{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T09:46:10Z","timestamp":1787737570219,"version":"build-2784847793"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2015,8,11]],"date-time":"2015-08-11T00:00:00Z","timestamp":1439251200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJERPH"],"abstract":"<jats:p>Model selection is an integral, yet contentious, component of epidemiologic research. Unfortunately, there remains no consensus on how to identify a single, best model among multiple candidate models. Researchers may be prone to selecting the model that best supports their a priori, preferred result; a phenomenon referred to as \u201cwish bias\u201d. Directed acyclic graphs (DAGs), based on background causal and substantive knowledge, are a useful tool for specifying a subset of adjustment variables to obtain a causal effect estimate.  In many cases, however, a DAG will support multiple, sufficient or minimally-sufficient adjustment sets. Even though all of these may theoretically produce unbiased effect estimates they may, in practice, yield somewhat distinct values, and the need to select between these models once again makes the research enterprise vulnerable to wish bias.  In this work, we suggest combining adjustment sets with model averaging techniques to obtain causal estimates based on multiple, theoretically-unbiased models. We use three techniques for averaging the results among multiple candidate models: information criteria weighting, inverse variance weighting, and bootstrapping. We illustrate these approaches with an example from the Pregnancy, Infection, and Nutrition (PIN) study. We show that each averaging technique returns similar, model averaged causal estimates. An a priori strategy of model averaging provides a means of integrating uncertainty in selection among candidate, causal models, while also avoiding the temptation to report the most attractive estimate from a suite of equally valid alternatives.<\/jats:p>","DOI":"10.3390\/ijerph120809391","type":"journal-article","created":{"date-parts":[[2015,8,11]],"date-time":"2015-08-11T10:38:57Z","timestamp":1439289537000},"page":"9391-9407","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Model Averaging for Improving Inference from  Causal Diagrams"],"prefix":"10.3390","volume":"12","author":[{"given":"Ghassan","family":"Hamra","sequence":"first","affiliation":[{"name":"Department of Environmental and Occupational Health, Drexel University School of Public Health, Philadelphia, PA 19104, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jay","family":"Kaufman","sequence":"additional","affiliation":[{"name":"Department of Epidemiology, Biostatistics, and Occupational Health, McGill University, Montreal, QC H3A 1A2, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anjel","family":"Vahratian","sequence":"additional","affiliation":[{"name":"Department of Obstetrics and Gynecology, University of Michigan, Ann Arbor, MI 48109, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,8,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1093\/oxfordjournals.aje.a114254","article-title":"The role of model selection in causal inference from nonexperimental data","volume":"123","author":"Robins","year":"1986","journal-title":"Am. J. Epidemiol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/0895-4356(90)90167-N","article-title":"The wish bias","volume":"43","author":"Wynder","year":"1990","journal-title":"J. Clin. Epidemiol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1038\/ijo.2009.239","article-title":"White hat bias: Examples of its presence in obesity research and a call for renewed commitment to faithfulness in research reporting","volume":"34","author":"Cope","year":"2010","journal-title":"Int. J. Obes."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1615","DOI":"10.1111\/j.1651-2227.2010.02006.x","article-title":"White hat bias: A threat to the integrity of scientific reporting","volume":"99","author":"Cope","year":"2010","journal-title":"Acta Paediatr."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1097\/00001648-199901000-00008","article-title":"Causal diagrams for epidemiologic research","volume":"10","author":"Greenland","year":"1999","journal-title":"Epidemiology"},{"key":"ref_6","unstructured":"Pearl, J. (2000). Causality: Models, Reasoning, and Inference, Cambridge University Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"111","DOI":"10.2307\/271063","article-title":"Bayesian model selection in social research","volume":"25","author":"Raftery","year":"1995","journal-title":"Sociol. Methodol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3215","DOI":"10.1002\/sim.976","article-title":"Variable selection and Bayesian model averaging in case-control studies","volume":"20","author":"Viallefont","year":"2001","journal-title":"Stat. Med."