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Epidemiological evidence supports an association between maternal exposure to air pollution and low birth weight. A popular method to estimate this association while identifying windows of susceptibility is a distributed lag model (DLM), which regresses an outcome onto exposure history observed at multiple time points. However, the standard DLM framework does not allow for modification of the association between repeated measures of exposure and the outcome. We propose a distributed lag interaction model that allows modification of the exposure\u2010time\u2010response associations across individuals by including an interaction between a continuous modifying variable and the exposure history. Our model framework is an extension of a standard DLM that uses a cross\u2010basis, or bi\u2010dimensional function space, to simultaneously describe both the modification of the exposure\u2010response relationship and the temporal structure of the exposure data. Through simulations, we showed that our model with penalization out\u2010performs a standard DLM when the true exposure\u2010time\u2010response associations vary by a continuous variable. Using a Colorado, USA birth cohort, we estimated the association between birth weight and ambient fine particulate matter air pollution modified by an area\u2010level metric of health and social adversities from Colorado EnviroScreen.<\/jats:p>","DOI":"10.1002\/env.2843","type":"journal-article","created":{"date-parts":[[2024,2,1]],"date-time":"2024-02-01T20:04:57Z","timestamp":1706817897000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Penalized distributed lag interaction model: Air pollution, birth weight, and neighborhood vulnerability"],"prefix":"10.1002","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-0785-3962","authenticated-orcid":false,"given":"Danielle","family":"Demateis","sequence":"first","affiliation":[{"name":"Department of Statistics Colorado State University  Fort Collins Colorado USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9423-2704","authenticated-orcid":false,"given":"Kayleigh P.","family":"Keller","sequence":"additional","affiliation":[{"name":"Department of Statistics Colorado State University  Fort Collins Colorado USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5854-2484","authenticated-orcid":false,"given":"David","family":"Rojas\u2010Rueda","sequence":"additional","affiliation":[{"name":"Department of Environmental and Radiological Health Sciences Colorado State University  Fort Collins Colorado USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5710-4992","authenticated-orcid":false,"given":"Marianthi\u2010Anna","family":"Kioumourtzoglou","sequence":"additional","affiliation":[{"name":"Department of Environmental Health Sciences Columbia University Mailman School of Public Health  New York NY USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4774-3883","authenticated-orcid":false,"given":"Ander","family":"Wilson","sequence":"additional","affiliation":[{"name":"Department of Statistics Colorado State University  Fort Collins Colorado USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2024,2]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/joc.3413"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.socscimed.2016.08.047"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1097\/01.ede.0000239732.50999.8f"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1136\/thx.54.12.1119"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/s13253-009-0004-z"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1164\/rccm.201702-0421OC"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.envres.2018.08.019"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00204-010-0514-z"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1093\/biostatistics\/kxu060"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.envint.2015.11.010"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.envres.2017.07.026"},{"key":"e_1_2_9_13_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41370-021-00403-8"},{"key":"e_1_2_9_14_1","volume-title":"Colorado EnviroScreen tool","author":"Colorado Department of Public Health and Environment (CDPHE)","year":"2022"},{"key":"e_1_2_9_15_1","volume-title":"Colorado EnviroScreen tool technical document","author":"Colorado EnviroScreen Tool Team","year":"2022"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.1186\/1471-2431-3-13"},{"key":"e_1_2_9_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.3940"},{"key":"e_1_2_9_18_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.12645"},{"key":"e_1_2_9_19_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11356-021-17762-2"},{"key":"e_1_2_9_20_1","volume-title":"Monographs on statistics and applied probability","author":"Green P. 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