have undetectable levels below a detection limit sometimes denoted as limit of detection (LOD). Ignoring observations below the LOD or substituting some replacement value as a method of correction has been shown to lead to negatively biased estimates of the area under the ROC curve for some distributions of single biomarkers. In this paper, we develop asymptotically unbiased estimators, via the maximum likelihood technique, of the area under the ROC curve of BLC of two bivariate normally distributed biomarkers affected by LODs. We also propose confidence intervals for this area under curve. Point and confidence interval estimates are scrutinized by simulation study, recording bias and root mean square error and coverage probability, respectively. An example using polychlorinated biphenyl (PCB) levels to classify women with and without endometriosis illustrates the potential benefits of our methods.

Exact confidence interval estimation for the difference in diagnostic accuracy with three ordinal diagnostic groups.

In the cases with three ordinal diagnostic groups, the important measures of diagnostic accuracy are the volume under surface (VUS) and the partial volume under surface (PVUS) which are the extended forms of the area under curve (AUC) and the partial area under curve (PAUC). This article addresses confidence interval estimation of the difference in paired VUS s and the difference in paired PVUS s. To focus especially on studies with small to moderate sample sizes, we propose an approach based on the concepts of generalized inference. A Monte Carlo study demonstrates that the proposed approach generally can provide confidence intervals with reasonable coverage probabilities even at small sample sizes. The proposed approach is compared to a parametric bootstrap approach and a large sample approach through simulation. Finally, the proposed approach is illustrated via an application to a data set of blood test results of anemia patients.

Linear regression with an independent variable subject to a detection limit.

BACKGROUND: Linear regression with a left-censored independent variable X due to limit of detection (LOD) was recently considered by 2 groups of researchers: Richardson and Ciampi (Am J Epidemiol. 2003;157:355-363), and Schisterman et al (Am J Epidemiol. 2006;163:374-383). METHODS: Both groups obtained consistent estimators for the regression slopes by replacing left-censored X with a constant, that is, the expectation of X given X below LOD E(X|X

Empirical and parametric likelihood interval estimation for populations with many zero values: application for assessing environmental chemical concentrations and reproductive health.

BACKGROUND: Understanding the health effects associated with environmental chemicals is challenging when individuals have concentrations at or below the laboratory limits of detection as well as when the values may round to zero or are presented in the form of 0 to substitute for missing values, which may result in many zeros in the database. Comparison of mean concentrations between individuals with and without disease necessitates estimation procedures that allow for data with many zero values. The main aim of this article is to propose and examine parametric and distribution-free methods for comparing data sets containing many zero observations. An important application of this approach is related to assessing environmental chemical concentrations and reproductive health. METHODS: We extended the empirical likelihood technique for estimating confidence intervals (CIs) in data sets with many zeros. We examined the proposed empirical likelihood interval estimations via a broad Monte Carlo study that compares the proposed method with parametric techniques. Certain characteristics of Monte Carlo simulations were chosen to be close to parameters of the real data set. We applied the method to a cohort study comprising 84 women aged 18-40 years who had undergone laparoscopy between 1999 and 2000 in whom serum concentrations of 2 organochlorine pesticides--Aldrin and beta-Benzene hexachloride (ß-BHC) were measured using gas chromatography with electron capture. RESULTS: When applied to the cohort study, the method produced efficient 95% CIs, allowing for the comparison of mean serum Aldrin concentrations for women with and without endometriosis (0.000338, 0.003561) and (0.000803, 0.004211), respectively. Mean ß-BHC concentrations also could be compared (0.000493, 0.005869) and (0.000680, 0.003807) based on individuals with and without the disease, respectively. Differences in mean concentrations for Aldrin and ß-BHC could be estimated (-0.001563, 0.003025) and (-0.003522, 0.002890), respectively. CONCLUSIONS: We found the empirical likelihood method for estimating CIs robust when data sets contain many zeros. In so doing, mean concentrations of Aldrin or ß-BHC did not differ by endometriosis diagnosis.

Analyzing incomplete data subject to a threshold using empirical likelihood methods: an application to a pneumonia risk study in an ICU setting.

The initial detection of ventilator-associated pneumonia (VAP) for inpatients at an intensive care unit needs composite symptom evaluation using clinical criteria such as the clinical pulmonary infection score (CPIS). When CPIS is above a threshold value, bronchoalveolar lavage (BAL) is performed to confirm the diagnosis by counting actual bacterial pathogens. Thus, CPIS and BAL results are closely related and both are important indicators of pneumonia whereas BAL data are incomplete. To compare the pneumonia risks among treatment groups for such incomplete data, we derive a method that combines nonparametric empirical likelihood ratio techniques with classical testing for parametric models. This technique augments the study power by enabling us to use any observed data. The asymptotic property of the proposed method is investigated theoretically. Monte Carlo simulations confirm both the asymptotic results and good power properties of the proposed method. The method is applied to the actual data obtained in clinical practice settings and compares VAP risks among treatment groups.

Hybrid pooled-unpooled design for cost-efficient measurement of biomarkers.

Evaluating biomarkers in epidemiological studies can be expensive and time consuming. Many investigators use techniques such as random sampling or pooling biospecimens in order to cut costs and save time on experiments. Commonly, analyses based on pooled data are strongly restricted by distributional assumptions that are challenging to validate because of the pooled biospecimens. Random sampling provides data that can be easily analyzed. However, random sampling methods are not optimal cost-efficient designs for estimating means. We propose and examine a cost-efficient hybrid design that involves taking a sample of both pooled and unpooled data in an optimal proportion in order to efficiently estimate the unknown parameters of the biomarker distribution. In addition, we find that this design can be used to estimate and account for different types of measurement and pooling error, without the need to collect validation data or repeated measurements. We show an example where application of the hybrid design leads to minimization of a given loss function based on variances of the estimators of the unknown parameters. Monte Carlo simulation and biomarker data from a study on coronary heart disease are used to demonstrate the proposed methodology.