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Marginalized random-effects models for clustered binomial data through innovative link functions. (English) Zbl 1480.62127

Summary: Random-effects models are frequently used to analyze clustered binomial data. The direct computation of the marginal mean response, when integrated over the distribution of random effects, is challenging due to taking nonclosed-form expressions of the marginal link function. This paper extends the marginalized modeling methodology using innovative link functions, where the marginal mean response is modeled in terms of covariates and random effects. To derive the explicit closed-form representation of both marginal and conditional means, the regression structure is designed through an original strategy to introduce particular random-effects distributions. It will consequently allow for a reasonable interpretation of covariate effects. A Bayesian approach is employed to make the statistical inference by implementing the Markov chain Monte Carlo scheme. We conducted simulation studies to show the usefulness of our methodology. Two real-life data sets, taken from the teratology and respiratory studies, have been analyzed for illustration. The findings confirm that our new modeling methodology offers convenient settings for analyzing binomial responses in practice.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
65C05 Monte Carlo methods

Software:

JAGS; SAS; BUGS
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Full Text: DOI

References:

[1] Albert, JH; Chib, S., Bayesian analysis of binary and polychotomous response data, J. Am. Stat. Assoc., 88, 422, 669-679 (1993) · Zbl 0774.62031
[2] Boehm, L.; Reich, BJ; Bandyopadhyay, D., Bridging conditional and marginal inference for spatially referenced binary data, Biometrics., 69, 2, 545-554 (2013) · Zbl 1274.62725
[3] Celeux, G.; Hurn, M.; Robert, CP, Computational and inferential difficulties with mixture posterior distributions, J. Am. Stat. Assoc., 95, 3, 957-979 (2000) · Zbl 0999.62020
[4] Chan, JS; Kuk, AY, Maximum likelihood estimation for probit-linear mixed models with correlated random effects, Biometrics, 53, 1, 86-97 (1997) · Zbl 1065.62503
[5] Chao, WH; Palta, M.; Young, T., Effect of omitted confounders on the analysis of correlated binary data, Biometrics, 53, 2, 678-89 (1997) · Zbl 0878.62078
[6] Coull, BA; Ryan, LM; El-Shaarawi, AH; Piegorsch, WW, Biological assay, Encyclopedia of Environmetrics, 189-192 (2002), Chichester: John Wiley and Sons, Chichester
[7] Czado, C.; Santner, TJ, The effect of link misspecification on binary regression inference, J. Stat. Plann. Infer., 33, 2, 213-231 (1992) · Zbl 0781.62037
[8] Demidenko, E., Mixed Models: Theory and Applications With R (2013), Hoboken: Wiley-Interscience, Hoboken · Zbl 1276.62049
[9] Dorazio, RM; Andrew Royle, J., Mixture models for estimating the size of a closed population when capture rates vary among individuals, Biometrics, 59, 2, 351-364 (2003) · Zbl 1210.62226
[10] Fitzmaurice, GM; Laird, NM, A likelihood based method for analysing longitudinal binary responses, Biometrika, 80, 1, 141-151 (1993) · Zbl 0775.62296
[11] Forcina, A.; Franconi, L., Regression analysis with the beta-binomial distribution, Rivista di Statistica Applicata., 21, 1, 7-12 (1998)
[12] Gelfand, AE; Dey, DK, Bayesian model choice: asymptotics and exact calculations, J. Roy. Stat. Soc. Ser. B (Stat. Methodol.), 56, 3, 501-514 (1994) · Zbl 0800.62170
[13] Gelman, A.; Rubin, DB, Inference from iterative simulation using multiple sequences, Stat. Sci., 7, 4, 457-472 (1992) · Zbl 1386.65060
