R Users, The d/p/q/r functions for the bridge distribution are now available in bridgedist. When a random intercept follows the bridge distribution, as detailed in Wang and Louis (2003) <doi:10.1093/biomet/90.4.765 <http://dx.doi.org/10.1093/biomet/90.4.765>>, a marginalized random-intercept logistic regression will still be a logistic regression with marginal coefficients that are scalar multiples of the conditional regression's coefficients. Another way to state the result is that the sum of a standard logistic random variable and a bridge random variable will follow a logistic distribution with scale > 1. Examples of use: dbridge(0)#> [1] 0.1591549 pbridge(0)#> [1] 0.5 qbridge(0.5)#> [1] 0 mean(rbridge(1e5)) ## approximately 0#> [1] -0.003490218 var(rbridge(1e5, scale = 1/sqrt(1+3/pi^2))) # approximately 1#> [1] 0.9983954 Vignette: https://cran.r-project.org/web/packages/bridgedist/vignettes/the_bridgedist_basics.html References: Wang, Z. and Louis, T.A. (2003) Matching conditional and marginal shapes in binary random intercept models using a bridge distribution function. Biometrika, 90(4), 765-775. <DOI:10.1093/biomet/90.4.765> See also: Swihart, B.J., Caffo, B.S., and Crainiceanu, C.M. (2013). A Unifying Framework for Marginalized Random-Intercept Models of Correlated Binary Outcomes. International Statistical Review, 82 (2), 275-295 1-22. <DOI: 10.1111/insr.12035> Griswold, M.E., Swihart, B.J., Caffo, B.S and Zeger, S.L. (2013). Practical marginalized multilevel models. Stat, 2(1), 129-142. <DOI: 10.1002/sta4.22> Heagerty, P.J. (1999). Marginally specified logistic-normal models for longitudinal binary data. Biometrics, 55(3), 688-698. <DOI: 10.1111/j.0006-341X.1999.00688.x> Heagerty, P.J. and Zeger, S.L. (2000). Marginalized multilevel models and likelihood inference (with comments and a rejoinder by the authors). Stat. Sci., 15(1), 1-26. <DOI: 10.1214/ss/1009212671> All the best, Bruce [[alternative HTML version deleted]]