{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T17:33:42Z","timestamp":1790357622461,"version":"4.1.0"},"reference-count":46,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["N000142412701"],"award-info":[{"award-number":["N000142412701"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,10,21]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>identified the issue of \u2032nonsense correlations\u2019 in time series data, where dependence within each of two random vectors causes overdispersion, i.e., variance inflation, for measures of dependence between the two. Since then much has been written about nonsense correlations, but nearly all of it confined to the time series literature. In this paper we provide the first, to our knowledge, rigorous study of this phenomenon for other forms of (positive) dependence, specifically for Markov random fields on lattices and graphs. We consider binary and continuous random vectors and three different measures of association: correlation, covariance and the ordinary least-squares coefficient from projecting one random vector onto the other. In some settings we find variance inflation consistent with Yule\u2019s nonsense correlation. Surprisingly, we also find variance deflation in some settings, and in others the variance is unchanged under dependence. Perhaps most notably, we find general conditions under which ordinary least-squares inference that ignores dependence is valid despite positive dependence in the regression errors, contradicting the presentation of ordinary least squares in countless textbooks and courses.<\/jats:p>","DOI":"10.1093\/biomet\/asaf041","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T07:31:35Z","timestamp":1748935895000},"source":"Crossref","is-referenced-by-count":1,"title":["Nonsense associations in Markov random fields with pairwise dependence"],"prefix":"10.1093","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7445-0096","authenticated-orcid":false,"given":"Sohom","family":"Bhattacharya","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Florida , 202 Griffin-Floyd Hall , Gainesville, Florida 32611,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rajarshi","family":"Mukherjee","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Harvard T.H. Chan School of Public Health , 655 Huntingdon Avenue , Boston, Massachusetts 02115,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7036-7973","authenticated-orcid":false,"given":"Elizabeth L","family":"Ogburn","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health , 615 North Wolfe Street , Baltimore, Maryland 21205,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,6,4]]},"reference":[{"key":"2026032607095765700_asaf041-B1","doi-asserted-by":"crossref","first-page":"11878","DOI":"10.1073\/pnas.0901910106","article-title":"Recovering time-varying networks of dependencies in social and biological studies","volume":"106","author":"Ahmed","year":"2009","journal-title":"Proc. Nat. Acad. Sci"},{"key":"2026032607095765700_asaf041-B2","doi-asserted-by":"crossref","first-page":"41174","DOI":"10.1038\/srep41174","article-title":"Enhancing gene regulatory network inference through data integration with Markov random fields","volume":"7","author":"Banf","year":"2017","journal-title":"Sci. Rep."},{"key":"2026032607095765700_asaf041-B3","author":"Bhattacharya","year":"2022"},{"key":"2026032607095765700_asaf041-B4","doi-asserted-by":"crossref","first-page":"1801","DOI":"10.1016\/j.aim.2008.07.008","article-title":"Convergent sequences of dense graphs I: subgraph frequencies, metric properties and testing","volume":"219","author":"Borgs","year":"2008","journal-title":"Adv. Math"},{"key":"2026032607095765700_asaf041-B5","doi-asserted-by":"crossref","first-page":"151","DOI":"10.4007\/annals.2012.176.1.2","article-title":"Convergent sequences of dense graphs II. Multiway cuts and statistical physics","volume":"176","author":"Borgs","year":"2012","journal-title":"Ann. Math."},{"key":"2026032607095765700_asaf041-B6","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/S0031-3203(02)00027-4","article-title":"EM procedures using mean field-like approximations for Markov model-based image segmentation","volume":"36","author":"Celeux","year":"2003","journal-title":"Pat. Recog"},{"key":"2026032607095765700_asaf041-B7","volume-title":"Handbook of Regression Analysis","author":"S.","year":"2013"},{"key":"2026032607095765700_asaf041-B8","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1109\/TFUZZ.2008.2005008","article-title":"A fuzzy clustering approach toward hidden Markov random field models for enhanced spatially constrained image