betabinomUC: The Beta-Binomial Distribution

In VGAM: Vector Generalized Linear and Additive Models

The Beta-Binomial Distribution

Description

Density, distribution function, and random generation for the beta-binomial distribution and the inflated beta-binomial distribution.

Usage

dbetabinom(x, size, prob, rho = 0, log = FALSE)
pbetabinom(q, size, prob, rho = 0, log.p = FALSE)
rbetabinom(n, size, prob, rho = 0)
dbetabinom.ab(x, size, shape1, shape2, log = FALSE,
              Inf.shape = exp(20), limit.prob = 0.5)
pbetabinom.ab(q, size, shape1, shape2, limit.prob = 0.5,
              log.p = FALSE)
rbetabinom.ab(n, size, shape1, shape2, limit.prob = 0.5,
              .dontuse.prob = NULL)
dzoibetabinom(x, size, prob, rho = 0, pstr0 = 0, pstrsize = 0,
              log = FALSE)
pzoibetabinom(q, size, prob, rho, pstr0 = 0, pstrsize = 0,
              lower.tail = TRUE, log.p = FALSE)
rzoibetabinom(n, size, prob, rho = 0, pstr0 = 0, pstrsize = 0)
dzoibetabinom.ab(x, size, shape1, shape2, pstr0 = 0, pstrsize = 0,
                 log = FALSE)
pzoibetabinom.ab(q, size, shape1, shape2, pstr0 = 0, pstrsize = 0,
              lower.tail = TRUE, log.p = FALSE)
rzoibetabinom.ab(n, size, shape1, shape2, pstr0 = 0, pstrsize = 0)

Arguments

x, q	vector of quantiles.
size	number of trials.
n	number of observations. Same as runif.
prob	the probability of success \mu. Must be in the unit closed interval [0,1].
rho	the correlation parameter \rho, which should be in the interval [0, 1). The default value of 0 corresponds to the usual binomial distribution with probability prob. Setting rho = 1 would set both shape parameters equal to 0, and the ratio 0/0, which is actually NaN, is interpreted by Beta as 0.5. See the warning below.
shape1, shape2	the two (positive) shape parameters of the standard beta distribution. They are called a and b in beta respectively. Note that shape1 = prob*(1-rho)/rho and shape2 = (1-prob)*(1-rho)/rho is an important relationship between the parameters, so that the shape parameters are infinite by default because rho = 0; hence limit.prob = prob is used to obtain the behaviour of the usual binomial distribution.
log, log.p, lower.tail	Same meaning as runif.
Inf.shape	Numeric. A large value such that, if shape1 or shape2 exceeds this, then special measures are taken, e.g., calling dbinom. Also, if shape1 or shape2 is less than its reciprocal, then special measures are also taken. This feature/approximation is needed to avoid numerical problem with catastrophic cancellation of multiple lbeta calls.
limit.prob	Numerical vector; recycled if necessary. If either shape parameters are Inf then the binomial limit is taken, with shape1 / (shape1 + shape2) as the probability of success. In the case where both are Inf this probability will be a NaN = Inf/Inf, however, the value limit.prob is used instead. Hence the default for dbetabinom.ab() is to assume that both shape parameters are equal as the limit is taken (indeed, Beta uses 0.5). Note that for [dpr]betabinom(), because rho = 0 by default, then limit.prob = prob so that the beta-binomial distribution behaves like the ordinary binomial distribution with respect to arguments size and prob.
.dontuse.prob	An argument that should be ignored and not used.
pstr0	Probability of a structual zero (i.e., ignoring the beta-binomial distribution). The default value of pstr0 corresponds to the response having a beta-binomial distribuion inflated only at size.
pstrsize	Probability of a structual maximum value size. The default value of pstrsize corresponds to the response having a beta-binomial distribution inflated only at 0.

Details

The beta-binomial distribution is a binomial distribution whose probability of success is not a constant but it is generated from a beta distribution with parameters shape1 and shape2. Note that the mean of this beta distribution is mu = shape1/(shape1+shape2), which therefore is the mean or the probability of success.