[14] Gradshteyn, IS; Ryzhik, IM, Table of Integrals, Series, and Products (2014), London: Academic Press, London · Zbl 0918.65002
[15] Griswold, ME; Swihart, BJ; Caffo, BS; Zeger, SL, Practical marginalized multilevel models, Stat., 2, 1, 129-142 (2013)
[16] Heagerty, PJ, Marginally specified logistic-normal models for longitudinal binary data, Biometrics, 55, 3, 688-698 (1999) · Zbl 1059.62566
[17] Heagerty, PJ; Zeger, SL, Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors), Stat. Sci., 15, 1, 1-26 (2000)
[18] Hedeker, D.; du Toit, SH; Demirtas, H.; Gibbons, RD, A note on marginalization of regression parameters from mixed models of binary outcomes, Biometrics, 74, 1, 354-361 (2018) · Zbl 1415.62098
[19] Hinde, J.; Demétrio, CGB, Overdispersion: models and estimation, Comput. Stat. Data Anal., 27, 2, 151-170 (1998) · Zbl 1042.62578
[20] Holgate, P., The modality of some compound Poisson distributions, Biometrika, 57, 3, 666-667 (1970) · Zbl 0203.52002
[21] Ibrahim, J.; Chen, MH; Sinha, D., Bayesian Survival Analysis (2001), New York: Springer-Verlag, New York · Zbl 0978.62091
[22] Iddi, S.; Molenberghs, G., A combined overdispersed and marginalized multilevel model, Comput. Stat. Anal., 56, 6, 1944-1951 (2012) · Zbl 1242.62118
[23] Inan, G., Comments on ongitudinal beta-binomial modeling using GEE for overdispersed binomial data, Stat. Med., 37, 3, 503-505 (2018)
[24] Kenward, MG; Molenberghs, G., A taxonomy of mixing and outcome distributions based on conjugacy and bridging, Commun. Stat. Theory Meth., 45, 7, 1953-1968 (2016) · Zbl 1381.62236
[25] Kim, S.; Chen, MH; Dey, DK, Flexible generalized t-link models for binary response data, Biometrika, 95, 1, 93-106 (2008) · Zbl 1437.62513
[26] Kim, G.; Lee, Y., Marginal versus conditional beta-binomial regression models, Stat. Meth. Med. Res., 28, 3, 761-769 (2019)
[27] Koenker, R.; Yoon, J., Parametric links for binary choice models: A Fisherian-Bayesian colloquy, J. Econ., 152, 2, 120-130 (2009) · Zbl 1431.62313
[28] Kupper, LL; El-Shaarawi, AH; Piegorsch, WW, Litter effect, Encyclopedia of Environmetrics, 1169-1172 (2002), Chichester: John Wiley and Sons, Chichester
[29] Lemonte, AJ; Bazán, JL, New links for binary regression: an application to coca cultivation in Peru, Test, 27, 3, 597-617 (2018) · Zbl 1417.62212
[30] Li, X.; Bandyopadhyay, D.; Lipsitz, S.; Sinha, D., Likelihood methods for binary responses of present components in a cluster, Biometrics, 67, 2, 629-635 (2011) · Zbl 1217.62178
[31] Lunn, D.; Spiegelhalter, D.; Thomas, A.; Best, N., The BUGS project: Evolution, critique and future directions, Stat. Med., 28, 25, 30-49 (2009)
[32] Luo, R.; Paul, S., Estimation for zero-inflated beta-binomial regression model with missing response data, Stat. Med., 37, 26, 3789-3813 (2018)
[33] Marquart, L.; Haynes, M., Misspecification of multimodal random-effect distributions in logistic mixed models, J. Roy. Stat. Soc. Ser. A (Stat. Soc.), 182, 1, 305-321 (2019)
[34] McCulloch, CE; Searle, SR, Generalized, Linear, and Mixed Models (2001), New York: John Wiley and Sons, New York · Zbl 0964.62061
[35] Millar, RB, Comparison of hierarchical Bayesian models for overdispersed count data using DIC and Bayes factors, Biometrics, 65, 3, 962-969 (2009) · Zbl 1172.62054
[36] Molenberghs, G.; Kenward, M.; Verbeke, G.; Iddi, S.; Efendi, A., On the connections between bridge distributions, marginalized multilevel models, and generalized linear mixed models, Int. J. Stat. Prob., 2, 24, 1-21 (2013)
[37] Molenberghs, G.; Verbeke, G.; Iddi, S.; Demétrio, CG, A combined beta and normal random-effects model for repeated, overdispersed binary and binomial data, J. Multivar. Anal., 111, 94-109 (2012) · Zbl 1294.62040
