A Diversity Index for Model Space Selection in the Estimation of Benchmark and Infectious Doses via Model Averaging

In chemical and microbial risk assessments, risk assessors fit dose‐response models to high‐dose data and extrapolate downward to risk levels in the range of 1–10%. Although multiple dose‐response models may be able to fit the data adequately in the experimental range, the estimated effective dose (ED) corresponding to an extremely small risk can be substantially different from model to model. In this respect, model averaging (MA) provides more robustness than a single dose‐response model in the point and interval estimation of an ED. In MA, accounting for both data uncertainty and model uncertainty is crucial, but addressing model uncertainty is not achieved simply by increasing the number of models in a model space. A plausible set of models for MA can be characterized by goodness of fit and diversity surrounding the truth. We propose a diversity index (DI) to balance between these two characteristics in model space selection. It addresses a collective property of a model space rather than individual performance of each model. Tuning parameters in the DI control the size of the model space for MA.     

An Integrated Risk Assessment Model of Township‐Scaled Land Subsidence Based on an Evidential Reasoning Algorithm and Fuzzy Set Theory

Land subsidence risk assessment (LSRA) is a multi‐attribute decision analysis (MADA) problem and is often characterized by both quantitative and qualitative attributes with various types of uncertainty. Therefore, the problem needs to be modeled and analyzed using methods that can handle uncertainty. In this article, we propose an integrated assessment model based on the evidential reasoning (ER) algorithm and fuzzy set theory. The assessment model is structured as a hierarchical framework that regards land subsidence risk as a composite of two key factors: hazard and vulnerability. These factors can be described by a set of basic indicators defined by assessment grades with attributes for transforming both numerical data and subjective judgments into a belief structure. The factor‐level attributes of hazard and vulnerability are combined using the ER algorithm, which is based on the information from a belief structure calculated by the Dempster‐Shafer (D‐S) theory, and a distributed fuzzy belief structure calculated by fuzzy set theory. The results from the combined algorithms yield distributed assessment grade matrices. The application of the model to the Xixi‐Chengnan area, China, illustrates its usefulness and validity for LSRA. The model utilizes a combination of all types of evidence, including all assessment information—quantitative or qualitative, complete or incomplete, and precise or imprecise—to provide assessment grades that define risk assessment on the basis of hazard and vulnerability. The results will enable risk managers to apply different risk prevention measures and mitigation planning based on the calculated risk states.     

A Tiered Framework for Risk‐Relevant Characterization and Ranking of Chemical Exposures: Applications to the National Children's Study (NCS)

A challenge for large‐scale environmental health investigations such as the National Children's Study (NCS), is characterizing exposures to multiple, co‐occurring chemical agents with varying spatiotemporal concentrations and consequences modulated by biochemical, physiological, behavioral, socioeconomic, and environmental factors. Such investigations can benefit from systematic retrieval, analysis, and integration of diverse extant information on both contaminant patterns and exposure‐relevant factors. This requires development, evaluation, and deployment of informatics methods that support flexible access and analysis of multiattribute data across multiple spatiotemporal scales. A new “Tiered Exposure Ranking” (TiER) framework, developed to support various aspects of risk‐relevant exposure characterization, is described here, with examples demonstrating its application to the NCS. TiER utilizes advances in informatics computational methods, extant database content and availability, and integrative environmental/exposure/biological modeling to support both “discovery‐driven” and “hypothesis‐driven” analyses. “Tier 1” applications focus on “exposomic” pattern recognition for extracting information from multidimensional data sets, whereas second and higher tier applications utilize mechanistic models to develop risk‐relevant exposure metrics for populations and individuals. In this article, “tier 1” applications of TiER explore identification of potentially causative associations among risk factors, for prioritizing further studies, by considering publicly available demographic/socioeconomic, behavioral, and environmental data in relation to two health endpoints (preterm birth and low birth weight). A “tier 2” application develops estimates of pollutant mixture inhalation exposure indices for NCS counties, formulated to support risk characterization for these endpoints. Applications of TiER demonstrate the feasibility of developing risk‐relevant exposure characterizations for pollutants using extant environmental and demographic/socioeconomic data.     

Effect of Inventory Information Discrepancy in a Drop‐Shipping Supply Chain

This article investigates the impact of information discrepancy between a drop‐shipper and an online retailer on the drop‐shipping supply chain performance. The inventory information misalignment between them contributes to the failure of order fulfillment and demand satisfaction, and hence the associated penalties are incurred. In this article, we first analyze the penalties of ignoring such information discrepancy on both the drop‐shipper and the online retailer. We then assess the impact of information discrepancy on both parties when the drop‐shipper understands the existence of the information discrepancy but is not able to eliminate the errors. The numerical experiments indicate that both parties can have significant amount of the percentage cost reductions if the information discrepancy can be eliminated, and the potential savings are substantial especially when the errors have large variability. Furthermore, we observe that the online retailer is more vulnerable to information discrepancy than the drop‐shipper, and the drop‐shipper is likely to suffer from the online retailer's underestimation of the physical inventory level more than the problem of its overestimation. Moreover, even if eliminating errors is not possible, both parties could still benefit from taking the possibility of errors into consideration in decision making.     

