{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:30:47Z","timestamp":1772253047336,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T00:00:00Z","timestamp":1663632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000066","name":"National Institute of Environmental Health Sciences (NIEHS)","doi-asserted-by":"publisher","award":["P30ES023515"],"award-info":[{"award-number":["P30ES023515"]}],"id":[{"id":"10.13039\/100000066","id-type":"DOI","asserted-by":"publisher"}]},{"name":"APC","award":["P30ES023515"],"award-info":[{"award-number":["P30ES023515"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Epidemiologists often study the associations between a set of exposures and multiple biologically relevant outcomes. However, the frequently used scale-and-context-dependent regression coefficients may not offer meaningful comparisons and could further complicate the interpretation if these outcomes do not have similar units. Additionally, when scaling up a hypothesis-driven study based on preliminary data, knowing how large to make the sample size is a major uncertainty for epidemiologists. Conventional p-value-based sample size calculations emphasize precision and might lead to a large sample size for small- to moderate-effect sizes. This asymmetry between precision and utility is costly and might lead to the detection of irrelevant effects. Here, we introduce the \u201c\u03b4-score\u201d concept, by modifying Cohen\u2019s f2. \u03b4-score is scale independent and circumvents the challenges of regression coefficients. Further, under a new hypothesis testing framework, it quantifies the maximum Cohen\u2019s f2 with certain optimal properties. We also introduced \u201cSufficient sample size\u201d, which is the minimum sample size required to attain a \u03b4-score. Finally, we used data on adults from a 2017\u20132018 U.S. National Health and Nutrition Examination Survey to demonstrate how the \u03b4-score and sufficient sample size reduced the asymmetry between precision and utility by finding associations between mixtures of per-and polyfluoroalkyl substances and metals with serum high-density and low-density lipoprotein cholesterol.<\/jats:p>","DOI":"10.3390\/sym14101962","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:08:09Z","timestamp":1663718889000},"page":"1962","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantifying the Effect Size of Exposure-Outcome Association Using \u03b4-Score: Application to Environmental Chemical Mixture Studies"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6643-5176","authenticated-orcid":false,"given":"Vishal","family":"Midya","sequence":"first","affiliation":[{"name":"Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA"},{"name":"Department of Public Health Sciences, Pennsylvania State College of Medicine, Hershey, PA 17033, USA"}]},{"given":"Jiangang","family":"Liao","sequence":"additional","affiliation":[{"name":"Division of Biostatistics and Bioinformatics, Pennsylvania State College of Medicine, Hershey, PA 17033, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6058-5907","authenticated-orcid":false,"given":"Chris","family":"Gennings","sequence":"additional","affiliation":[{"name":"Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA"}]},{"given":"Elena","family":"Colicino","sequence":"additional","affiliation":[{"name":"Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA"}]},{"given":"Susan L.","family":"Teitelbaum","sequence":"additional","affiliation":[{"name":"Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA"}]},{"given":"Robert O.","family":"Wright","sequence":"additional","affiliation":[{"name":"Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4633-229X","authenticated-orcid":false,"given":"Damaskini","family":"Valvi","sequence":"additional","affiliation":[{"name":"Department of Environmental Medicine and Public Health, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/j.jhazmat.2014.12.012","article-title":"Why endocrine disrupting chemicals (edcs) challenge traditional risk assessment and how to respond","volume":"286","author":"Tal","year":"2015","journal-title":"J. Hazard. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cano, R., P\u00e9rez, J.L., D\u00e1vila, L.A., Ortega, A., G\u00f3mez, Y., Valero-Cede\u00f1o, N.J., Parra, H., Manzano, A., V\u00e9liz Castro, T.I., and Albornoz, M.P.D. (2021). Role of endocrine-disrupting chemicals in the pathogenesis of non-alcoholic fatty liver disease: A comprehensive review. Int. J. Mol. Sci., 22.","DOI":"10.3390\/ijms22094807"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"e2220176","DOI":"10.1001\/jamanetworkopen.2022.20176","article-title":"Association of prenatal exposure to endocrine-disrupting chemicals with liver injury in children","volume":"5","author":"Midya","year":"2022","journal-title":"JAMA Netw. Open"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/S0140-6736(13)62227-8","article-title":"Increasing value and reducing waste in research design, conduct, and analysis","volume":"383","author":"Ioannidis","year":"2014","journal-title":"Lancet"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1080\/00031305.2016.1154108","article-title":"The asa statement on p-values: Context, process, and purpose","volume":"70","author":"Wasserstein","year":"2016","journal-title":"Am. Stat."