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
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