Bridge {bridgedist}

The Bridge Distribution

Description

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

Usage

dbridge(x, phi = 1/2, log = FALSE)

pbridge(q, phi = 1/2, lower.tail = TRUE, log.p = FALSE)

qbridge(p, phi = 1/2, lower.tail = TRUE, log.p = FALSE)

rbridge(n, phi = 1/2)

Arguments

x, q

vector of quantiles.

phi

phi parameter. The phi must be between 0 and 1. A phi 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 \le 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 phi is omitted, the default value 1/2 is assumed.

The Bridge distribution parameterized by phi has distribution function

and density

The mean is \mu and the variance is \pi^2 (\phi^{-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>

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>

See Also

Distributions for other standard distributions.

Examples

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