},{"key":"ref_6","unstructured":"Cohen, J. (1976). Statistical Power Analysis for the Behavioral Sciences, Lawrence Erlbaum Associates."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"813","DOI":"10.3389\/fpsyg.2019.00813","article-title":"The meaningfulness of effect sizes in psychological research: Differences between sub-disciplines and the impact of potential biases","volume":"10","author":"Schwarz","year":"2019","journal-title":"Front. Psychol."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Smithson, M. (2003). Confidence Intervals, SAGE Publications. Number No. 140 in Confidence Intervals.","DOI":"10.4135\/9781412983761"},{"key":"ref_9","unstructured":"Grissom, R., and Kim, J. (2005). Effect Sizes for Research: A Broad Practical Approach, Lawrence Erlbaum Associates."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1038\/ng.608","article-title":"Common snps explain a large proportion of the heritability for human height","volume":"42","author":"Yang","year":"2010","journal-title":"Nat. Genet."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Joubert, B.R., Kioumourtzoglou, M.A., Chamberlain, T., Chen, H.Y., Gennings, C., Turyk, M.E., Miranda, M.L., Webster, T.F., Ensor, K.B., and Dunson, D.B. (2022). Powering research through innovative methods for mixtures in epidemiology (prime) program: Novel and expanded statistical methods. Int. J. Environ. Res. Public Health, 19.","DOI":"10.3390\/ijerph19031378"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Chen, H.Y., Li, H., Argos, M., Persky, V.W., and Turyk, M.E. (2022). Statistical methods for assessing the explained variation of a health outcome by a mixture of exposures. Int. J. Environ. Res. Public Health, 19.","DOI":"10.3390\/ijerph19052693"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1214\/aoms\/1177732360","article-title":"The large-sample distribution of the likelihood ratio for testing composite hypotheses","volume":"9","author":"Wilks","year":"1938","journal-title":"Ann. Math. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3137","DOI":"10.1002\/(SICI)1097-0258(19991130)18:22<3137::AID-SIM239>3.0.CO;2-O","article-title":"Asymptotic power calculations: Description, examples, computer code","volume":"18","author":"Brown","year":"1999","journal-title":"Stat. Med."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"111","DOI":"10.3389\/fpsyg.2012.00111","article-title":"A practical guide to calculating cohen\u2019s f2, a measure of local effect size, from proc mixed","volume":"3","author":"Selya","year":"2012","journal-title":"Front. Psychol."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Parzen, E., Tanabe, K., and Kitagawa, G. (1998). Information theory and an extension of the maximum likelihood principle. Selected Papers of Hirotugu Akaike, Springer.","DOI":"10.1007\/978-1-4612-1694-0"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1093\/bib\/bbz016","article-title":"Sensitivity and specificity of information criteria","volume":"21","author":"Dziak","year":"2019","journal-title":"Brief. Bioinform."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"307","DOI":"10.2307\/1912557","article-title":"Likelihood ratio tests for model selection and non-nested hypotheses","volume":"57","author":"Vuong","year":"1989","journal-title":"Econometrica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1037\/met0000061","article-title":"Sequential hypothesis testing with bayes factors: Efficiently testing mean differences","volume":"22","author":"Wagenmakers","year":"2017","journal-title":"Psychol. Methods"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1037\/a0024377","article-title":"Bayes factor approaches for testing interval null hypotheses","volume":"16","author":"Morey","year":"2011","journal-title":"Psychol. Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1037\/a0029146","article-title":"Bayesian estimation supersedes the t test","volume":"142","author":"Kruschke","year":"2013","journal-title":"J. Exp. Psychol. Gen."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1080\/00031305.2019.1701550","article-title":"Connecting and contrasting the bayes factor and a modified rope procedure for testing interval null hypotheses","volume":"75","author":"Liao","year":"2020","journal-title":"Am. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Midya, V., and Liao, J. (2021). Systematic deviation in mean of log bayes factor: Implication and application. Commun. Stat.-Theory Methods, 1\u201310.","DOI":"10.1080\/03610926.2021.1970768"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/BF01068419","article-title":"A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability","volume":"15","author":"Schuirmann","year":"1987","journal-title":"J. Pharmacokinet. Biopharm."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF01063556","article-title":"Power of the two one-sided tests procedure in bioequivalence","volume":"18","author":"Phillips","year":"1990","journal-title":"J. Pharmacokinet. Biopharm."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2202\/1557-4679.1169","article-title":"Power for testing multiple instances of the two one-sided tests procedure","volume":"5","author":"Phillips","year":"2009","journal-title":"Int. J. Biostat."