NegBinomial {stats} | R Documentation |
Density, distribution function, quantile function and random
generation for the negative binomial distribution with parameters
size
and prob
.
dnbinom(x, size, prob, mu, log = FALSE) pnbinom(q, size, prob, mu, lower.tail = TRUE, log.p = FALSE) qnbinom(p, size, prob, mu, lower.tail = TRUE, log.p = FALSE) rnbinom(n, size, prob, mu)
x |
vector of (non-negative integer) quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If length(n) > 1 , the length
is taken to be the number required. |
size |
target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). |
prob |
probability of success in each trial. |
mu |
alternative parametrization via mean: see Details |
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]. |
The negative binomial distribution with size
= n and
prob
= p has density
p(x) = Gamma(x+n)/(Gamma(n) x!) p^n (1-p)^x
for x = 0, 1, 2, ...
This represents the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached.
A negative binomial distribution can arise as a mixture of Poisson
distributions with mean distributed as a
Γ (pgamma
) distribution with scale parameter
(1 - prob)/prob
and shape parameter size
. (This
definition allows non-integer values of size
.)
In this model prob
= scale/(1+scale)
, and the mean is
size * (1 - prob)/prob
.
The alternative parametrization (often used in ecology) is by the
mean mu
, and size
, the dispersion parameter,
where prob
= size/(size+mu)
.
The variance is mu + mu^2/size
in this parametrization or
n (1-p)/p^2 in the first one.
If an element of x
is not integer, the result of dnbinom
is zero, with a warning.
The quantile is defined as the smallest value x such that F(x) >= p, where F is the distribution function.
dnbinom
gives the density,
pnbinom
gives the distribution function,
qnbinom
gives the quantile function, and
rnbinom
generates random deviates.
dbinom
for the binomial, dpois
for the
Poisson and dgeom
for the geometric distribution, which
is a special case of the negative binomial.
x <- 0:11 dnbinom(x, size = 1, prob = 1/2) * 2^(1 + x) # == 1 126 / dnbinom(0:8, size = 2, prob = 1/2) #- theoretically integer ## Cumulative ('p') = Sum of discrete prob.s ('d'); Relative error : summary(1 - cumsum(dnbinom(x, size = 2, prob = 1/2)) / pnbinom(x, size = 2, prob = 1/2)) x <- 0:15 size <- (1:20)/4 persp(x,size, dnb <- outer(x,size,function(x,s)dnbinom(x,s, pr= 0.4)), xlab = "x", ylab = "s", zlab="density", theta = 150) title(tit <- "negative binomial density(x,s, pr = 0.4) vs. x & s") image (x,size, log10(dnb), main= paste("log [",tit,"]")) contour(x,size, log10(dnb),add=TRUE) ## Alternative parametrization x1 <- rnbinom(500, mu = 4, size = 1) x2 <- rnbinom(500, mu = 4, size = 10) x3 <- rnbinom(500, mu = 4, size = 100) h1 <- hist(x1, breaks = 20, plot = FALSE) h2 <- hist(x2, breaks = h1$breaks, plot = FALSE) h3 <- hist(x3, breaks = h1$breaks, plot = FALSE) barplot(rbind(h1$counts, h2$counts, h3$counts), beside = TRUE, col = c("red","blue","cyan"), names.arg = round(h1$breaks[-length(h1$breaks)]))