VGAM (version 1.0-4)

## Description

Computes the logit transformation, including its inverse and the first two derivatives.

## Usage

logit(theta, bvalue = NULL, inverse = FALSE, deriv = 0,
short = TRUE, tag = FALSE)
extlogit(theta, min = 0, max = 1, bminvalue = NULL, bmaxvalue = NULL,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)

## Arguments

theta

Numeric or character. See below for further details.

bvalue, bminvalue, bmaxvalue

See Links. These are boundary values. For extlogit, values of theta less than or equal to $$A$$ or greater than or equal to $$B$$ can be replaced by bminvalue and bmaxvalue.

min, max

For extlogit, min gives $$A$$, max gives $$B$$, and for out of range values, bminvalue and bmaxvalue.

inverse, deriv, short, tag

Details at Links.

## Value

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 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.

## Details

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 extlogit 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.

## References

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Links, logitoffsetlink, probit, cloglog, cauchit, logistic1, loge, plogis, multilogit.

## Examples

Run this code
# NOT RUN {
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, bvalue = .Machine$double.eps) # Has no NAs p <- seq(0.9, 2.2, by = 0.1) extlogit(p, min = 1, max = 2, bminvalue = 1 + .Machine$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( logit(p, deriv = d, inverse = myinv),
probit(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,   logit(p, deriv = d, inverse = myinv), col = "limegreen")
lines(p,  probit(p, deriv = d, inverse = myinv), col = "purple")
lines(p, cloglog(p, deriv = d, inverse = myinv), col = "chocolate")
lines(p, cauchit(p, deriv = d, inverse = myinv), col = "tan")
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 = mylwd)
} 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)), las = 1,
type = "n", col = "purple", xlab = "transformation", ylab = "p",
main = if (d ==  0) "Some inverse probability link functions"
else "First derivative")
lines(y,   logit(y, deriv = d, inverse = TRUE), col = "limegreen")
lines(y,  probit(y, deriv = d, inverse = TRUE), col = "purple")
lines(y, cloglog(y, deriv = d, inverse = TRUE), col = "chocolate")
lines(y, cauchit(y, deriv = d, inverse = TRUE), col = "tan")
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 = mylwd)
}
}

p <- seq(0.21, 0.59, by = 0.01)
plot(p, extlogit(p, min = 0.2, max = 0.6),
type = "l", col = "black", ylab = "transformation", xlim = c(0, 1),
las = 1, main = "extlogit(p, min = 0.2, max = 0.6)")
par(lwd = 1)
# }


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