cauchit {VGAM} | R Documentation |
Computes the cauchit (tangent) link transformation, including its inverse and the first two derivatives.
cauchit(theta, earg = list(bvalue= .Machine$double.eps), inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)
theta |
Numeric or character. See below for further details. |
earg |
List. Extra argument for passing in additional information.
Values of |
inverse |
Logical. If |
deriv |
Order of the derivative. Integer with value 0, 1 or 2. |
short |
Used for labelling the |
tag |
Used for labelling the linear/additive predictor in the
|
This link function is an alternative link function for parameters that lie in the unit interval. This type of link bears the same relation to the Cauchy distribution as the probit link bears to the Gaussian. One characteristic of this link function is that the tail is heavier relative to the other links (see examples below).
Numerical values of theta
close to 0 or 1 or out of range result
in Inf
, -Inf
, NA
or NaN
. The arguments
short
and tag
are used only if theta
is character.
For deriv = 0
, the tangent of theta
, i.e.,
tan(pi * (theta-0.5))
when inverse = FALSE
,
and if inverse = TRUE
then
0.5 + atan(theta)/pi
.
For deriv = 1
, then the function returns
d theta
/ d eta
as a function of
theta
if inverse = FALSE
, else if inverse = TRUE
then it returns the reciprocal.
Numerical instability may occur when theta
is close to 1 or 0.
One way of overcoming this is to use earg
.
As mentioned above,
in terms of the threshold approach with cumulative probabilities for
an ordinal response this link function corresponds to the
Cauchy distribution (see cauchy1
).
Thomas W. Yee
McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.
logit
,
probit
,
cloglog
,
loge
,
cauchy
,
cauchy1
.
p = seq(0.01, 0.99, by=0.01) cauchit(p) max(abs(cauchit(cauchit(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)) cauchit(p) # Has no NAs ## Not run: par(mfrow=c(2,2)) y = seq(-4, 4, length=100) p = seq(0.01, 0.99, by=0.01) for(d in 0:1) { matplot(p, cbind(logit(p, deriv=d), probit(p, deriv=d)), type="n", col="purple", ylab="transformation", lwd=2, las=1, main = if (d == 0) "Some probability link functions" else "First derivative") lines(p, logit(p, deriv=d), col="limegreen", lwd=2) lines(p, probit(p, deriv=d), col="purple", lwd=2) lines(p, cloglog(p, deriv=d), col="chocolate", lwd=2) lines(p, cauchit(p, deriv=d), col="tan", lwd=2) if (d == 0) { abline(v=0.5, h=0, lty="dashed") legend(0, 4.5, c("logit", "probit", "cloglog", "cauchit"), col=c("limegreen","purple","chocolate", "tan"), lwd=2) } else abline(v=0.5, lty="dashed") } for(d in 0) { matplot(y, cbind(logit(y, deriv=d, inverse=TRUE), probit(y, deriv=d, inverse=TRUE)), type ="n", col="purple", xlab="transformation", ylab="p", main = if (d == 0) "Some inverse probability link functions" else "First derivative", lwd=2, las=1) lines(y, logit(y, deriv=d, inverse=TRUE), col="limegreen", lwd=2) lines(y, probit(y, deriv=d, inverse=TRUE), col="purple", lwd=2) lines(y, cloglog(y, deriv=d, inverse=TRUE), col="chocolate", lwd=2) lines(y, cauchit(y, deriv=d, inverse=TRUE), col="tan", lwd=2) if (d == 0) { abline(h=0.5, v=0, lty="dashed") legend(-4, 1, c("logit", "probit", "cloglog", "cauchit"), col=c("limegreen","purple","chocolate", "tan"), lwd=2) } } ## End(Not run)