gordlink: 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.

Value

See Yee (2019) for details.

Warning

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

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.

References

Yee, T. W. (2019) Ordinal ordination with normalizing link functions for count data, (in preparation).

Examples

Run this code

# 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
# }

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