Minge Xie

Temporal Relationship between Vitamin D Status and Parathyroid Hormone in the United States

Background Interpretation of parathyroid hormone (iPTH) requires knowledge of vitamin D status that is influenced by season. Objective Characterize the temporal relationship between 25-hydroxyvitamin D3 levels [25(OH)D3] and intact iPTH for several seasons, by gender and latitude in the U.S. and relate 25-hydrovitamin D2 [25(OH)D2] levels with PTH levels and total 25(OH)D levels. Method We retrospectively determined population weekly-mean concentrations of unpaired [25(OH)D2 and 25(OH)D3] and iPTH using 3.8 million laboratory results of adults. The 25(OH)D3 and iPTH distributions were normalized and the means fit with a sinusoidal function for both gender and latitudes: North >40, Central 32–40 and South <32 degrees. We analyzed PTH and total 25(OH)D separately in samples with detectable 25(OH)D2 (≥4 ng/mL). Findings Seasonal variation was observed for all genders and latitudes. 25(OH)D3 peaks occurred in September and troughs in March. iPTH levels showed an inverted pattern of peaks and troughs relative to 25(OH)D3, with a delay of 4 weeks. Vitamin D deficiency and insufficiency was common (33% <20 ng/mL; 60% <30 ng/mL) as was elevated iPTH levels (33%>65 pg/mL). The percentage of patients deficient in 25(OH)D3 seasonally varied from 21% to 48% and the percentage with elevated iPTH reciprocally varied from 28% to 38%. Patients with detectable 25(OH)D2 had higher PTH levels and 57% of the samples with a total 25(OH)D > 50 ng/mL had detectable 25(OH)D2. Interpretation 25(OH)D3 and iPTH levels vary in a sinusoidal pattern throughout the year, even in vitamin D2 treated patients; 25(OH)D3, being higher in the summer and lower in the winter months, with iPTH showing the reverse pattern. A large percentage of the tested population showed vitamin D deficiency and secondary hyperparathyroidism. These observations held across three latitudinal regions, both genders, multiple-years, and in the presence or absence of detectable 25(OH)D2, and thus are applicable for patient care.

Confidence Distributions and a Unifying Framework for Meta-Analysis

This article develops a unifying framework, as well as robust meta-analysis approaches, for combining studies from independent sources. The device used in this combination is a confidence distribution (CD), which uses a distribution function, instead of a point (point estimator) or an interval (confidence interval), to estimate a parameter of interest. A CD function contains a wealth of information for inferences, and it is a useful device for combining studies from different sources. The proposed combining framework not only unifies most existing meta-analysis approaches, but also leads to development of new approaches. We illustrate in this article that this combining framework can include both the classical methods of combining p-values and modern model-based meta-analysis approaches. We also develop, under the unifying framework, two new robust meta-analysis approaches, with supporting asymptotic theory. In one approach each study size goes to infinity, and in the other approach the number of studies goes to infinity. Our theoretical development suggests that both these robust meta-analysis approaches have high breakdown points and are highly efficient for normal models. The new methodologies are applied to study-level data from publications on prophylactic use of lidocaine in heart attacks and a treatment of stomach ulcers. The robust methods performed well when data are contaminated and have realistic sample sizes and number of studies.

A Split-and-Conquer Approach for Analysis of Extraordinarily Large Data

If there are datasets, too large to fit into a single computer or too expen- sive for a computationally intensive data analysis, what should we do? We propose a split-and-conquer approach and illustrate it using several computationally inten- sive penalized regression methods, along with a theoretical support. We show that the split-and-conquer approach can substantially reduce computing time and com- puter memory requirements. The proposed methodology is illustrated numerically using both simulation and data examples.

Confidence distribution (CD) -- distribution estimator of a parameter

The notion of confidence distribution (CD), an entirely frequentist concept, is in essence a Neymanian interpretation of Fishers Fiducial distri- bution. It contains information related to every kind of frequentist inference. In this article, a CD is viewed as a distribution estimator of a parameter. This leads naturally to consideration of the information contained in CD, com- parison of CDs and optimal CDs, and connection of the CD concept to the (profile) likelihood function. A formal development of a multiparameter CD is also presented.

Multivariate Meta-Analysis of Heterogeneous Studies Using Only Summary Statistics: Efficiency and Robustness

Meta-analysis has been widely used to synthesize evidence from multiple studies for common hypotheses or parameters of interest. However, it has not yet been fully developed for incorporating heterogeneous studies, which arise often in applications due to different study designs, populations, or outcomes. For heterogeneous studies, the parameter of interest may not be estimable for certain studies, and in such a case, these studies are typically excluded from conventional meta-analysis. The exclusion of part of the studies can lead to a nonnegligible loss of information. This article introduces a meta-analysis for heterogeneous studies by combining the confidence density functions derived from the summary statistics of individual studies, hence referred to as the CD approach. It includes all the studies in the analysis and makes use of all information, direct as well as indirect. Under a general likelihood inference framework, this new approach is shown to have several desirable properties, including: (i) it is asymptotically as efficient as the maximum likelihood approach using individual participant data (IPD) from all studies; (ii) unlike the IPD analysis, it suffices to use summary statistics to carry out the CD approach. Individual-level data are not required; and (iii) it is robust against misspecification of the working covariance structure of parameter estimates. Besides its own theoretical significance, the last property also substantially broadens the applicability of the CD approach. All the properties of the CD approach are further confirmed by data simulated from a randomized clinical trials setting as well as by real data on aircraft landing performance. Overall, one obtains a unifying approach for combining summary statistics, subsuming many of the existing meta-analysis methods as special cases.