},{"key":"ref_9","unstructured":"Burnham, K.P., and Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-theoretic Approach, Springer. [2nd ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1097\/EDE.0b013e318127181b","article-title":"Four types of effect modification: A classification based on directed acyclic graphs","volume":"18","author":"VanderWeele","year":"2007","journal-title":"Epidemiology"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1186\/1471-2288-8-70","article-title":"Reducing bias through directed acyclic graphs","volume":"8","author":"Shrier","year":"2008","journal-title":"BMC Med. Res. Methodol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1969","DOI":"10.1093\/ije\/dyu149","article-title":"Good practices for quantitative bias analysis","volume":"43","author":"Lash","year":"2014","journal-title":"Int. J. Epidemiol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1097\/EDE.0b013e3181a819a1","article-title":"Overadjustment bias and unnecessary adjustment in epidemiologic studies","volume":"20","author":"Schisterman","year":"2009","journal-title":"Epidemiology"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1097\/EDE.0b013e31822549e8","article-title":"A comparison of methods to estimate the hazard ratio under conditions of time-varying confounding and nonpositivity","volume":"22","author":"Naimi","year":"2011","journal-title":"Epidemiology"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1097\/EDE.0b013e31818ef366","article-title":"The consistency statement in causal inference: A definition or an assumption?","volume":"20","author":"Cole","year":"2009","journal-title":"Epidemiology"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1398","DOI":"10.1198\/016214506000000636","article-title":"What do randomized studies of housing mobility demonstrate? Causal inference in the face of interference","volume":"101","author":"Sobel","year":"2006","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1097\/00001648-199011000-00003","article-title":"Randomization, statistics, and causal inference","volume":"1","author":"Greenland","year":"1990","journal-title":"Epidemiology"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1093\/ije\/dyp334","article-title":"Illustrating bias due to conditioning on a collider","volume":"39","author":"Cole","year":"2010","journal-title":"Int. J. Epidemiol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1046\/j.1365-3016.1999.00156.x","article-title":"Determinants of participation in an epidemiological study of preterm delivery","volume":"13","author":"Savitz","year":"1999","journal-title":"Paediatr. Perinat. Epidemiol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.annepidem.2005.02.005","article-title":"Maternal pre-pregnancy overweight and obesity and the risk of cesarean delivery in nulliparous women","volume":"15","author":"Vahratian","year":"2005","journal-title":"Ann. Epidemiol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"603","DOI":"10.2307\/2533961","article-title":"Model selection: An integral part of inference","volume":"53","author":"Buckland","year":"1997","journal-title":"Biometrics"},{"key":"ref_22","first-page":"382","article-title":"Bayesian model averaging: A tutorial","volume":"14","author":"Hoeting","year":"1999","journal-title":"Statist. Sci."},{"key":"ref_23","unstructured":"Rothman, K.J., Greenland, S., and Lash, T.L. (2008). Modern Epidemiology, Lippincott Williams & Wilkins. [3rd ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Higgins, J.P.T., and Green, S. (2008). Cochrane Handbook for Systematic Reviews of Interventions, Wiley-Blackwell.","DOI":"10.1002\/9780470712184"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R. (1993). An Introduction to the Bootstrap, CRC Press.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1097\/00001648-200105000-00005","article-title":"Low P-values or narrow confidence intervals: Which are more durable?","volume":"12","author":"Poole","year":"2001","journal-title":"Epidemiology"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1097\/EDE.0b013e31817307dc","article-title":"Model selection and health effect estimation in environmental epidemiology","volume":"19","author":"Dominici","year":"2008","journal-title":"Epidemiology"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00411-011-0395-4","article-title":"Model averaging in the analysis of leukemia mortality among Japanese A-bomb survivors","volume":"51","author":"Richardson","year":"2012","journal-title":"Radiat. Environ. Biophys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1093\/aje\/kwm355","article-title":"Invited commentary: Variable selection versus shrinkage in the control of multiple confounders","volume":"167","author":"Greenland","year":"2008","journal-title":"Am. J. Epidemiol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1002\/sim.2739","article-title":"The design versus the analysis of observational studies for causal effects: Parallels with the design of randomized trials","volume":"26","author":"Rubin","year":"2007","journal-title":"Stat. Med."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1093\/aje\/kwj149","article-title":"Variable selection for propensity score models","volume":"163","author":"Brookhart","year":"2006","journal-title":"Am. J. Epidemiol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1214\/ss\/1009211805","article-title":"Confounding and collapsibility in causal inference","volume":"14","author":"Greenland","year":"1999","journal-title":"Stat. Sci."}],"container-title":["International Journal of Environmental Research and Public Health"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1660-4601\/12\/8\/9391\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:50:30Z","timestamp":1760215830000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1660-4601\/12\/8\/9391"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,11]]},"references-count":32,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2015,8]]}},"alternative-id":["ijerph120809391"],"URL":"https:\/\/doi.org\/10.3390\/ijerph120809391","relation":{},"ISSN":["1660-4601"],"issn-type":[{"value":"1660-4601","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,8,11]]}}}