Missing Exposure Data in Stereotype Regression Model: Application to Matched Case–Control Study with Disease Subclassification

Summary With advances in modern medicine and clinical diagnosis, case–control data with characterization of finer subtypes of cases are often available. In matched case–control studies, missingness in exposure values often leads to deletion of entire stratum, and thus entails a significant loss in information. When subtypes of cases are treated as categorical outcomes, the data are further stratified and deletion of observations becomes even more expensive in terms of precision of the category‐specific odds‐ratio parameters, especially using the multinomial logit model. The stereotype regression model for categorical responses lies intermediate between the proportional odds and the multinomial or baseline category logit model. The use of this class of models has been limited as the structure of the model implies certain inferential challenges with nonidentifiability and nonlinearity in the parameters. We illustrate how to handle missing data in matched case–control studies with finer disease subclassification within the cases under a stereotype regression model. We present both Monte Carlo based full Bayesian approach and expectation/conditional maximization algorithm for the estimation of model parameters in the presence of a completely general missingness mechanism. We illustrate our methods by using data from an ongoing matched case–control study of colorectal cancer. Simulation results are presented under various missing data mechanisms and departures from modeling assumptions.     

Application of Markov Chain Model to Calculate the Average Number of Times of Use of a Material in Society. An Allocation Methodology for Open-Loop Recycling. Part 1: Methodology Development (7 pp)

- Preamble. In this series of two papers, a methodology to calculate the average number of times a material is used in a society from cradle to grave is presented and applied to allocation of environmental impact of virgin material. Part 1 focuses on methodology development and shows how the methodology works with hypothetical examples of material flows. Part 2 presents case studies for steel recycling in Japan, in which the methodology is applied and allocation of environmental impact of virgin steel is conducted. - Abstract Goal, Scope and Background. It has been recognized that LCA has a limitation in assessing open cycle recycling of materials because of inevitable subjective judgments in setting system boundary. According with the enforcement of recycling laws, there has been a rapid increase in recycling ratio of materials at the end-of-life of products in many industrialized countries. So, materials' life cycle is getting more complicated, which makes it difficult to quantify the environmental impacts of materials used in a product in an appropriate way. The purpose of this paper is to develop a methodology to calculate the average number of times a material is used in a society from cradle to grave. The method developed in this paper derives the average number of times material is used; this value could be used for allocation of environmental burdens of virgin material as well as an indicator for assessing the state of material use in a certain year, based on material flow of material in that year. Main Features Our methodology is based on Markov chain model using matrix-based numerical analysis. A major feature of this method is that it creates transition probability matrices for a material from the way in which the material is produced, consumed, and recycled, making it possible to simply elicit indicators that assess the status of material use in products in society. Our methodology could be an alternative method to derive the average number of times material is used, which could be used for allocation of environmental burdens of virgin material. Results and Discussions The methodology was applied to hypothetical examples of material flows, in which a virgin material was produced and used in products, recycled and finally landfilled. In some cases, closed loop and open loop recycling of materials existed. The transition probability matrix was created for each material flow, and how many times a virgin material is used in products until all of the elements are ultimately landfilled. Conclusions This methodology is applicable to a complicated material flow if the status of residence of a material and its flow in a society can be figured out. All the necessary data are the amount of virgin material production, amount of the material used in products, recycling rate of the material at the end of life of each product, the amount of scrap of the material that are used for products. In Part 2 of this paper, case studies for steel were conducted.     

Application of Markov Chain Model to Calculate the Average Number of Times of Use of a Material in Society. An Allocation Methodology for Open-Loop Recycling. Part 2: Case Study for Steel (6 pp)

- Preamble. In this series of two papers, a methodology to calculate the average number of times a material is used in a society from cradle to grave is presented and applied to allocation of environmental impact of virgin material. Part 1 focused on methodology development and showed how the methodology works with hypothetical examples of material flows. Part 2 presents case studies for steel recycling in Japan, in which the methodology is applied and allocation of environmental impact of virgin steel is conducted. - Abstract Goal, Scope and Background. The life cycle of steel begins with the mining of iron ore from the earth. Steel is produced in steel works and used in various products. Some of the steels are recycled at the products' end of life and used as a resource for the production of new steel in electric furnaces, while the remaining steel is used just once in products before being discarded (landfilled). In this paper, case studies were conducted to analyze the average number of times the element of iron is used and its residence time in society, in which the methodology developed in Part 1 of the paper was applied. CO2 emissions caused by steel productions and recycling were allocated by the number of times the element of steel is used in a society. Results and Discussion On the basis of the material flows of steel in Japan in 2000, it was calculated that at least 70% of the BF crude iron produced in Japan in 2000 was ultimately exported. On the assumption that steel is used in other countries in the same way as it is in Japan, the average number of times of use and the residence time of elemental iron in society are 2.67 and 62.9 years, respectively. Both of these values depend significantly on the recycling ratios of steel from construction and automobiles. Our model indicated that if the recycling ratio of steel from civil engineering and construction increased from 50% to 60%, the average number of times used would increase to 3.17 and the residence time of elemental iron in society would increase to 75.8 years. If CO2 emissions caused by steel productions and recycling are allocated by the number of times the element of steel is used in a society, it was calculated that steel use of one time generates in average an environmental burden of 1.03 t-CO2/t. Conclusion A method was developed to calculate the average number of times a material is used in a society from cradle to grave. Our methodology is based on Markov chain model using matrix-based numerical analysis, and has been successfully applied to steel. The results obtained by this methodology, i.e. the average number of times the element of iron is used in society, could be used for allocation of environmental burdens of virgin material as well as an indicator for assessing the state of material use in a certain year, based on material flow of material in that year. Recommendation and Perspective It is recognized that further researches must be conducted to gather data on steel production, use, and recovery in other countries and incorporate them into the transition probability matrix to obtain more precise results. Although this paper deals only with steel, this method can also be applied to other materials.