2.3. Bayesian estimation in IRT

Considering the distribution Bernoulli for the response variable, the likelihood function for IRT model in the three-parameter IRT model is given by

Where is the cdf of an asymmetric distribution indexed by the parameter , associated with the asymmetric ICC.

The Logistic (logit), Normal ogive (probit), LPE and RLPE consider this likelihood function; however skew probit IRT consider other version of the likelihood function considering augmented version that is discussed in the specific references of this model. In WinBUGS, the implementation of this procedure is not direct because it requires of a correct specification of the indicator variables. Main details can find in Bazán, Branco and Bolfarine (2006).

In the Bayesian Inference, the parameters of interest are assumed like random variables and so is need establishes a priori probability distributions that reflects our previous knowledge of its behavior. Combining the likelihood function and the priori distributions we can obtain the posteriori distributions of the parameters of interest. In the present work, we consider priors that they are vague proper priors with known distributions but variance big as well as independence between priors (see Nzoufras, 2009). In traditional IRT models priors are discussed in Albert (1992), Johnson and Albert (1999), Patz and Junker (1999), Sahu (2002), Rupp, Dey and Zumbo (2004), Bazán, Bolfarine and Leandro (2006), Fox (2010).

In IRT models there is consensus about the prior specification for latent trait, thus is assumed for . However about item parameter there is several proposals. Here is assumed independent priors as

Where is a pdf of normal distribution and correspond to the prior distributions of item parameters and , respectively.

In the special case of size of small samples we suggested the use of the following prior specification

- with and ( and ) where correspond the positive normal or Half-normal distribution.

- with and , ( and )

- (, ).

- For LPE and RLPE models (, )

- For Ogive skew normal model , where and .

Bayesian Inference in IRT models is facilitated with the use of different methods MCMC implemented in WinBUGS or OpenBUGS software. An introduction to MCMC methods is given in Gilks, Richardson, and Spiegelhalter (1996). For more details about the use of these softwares for Bayesian Inference we suggest the book of Congdon (2005), Congdon (2010) and Ntzoufras (2009). For traditional IRT models Bugs codes are available by example in Curtis (2010), Fox (2010), Bazan, Valdivieso and Calderón (2010).

Also, Bayesian Inference for some traditional IRT models using R package (MCMCpack: Martin and Quinn, 2002) and Matlab package: (IRTuno: Sheng, 2008a, IRTmu2no: Sheng, 2008b, IRTm2noHA: Sheng, 2010) are available.

However, for most of the models presented here, there is no program that generates codes for WinBUGS or OpenBUGS. This is facilitated using BayesianModeling. The IRT models implemented in BayesianModeling classified according to its links are:

• Symmetric: logistic (1L, 2L, 3L), probit (1P, 2P, 3P).

• Asymmetric: LPE (1LPE, 2LPE, 3LPE), RLPE (1RLPE, 2RLPE, 3RLPE), skew probit (1SP, 2SP).

All codes for IRT models are established considering the likelihood function presented here and considering the priors suggested with the exception of the skew probit IRT models which use an augmented likelihood function version. In BayesianModeling, when a specific code is generated for a IRT model, also References to justify the model and the choices of priors are showed.