{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T16:00:56Z","timestamp":1747152056565,"version":"3.40.5"},"reference-count":22,"publisher":"Wiley","issue":"9","license":[{"start":{"date-parts":[[2021,2,15]],"date-time":"2021-02-15T00:00:00Z","timestamp":1613347200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Statistics in Medicine"],"published-print":{"date-parts":[[2021,4,30]]},"abstract":"<jats:p>Rathouz and Gao [2] and Luo and Tsai [3] proposed valuable extensions to the generalized linear model for modeling a nonlinear monotonic relationship between the mean response and a set of covariates. In their extensions for discrete data the baseline response distribution is unspecified and is estimated from the data. We propose to extend this model for the analysis of longitudinal data by incorporating random effects into the linear predictor, and using maximum likelihood for estimation and inference. Motivated in particular by longitudinal studies of clinical scale outcomes, we developed an estimation procedure for a finite\u2010support response using a generalized expectation\u2010maximization algorithm where Gauss\u2010Hermite quadrature is employed to approximate the integrals in the E step of the algorithm. Upon convergence, the observed information matrix is estimated through second\u2010order numerical differentiation of the log\u2010likelihood function. Asymptotic properties of the maximum likelihood estimates follow under the usual regularity conditions. Simulation studies are conducted to assess its finite\u2010sample properties and compare the proposed model to the generalized linear mixed model. The proposed method is illustrated in an analysis of data from a longitudinal study of Huntington's disease.<\/jats:p>","DOI":"10.1002\/sim.8902","type":"journal-article","created":{"date-parts":[[2021,2,18]],"date-time":"2021-02-18T13:56:34Z","timestamp":1613656594000},"page":"2272-2285","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Proportional likelihood ratio mixed model for discrete longitudinal data"],"prefix":"10.1002","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7782-2510","authenticated-orcid":false,"given":"Hongqian","family":"Wu","sequence":"first","affiliation":[{"name":"BeiGene  Ridgefield Park New Jersey USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael P.","family":"Jones","sequence":"additional","affiliation":[{"name":"Department of Biostatistics University of Iowa  Iowa City Iowa USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2021,2,15]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.2307\/2344614"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1093\/biostatistics\/kxn030"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asr060"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1136\/jnnp.2007.128728"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1097\/MLR.0000000000000120"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1097\/MLR.0000000000000338"},{"key":"e_1_2_8_8_1","doi-asserted-by":"publisher","DOI":"10.1186\/s12913-016-1734-6"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.2307\/2529876"},{"key":"e_1_2_8_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/9780470594001"},{"key":"e_1_2_8_11_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/73.1.13"},{"key":"e_1_2_8_12_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asu055"},{"volume-title":"Theory of Point Estimation","year":"1998","author":"Lehmann EL","key":"e_1_2_8_13_1"},{"volume-title":"The EM Algorithm and Extensions","year":"2007","author":"McLachlan G","key":"e_1_2_8_14_1"},{"volume-title":"Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables","year":"1972","author":"Abramowitz M","key":"e_1_2_8_15_1"},{"key":"e_1_2_8_16_1","unstructured":"GilbertP VaradhanR. numDeriv: accurate numerical derivatives. 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