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.