The Consistency Formula

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Obata, T.; Shiraishi, S.; Daigo, M.; Nobuyuki, N. Assessment for an incomplete comparison matrix and improvement of an inconsistent comparison: Computational experiments. In Proceedings of the ISAHP, Kobe, Japan, 12–14 August 1999; pp. 200–205. [ Google Scholar] Rasch framework provides methods and procedures for identifying and handling unexpected response behaviors. Mean-square fit statistics are computed for each individual and each item. Their expected value is 1. Values greater than 1 indicate underfit to the model (i.e., the responses are less predictable than the Rasch model expects), whereas values smaller than 1 indicate overfit (i.e., the responses are more predictable than the model expects; Linacre, 2002b). There are two types of mean-square fit statistics: outfit and infit. Outfit is mostly influenced by unexpected responses of high entity, whereas infit is mostly influenced by unexpected responses of small entity. An example of unexpected response is an incorrect response to an item for which a correct response is expected (i.e., an item for which, according to the Rasch model, the probability of a correct response is larger than that of an incorrect response). If the probability of the correct response is much larger than that of the incorrect response, the unexpected response mainly influences outfit. If the probability of the correct response is slightly larger than that of the incorrect response, the unexpected response mainly influences infit. Smith E. V., Jr. (2001). Evidence for the reliability of measures and validity of measure interpretation: a Rasch measurement perspective. J. Appl. Meas. 2 Mazurek, J.; Perzina, R. On the Inconsistency of Pairwise Comparisons: An Experimental Study. Ph.D. Thesis, Faculty of Economics and Administration, University of Pardubice, Pardubice, Czech Republic, 2017. [ Google Scholar] Vidotto G., Bertolotti G., Carone M., Arpinelli F., Bellia V., Jones P. W., et al. (2006). A new questionnaire specifically designed for patients affected by chronic obstructive pulmonary disease. the Italian health status questionnaire. Respir. Med. 100Salo, A.; Hamalainen, R. On the measurement of preference in the analytic hierarchy process. J. Multi-Criteria Decis. Anal. 1997, 6, 309–319. [ Google Scholar] [ CrossRef] Ramík, J.; Korviny, P. Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 2010, 161, 1604–1613. [ Google Scholar] [ CrossRef] Murphy, C.K. Limits on the analytic hierarchy process from its consistency index. Eur. J. Oper. Res. 1993, 65, 138–139. [ Google Scholar] [ CrossRef] Wedley, W.C. Relative measure of Consistency Ratio. In Proceedings of the 2nd International Symposium on the Analytic Hierarchy Process; University of Pittsburgh: Pittsburgh, PA, USA, February–March 1991; pp. 185–196. [ Google Scholar] Vidotto G., Moroni L., Burro R., Filipponi L., Balestroni G., Bettinardi O., et al. (2010). A revised short version of the depression questionnaire. Eur. J. Cardiovasc. Prev. Rehabil. 17