## Description

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

## Usage

 ```1 2``` ```cauchitlink(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`.

## 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`.

## 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.

## Note

Numerical instability may occur when `theta` is close to 1 or 0. One way of overcoming this is to use `bvalue`.

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

## References

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

`logitlink`, `probitlink`, `clogloglink`, `loglink`, `cauchy`, `cauchy1`, `Cauchy`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50``` ```p <- seq(0.01, 0.99, by = 0.01) cauchitlink(p) max(abs(cauchitlink(cauchitlink(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)) cauchitlink(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(logitlink(p, deriv = d), probitlink(p, deriv = d)), type = "n", col = "purple", ylab = "transformation", las = 1, main = if (d == 0) "Some probability link functions" else "First derivative") lines(p, logitlink(p, deriv = d), col = "limegreen") lines(p, probitlink(p, deriv = d), col = "purple") lines(p, clogloglink(p, deriv = d), col = "chocolate") lines(p, cauchitlink(p, deriv = d), col = "tan") if (d == 0) { abline(v = 0.5, h = 0, lty = "dashed") legend(0, 4.5, c("logitlink", "probitlink", "clogloglink", "cauchitlink"), lwd = mylwd, col = c("limegreen", "purple", "chocolate", "tan")) } else abline(v = 0.5, lty = "dashed") } for (d in 0) { matplot(y, cbind( logitlink(y, deriv = d, inverse = TRUE), probitlink(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, logitlink(y, deriv = d, inverse = TRUE), col = "limegreen") lines(y, probitlink(y, deriv = d, inverse = TRUE), col = "purple") lines(y, clogloglink(y, deriv = d, inverse = TRUE), col = "chocolate") lines(y, cauchitlink(y, deriv = d, inverse = TRUE), col = "tan") if (d == 0) { abline(h = 0.5, v = 0, lty = "dashed") legend(-4, 1, c("logitlink", "probitlink", "clogloglink", "cauchitlink"), lwd = mylwd, col = c("limegreen", "purple", "chocolate", "tan")) } } par(lwd = 1) ## End(Not run) ```