Logit Link Function
Computes the logit transformation, including its inverse and the first two derivatives.
logitlink(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
short = TRUE, tag = FALSE)
extlogitlink(theta, min = 0, max = 1, bminvalue = NULL, bmaxvalue = NULL,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)theta |
Numeric or character. See below for further details. |
bvalue, bminvalue, bmaxvalue |
See |
min, max |
For |
inverse, deriv, short, tag |
Details at |
The logit link function is very commonly used for parameters that
lie in the unit interval.
It is the inverse CDF of the logistic distribution.
Numerical values of theta close to 0 or 1 or out of range
result in
Inf, -Inf, NA or NaN.
The extended logit link function extlogitlink should be used
more generally for parameters that lie in the interval (A,B),
say.
The formula is
log((theta-A)/(B-theta))
and the default values for A and B correspond to
the ordinary
logit function.
Numerical values of theta close to A or B or
out of range
result in
Inf, -Inf, NA or NaN.
However these can be replaced by values bminvalue and
bmaxvalue first before computing the link function.
For logitlink with deriv = 0, the logit of theta, i.e.,
log(theta/(1-theta)) when inverse = FALSE,
and if inverse = TRUE then
exp(theta)/(1+exp(theta)).
For deriv = 1, then the function returns
d eta / d theta as a function of theta
if inverse = FALSE,
else if inverse = TRUE then it returns the reciprocal.
Here, all logarithms are natural logarithms, i.e., to base e.
Numerical instability may occur when theta is
close to 1 or 0 (for logitlink), or close to A or B for
extlogitlink.
One way of overcoming this is to use, e.g., bvalue.
In terms of the threshold approach with cumulative probabilities for
an ordinal response this link function corresponds to the univariate
logistic distribution (see logistic).
Thomas W. Yee
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
p <- seq(0.01, 0.99, by = 0.01)
logitlink(p)
max(abs(logitlink(logitlink(p), inverse = TRUE) - p)) # Should be 0
p <- c(seq(-0.02, 0.02, by = 0.01), seq(0.97, 1.02, by = 0.01))
logitlink(p) # Has NAs
logitlink(p, bvalue = .Machine$double.eps) # Has no NAs
p <- seq(0.9, 2.2, by = 0.1)
extlogitlink(p, min = 1, max = 2,
bminvalue = 1 + .Machine$double.eps,
bmaxvalue = 2 - .Machine$double.eps) # Has no NAs
## Not run: par(mfrow = c(2,2), lwd = (mylwd <- 2))
y <- seq(-4, 4, length = 100)
p <- seq(0.01, 0.99, by = 0.01)
for (d in 0:1) {
myinv <- (d > 0)
matplot(p, cbind( logitlink(p, deriv = d, inverse = myinv),
probitlink(p, deriv = d, inverse = myinv)),
type = "n", col = "purple", ylab = "transformation", las = 1,
main = if (d == 0) "Some probability link functions"
else "1 / first derivative")
lines(p, logitlink(p, deriv = d, inverse = myinv), col = "limegreen")
lines(p, probitlink(p, deriv = d, inverse = myinv), col = "purple")
lines(p, clogloglink(p, deriv = d, inverse = myinv), col = "chocolate")
lines(p, cauchitlink(p, deriv = d, inverse = myinv), col = "tan")
if (d == 0) {
abline(v = 0.5, h = 0, lty = "dashed")
legend(0, 4.5, c("logitlink", "probitlink", "clogloglink", "cauchitlink"),
col = c("limegreen", "purple", "chocolate", "tan"), lwd = mylwd)
} else
abline(v = 0.5, lty = "dashed")
}
for (d in 0) {
matplot(y, cbind(logitlink(y, deriv = d, inverse = TRUE),
probitlink(y, deriv = d, inverse = TRUE)), las = 1,
type = "n", col = "purple", xlab = "transformation", ylab = "p",
main = if (d == 0) "Some inverse probability link functions"
else "First derivative")
lines(y, logitlink(y, deriv = d, inverse = TRUE), col = "limegreen")
lines(y, probitlink(y, deriv = d, inverse = TRUE), col = "purple")
lines(y, clogloglink(y, deriv = d, inverse = TRUE), col = "chocolate")
lines(y, cauchitlink(y, deriv = d, inverse = TRUE), col = "tan")
if (d == 0) {
abline(h = 0.5, v = 0, lty = "dashed")
legend(-4, 1, c("logitlink", "probitlink", "clogloglink", "cauchitlink"),
col = c("limegreen", "purple", "chocolate", "tan"), lwd = mylwd)
}
}
p <- seq(0.21, 0.59, by = 0.01)
plot(p, extlogitlink(p, min = 0.2, max = 0.6),
type = "l", col = "black", ylab = "transformation", xlim = c(0, 1),
las = 1, main = "extlogitlink(p, min = 0.2, max = 0.6)")
par(lwd = 1)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.