{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/pseudo-r-2-statistics-under-complex-sampling","title":"Pseudo-$R^2$ statistics under complex sampling","arxiv_id":"1701.07745","date":"2017-01-26","proceeding":null,"authors":["Thomas Lumley"],"abstract":"Model summaries based on the ratio of fitted and null likelihoods have been proposed for generalised linear models, reducing to the familiar $R^2$ coefficient of determination in the Gaussian model with identity link. In this note I show how to define the Cox--Snell and Nagelkerke summaries under arbitrary probability sampling designs, giving a design-consistent estimator of the population model summary. I also show that for logistic regression models under case--control sampling the usual Cox--Snell and Nagelkerke $R^2$ are not design-consistent, but are systematically larger than would be obtained with a cross-sectional or cohort sample, even in settings where the weighted and unweighted logistic regression estimators are similar or identical.","url_abs":"http://arxiv.org/abs/1701.07745v1","url_pdf":"http://arxiv.org/pdf/1701.07745v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"pseudo-r-2-statistics-under-complex-sampling","repo_url":"https://github.com/tslumley/pseudorsq","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}