{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T21:43:56Z","timestamp":1770673436538,"version":"3.49.0"},"reference-count":42,"publisher":"SAGE Publications","issue":"2","license":[{"start":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T00:00:00Z","timestamp":1769731200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"name":"Karmanos Cancer Insitute","award":["Cancer Research Horizons Fund"],"award-info":[{"award-number":["Cancer Research Horizons Fund"]}]},{"name":"NIH\/NCI","award":["P30 CA022453"],"award-info":[{"award-number":["P30 CA022453"]}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational Biology"],"published-print":{"date-parts":[[2026,2]]},"abstract":"<jats:p>\n                    Quadratic forms of multivariate normal variables play a critical role in statistical applications, particularly in genomics and bioinformatics. However, accurately computing small right-tail probabilities (\n                    <jats:italic toggle=\"yes\">p<\/jats:italic>\n                    -values) for large-scale quadratic forms is computationally challenging due to the intractability of their probability distributions, as well as significant numerical constraints and computational burdens. To address these problems, we propose Markov chain Monte Carlo cross-entropy (MCMC-CE), an innovative algorithm that integrates MCMC sampling with the CE method, coupled with leading eigenvalue extraction and Satterthwaite-type approximation techniques. Our approach efficiently estimates small\n                    <jats:italic toggle=\"yes\">p<\/jats:italic>\n                    -values for quadratic forms with their ranks exceeding 10,000. Through extensive simulation studies and real-world applications in genomics, including genome-wide association studies and pathway enrichment analyses, our method demonstrates advantageous numerical accuracy and computational reliability compared with existing approaches such as Davies\u2019, Imhof\u2019s, Farebrother\u2019s, Liu\u2013Tang\u2013Zhang\u2019s, and saddlepoint approximation methods. MCMC-CE provides a robust and scalable solution for accurately computing small\n                    <jats:italic toggle=\"yes\">p<\/jats:italic>\n                    -values for quadratic forms, facilitating more precise statistical inference in large-scale genomic studies.\n                  <\/jats:p>","DOI":"10.1177\/15578666251406305","type":"journal-article","created":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T13:50:37Z","timestamp":1769781037000},"page":"236-254","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["MCMC-CE: A Novel and Efficient Algorithm for Estimating Small Right-Tail Probabilities of Quadratic Forms with Applications in Genomics"],"prefix":"10.1177","volume":"33","author":[{"given":"Vy Q.","family":"Ong","sequence":"first","affiliation":[{"name":"Biostatistics and Bioinformatics Core, Karmanos Cancer Institute, Department of Oncology, Wayne State University School of Medicine, Detroit, Michigan, USA."},{"name":"Department of Biostatistics, Data Science and Epidemiology, School of Public Health, Augusta University, Augusta, Georgia, USA."}]},{"given":"Bich N.","family":"Choi","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Data Science and Epidemiology, School of Public Health, Augusta University, Augusta, Georgia, USA."},{"name":"Institute for Clinical Research and Learning Health Care, Department of Pediatrics, McGovern Medical School, University of Texas Health Science Center at Houston, Houston, Texas, USA."}]},{"given":"Devin P.","family":"Lundy","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Data Science and Epidemiology, School of Public Health, Augusta University, Augusta, Georgia, USA."}]},{"given":"Yingnan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Biostatistics and Bioinformatics Core, Karmanos Cancer Institute, Department of Oncology, Wayne State University School of Medicine, Detroit, Michigan, USA."}]},{"given":"Yin","family":"Wan","sequence":"additional","affiliation":[{"name":"Biostatistics and Bioinformatics Core, Karmanos Cancer Institute, Department of Oncology, Wayne State University School of Medicine, Detroit, Michigan, USA."}]},{"given":"Hongyan","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Data Science and Epidemiology, School of Public Health, Augusta University, Augusta, Georgia, USA."}]},{"given":"Santu","family":"Ghosh","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Data Science and Epidemiology, School of Public Health, Augusta University, Augusta, Georgia, USA."}]},{"given":"Hui","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Center for Computational Medicine and Bioinformatics and University of Michigan Rogel Cancer Center, University of Michigan, Ann Arbor, Michigan, USA."}]},{"given":"Yang","family":"Shi","sequence":"additional","affiliation":[{"name":"Biostatistics and Bioinformatics Core, Karmanos Cancer Institute, Department of Oncology, Wayne State University School of Medicine, Detroit, Michigan, USA."},{"name":"Department of Biostatistics, Data Science and Epidemiology, School of Public Health, Augusta University, Augusta, Georgia, USA."}]}],"member":"179","published-online":{"date-parts":[[2026,1,30]]},"reference":[{"key":"e_1_3_4_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2008.11.028"},{"key":"e_1_3_4_3_1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/46\/50\/505202"},{"key":"e_1_3_4_4_1","unstructured":"Brubaker MA Salzmann M Urtasun R. 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