2.2. IRT Models with asymmetric links

In the traditional IRT models, the asymmetric ICC are considered symmetrical; this is the case of the logistic and of normal ogive models. However, as it has observed Samejima (2000), Bazán et al (2006) and Bolfarine and Bazán (2010) asymmetric ICC can be incorporated considering a new parameter of item that controls the shape of the curve. This asymmetry is necessary in many cases for a better modelization of answers with a low proportion of 0´s or 1's. Then will show three of these models.

The skew normal ogive model

The skew normal ogive model was proposed by Bazán et al (2006) assuming that the probability of success considering the abilities and the item parameters associates it given by:

where is a parameter of asymmetry, is the latent linear predictor and denote the cdf of a skew normal distributions with function of density being the pdf of a standard normal variable.

Notice that if , the normal ogive model (2P) is obtained, but as indicated in Bazán et al (2006) if , the probability of correct response has a slow growth for low values of latent variable . On the other hand, if , the probability of correct response has a quick growth for low values of the latent variable . Is because this behavior that this parameter is interpreted as a penalization parameter for item. Main details about this model can be reviewed in Bazán et al (2006). In this formulation the link considered is the BBB skew probit link (see Bazán, Bolfarine and Branco, 2010) and for this reason the model can be named also two-parameter skew probit or 2SP IRT model. When only difficulty parameter is considered we have the 1SP IRT model.

The LPE and RLPE models

Logistic positive exponent (LPE) was proposed by Samejima (2000). A reversal version, named Reflection Logistic positive exponent (RLPE) was formulated by Bolfarine and Bazán (2010). These models, studied in Bolfarine and Bazán (2010), assume that the probability of correct response considering the abilities and the item parameters associates it given by

where is the cdf of the logistic distribution indexed by the parameter and evaluated in . For LPE model and for RLPE model .

Depending that function of distribution specifies will have 2LPE or 2RLPE IRT models. In the first case, this characterize by

And the second case by:

These correspond to the cdf of the Scobit distribution and Burr of type II, respectively.

Note that or and y are asymmetric but it holds that or

In both models can also interpret like a parameter of penalty or bonus of similar way to the case of the model of skew normal ogive model given by Bazán, Branco and Bolfarine (2006). More details of this model can review in Bolfarine y Bazán (2010). Particular cases and extensions considering one-parameter or three-parameters are possible. Thus, 1LPE, 1RLPE, 3LPE and 3RLPE are another IRT models implemented in BayesianModeling.