Averaged rank

This is a nonparametric test in which the data in each group are first ordered from lowest to highest values, then the entire set (both control and treated values) is ranked, with the average rank being assigned to tied values. The ranks are then summed for each group and U is determined according to... 

Rank - the position of a data value when all the data are placed in order of ascending magnitude. If ties occur, an average rank of the tied variates is used. Thus, the rank pf the datum 6 in the sequence 1,3,5,6,8,8,10 is 4 the rank of eaehdatum with value 8 is 5.5. 

The set of linear extensions is the basis for probability considerations Dividing A,k(x) by LT the quantity prob (rk(x) = k) = A,k(x)/LT can be interpreted as (ordinal) probability to get the rank k, sometimes also called absolute rank . Hence, an averaged rank, Rkav can be derived by... 

A rather good approximation for an element of interest, x, may be obtained, if the successors (all elements "below" x) and predecessors (all elements "above" x), respectively, are organized into a so-called S-x-P chain, all remaining elements, i.e. those incomparable to x being considered as isolated. From a combinatorial study follows that the averaged rank of an element x can be expressed as... 

Since N = S+P+U+l the averaged rank of an element x may be expressed by the following simple relation 1... 

The principle is illustrative demonstrated by determining the averaged rank of element b in the Hasse diagram depicted in Fig. 15. 

Fig. 15. Example for application of equation 10. The averaged rank of element b is to be estimated...

As stated earlier, it is possible to calculate averaged ranks the full list of information is given in Table 5, where the minimum, maximum rank and the local incomparabilities are displayed. 

A priori this is fine as the first attribute definitely should compensate the second one. However, the sites, which belong to (1,0),(0,1) on the one side and (1,0), (0,5) on the other side will get the same averaged rank Thus, the analysis by linear extensions alone should be carried out with appropriate care. We continue the analysis of the poset and discuss the attribute profiles. 

Briiggemann R, Simon U, Mey S (2005) Estimation of averaged ranks by extended local partial order models. Math Commun Math Comput Chem 54(3) 489-518... 

In the first chapter written by Carlsen an important step is the introduction of "noise - deficient" QSARs. The basics of HDT are briefly repeated before the central item how to get information about unknown properties is discussed. In comparison to usual methods, which are well known and widespread in the scientific literature, the approach shown by Carlsen does neither assume linearity in the property-property relationships nor any distribution properties. The method can be considered as parameter free. "Giving the molecules an identity" is the very idea how to rank a chemical whose environmental relevant data are unknown but some structural descriptors. Although the incomparabilities may be considered as "Achilles heel" Carlsen shows in his chapter how to handle such situations by means of linear extensions, probability distributions and the concept of averaged ranks. 

The average rank is simply the average of the ranks in all the linear extensions. On this basis the most probably rank for each element can be obtained leading to the most probably linear rank of the substances studied. 

The generation of the averaged rank of the single compounds in the Hasse diagram is obtained applying the simple relation recently reported by Briiggemann et al. (2004) (see also p. 86). The averaged rank of a specific compound, c , can be obtained by the simple relation... 

Thus, from the above discussion on the probabilities for specific ranks, it was concluded that the new compound X must be located between the compounds 4 and 7 and compound 10 (cf. Fig. 2B), which is further substantiated by the figures in Table 1. Assuming that if the averaged ranks, Rkav, of two compounds are close, the two compounds will on an average basis display similar characteristics as being determined by the set of descriptors applied, the analysis of average rank discloses, cf. Table 1, that compounds X most closely resembles compounds 6 and 9. Consequently, compound X has now obtained an identity compared to the basis set of compounds, i.e., compounds 1-10. 

Table 1. Averaged ranks of the 11 compounds included in the Hasse diagram displayed in Figure 2B...

Briiggemann R, Lerche D, Sorensen PB, Carlsen L (2004) Estimation of averaged ranks by a local partial order model, J Chem Inf Comput Sci 44 618-625... 

Fig. lb. Calculation of the Ranking frequencies (left table) and ordinal ranking probabilities (right table) and of averaged ranks... 

For the substance x3 the averaged rank analogously is found to be 3.125. Therefore in the linear rank, x3 will be given a higher rank than x2. In the example given in Fig. 1 the most probable sequence of ranks is ... 

Using all linear extensions the probability distribution of ranks for each substance can be found. Table 3 gives the ordinal probability for the individual substances to occupy a certain rank. The ranking probabilities are spread out in the interval of possible ranking positions. Clearly the probabilities of isolated objects, like malathion (MAL) (compare Fig. 2) will be spread equally out over the whole interval providing an averaged rank of 6.5. 

The highest probability for a position is only 0.553 for chlormequat chlorid (CHL) on rank no. 1. The averaged rank is presented in the HDT-column in Table 4. 

Rank HDT All linear extensions HDT average rank The Utility Utility Function r PRO.a) PROMETHEE r ... 

---