Port-of-Entry Inspection: Sensor Deployment Policy Optimization

This paper considers the problem of container inspection at a port-of-entry. Containers are inspected through a specific sequence to detect the presence of nuclear materials, biological and chemical agents, and other illegal shipments. The threshold levels of sensors at inspection stations of the port-of-entry affect the probabilities of incorrectly accepting or rejecting a container. In this paper, we present several optimization approaches for determining the optimum sensor threshold levels, while considering misclassification errors, total cost of inspection, and budget constraints. In contrast to previous work which determines threshold levels and sequence separately, we consider an integrated system and determine them simultaneously. Examples applying the approaches in different sensor arrangements are demonstrated.

Exact Meta-Analysis Approach for Discrete Data and its Application to 2 × 2 Tables With Rare Events

This article proposes a general exact meta-analysis approach for synthesizing inferences from multiple studies of discrete data. The approach combines the p-value functions (also known as significance functions) associated with the exact tests from individual studies. It encompasses a broad class of exact meta-analysis methods, as it permits broad choices for the combining elements, such as tests used in individual studies, and any parameter of interest. The approach yields statements that explicitly account for the impact of individual studies on the overall inference, in terms of efficiency/power and the Type I error rate. Those statements also give rises to empirical methods for further enhancing the combined inference. Although the proposed approach is for general discrete settings, for convenience, it is illustrated throughout using the setting of meta-analysis of multiple 2 × 2 tables. In the context of rare events data, such as observing few, zero, or zero total (i.e., zero events in both arms) outcomes in binomial trials or 2 × 2 tables, most existing meta-analysis methods rely on the large-sample approximations which may yield invalid inference. The commonly used corrections to zero outcomes in rare events data, aiming to improve numerical performance can also incur undesirable consequences. The proposed approach applies readily to any rare event setting, including even the zero total event studies without any artificial correction. While debates continue on whether or how zero total event studies should be incorporated in meta-analysis, the proposed approach has the advantage of automatically including those studies and thus making use of all available data. Through numerical studies in rare events settings, the proposed exact approach is shown to be efficient and, generally, outperform commonly used meta-analysis methods, including Mantel-Haenszel and Peto methods.

A note on Dempster-Shafer recombination of confidence distributions

It is often the case that there are several studies measuring the same parameter. Naturally, it is of interest to provide a systematic way to combine the information from these studies. Examples of such situa- tions include clinical trials, key comparison trials and other problems of practical importance. Singh et al. (2005) provide a compelling framework for combining information from multiple sources using the framework of confidence distributions. In this paper we investigate the feasibility of us- ing the Dempster-Shafer recombination rule on this problem. We derive a practical combination rule and show that under assumption of asymptotic normality, the Dempster-Shafer combined confidence distribution is asymp- totically equivalent to one of the method proposed in Singh et al. (2005). Numerical studies and comparisons for the common mean problem and the odds ratio in 2 × 2 tables are included.

Multiobjective Optimization of a Port-of-Entry Inspection Policy

At the port-of-entry, containers are inspected through a specific sequence of sensor stations to detect the presence of radioactive materials, biological and chemical agents, and other illegal cargo. The inspection policy, which includes the sequence in which sensors are applied and the threshold levels used at the inspection stations, affects the probability of misclassifying a container as well as the cost and time spent in inspection. This work is an extension of a paper by Elsayed et al., which considers an inspection system operating with a Boolean decision function combining station results. In this paper, we present a multiobjective optimization approach to determine the optimal sensor arrangement and threshold levels, while considering cost and time. The total cost includes cost incurred by misclassification errors and the total expected cost of inspection, while the time represents the total expected time a container spends in the inspection system. Examples which apply the approach in various systems are presented.

Confidence Intervals for Population Ranks in the Presence of Ties and Near Ties

Frequentist confidence intervals for population ranks and their statistical justifications are not well established, even though there is a great need for such procedures in practice. How do we assign confidence bounds for the ranks of health care facilities, schools, and financial institutions based on data that do not clearly separate the performance of different entities apart? The commonly used bootstrap-based frequentist confidence intervals and Bayesian intervals for population ranks may not achieve the intended coverage probability in the frequentist sense, especially in the presence of unknown ties or “near ties” among the populations to be ranked. Given random samples from k populations, we propose confidence bounds for population ranking parameters and develop rigorous frequentist theory and nonstandard bootstrap inference for population ranks, which allow ties and near ties. In the process, a notion of modified population rank is introduced that appears quite suitable for dealing with the population ranking problem. The proposed methodology and theoretical results are illustrated through simulations and a real dataset from a health research study involving 79 Veterans Health Administration (VHA) facilities. The results are extended to general risk adjustment models.