segmentation","volume":"16","author":"Chatzis","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"2026032607095765700_asaf041-B9","doi-asserted-by":"crossref","first-page":"123","DOI":"10.2307\/2532039","article-title":"Assessing the significance of the correlation between two spatial processes","volume":"45","author":"Clifford","year":"1989","journal-title":"Biometrics"},{"key":"2026032607095765700_asaf041-B10","doi-asserted-by":"crossref","first-page":"104027","DOI":"10.1016\/j.jinteco.2024.104027","article-title":"The standard errors of persistence","volume":"153","author":"Conley","year":"2019","journal-title":"J. Int. Econ"},{"key":"2026032607095765700_asaf041-B11","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1145\/3313276.3316362","volume-title":"Proc. 51st Ann. ACM SIGACT Symp. Theory Comp","author":"Daskalakis","year":"2019"},{"key":"2026032607095765700_asaf041-B12","author":"Daskalakis","year":"2020"},{"key":"2026032607095765700_asaf041-B13","doi-asserted-by":"crossref","first-page":"1473","DOI":"10.1086\/681254","article-title":"Why do liberals drink lattes?","volume":"120","author":"DellaPosta","year":"2015","journal-title":"Am. J. Sociol"},{"key":"2026032607095765700_asaf041-B14","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1162\/00335530360698469","article-title":"Persuasion bias, social influence, and unidimensional opinions","volume":"118","author":"DeMarzo","year":"2003","journal-title":"Q. J. Econ"},{"key":"2026032607095765700_asaf041-B15","first-page":"57","volume-title":"Proc. 7th ACM SIGKDD Int. Conf. Know. Disc. Data Mining","author":"Domingos","year":"2001"},{"key":"2026032607095765700_asaf041-B16","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1525\/ae.1984.11.4.02a00080","article-title":"Galton\u2019s problem as network autocorrelation","volume":"11","author":"Dow","year":"1984","journal-title":"Am. Ethnol"},{"key":"2026032607095765700_asaf041-B17","first-page":"35","volume-title":"Random Graphs, Phase Transitions, and the Gaussian Free Field","author":"Duminil-Copin","year":"2017"},{"key":"2026032607095765700_asaf041-B18","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2532625","article-title":"Modifying the t test for assessing the correlation between two spatial processes","volume":"49","author":"Dutilleul","year":"1993","journal-title":"Biometrics"},{"key":"2026032607095765700_asaf041-B19","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781316576533","volume-title":"Computer Age Statistical Inference","author":"Efron","year":"2016"},{"key":"2026032607095765700_asaf041-B20","doi-asserted-by":"crossref","first-page":"1789","DOI":"10.1214\/16-AOS1509","article-title":"Yule\u2019s \u2032nonsense correlation' solved!","volume":"45","author":"Ernst","year":"2017","journal-title":"Ann. Statist."},{"key":"2026032607095765700_asaf041-B21","doi-asserted-by":"crossref","DOI":"10.1017\/9781316882603","volume-title":"Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction","author":"Friedli","year":"2017"},{"key":"2026032607095765700_asaf041-B22","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-55864-1","volume-title":"Linear Regression (Lecture Notes Statist. 175).","author":"Gro\u00df","year":"2003"},{"key":"2026032607095765700_asaf041-B23","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1214\/14-AOAS794","article-title":"Statistical paleoclimate reconstructions via Markov random fields","volume":"9","author":"Guillot","year":"2015","journal-title":"Ann. Appl. Statist"},{"key":"2026032607095765700_asaf041-B24","author":"Harris","year":"2020"},{"key":"2026032607095765700_asaf041-B25","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/BF02980577","article-title":"Beitrag zur theorie des ferromagnetismus","volume":"31","author":"Ising","year":"1925","journal-title":"Z. Phys. A: Hadrons Nucl"},{"key":"2026032607095765700_asaf041-B26","author":"Kandiros","year":"2021"},{"key":"2026032607095765700_asaf041-B27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-9602-1","volume-title":"Analysis of Neural Data","author":"Kass","year":"2014"},{"key":"2026032607095765700_asaf041-B28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/0022250X.1980.9989895","article-title":"On the relation between Markov random fields and social networks","volume":"7","author":"Kindermann","year":"1980","journal-title":"J. Math. Sociol"},{"key":"2026032607095765700_asaf041-B29","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/BF01152284","article-title":"Coexistence of phases in Ising ferromagnets","volume":"16","author":"Lebowitz","year":"1977","journal-title":"J. Statist. Phys"},{"key":"2026032607095765700_asaf041-B30","doi-asserted-by":"crossref","first-page":"1060","DOI":"10.1080\/01621459.2020.1782219","article-title":"Network dependence can lead to spurious associations and invalid