On evaluating how well a biomarker can predict treatment response with survival data

One of the objectives of personalized medicine is to take treatment decisions based on a biomarker measurement. Therefore, it is often interesting to evaluate how well a biomarker can predict the response to a treatment. To do so, a popular methodology consists of using a regression model and testing for an interaction between treatment assignment and biomarker. However, the existence of an interaction is not sufficient for a biomarker to be predictive. It is only necessary. Hence, the use of the marker‐by‐treatment predictiveness curve has been recommended. In addition to evaluate how well a single continuous biomarker predicts treatment response, it can further help to define an optimal threshold. This curve displays the risk of a binary outcome as a function of the quantiles of the biomarker, for each treatment group. Methods that assume a binary outcome or rely on a proportional hazard model for a time‐to‐event outcome have been proposed to estimate this curve. In this work, we propose some extensions for censored data. They rely on a time‐dependent logistic model, and we propose to estimate this model via inverse probability of censoring weighting. We present simulations results and three applications to prostate cancer, liver cirrhosis, and lung cancer data. They suggest that a large number of events need to be observed to define a threshold with sufficient accuracy for clinical usefulness. They also illustrate that when the treatment effect varies with the time horizon which defines the outcome, then the optimal threshold also depends on this time horizon.     

A strong law of large numbers for independent random variables under non-additive probabilities

Abstract

Under non‐additive probabilities, cluster points of the empirical average have been proved to quasi-surely fall into the interval constructed by either the lower and upper expectations or the lower and upper Choquet expectations. In this paper, based on the initiated notion of independence, we obtain a different Marcinkiewicz-Zygmund type strong law of large numbers. Then the Kolmogorov type strong law of large numbers can be derived from it directly, stating that the closed interval between the lower and upper expectations is the smallest one that covers cluster points of the empirical average quasi-surely.     

Exploring Parameter Relations for Multi‐Stage Models in Stage‐Wise Constant and Time Dependent Hazard Rates

Single cohort stage‐frequency data are considered when assessing the stage reached by individuals through destructive sampling. For this type of data, when all hazard rates are assumed constant and equal, Laplace transform methods have been applied in the past to estimate the parameters in each stage‐duration distribution and the overall hazard rates. If hazard rates are not all equal, estimating stage‐duration parameters using Laplace transform methods becomes complex. In this paper, two new models are proposed to estimate stage‐dependent maturation parameters using Laplace transform methods where non‐trivial hazard rates apply. The first model encompasses hazard rates that are constant within each stage but vary between stages. The second model encompasses time‐dependent hazard rates within stages. Moreover, this paper introduces a method for estimating the hazard rate in each stage for the stage‐wise constant hazard rates model. This work presents methods that could be used in specific types of laboratory studies, but the main motivation is to explore the relationships between stage maturation parameters that, in future work, could be exploited in applying Bayesian approaches. The application of the methodology in each model is evaluated using simulated data in order to illustrate the structure of these models.     

Inference in Semi‐Parametric Dynamic Models for Repeated Count Data

This paper deals with a longitudinal semi‐parametric regression model in a generalised linear model setup for repeated count data collected from a large number of independent individuals. To accommodate the longitudinal correlations, we consider a dynamic model for repeated counts which has decaying auto‐correlations as the time lag increases between the repeated responses. The semi‐parametric regression function involved in the model contains a specified regression function in some suitable time‐dependent covariates and a non‐parametric function in some other time‐dependent covariates. As far as the inference is concerned, because the non‐parametric function is of secondary interest, we estimate this function consistently using the independence assumption‐based well‐known quasi‐likelihood approach. Next, the proposed longitudinal correlation structure and the estimate of the non‐parametric function are used to develop a semi‐parametric generalised quasi‐likelihood approach for consistent and efficient estimation of the regression effects in the parametric regression function. The finite sample performance of the proposed estimation approach is examined through an intensive simulation study based on both large and small samples. Both balanced and unbalanced cluster sizes are incorporated in the simulation study. The asymptotic performances of the estimators are given. The estimation methodology is illustrated by reanalysing the well‐known health care utilisation data consisting of counts of yearly visits to a physician by 180 individuals for four years and several important primary and secondary covariates.     

Nonparametric Estimation of the Number of Drug Users in Hong Kong Using Repeated Multiple Lists

We update a previous approach to the estimation of the size of an open population when there are multiple lists at each time point. Our motivation is 35 years of longitudinal data on the detection of drug users by the Central Registry of Drug Abuse in Hong Kong. We develop a two‐stage smoothing spline approach. This gives a flexible and easily implemented alternative to the previous method which was based on kernel smoothing. The new method retains the property of reducing the variability of the individual estimates at each time point. We evaluate the new method by means of a simulation study that includes an examination of the effects of variable selection. The new method is then applied to data collected by the Central Registry of Drug Abuse. The parameter estimates obtained are compared with the well known Jolly–Seber estimates based on single capture methods.     

A win ratio approach to comparing continuous non‐normal outcomes in clinical trials

Clinical trials are often designed to compare continuous non‐normal outcomes. The conventional statistical method for such a comparison is a non‐parametric Mann–Whitney test, which provides a P‐value for testing the hypothesis that the distributions of both treatment groups are identical, but does not provide a simple and straightforward estimate of treatment effect. For that, Hodges and Lehmann proposed estimating the shift parameter between two populations and its confidence interval (CI). However, such a shift parameter does not have a straightforward interpretation, and its CI contains zero in some cases when Mann–Whitney test produces a significant result. To overcome the aforementioned problems, we introduce the use of the win ratio for analysing such data. Patients in the new and control treatment are formed into all possible pairs. For each pair, the new treatment patient is labelled a ‘winner’ or a ‘loser’ if it is known who had the more favourable outcome. The win ratio is the total number of winners divided by the total numbers of losers. A 95% CI for the win ratio can be obtained using the bootstrap method. Statistical properties of the win ratio statistic are investigated using two real trial data sets and six simulation studies. Results show that the win ratio method has about the same power as the Mann–Whitney method. We recommend the use of the win ratio method for estimating the treatment effect (and CI) and the Mann–Whitney method for calculating the P‐value for comparing continuous non‐Normal outcomes when the amount of tied pairs is small. Copyright © 2016 John Wiley & Sons, Ltd.