See betabinomial and betabinomialff, the VGAM family functions for estimating the parameters, for the formula of the probability density function and other details.

For the inflated beta-binomial distribution, the probability mass function is

P(Y = y) = (1 - pstr0 - pstrsize) \times BB(y) + pstr0 \times I[y = 0] + pstrsize \times I[y = size]

where BB(y) is the probability mass function of the beta-binomial distribution with the same shape parameters (pbetabinom.ab), pstr0 is the inflated probability at 0 and pstrsize is the inflated probability at 1. The default values of pstr0 and pstrsize mean that these functions behave like the ordinary Betabinom when only the essential arguments are inputted.

Value

dbetabinom and dbetabinom.ab give the density, pbetabinom and pbetabinom.ab give the distribution function, and rbetabinom and rbetabinom.ab generate random deviates.

dzoibetabinom and dzoibetabinom.ab give the inflated density, pzoibetabinom and pzoibetabinom.ab give the inflated distribution function, and rzoibetabinom and rzoibetabinom.ab generate random inflated deviates.

Warning

Setting rho = 1 is not recommended, however the code may be modified in the future to handle this special case.

Note

pzoibetabinom, pzoibetabinom.ab, pbetabinom and pbetabinom.ab can be particularly slow. The functions here ending in .ab are called from those functions which don't. The simple transformations \mu=\alpha / (\alpha + \beta) and \rho=1/(1 + \alpha + \beta) are used, where \alpha and \beta are the two shape parameters.

Author(s)

T. W. Yee and Xiangjie Xue

See Also

Extbetabinom, betabinomial, betabinomialff, Zoabeta, Beta.

Examples

set.seed(1); rbetabinom(10, 100, prob = 0.5)
set.seed(1);     rbinom(10, 100, prob = 0.5)  # The same as rho = 0

## Not run:  N <- 9; xx <- 0:N; s1 <- 2; s2 <- 3
dy <- dbetabinom.ab(xx, size = N, shape1 = s1, shape2 = s2)
barplot(rbind(dy, dbinom(xx, size = N, prob = s1 / (s1+s2))),
        beside = TRUE, col = c("blue","green"), las = 1,
        main = paste("Beta-binomial (size=",N,", shape1=", s1,
                   ", shape2=", s2, ") (blue) vs\n",
        " Binomial(size=", N, ", prob=", s1/(s1+s2), ") (green)",
                     sep = ""),
        names.arg = as.character(xx), cex.main = 0.8)
sum(dy * xx)  # Check expected values are equal
sum(dbinom(xx, size = N, prob = s1 / (s1+s2)) * xx)
# Should be all 0:
cumsum(dy) - pbetabinom.ab(xx, N, shape1 = s1, shape2 = s2)

y <- rbetabinom.ab(n = 1e4, size = N, shape1 = s1, shape2 = s2)
ty <- table(y)
barplot(rbind(dy, ty / sum(ty)),
        beside = TRUE, col = c("blue", "orange"), las = 1,
        main = paste("Beta-binomial (size=", N, ", shape1=", s1,
                     ", shape2=", s2, ") (blue) vs\n",
        " Random generated beta-binomial(size=", N, ", prob=",
        s1/(s1+s2), ") (orange)", sep = ""), cex.main = 0.8,
        names.arg = as.character(xx))

N <- 1e5; size <- 20; pstr0 <- 0.2; pstrsize <- 0.2
kk <- rzoibetabinom.ab(N, size, s1, s2, pstr0, pstrsize)
hist(kk, probability = TRUE, border = "blue", ylim = c(0, 0.25),
     main = "Blue/green = inflated; orange = ordinary beta-binomial",
     breaks = -0.5 : (size + 0.5))
sum(kk == 0) / N  # Proportion of 0
sum(kk == size) / N  # Proportion of size
lines(0 : size,
      dbetabinom.ab(0 : size, size, s1, s2), col = "orange")
lines(0 : size, col = "green", type = "b",
      dzoibetabinom.ab(0 : size, size, s1, s2, pstr0, pstrsize))

## End(Not run)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.