inference","volume":"116","author":"Lee","year":"2021","journal-title":"J. Am. Statist. Assoc"},{"key":"2026032607095765700_asaf041-B31","doi-asserted-by":"crossref","first-page":"105335","DOI":"10.1016\/j.jmva.2024.105335","article-title":"Tensor recovery in high-dimensional Ising models","volume":"203","author":"Liu","year":"2024","journal-title":"J. Mult. Anal"},{"key":"2026032607095765700_asaf041-B32","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/060","volume-title":"Large Networks and Graph Limits","author":"Lov\u00e1sz","year":"2012"},{"key":"2026032607095765700_asaf041-B33","first-page":"1457","article-title":"Estimation in tensor Ising models","volume":"11","author":"Mukherjee","year":"2022","journal-title":"Inf. Inference"},{"key":"2026032607095765700_asaf041-B34","doi-asserted-by":"crossref","first-page":"2472","DOI":"10.1214\/18-AOS1754","article-title":"Property testing in high-dimensional Ising models","volume":"47","author":"Neykov","year":"2019","journal-title":"Ann. Statist."},{"key":"2026032607095765700_asaf041-B35","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1111\/rssa.12594","article-title":"Causal inference, social networks and chain graphs","volume":"183","author":"Ogburn","year":"2020","journal-title":"J. R. Statist. Soc. A"},{"key":"2026032607095765700_asaf041-B36","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","article-title":"Crystal statistics. I. A two-dimensional model with an order-disorder transition","volume":"65","author":"Onsager","year":"1944","journal-title":"Phys. Rev."},{"key":"2026032607095765700_asaf041-B37","first-page":"1116","article-title":"Non-separable spatio-temporal models via transformed multivariate Gaussian Markov random fields","volume":"71","author":"Prates","year":"2022","journal-title":"Appl. Statist"},{"key":"2026032607095765700_asaf041-B38","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1080\/00031305.1989.10475644","article-title":"The equality of the ordinary least squares estimator and the best linear unbiased estimator","volume":"43","author":"Puntanen","year":"1989","journal-title":"Am. Statistician"},{"key":"2026032607095765700_asaf041-B39","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1890\/04-0770","article-title":"Moran effect on nonlinear population processes","volume":"75","author":"Royama","year":"2005","journal-title":"Ecol. Monogr"},{"key":"2026032607095765700_asaf041-B40","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1007\/s11004-013-9513-y","article-title":"A mathematical view of weights-of-evidence, conditional independence, and logistic regression in terms of Markov random fields","volume":"46","author":"Schaeben","year":"2014","journal-title":"Math. Geosci"},{"key":"2026032607095765700_asaf041-B41","first-page":"179","article-title":"IV. The elimination of spurious correlation due to position in time or space","volume":"10","author":"\u2018STUDENT\u2019","year":"1914","journal-title":"Biometrika"},{"key":"2026032607095765700_asaf041-B42","volume-title":"Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models","author":"Vittinghoff","year":"2006"},{"key":"2026032607095765700_asaf041-B43","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1093\/bioinformatics\/btm129","article-title":"A Markov random field model for network-based analysis of genomic data","volume":"23","author":"Wei","year":"2007","journal-title":"Bioinformatics"},{"key":"2026032607095765700_asaf041-B44","author":"West","year":"2014"},{"key":"2026032607095765700_asaf041-B45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/2341482","article-title":"Why do we sometimes get nonsense-correlations between time-series?\u2013A study in sampling and the nature of time-series","volume":"89","author":"Yule","year":"1926","journal-title":"J. R. Statist. Soc"},{"key":"2026032607095765700_asaf041-B46","doi-asserted-by":"crossref","first-page":"asae054","DOI":"10.1093\/biomet\/asae054","article-title":"With random regressors, least squares inference is robust to correlated errors with unknown correlation structure","volume":"112","author":"Zhang","year":"2024","journal-title":"Biometrika"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asaf041\/63436223\/asaf041.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/112\/4\/asaf041\/63436223\/asaf041.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/112\/4\/asaf041\/63436223\/asaf041.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T11:10:08Z","timestamp":1774523408000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asaf041\/8156770"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,10,21]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asaf041","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025]]},"published":{"date-parts":[[2025]]},"article-number":"asaf041"}}