logit {VGAM} | R Documentation |
Computes the logit transformation, including its inverse and the first two derivatives.
logit(theta, earg = list(), inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE) elogit(theta, earg = list(min=0, max=1), 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 .
Similarly, for elogit , values of theta less than or equal
to A or greater than or equal to B can be replaced
by the bminvalue and bmaxvalue components of the list earg .
For elogit , earg should be a list with components
min giving A,
max giving B, and for out of range values,
bminvalue and bmaxvalue .
If earg is used, these
component names should not be abbreviated.
|
inverse |
Logical. If TRUE the inverse function is computed.
The inverse logit function is known as the expit function.
|
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 logit link function is very 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 extended logit link function elogit
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.
The arguments short
and tag
are used only if
theta
is character.
For logit
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 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 (for logit
), or close to A or B for
elogit
.
One way of overcoming this is to use earg
.
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.
Links
,
probit
,
cloglog
,
cauchit
,
loge
.
p = seq(0.01, 0.99, by=0.01) logit(p) max(abs(logit(logit(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)) logit(p) # Has NAs logit(p, earg=list(bvalue= .Machine$double.eps)) # Has no NAs p = seq(0.9, 2.2, by=0.1) elogit(p, earg=list(min=1, max=2, bminvalue = 1 + .Machine$double.eps, bmaxvalue = 2 - .Machine$double.eps)) # Has no NAs ## Not run: par(mfrow=c(2,2)) y = seq(-4, 4, length=100) 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", lwd=2, las=1, main=if(d==0) "Some inverse probability link functions" else "First derivative") 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) } } p = seq(0.21, 0.59, by=0.01) plot(p, elogit(p, earg=list(min=0.2, max=0.6)), lwd=2, type="l", col="black", ylab="transformation", xlim=c(0,1), las=1, main="elogit(p, earg=list(min=0.2, max=0.6)") ## End(Not run)