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1017\/S0269964820000625","article-title":"Ratio estimation of the population mean using auxiliary information under the optimal sampling design","volume":"36","author":"Long","year":"2022","journal-title":"Probab. Eng. Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.envpol.2017.09.019","article-title":"Association among total serum isomers of perfluorinated chemicals, glucose homeostasis, lipid profiles, serum protein and metabolic syndrome in adults: Nhanes, 2013\u20132014","volume":"232","author":"Liu","year":"2018","journal-title":"Environ. Pollut."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.envpol.2018.08.060","article-title":"Associations between lipid\/lipoprotein levels and perfluoroalkyl substances among us children aged 6\u201311 years","volume":"243","author":"Jain","year":"2018","journal-title":"Environ. Pollut."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"850","DOI":"10.3389\/fendo.2021.706352","article-title":"Exposure to perfluoroalkyl chemicals and cardiovascular disease: Experimental and epidemiological evidence","volume":"12","author":"Meneguzzi","year":"2021","journal-title":"Front. Endocrinol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1177\/0960327119889654","article-title":"The association between heavy metal and serum cholesterol levels in the us population: National health and nutrition examination survey 2009\u20132012","volume":"39","author":"Buhari","year":"2020","journal-title":"Hum. Exp. Toxicol."},{"key":"ref_33","unstructured":"CDC, and NCHS (2022, September 01). US National Health and Nutrition Examination Survey Data, 2017\u20132018, Available online: https:\/\/wwwn.cdc.gov\/nchs\/nhanes\/search\/datapage.aspx?Component=Laboratory&Cycle=2017-2018."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/s13253-014-0180-3","article-title":"Characterization of weighted quantile sum regression for highly correlated data in a risk analysis setting","volume":"20","author":"Carrico","year":"2015","journal-title":"J. Agric. Biol. Environ. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1093\/biostatistics\/kxu058","article-title":"Bayesian kernel machine regression for estimating the health effects of multi-pollutant mixtures","volume":"16","author":"Bobb","year":"2014","journal-title":"Biostatistics"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"e092","DOI":"10.1097\/EE9.0000000000000092","article-title":"Per- and poly-fluoroalkyl substances and bone mineral density","volume":"4","author":"Colicino","year":"2020","journal-title":"Environ. Epidemiol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"047004","DOI":"10.1289\/EHP5838","article-title":"A quantile-based g-computation approach to addressing the effects of exposure mixtures","volume":"128","author":"Keil","year":"2020","journal-title":"Environ. Health Perspect."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1186\/s12940-019-0515-1","article-title":"An overview of methods to address distinct research questions on environmental mixtures: An application to persistent organic pollutants and leukocyte telomere length","volume":"18","author":"Gibson","year":"2019","journal-title":"Environ. Health"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1093\/ije\/dyaa259","article-title":"Reflection on modern methods: Good practices for applied statistical learning in epidemiology","volume":"50","author":"Nunez","year":"2021","journal-title":"Int. J. Epidemiol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.ecolmodel.2018.07.002","article-title":"Bayesian modeling of individual growth variability using back-calculation: Application to pink cusk-eel (genypterus blacodes) off chile","volume":"385","author":"Wiff","year":"2018","journal-title":"Ecol. Model."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Vincenzi, S., Mangel, M., Crivelli, A.J., Munch, S., and Skaug, H.J. (2014). Determining individual variation in growth and its implication for life-history and population processes using the empirical bayes method. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003828"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"107422","DOI":"10.1016\/j.envint.2022.107422","article-title":"State-of-the-art methods for exposure-health studies: Results from the exposome data challenge event","volume":"168","author":"Maitre","year":"2022","journal-title":"Environ. Int."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Cui, Y., Eccles, K.M., Kwok, R.K., Joubert, B.R., Messier, K.P., and Balshaw, D.M. (2022). Integrating multiscale geospatial environmental data into large population health studies: Challenges and opportunities. Toxics, 10.","DOI":"10.3390\/toxics10070403"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/1962\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:35:48Z","timestamp":1760142948000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/1962"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,20]]},"references-count":43,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14101962"],"URL":"https:\/\/doi.org\/10.3390\/sym14101962","relation":{"has-preprint":[{"id-type":"doi","id":"10.1101\/2022.03.02.22271732","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,20]]}}}