VGAM (version 1.0-4)

## Description

Computes the cauchit (tangent) link transformation, including its inverse and the first two derivatives.

## Usage

```cauchit(theta, bvalue = .Machine\$double.eps,
inverse = FALSE, deriv = 0, short = TRUE, tag = FALSE)```

## Arguments

theta

Numeric or character. See below for further details.

bvalue

See `Links`.

inverse, deriv, short, tag

Details at `Links`.

## Value

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 `eta` / d `theta` as a function of `theta` if `inverse = FALSE`, else if `inverse = TRUE` then it returns the reciprocal.

## Details

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`.

## References

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

`logit`, `probit`, `cloglog`, `loge`, `cauchy`, `cauchy1`.

## Examples

Run this code
```# NOT RUN {
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), lwd = (mylwd <- 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",
las = 1, main = if (d == 0) "Some probability link functions"
else "First derivative")
lines(p,   logit(p, deriv = d), col = "limegreen")
lines(p,  probit(p, deriv = d), col = "purple")
lines(p, cloglog(p, deriv = d), col = "chocolate")
lines(p, cauchit(p, deriv = d), col = "tan")
if (d == 0) {
abline(v = 0.5, h = 0, lty = "dashed")
legend(0, 4.5, c("logit", "probit", "cloglog", "cauchit"), lwd = mylwd,
col = c("limegreen","purple","chocolate", "tan"))
} 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", las=1)
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"), lwd = mylwd,
col = c("limegreen", "purple", "chocolate", "tan"))
}
}
par(lwd = 1)
# }
```

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