Extending the roughness of the data via transitive closures of similarity indexes

One main assumption in the theory of rough sets applied to information tables is that the elements that exhibit the same information are indiscernible (similar) and form blocks that can be understood as elementary granules of knowledge about the universe. We propose a variant of this concept defining a measure of similarity between the elements of the universe in order to consider that two objects can be indiscernible even though they do not share all the attribute values because the knowledge is partial or uncertain. The set of similarities define a matrix of a fuzzy relation satisfying reflexivity and symmetry but transitivity thus a partition of the universe is not attained. This problem can be solved calculating its transitive closure what ensure a partition for each level belonging to the unit interval [0,1]. This procedure allows generalizing the theory of rough sets depending on the minimum level of similarity accepted. This new point of view increases the rough character of the data because increases the set of indiscernible objects. Finally, we apply our results to a not real application to be capable to remark the differences and the improvements between this methodology and the classical one

Bias and standard error for social reciprocity measurements

The directional consistency and skew-symmetry statistics have been proposed as global measurements of social reciprocity. Although both measures can be useful for quantifying social reciprocity, researchers need to know whether these estimators are biased in order to assess descriptive results properly. That is, if estimators are biased, researchers should compare actual values with expected values under the specified null hypothesis. Furthermore, standard errors are needed to enable suitable assessment of discrepancies between actual and expected values. This paper aims to derive some exact and approximate expressions in order to obtain bias and standard error values for both estimators for round-robin designs, although the results can also be extended to other reciprocal designs.