Ranking of chemical substances based on the Japanese Pollutant Release and Transfer Register using partial order theory and random linear extensions.

In 1997 a Pollutant Release and Transfer Register (PRTR) pilot project was initiated in Japan. In 1998 the project was expanded and in 1999 a law concerning the establishment of a national PRTR was adopted. Data on the emissions of chemical substances are therefore now being reported on a continuous base. In relation to the PRTR project data on toxicity have been collected. In order to make efficient use of the collected information on emission and toxicity it is useful to group or rank the chemical substances according to the impact on human health and the environment. It has recently been argued that partial order theory (POT) in combination with the use of linear extensions (LE) may be the most objective way to create a linear rank. The methodology has been further expanded to handle larger data sets by the use of random linear extensions (RLE). In this paper the Japanese PRTR data are ranked using the POT/RLE methodology. An average rank is established for chemical substances in the 1998 and 1999 PRTR in Japan. The top 10 chemical substances in the 1998 PRTR are: dichlorvos, inorganic arsenic compounds, cobalt compounds, beryllium compounds, fenitrothion, disulfoton, parathion, diazinon, 4,4'-diamino-3,3'-dichlorodiphenylmethane and antimony compounds. The top 10 chemical substances from the 1999 PRTR are PCBs, lead compounds, fenitrothion, dichlorvos, disulfoton, inorganic arsenic compounds, chlorothalonil, thiobencarb, chromium and HCFC-141b. The descriptor having the highest influence on the ranking of the 1998 PRTR data is the production volume, which, however, is not given in the 1999 PRTR. Further, the disagreement between the ranking with the lack of toxicity data substituted with mean and maximum values, respectively, strongly indicates a general need for further toxicological investigations.

Estimation of averaged ranks by a local partial order model.

This paper continues the series of publications about applications of partial ordering. The focus of this publication is the derivation of approximate analytical expressions for the averaged rank and the ranking probabilities. To derive such combinatorial formulas a local partial order is suggested as an approximation. The performance of the approximation is rather high; we therefore conclude that three very simple descriptors of the local partial order seem to be sufficient to get a rough impression of the linear order, induced by the averaged ranks and the ranking probabilities of empirical partially ordered sets. Linear order derived from the partial order, ranking probabilities, and other characteristics are considered as parts of a so-called "General Ranking Model" (GRM). Following the local partial order, the averaged rank of an object x can be estimated applying the following simple formula: Rk(av) = (S+1)*(N+1)/(N+1-U). S is the number of successors of the object x, N is the total number of objects (of the quotient set), and U is the number of objects incomparable with x. More complex formulas for the ranking probabilities are given in the text. A list of abbreviations and symbols can be found in Tables 3 and 4.

Improved estimation of the ranking probabilities in partial orders using random linear extensions by approximation of the mutual ranking probability.

The application of partial order theory and Hasse diagram technique in environmental science is getting increasing attention. One of the latest developments in the field of Hasse diagram technique is the use of random linear extensions to estimate ranking probabilities. In the original algorithm for estimating the ranking probability it is assumed that the order between two incomparable pair of objects can be chosen randomly. However, if the total set of linear extensions is considered there is a specific probability that one object will be larger than another, which can be far from 50%. In this study it is investigated if an approximation of the mutual ranking probability can improve the algorithm. Applying an approximation of the mutual ranking probability the estimation of the ranking probabilities are significantly improved. Using a test set of 39 partial orders with randomly chosen values the relative mean root square difference (MRSD) decrease in average from 7.9% to 2.2% and a maximum relative improvement of 90% can be found. In the most successful case the relative MRSD goes as low as 0.77%.

Evaluation of the ranking probabilities for partial orders based on random linear extensions.

Partial order theory and Hasse diagrams appears to be a promising tool for decision-making in environmental issues. Alternatives or objects are said to be partial ordered when it is impossible to find a mutual relationship (< or >) for all criteria. This is often the case in complicated real life situations. However, sometimes it is attractive to apply a total order, i.e. linear rank, and not just the partial order. Based on ranking probabilities and linear extensions it is possible to derive a total order. A linear extension is a projection of the partial order into a total order that comply with all the relations in the partial order. When all linear extensions are known the ranking probabilities can be found as the probability for an object to occupy a specific rank. However, the total number of linear extensions is proportional with the faculty of the number of objects in the partial order. Therefore it is practically impossible to identify all possible linear extensions for partial orders with more than around 20 objects. This study reviews and evaluates a method which estimates the ranking probability based on sampling of a minor random fraction of the linear extensions. Using standard statistics the necessary number of random linear extensions is described as a function of the ranking probability estimate and the restrictions on the confidence interval around the ranking probability. The analysis reveals a smaller systematic uncertainty, which occurs due to the random selection of ranking between two incomparable objects. The discrepancy appears to be dependent on the structure of the partial order. The method using random linear extensions thus appears as a valuable tool for analysing larger partially ordered sets, which are practically impossible to handle using the total set of linear extensions.

