cloglog {VGAM} | R Documentation |
Computes the complementary log-log transformation, including its inverse and the first two derivatives.
cloglog(theta, earg = list(), inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)
theta |
Numeric or character.
See below for further details.
|
earg |
Optional list. Extra argument for passing in additional information.
Values of theta which are less than or equal to 0 can be
replaced by the bvalue component of the list earg
before computing the link function value.
Values of theta which are greater than or equal to 1 can be
replaced by 1 minus the bvalue component of the list earg
before computing the link function value.
The component name bvalue stands for ``boundary value''.
See Links for general information about earg .
|
inverse |
Logical. If TRUE the inverse function is computed.
|
deriv |
Order of the derivative. Integer with value 0, 1 or 2.
|
short |
Used for labelling the blurb slot of a
vglmff-class object.
|
tag |
Used for labelling the linear/additive predictor in the
initialize slot of a vglmff-class object.
Contains a little more information if TRUE .
|
The complementary log-log link function is commonly used for parameters
that lie in the unit interval. 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 complimentary log-log of theta
,
i.e., log(-log(1 - theta))
when inverse = FALSE
, and if
inverse = TRUE
then 1-exp(-exp(theta))
,.
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.
Here, all logarithms are natural logarithms, i.e., to base e.
Numerical instability may occur when theta
is close to 1 or 0.
One way of overcoming this is to use earg
.
With constrained ordination (e.g., cqo
and
cao
) used with binomialff
, a complementary
log-log link function is preferred over the default logit
link, for a good reason. See the example below.
In terms of the threshold approach with cumulative probabilities for an ordinal response this link function corresponds to the extreme value distribution.
Thomas W. Yee
McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.
Links
,
logit
,
probit
,
cauchit
.
p = seq(0.01, 0.99, by=0.01) cloglog(p) max(abs(cloglog(cloglog(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)) cloglog(p) # Has NAs cloglog(p, earg=list(bvalue= .Machine$double.eps)) # Has no NAs ## Not run: plot(p, logit(p), type="l", col="limegreen", ylab="transformation", lwd=2, las=1, main="Some probability link functions") lines(p, probit(p), col="purple", lwd=2) lines(p, cloglog(p), col="chocolate", lwd=2) lines(p, cauchit(p), col="tan", lwd=2) abline(v=0.5, h=0, lty="dashed") legend(0.1, 4, c("logit", "probit", "cloglog", "cauchit"), col=c("limegreen","purple","chocolate", "tan"), lwd=2) ## End(Not run) # This example shows that a cloglog link is preferred over the logit n = 500; p = 5; S = 3; Rank = 1 # Species packing model: mydata = rcqo(n, p, S, EqualTol=TRUE, ESOpt=TRUE, EqualMax=TRUE, family="binomial", hiabundance=5, seed=123, Rank=Rank) fitc = cqo(attr(mydata, "formula"), ITol=TRUE, data=mydata, fam=binomialff(mv=TRUE, link="cloglog"), Rank=Rank) fitl = cqo(attr(mydata, "formula"), ITol=TRUE, data=mydata, fam=binomialff(mv=TRUE, link="logit"), Rank=Rank) # Compare the fitted models (cols 1 and 3) with the truth (col 2) cbind(ccoef(fitc), attr(mydata, "ccoefficients"), ccoef(fitl))