# posbinomUC: Positive-Binomial Distribution In VGAMdata: Data Supporting the 'VGAM' Package

 Posbinom R Documentation

## Positive-Binomial Distribution

### Description

Density, distribution function, quantile function and random generation for the positive-binomial distribution.

### Usage

```dposbinom(x, size, prob, log = FALSE)
pposbinom(q, size, prob)
qposbinom(p, size, prob)
rposbinom(n, size, prob)
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. Fed into `runif`. `size` number of trials. It is the N symbol in the formula given in `posbinomial` and should be positive. `prob` probability of success on each trial. Should be in (0,1).
 `log` See `dbinom`.

### Details

The positive-binomial distribution is a binomial distribution but with the probability of a zero being zero. The other probabilities are scaled to add to unity. The mean therefore is

mu / (1-(1-mu)^N)

where mu is the argument `prob` above. As mu increases, the positive-binomial and binomial distributions become more similar. Unlike similar functions for the binomial distribution, a zero value of `prob` is not permitted here.

### Value

`dposbinom` gives the density, `pposbinom` gives the distribution function, `qposbinom` gives the quantile function, and `rposbinom` generates random deviates.

### Note

These functions are or are likely to be deprecated. Use `Gaitdbinom` instead.

For `dposbinom()`, if arguments `size` or `prob` equal 0 then a `NaN` is returned.

The family function `posbinomial` estimates the parameters by maximum likelihood estimation.

### Author(s)

T. W. Yee.

`posbinomial`, `dposbern`, `Gaitdbinom`, `zabinomial`, `zibinomial`, `Binomial`.

### Examples

```prob <- 0.2; size <- 10
table(y <- rposbinom(n = 1000, size, prob))
mean(y)  # Sample mean
size * prob / (1 - (1 - prob)^size)  # Population mean

(ii <- dposbinom(0:size, size, prob))
cumsum(ii) - pposbinom(0:size, size, prob)  # Should be 0s
table(rposbinom(100, size, prob))

table(qposbinom(runif(1000), size, prob))
round(dposbinom(1:10, size, prob) * 1000)  # Should be similar

## Not run:  barplot(rbind(dposbinom(x = 0:size, size, prob),
dbinom(x = 0:size, size, prob)),
beside = TRUE, col = c("blue", "green"),
main = paste("Positive-binomial(", size, ",",
prob, ") (blue) vs",
" Binomial(", size, ",", prob, ") (green)", sep = ""),
names.arg = as.character(0:size), las = 1)
## End(Not run)

# Simulated data example
nn <- 1000; sizeval1 <- 10; sizeval2 <- 20
pdata <- data.frame(x2 = seq(0, 1, length = nn))
pdata <- transform(pdata, prob1  = logitlink(-2 + 2 * x2, inverse = TRUE),
prob2  = logitlink(-1 + 1 * x2, inverse = TRUE),
sizev1 = rep(sizeval1, len = nn),
sizev2 = rep(sizeval2, len = nn))
pdata <- transform(pdata, y1 = rposbinom(nn, size = sizev1, prob = prob1),
y2 = rposbinom(nn, size = sizev2, prob = prob2))
with(pdata, table(y1))
with(pdata, table(y2))
# Multiple responses
fit2 <- vglm(cbind(y1, y2) ~ x2, posbinomial(multiple.responses = TRUE),
trace  = TRUE, data = pdata, weight = cbind(sizev1, sizev2))
coef(fit2, matrix = TRUE)
```

VGAMdata documentation built on March 18, 2022, 8:03 p.m.