Ranking contaminated sites using a partial ordering method.

In this project, we apply the method of partial ordering on the ranking of 74 contaminated sites located in the county of West Zealand (Denmark). The method is based on the concept that the parameters are kept separated through the ranking analysis, and thus no weighing of the different parameter values is necessary. The ranking is displayed in a graphical form by the Hasse diagram technique to ease the interpretation. A critical comparison is made of the ranking of contaminated sites by the partial ordering method and an index function used by the county of West Zealand. Comparing the ranking by the partial ordering method to the index function shows that the choice of score points and index function highly influences the ranking result, as only four sites are equally ranked. The importance of the parameters used to identify the environmental hazard of the contaminated sites is analyzed in order to evaluate the influence of each parameter on the ranking. From among a total of six different parameters, two have high influence, two medium, and two low because of both the construction of the scoring system and the characteristics of the data.

Comparison of the combined monitoring-based and modelling-based priority setting scheme with partial order theory and random linear extensions for ranking of chemical substances.

The combined monitoring-based and modelling-based priority setting scheme (COMMPS) used to establish a priority setting list within the EU Water Framework Directive plays a major role in the European environmental policy on chemical substances. The COMMPS procedure can be classified as a so-called scoring method. The applied functional relationship and weight factors are established based on expert judgement, which unfortunately appears to be vulnerable to subjective inputs. In this study an alternative priority setting methods based on partial order theory (POT) and random linear extensions (RLE) is suggested and compared to the COMMPS procedure. The POT/RLE is characterised as being based on fewer assumptions concerning functional relationships and does not apply weighting factors. Using the POT/RLE methodology a different ranking result occur than when using the COMMPS procedure. Eight of the top 20 substances from the COMMPS procedure are not ranked within the top 20 when using POT/RLE. From the viewpoint of environmental protection, especially the substances that have been given low priority in the COMMPS procedure, but a high rank in POT/RLE, are of interest in a regulatory context. These substances are naphthalene, trichloromethane, isoproturon, metolachlor, endosulfan, acenaphthene, alachlor and dichloromethane. An analysis of the ability of the descriptors to separate the single substance discloses that the most significant descriptor is the concentrations detected in the environment. Further, the frequency of detection is not applied as a descriptor in the COMMPS procedure. However, if this descriptor was to be applied the analysis revealed that it would have been the third most significant descriptor.

A comparison of partial order technique with three methods of multi-criteria analysis for ranking of chemical substances.

An alternative to the often cumbersome and time-consuming risk assessments of chemical substances could be more reliable and advanced priority setting methods. An elaboration of the simple scoring methods is provided by Hasse Diagram Technique (HDT) and/or Multi-Criteria Analysis (MCA). The present study provides an in depth evaluation of HDT relative to three MCA techniques. The new and main methodological step in the comparison is the use of probability concepts based on mathematical tools such as linear extensions of partially ordered sets and Monte Carlo simulations. A data set consisting of 12 High Production Volume Chemicals (HPVCs) is used for illustration. It is a paradigm in this investigation to claim that the need of external input (often subjective weightings of criteria) should be minimized and that the transparency should be maximized in any multicriteria prioritisation. The study illustrates that the Hasse diagram technique (HDT) needs least external input, is most transparent and is least subjective. However, HDT has some weaknesses if there are criteria which exclude each other. Then weighting is needed. Multi-Criteria Analysis (i.e. Utility Function approach, PROMETHEE and concordance analysis) can deal with such mutual exclusions because their formalisms to quantify preferences allow participation e.g. weighting of criteria. Consequently MCA include more subjectivity and loose transparency. The recommendation which arises from this study is that the first step in decision making is to run HDT and as the second step possibly is to run one of the MCA algorithms.

Improving the predicting power of partial order based QSARs through linear extensions.

Partial order theory (POT) is an attractive and operationally simple method that allows ordering of compounds, based on selected structural and/or electronic descriptors (modeled order), or based on their end points, e.g., solubility (experimental order). If the modeled order resembles the experimental order, compounds that are not experimentally investigated can be assigned a position in the model that eventually might lead to a prediction of an end-point value. However, in the application of POT in quantitative structure-activity relationship modeling, only the compounds directly comparable to the noninvestigated compounds are applied. To explore the possibilities of improving the methodology, the theory is extended by application of the so-called linear extensions of the model order. The study show that partial ordering combined with linear extensions appears as a promising tool providing probability distribution curves in the range of possible end-point values for compounds not being experimentally investigated.