# plinks

##### Parametric Links for Binomial Generalized Linear Models

Various symmetric and asymmetric parametric links for use as link function for binomial generalized linear models.

- Keywords
- regression

##### Usage

```
gj(phi, verbose = FALSE)
foldexp(phi, verbose = FALSE)
ao1(phi, verbose = FALSE)
ao2(phi, verbose = FALSE)
talpha(alpha, verbose = FALSE, splineinv = TRUE,
eps = 2 * .Machine$double.eps, maxit = 100)
rocke(shape1, shape2, verbose = FALSE)
``` gosset(nu, verbose = FALSE)
pregibon(a, b)
nblogit(theta)

angular(verbose = FALSE)
loglog()

##### Arguments

- phi, a, b
- numeric.
- alpha
- numeric. Parameter in $[0,2]$.
- shape1, shape2, nu, theta
- numeric. Non-negative parameter.
- splineinv
- logical. Should a (quick and dirty) spline function be used for computing the inverse link function? Alternatively, a more precise but somewhat slower Newton algorithm is used.
- eps
- numeric. Desired convergence tolerance for Newton algorithm.
- maxit
- integer. Maximal number of steps for Newton algorithm.
- verbose
- logical. Should warnings about numerical issues be printed?

##### Details

Symmetric and asymmetric families parametric link functions are available.
Many families contain the logit for some value(s) of their parameter(s).
The symmetric Aranda-Ordaz (1981) transformation
$$y = \frac{2}{\phi}\frac{x^\phi-(1-x)^\phi}{x^\phi+(1-x)^\phi}$$
and the asymmetric Aranda-Ordaz (1981) transformation
$$y = \log([(1-x)^{-\phi}-1]/\phi)$$
both contain the logit for $\phi = 0$ and
$\phi = 1$ respectively, where the latter also includes the
complementary log-log for $\phi = 0$.
The Pregibon (1980) two parameter family is the link given by
$$y = \frac{x^{a-b}-1}{a-b}-\frac{(1-x)^{a+b}-1}{a+b}.$$
For $a = b = 0$ it is the logit. For $b = 0$ it is symmetric and
$b$ controls the skewness; the heavyness of the tails is controlled by
$a$. The implementation uses the generalized lambda distribution
`gl`

.
The Guerrero-Johnson (1982) family
$$y = \frac{1}{\phi}\left(\left[\frac{x}{1-x}\right]^\phi-1\right)$$
is symmetric and contains the logit for $\phi = 0$.
The Rocke (1993) family of links is, modulo a linear transformation, the
cumulative density function of the Beta distribution. If both parameters are
set to $0$ the logit link is obtained. If both parameters equal
$0.5$ the Rocke link is, modulo a linear transformation, identical to the
angular transformation. Also for `shape1`

= `shape2`

$= 1$, the
identity link is obtained. Note that the family can be used as a one and a two
parameter family.
The folded exponential family (Piepho, 2003) is symmetric and given by
$$y = \left{\begin{array}{ll}
\frac{\exp(\phi x)-\exp(\phi(1-x))}{2\phi} &(\phi \neq 0) \
x- \frac{1}{2} &(\phi = 0)
\end{array}\right.$$
The $t_\alpha$ family (Doebler, Holling & Boehning, 2011) given by
$$y = \alpha\log(x)-(2-\alpha)\log(1-x)$$
is asymmetric and contains the logit for $\phi = 1$.
The Gosset family of links is given by the inverse of the cumulative
distribution function of the t-distribution. The degrees of freedom $\nu$
control the heavyness of the tails and is restricted to values $>0$. For
$\nu = 1$ the Cauchy link is obtained and for $\nu \to \infty$ the link
converges to the probit. The implementation builds on `qf`

and is
reliable for $\nu \geq 0.2$. Liu (2004) reports that the Gosset link
approximates the logit well for $\nu = 7$.
Also the (parameterless) angular (arcsine) transformation
$y = \arcsin(\sqrt{x})$ is available as a link
function.

##### Value

- An object of the class
`link-glm`

, see the documentation of`make.link`

.

##### concept

- parametric link
- transformation

##### References

Aranda-Ordaz F (1981). *Biometrika*, **68**, 357--363.

Doebler P, Holling H, Boehning D (2012). *Psychological Methods*, **17**(3), 418--436.

Guerrero V, Johnson R (1982). *Biometrika*, **69**, 309--314.

Koenker R (2006). *R News*, **6**(4), 32--34.

Koenker R, Yoon J (2009). *Journal of Econometrics*, **152**, 120--130.

Liu C (2004). *Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives*, Chapter 21,
pp. 227--238. John Wiley & Sons.

Piepho H (2003). The Folded Exponential Transformation for Proportions.
*Journal of the Royal Statistical Society D*, **52**, 575--589.
Pregibon D (1980). *Journal of the Royal Statistical Society C*, **29**, 15--23.
Rocke DM (1993). *Technometrics*, **35**, 73--81.

##### See Also

*Documentation reproduced from package glmx, version 0.1-1, License: GPL-2 | GPL-3*