[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85790-en":3,"doc-seo-85790-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85790,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Geometric mean-based pairwise comparison method with the reference values—statistical approach","For many years, pairwise comparison methods have supported expert decision-making, with AHP as a widely known benchmark relying on a principal eigenvector to derive alternative weights. This paper introduces a statistical perspective that uses reference values and the geometric mean to compute alternative priorities. The approach simultaneously reflects inconsistency in judgments and preference distance. Quality indicators for the resulting weight vector are proposed, delivering interpretable measures tied to the statistical formulation.","arXiv :2607 . 10038v1 [ stat .ME] 10 Jul 2026  \nGeometric mean-based pairwise comparison method with the reference values-statistical approach  \nKonrad Kułakowskia,∗, Jacek Szybowskib  \naAGH University of Krakow, WEAiIB, al. Mickiewicza 30, 30-059 Poland bAGH University of Krakow, WMS, al. Mickiewicza 30, 30-059 Poland  \nAbstract  \nFor many years, the pairwise comparison method has been widely used for decision-making involving experts. The best-known example of this method is the Analytic Hierarchy Process (AHP) . In this now classic approach, the weights of alternatives are calculated using the principal eigenvector of the comparison matrix.  \nIn this paper, we present a statistical view of the pairwise comparison method, using reference values and the geometric mean to calculate alternative priorities. Thanks to this approach, we can simultaneously capture both the phenomenon of inconsistency in pairwise comparisons and the preference distance between alternatives. In this paper, we define indicators that measure the quality of the obtained weight vector, which, thanks to the statistical approach, have a clear and intuitive interpretation.  \nKeywords: pairwise comparisons, decision-making, heuristic rating estimation, AHP, GMM, inconsistency  \n1. Introduction  \nPairwise comparisons are one of the oldest approaches to decision-making. Their first well-documented application is generally attributed to Ramon Llull, although the underlying idea is likely much older. Early applications by Llull [11] and later thinkers, including Nicholas of Cusa, the Marquis de Condorcet, and Arthur Copeland [12, 5], were based on qualitative comparisons that allowed only for the identification of a preferred alternative. Quantitative pairwise comparisons emerged with the work of Louis Thurstone [45] and gained widespread recognition thanks to the Analytic Hierarchy Process (AHP) proposed by Thomas L. Saaty [41] . Currently, pairwise comparisons are used not only in AHP but also in methods such as the Best-Worst Method (BWM) [7, 37],  \n∗ Corresponding author  \nEmail addresses: [konrad.kulakowski@agh.edu.pl](konrad.kulakowski@agh.edu.pl) (Konrad Kułakowski), [szybowsk@agh.edu.pl](szybowsk@agh.edu.pl) (Jacek Szybowski)  \n2  \nPROMETHEE, ELECTRE [17], and MACBETH [2] . Despite the long history of the pairwise comparison method, it remains an active area of research, particularly in measuring inconsistency [6, 4, 9, 21], modeling uncertainty [46, 29], handling incomplete comparisons [20] or studying the properties of the method [43, 19] . Some decision-making methods use reference values or have variants that allow for such use. Examples of such methods include TOPSIS [33], VIKOR [32], the Helwig method [38], BWM [34, 37], COMET [42], DARP, BIPOLAR [39], or the Parsimonious Analytic Hierarchy Process [1] . For quantitative pairwise comparison methods, one such method is HRE (heuristic rating estimation)[27, 23, 22, 26] . In the latter, it is possible to specify virtually any number of alternatives with reference values, depending on data availability.  \nThe result of a single pairwise comparison can be interpreted as a single measurement of a decision-maker’s (or expert’s) preference. This understanding of comparison makes it possible to use statistical methods to assess the quality of a set of pairwise comparisons. This line of research—which involves pairwise comparisons but does not use reference data—appears in a number of scholarly works. In particular Carriere and Finster [8] developed a statistical framework for the ratio model of paired comparisons, treating observed comparison ratios as random variables generated by underlying preference parameters. The authors investigated the estimation and inferential properties of the model, deriving asymptotic results and statistical procedures for assessing preference intensities. In their subsequent paper, Genest and Rivest [16] interpreted pairwise judgments as noisy observations of underlying prefer","cbCaicGB5ARTAwF1","https://ap.wps.com/l/cbCaicGB5ARTAwF1","pdf",598722,1,30,"English","en",105,"# Introduction\n## Historical background and related decision methods\n## Statistical interpretation and prior work\n# Proposed statistical framework (geometric mean and reference values)","[{\"question\":\"What problem does the paper address in pairwise comparison methods?\",\"answer\":\"It reframes pairwise comparison using a statistical view, aiming to compute alternative priorities while capturing both inconsistency and preference distance in experts’ judgments.\"},{\"question\":\"How are alternative priorities calculated in the proposed approach?\",\"answer\":\"The method uses reference values together with the geometric mean to derive alternative priorities from pairwise comparisons.\"},{\"question\":\"What outputs or metrics does the paper introduce to assess results?\",\"answer\":\"It defines indicators that measure the quality of the obtained weight vector, with interpretations grounded in the statistical approach.\"}]",1784206300,76,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"geometric-mean-based-pairwise-comparison-method-with-the-reference-valuesstatistical-approach","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/geometric-mean-based-pairwise-comparison-method-with-the-reference-valuesstatistical-approach/85790/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper address in pairwise comparison methods?","Question",{"text":74,"@type":75},"It reframes pairwise comparison using a statistical view, aiming to compute alternative priorities while capturing both inconsistency and preference distance in experts’ judgments.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How are alternative priorities calculated in the proposed approach?",{"text":79,"@type":75},"The method uses reference values together with the geometric mean to derive alternative priorities from pairwise comparisons.",{"name":81,"@type":72,"acceptedAnswer":82},"What outputs or metrics does the paper introduce to assess results?",{"text":83,"@type":75},"It defines indicators that measure the quality of the obtained weight vector, with interpretations grounded in the statistical 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