BB: Beta Binomial Distribution For Fitting a GAMLSS Model

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

Beta Binomial Distribution For Fitting a GAMLSS Model

Description

This function defines the beta binomial distribution, a two parameter distribution, for a gamlss.family object to be used in a GAMLSS fitting using the function gamlss()

Usage

BB(mu.link = "logit", sigma.link = "log")
dBB(x, mu = 0.5, sigma = 1, bd = 10, log = FALSE)
pBB(q, mu = 0.5, sigma = 1, bd = 10, lower.tail = TRUE,
      log.p = FALSE)
qBB(p, mu = 0.5, sigma = 1, bd = 10, lower.tail = TRUE, 
       log.p = FALSE, fast = FALSE)
rBB(n, mu = 0.5, sigma = 1, bd = 10, fast = FALSE)

Arguments

mu.link	Defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log)
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "sqrt"
mu	vector of positive probabilities
sigma	the dispersion parameter
bd	vector of binomial denominators
p	vector of probabilities
x,q	vector of quantiles
n	number of random values to return
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]
fast	a logical variable if fast=TRUE the dBB function is used in the calculation of the inverse c.d.f function. This is faster to the default fast=FALSE, where the pBB{} is used, but not always consistent with the results obtained from pBB(), for example if p <- pBB(c(0,1,2,3,4,5), mu=.5 , sigma=1, bd=5) do not ensure that qBB(p, mu=.5 , sigma=1, bd=5) will be c(0,1,2,3,4,5)

Details

Definition file for beta binomial distribution.

f(y|\mu,\sigma)=\frac{\Gamma(n+1)} {\Gamma(y+1)\Gamma(n-y+1)} \frac{\Gamma(\frac{1}{\sigma}) \Gamma(y+\frac{\mu}{\sigma}) \Gamma[n+\frac{(1-\mu)}{\sigma}-y]}{\Gamma(n+\frac{1}{\sigma}) \Gamma(\frac{\mu}{\sigma}) \Gamma(\frac{1-\mu}{\sigma})}

for y=0,1,2,\ldots,n, 0<\mu<1 and \sigma>0, see pp. 523-524 of Rigby et al. (2019). . For \mu=0.5 and \sigma=0.5 the distribution is uniform.

Value

Returns a gamlss.family object which can be used to fit a Beta Binomial distribution in the gamlss() function.

Warning

The functions pBB and qBB are calculated using a laborious procedure so they are relatively slow.

Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures. The parameter mu represents a probability parameter with limits 0 < \mu < 1. n \mu is the mean of the distribution where n is the binomial denominator. \{n \mu (1-\mu)[1+(n-1) \sigma/(\sigma+1)]\}^{0.5} is the standard deviation of the Beta Binomial distribution. Hence \sigma is a dispersion type parameter

Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.i07")}.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

(see also https://www.gamlss.com/).

See Also

gamlss.family, BI,

Examples

# BB()# gives information about the default links for the Beta Binomial distribution 
#plot the pdf
plot(function(y) dBB(y, mu = .5, sigma = 1, bd =40), from=0, to=40, n=40+1, type="h")
#calculate the cdf and plotting it
ppBB <- pBB(seq(from=0, to=40), mu=.2 , sigma=3, bd=40)
plot(0:40,ppBB, type="h")
#calculating quantiles and plotting them  
qqBB <- qBB(ppBB, mu=.2 , sigma=3, bd=40)
plot(qqBB~ ppBB)
# when the argument fast is useful
p <- pBB(c(0,1,2,3,4,5), mu=.01 , sigma=1, bd=5)
qBB(p, mu=.01 , sigma=1, bd=5, fast=TRUE)
#  0 1 1 2 3 5
qBB(p, mu=.01 , sigma=1, bd=5, fast=FALSE)
#  0 1 2 3 4 5
# generate random sample
tN <- table(Ni <- rBB(1000, mu=.2, sigma=1, bd=20))
r <- barplot(tN, col='lightblue')
# fitting a model 
# library(gamlss)
#data(aep)   
# fits a Beta-Binomial model
#h<-gamlss(y~ward+loglos+year, sigma.formula=~year+ward, family=BB, data=aep) 

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.