[38] Muff, S.; Held, L.; Keller, LF, Marginal or conditional regression models for correlated non-normal data?, Meth. Ecol. Evol., 7, 12, 1514-1524 (2016)
[39] Najera-Zuloaga, J.; Lee, DJ; Arostegui, I., A beta-binomial mixed-effects model approach for analyzing longitudinal discrete and bounded outcomes, Biometric. Jl., 61, 3, 500-519 (2019) · Zbl 1429.62576
[40] Najera-Zuloaga, J.; Lee, DJ; Arostegui, I., Comparison of beta-binomial regression model approaches to analyze health-related quality of life data, Stat. Meth. Med. Res., 27, 10, 2989-3009 (2018)
[41] Naranjo, L.; Martín, J.; Pérez, CJ, Bayesian binary regression with exponential power link, Comput. Stat. Data Anal., 71, 464-476 (2014) · Zbl 1471.62147
[42] Neuhaus, JM; Hauck, WW; Kalbfleisch, JD, The effects of mixture distribution misspecification when fitting mixed-effects logistic models, Biometrika, 79, 4, 755-762 (1992)
[43] Parzen, M.; Souparno, G.; Stuart, L.; Debajyoti, S.; Garrett, MF; Bani, KM; Joseph, GI, A generalized linear mixed model for longitudinal binary data with a marginal logit link function, Annal. Appl. Stat., 5, 1, 449-467 (2011) · Zbl 1220.62093
[44] Piegorsch, WW; Bailer, AJ, Analyzing Environmental Data (2005), New York: John Wiley and Sons, New York · Zbl 0997.62523
[45] Plummer, M.: JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International Workshop on Distributed Statistical Computing 124(125.10), pp. 1-10 (2003)
[46] Ritz, J.; Spiegelman, D., Equivalence of conditional and marginal regression models for clustered and longitudinal data, Stat. Meth. Med. Res., 13, 4, 309-323 (2004) · Zbl 1121.62657
[47] Robert, C.; Casella, G., Monte Carlo Statistical Methods (2004), Berlin: Springer, Berlin · Zbl 1096.62003
[48] Spiegelhalter, DJ; Best, NG; Carlin, BP; Van Der Linde, A., Bayesian measures of model complexity and fit, J. Roy. Stat. Soc. Ser. B (Stat. Methodol.), 64, 4, 583-639 (2002) · Zbl 1067.62010
[49] Stokes, ME; Davisc, S.; Koch, GG, Categorical Data Analysis Using the SAS System (1995), Cary NC: SAS Institute Inc, Cary NC
[50] Swihart, BJ; Caffo, BS; Crainiceanu, CM, A unifying framework for marginalised random-intercept models of correlated binary outcomes, Int. Stat. Rev., 82, 2, 275-295 (2014) · Zbl 1416.62288
[51] Vangeneugden, T.; Molenberghs, G.; Verbeke, G.; Demétrio, CGB, Marginal Correlation from Logit- and Probit-Beta-Normal Models for Hierarchical Binary Data, Communications in Statistics - Theory and Methods., 43, 19, 4164-4178 (2014) · Zbl 1309.62017
[52] Wang, X.; Dey, DK, Generalized extreme value regression for binary response data: An application to B2B electronic payments system adoption, The Annals of Applied Statistics., 4, 4, 2000-2023 (2010) · Zbl 1220.62165
[53] Wang, Z.; Louis, T., Marginalized binary mixed-effects models with covariate-dependent random effects and likelihood inference, Biometrics, 60, 4, 884-891 (2004) · Zbl 1274.62182
[54] Wang, Z.; Louis, T., Matching conditional and marginal shapes in binary random intercept models using a bridge distribution function, Biometrika, 90, 4, 765-775 (2003) · Zbl 1436.62294
[55] Wu, H.; Zhang, Y.; Long, JD, Longitudinal beta-binomial modeling using GEE for over-dispersed binomial data, Stat. Med., 36, 6, 1029-1040 (2017)
[56] Yu, S.; Huang, X., Link misspecification in generalized linear mixed models with a random intercept for binary responses, TEST, 28, 3, 827-843 (2019) · Zbl 1420.62321
[57] Zeger, SL; Liang, KY; Albert, PA, Models for longitudinal data: a generalized estimating equation approach, Biometrics, 44, 4, 1049-1060 (1998) · Zbl 0715.62136
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