1.2. Asymmetric Links in Binary Regression

As reported in the literature, symmetric links are not always appropriate for modeling this kind of data. Nagler (1994), Chen, Dey and Shao (1999) among others, showed the importance of appropriately choosing the link function and how sensitive is the inference if a symmetric link function is incorrectly used in the place of an asymmetric link. The problem appears when the probability of a given binary response approaches at a different rate than it approaches. Moreover, examples are listed in different textbooks (see, for example, Collet, 2003) reporting situations where an asymmetric link is more appropriate than a symmetric one.

In this case, it is necessary to consider asymmetric links. A very popular example of asymmetric link is the complementary log-log link or cloglog, where the cdf of the Gumbel distribution is considered as defined by

Where the cdf is completely specified and, it does not depend on any unknown additional parameter and it does not include any particular case as a symmetrical link. This link is considered in Bayesian Modeling.

Information of how to implement the Bayesian estimation of the binary regression using the cloglog, probit and logit links in WinBUGS or OpenBUGS can be seeing in the Example Beetles: logistic, probit and carries far estimates models of the Manual. Nevertheless Bayesian approach to binary regression models considering other links as the discussed by Bazán, Bolfarine and Branco (2006 and 2010), Prentice (1976), Nagler (1994), Chen, Dey and Shao (1999, 2001) are not available at the moment.

Asymmetric links considered in BayesianModeling are those that are obtained considering other cdf like the following:

these links are asymmetric logit and are known as scobit and power logit, respectively, and include the logit link as special case when the parameter . For a review of these links see Prentice (1976) and Nagler (1994).

In BayesianModeling also are implemented three links that are based in the cdf of a skew normal distribution (see Azzalini, 1985), this cdf can be represented in general by the following way:

Where ; ; it represents the distribution accumulated of a normal distribution bivariate with parameters and ; and .

This links were proposed by Bazán, Bolfarine y Branco (2006 and 2010) and as especial cases of this general formulation, the implemented links in the BayesianModeling are the following:

· If , obtains the asymmetric probit link proposed in Chen et al (1999) named as CDS skew probit.

· If , obtain the asymmetric probit link proposed by Bazán, Branco and Bolfarine (2006) named as BBB skew probit.

· If , with obtain the standard asymmetric probit link (Bazán, Bolfarine y Branco 2006 y 2010), named here as Standard skew probit.

In these three links, is the shape parameter that controls the asymmetry, so we have for negative values (positive) of has negative asymmetry (positive).

These three models can see also as belonging to the kind of mixes of eliptic distributions proposed by Basu and Mukhopadhyay (2000) given by:

,

where G is the cumulative distribution function [0, ¥> and H is an eliptic distribution. For instance, the CDS skew probit considers a kind of mixes of normal where the measure of mix is the positive normal distribution with density function given by g(x)=2 f (x), x>0, with f (.) being the function of density of the standard normal.

Another interesting case when mixture of the positive normal with H the cumulative distribution function of the logistic distribution it is considered as skew logistic or skew logit (see Chen, Dey and Shao, 2001) that also is implemented in BayesianModeling.