VGAM (version 1.0-4)

## Description

Computes the Poisson-ordinal transformation, including its inverse and the first two derivatives.

## Usage

```polf(theta, cutpoint = NULL,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)```

## Arguments

theta

Numeric or character. See below for further details.

cutpoint

The cutpoints should be non-negative integers. If `polf()` is used as the link function in `cumulative` then one should choose `reverse = TRUE, parallel = TRUE`.

inverse, deriv, short, tag

Details at `Links`.

## Value

See Yee (2012) for details.

## Warning

Prediction may not work on `vglm` or `vgam` etc. objects if this link function is used.

## Details

The Poisson-ordinal link function (POLF) can be applied to a parameter lying in the unit interval. Its purpose is to link cumulative probabilities associated with an ordinal response coming from an underlying Poisson distribution. If the cutpoint is zero then a complementary log-log link is used.

See `Links` for general information about VGAM link functions.

## References

Yee, T. W. (2012) Ordinal ordination with normalizing link functions for count data, (in preparation).

`Links`, `ordpoisson`, `poissonff`, `nbolf`, `golf`, `cumulative`.

## Examples

Run this code
```# NOT RUN {
polf("p", cutpoint = 2, short = FALSE)
polf("p", cutpoint = 2, tag = TRUE)

p <- seq(0.01, 0.99, by = 0.01)
y <- polf(p, cutpoint = 2)
y. <- polf(p, cutpoint = 2, deriv = 1)
max(abs(polf(y, cutpoint = 2, inv = TRUE) - p))  # Should be 0

#\ dontrun{ par(mfrow = c(2, 1), las = 1)
#plot(p, y, type = "l", col = "blue", main = "polf()")
#abline(h = 0, v = 0.5, col = "orange", lty = "dashed")
#
#plot(p, y., type = "l", col = "blue",
#     main = "(Reciprocal of) first POLF derivative")
#}

# Rutherford and Geiger data
ruge <- data.frame(yy = rep(0:14,
times = c(57,203,383,525,532,408,273,139,45,27,10,4,0,1,1)))
with(ruge, length(yy))  # 2608 1/8-minute intervals
cutpoint <- 5
ruge <- transform(ruge, yy01 = ifelse(yy <= cutpoint, 0, 1))
fit <- vglm(yy01 ~ 1, binomialff(link = polf(cutpoint = cutpoint)), ruge)
coef(fit, matrix = TRUE)
exp(coef(fit))

# Another example
pdata <- data.frame(x2 = sort(runif(nn <- 1000)))
pdata <- transform(pdata, x3 = runif(nn))
pdata <- transform(pdata, mymu = exp( 3 + 1 * x2 - 2 * x3))
pdata <- transform(pdata, y1 = rpois(nn, lambda = mymu))
cutpoints <- c(-Inf, 10, 20, Inf)
pdata <- transform(pdata, cuty = Cut(y1, breaks = cutpoints))
#\ dontrun{ with(pdata, plot(x2, x3, col = cuty, pch = as.character(cuty))) }
with(pdata, table(cuty) / sum(table(cuty)))
fit <- vglm(cuty ~ x2 + x3, data = pdata, trace = TRUE,
cumulative(reverse = TRUE,
parallel = TRUE,