Beta Binomial Distribution For Fitting a GAMLSS Model
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()
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)mu.link |
Defines the |
sigma.link |
Defines the |
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 |
Definition file for beta binomial distribution.
f(y|mu,sigma)=[Gamma(n+1)/Gamma(y+1)/Gamma(n-y+1)]*[Gamma(y+mu/sigma)*Gamma(1/sigma)*Gamma[n+(1-mu)/sigma-y]]/[Gamma(n+(1/sigma)) * Gamma(mu/sigma) * Gamma((1-mu)/sigma)]
for y=0,1,2,…,n, 0<μ<1 and σ>0. For μ=0.5 and σ=0.5 the distribution is uniform.
Returns a gamlss.family object which can be used to fit a Beta Binomial distribution in the gamlss() function.
The functions pBB and qBB are calculated using a laborious procedure so they are relatively slow.
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
Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou
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. 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, https://www.jstatsoft.org/v23/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.
(see also https://www.gamlss.com/).
# 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)
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