# Parametric Links for Binomial Generalized Linear Models

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### 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 [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 = \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.

make.link, family, glmx, WECO