# Parametric Links for Binomial Generalized Linear Models

### Description

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

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
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 |

`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 = \tfrac{2}{φ}\tfrac{x^φ-(1-x)^φ}{x^φ+(1-x)^φ}*

and the asymmetric Aranda-Ordaz (1981) transformation

*y = \log([(1-x)^{-φ}-1]/φ)*

both contain the logit for *φ = 0* and
*φ = 1* respectively, where the latter also includes the
complementary log-log for *φ = 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}{φ}≤ft(≤ft[\frac{x}{1-x}\right]^φ-1\right)*

is symmetric and contains the logit for *φ = 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 = ≤ft\{\begin{array}{ll}
\frac{\exp(φ x)-\exp(φ(1-x))}{2φ} &(φ \neq 0) \\
x- \frac{1}{2} &(φ = 0)
\end{array}\right.*

The *t_α* family (Doebler, Holling & Boehning, 2011) given by

*y = α\log(x)-(2-α)\log(1-x)*

is asymmetric and contains the logit for *φ = 1*.

The Gosset family of links is given by the inverse of the cumulative
distribution function of the t-distribution. The degrees of freedom *ν*
control the heavyness of the tails and is restricted to values *>0*. For
*ν = 1* the Cauchy link is obtained and for *ν \to ∞* the link
converges to the probit. The implementation builds on `qf`

and is
reliable for *ν ≥q 0.2*. Liu (2004) reports that the Gosset link
approximates the logit well for *ν = 7*.

Also the (parameterless) angular (arcsine) transformation
*y = \arcsin(√{x})* is available as a link
function.

### Value

An object of the class `link-glm`

, see the documentation of `make.link`

.

### References

Aranda-Ordaz F (1981). “On Two Families of Transformations to Additivity for Binary Response Data.”
*Biometrika*, **68**, 357–363.

Doebler P, Holling H, Boehning D (2012). “A Mixed Model Approach to Meta-Analysis of Diagnostic Studies with Binary Test Outcome.”
*Psychological Methods*, **17**(3), 418–436.

Guerrero V, Johnson R (1982). “Use of the Box-Cox Transformation with Binary Response Models.”
*Biometrika*, **69**, 309–314.

Koenker R (2006). “Parametric Links for Binary Response.”
*R News*, **6**(4), 32–34.

Koenker R, Yoon J (2009). “Parametric Links for Binary Choice Models: A Fisherian-Bayesian Colloquy.”
*Journal of Econometrics*, **152**, 120–130.

Liu C (2004). “Robit Regression: A Simple Robust Alternative to Logistic and Probit Regression.”
In Gelman A, Meng X-L (Eds.),
*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). “Goodness of Link Tests for Generalized Linear Models.”
*Journal of the Royal Statistical Society C*, **29**, 15–23.

Rocke DM (1993). “On the Beta Transformation Family.”
*Technometrics*, **35**, 73–81.

### See Also

`make.link`

, `family`

, `glmx`

, `WECO`

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