Betabinom: 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, 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, log.p = FALSE)
rbetabinom.ab(n, size, shape1, shape2, .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.

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 $\rho$. Usually must be in the unit open interval $(0,1)$, however, the value 0 is sometimes supported (if so then it corresponds to the usual binomial distribution).

shape1, shape2

the two (positive) shape parameters of the standard beta distribution. They are called a and b in beta respectively.

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

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 is to assume that both shape parameters are equal as the limit is taken. Purists may assign NaN to this argument.

.dontuse.prob

An argument that should be ignored and unused.

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.

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.

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.

Examples

Run this code

# NOT RUN {
set.seed(1); rbetabinom(10, 100, prob = 0.5)
set.seed(1);     rbinom(10, 100, prob = 0.5)  # The same since 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)
cumsum(dy) - pbetabinom.ab(xx, N, shape1 = s1, shape2 = s2)  # Should be all 0

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))
# }

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