# Bridge: The Bridge Distribution In bridgedist: An Implementation of the Bridge Distribution with Logit-Link as in Wang and Louis (2003)

## Description

Density, distribution function, quantile function and random generation for the bridge distribution with parameter `scale`. See Wang and Louis (2003).

## Usage

 ```1 2 3 4 5 6 7``` ```dbridge(x, scale = 1/2, log = FALSE) pbridge(q, scale = 1/2, lower.tail = TRUE, log.p = FALSE) qbridge(p, scale = 1/2, lower.tail = TRUE, log.p = FALSE) rbridge(n, scale = 1/2) ```

## Arguments

 `x, q` vector of quantiles. `scale` scale parameter. The scale must be between 0 and 1. A scale of 1/sqrt(1+3/pi^2) gives unit variance. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

## Details

If `scale` is omitted, the default value `1/2` is assumed.

The Bridge distribution parameterized by `scale` has distribution function

F(q) = 1 - 1/(pi*scale) * (pi/2 - atan( (exp(scale*q) + cos(scale*pi)) / sin(scale*pi) ))

and density

f(x) = 1/(2*pi) * sin(scale*pi) / (cosh(scale*x) + cos(scale*pi)).

The mean is 0 and the variance is pi^2 * (scale^{-2} - 1) / 3 .

## Value

`dbridge` gives the density, `pbridge` gives the distribution function, `qbridge` gives the quantile function, and `rbridge` generates random deviates.

The length of the result is determined by `n` for `rbridge`, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than `n` are recycled to the length of the result. Only the first elements of the logical arguments are used.

## Note

Consult the vignette for some figures comparing the normal, logistic, and bridge distributions.

## Source

`[dpq]bridge` are calculated directly from the definitions.

`rbridge` uses inversion.

## 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>

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>

 ```1 2``` ``` ## Confirm unit variance for scale = 1/sqrt(1+3/pi^2) var(rbridge(1e5, scale = 1/sqrt(1+3/pi^2))) # approximately 1 ```