gordlink: Gamma-Ordinal Link Function

In VGAM: Vector Generalized Linear and Additive Models

Gamma-Ordinal Link Function

Description

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

Usage

gordlink(theta, lambda = 1, cutpoint = NULL,
         inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta	Numeric or character. See below for further details.
lambda, cutpoint	The former is the shape parameter in gamma2. cutpoint is optional; if NULL then cutpoint is ignored from the GOLF definition. If given, the cutpoints should be non-negative integers. If gordlink() is used as the link function in cumulative then, if the cutpoints are known, then one should choose reverse = TRUE, parallel = FALSE ~ -1. If the cutpoints are unknown, then choose reverse = TRUE, parallel = TRUE.
inverse, deriv, short, tag	Details at Links.

Details

The gamma-ordinal link function (GOLF) 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 2-parameter gamma distribution.

See Links for general information about VGAM link functions.

Value

See Yee (2019) for details.

Warning

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

Note

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 gamma distribution (see gamma2) that has been recorded as an ordinal response using known cutpoints.

Author(s)

Thomas W. Yee

References

Yee, T. W. (2020). Ordinal ordination with normalizing link functions for count data, (in preparation).

See Also

Links, gamma2, pordlink, nbordlink, cumulative.

Examples

## Not run: 
gordlink("p", lambda = 1, short = FALSE)
gordlink("p", lambda = 1, tag = TRUE)

p <- seq(0.02, 0.98, len = 201)
y <- gordlink(p, lambda = 1)
y. <- gordlink(p, lambda = 1, deriv = 1, inverse = TRUE)
max(abs(gordlink(y, lambda = 1, inverse = TRUE) - p))  # Should be 0

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

# Another example
gdata <- data.frame(x2 = sort(runif(nn <- 1000)))
gdata <- transform(gdata, x3 = runif(nn))
gdata <- transform(gdata, mymu = exp( 3 + 1 * x2 - 2 * x3))
lambda <- 4
gdata <- transform(gdata,
         y1 = rgamma(nn, shape = lambda, scale = mymu / lambda))
cutpoints <- c(-Inf, 10, 20, Inf)
gdata <- transform(gdata, cuty = Cut(y1, breaks = cutpoints))

#\ dontrun{ par(mfrow = c(1, 1), las = 1)
#with(gdata, plot(x2, x3, col = cuty, pch = as.character(cuty))) }
with(gdata, table(cuty) / sum(table(cuty)))
fit <- vglm(cuty ~ x2 + x3, cumulative(multiple.responses = TRUE,
           reverse = TRUE, parallel = FALSE ~ -1,
           link = gordlink(cutpoint = cutpoints[2:3], lambda = lambda)),
           data = gdata, trace = TRUE)
head(depvar(fit))
head(fitted(fit))
head(predict(fit))
coef(fit)
coef(fit, matrix = TRUE)
constraints(fit)
fit@misc

## End(Not run)

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.