Poisson-Ordinal Link Function
Computes the Poisson-ordinal transformation, including its inverse and the first two derivatives.
pordlink(theta, cutpoint = NULL,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)theta |
Numeric or character. See below for further details. |
cutpoint |
The cutpoints should be non-negative integers.
If |
inverse, deriv, short, tag |
Details at |
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.
See Yee (2012) for details.
Numerical values of theta too close to 0 or 1 or out of range
result in large positive or negative values, or maybe 0 depending on
the arguments.
Although measures have been taken to handle cases where
theta is too close to 1 or 0,
numerical instabilities may still arise.
In terms of the threshold approach with cumulative probabilities for
an ordinal response this link function corresponds to the
Poisson distribution (see poissonff) that has been
recorded as an ordinal response using known cutpoints.
Thomas W. Yee
Yee, T. W. (2020). Ordinal ordination with normalizing link functions for count data, (in preparation).
## Not run:
pordlink("p", cutpoint = 2, short = FALSE)
pordlink("p", cutpoint = 2, tag = TRUE)
p <- seq(0.01, 0.99, by = 0.01)
y <- pordlink(p, cutpoint = 2)
y. <- pordlink(p, cutpoint = 2, deriv = 1)
max(abs(pordlink(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 = "pordlink()")
#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=pordlink(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,
link = pordlink(cutpoint = cutpoints[2:3]),
multiple.responses = TRUE))
head(depvar(fit))
head(fitted(fit))
head(predict(fit))
coef(fit)
coef(fit, matrix = TRUE)
constraints(fit)
fit@misc$earg
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