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Median levels of regulated flame retardants, i.e., polybrominated diphenyl ethers (PBDEs), brominated biphenyl (BB)-153, and hexabromocyclododecane (HBCD), in 33-48 eggs of peregrine falcons (Falco peregrinus) from Greenland were 1909, 359, and 5.98 ng/g lipid weight (lw), respectively, and generally intermediate to levels in North America and Europe. Unregulated flame retardants had lower median concentrations of 1.06 (2-ethylhexyl-2,3,4,5-tetrabromobenzoate, EH-TBB), 2.42 (1,2-bis(2,4,6-tribromophenoxy)-ethane, BTBPE), 0.52 (2,4,6-tribromophenyl 2,3-dibromopropyl ether, DPTE), and 4.78 (dechlorane plus) ng/g lw. Although these compounds are often described as recent replacements for PBDEs, they were also present in eggs from the 1980s. BDE-209 was the only compound with a significant increase (+7.2% annual change) between 1986 and 2014, while BB-153 and DPTE decreased significantly (-8.0% and -2.8% annual change, respectively). Dechlorane plus showed a nonsignificant increase. Individual birds, equipped with light-logging geolocators, confirmed the contaminant exposure over a large geographical area as the birds spent nearly equal time periods in their breeding and wintering grounds in Greenland and Central/South America, respectively, interrupted by 5-6 weeks of migration through North America.

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.

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%.

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.