2. Item Response Theory

2. 1. Item Response Theory models

Consider data collected of persons who have each given responses on different items of a test. A Two-Parameter Item Response Theory (IRT) model one-dimensional and binary is a system in which for each person has a unidimensional monotone latent variable model , defined by the following expressions:

where

- is the manifest variable which model the binary response of the person that answer to the item . The items have binary outcomes, i.e., the items are scored as 1 if correct and 0 if no.

- are two parameters that represent, respectively, to the discrimination and the difficulty of the item .
- is the value of the latent variable or trait latent for the person , and some occasions it is interpreted as the latent ability of the person .
- is the conditional probability given to r espond correctly to item .
- is called the item characteristic curve (ICC) and
- is a latent lineal predictor associated with the latent trait of the person and the item parameters for the item .

Observations

The Two-Parameter IRT model

- satisfies the property of latent conditional independence; it is, for a person the response to the different items are conditionally independent given the latent variable
- satisfies the property of latent monotonicity, because is a function strictly no decreasing of ,
- is one-dimensional latent.
- , where and , is the same for each case and is called the link function.

- Also is assumed that responses are independent between persons.
- The parameters of difficulty and of discrimination represent the location and inclination of the item, respectively, being a proportional value to the inclination of the ICC in the point and is the point on where the ICC has its maximum slope. Values as are not expected. The parametric space for the parameter is arbitrary and to be the same as than by the usual take values in the line real.

Another parameterization very common for the predictor linear latent is . This parametrización is very important from the computational point of view since it facilitates the computational time of convergence. When it used this parameterization, the previous parameter of difficulty can be obtained doing in the obtained result. Generally, this parameterization is preferred in the Bayesian Inference and also in BayesianModeling.

In The Two-Parameter IRT model, the conjoint density of the vector of multivariate responses , with given the vector of latent variables and the vector of parameters of the items can be written as:

The proof of this result is direct by considering the latent conditional independence.

The first IRT binary model was introduced by Lord (1952) with an ICC given by being the cdf of a standard normal variable. This model is known in the psychometric literature as normal ogive model which corresponds in the context of Generalize Linear Models, for a probit link function and empathizing this can be named as 2P model.

On the other hand, Birbaum (1968) considered a ICC given by , where denotes the cdf of a logistic variable. This induces, in the language of the generalized linear models, to a logit link function. This model is known as the logistic model and empathizing the link is named here as 2L model.

Particular cases

The IRT model admits diverse formulations, which depend basically of as it considers the ICC. In its simplest version could take and consider an ICC of the form

This is called of one-parameter IRT model and when probit or logit links are considered we have 1P or 1L IRT model respectively.

In a general way we could consider an ICC of the form

Where the parameter indicates the probability that very low ability individuals will get this item correct by chance, and is the distribution function. This is known as the three-parameter IRT model. If , the model is reduced to the two- parameters IRT model. Again, when probit or logit links are considered we have 3P or 3L IRT model respectively.

The IRT model with logit link

The IRT model with logit link or logistic model is probably the model more used in IRT. The version of three parameters for this model establishes that the probability that the person hit the item is given by:

where usually is assumed that although some authors consider also the value for approximating this model to the normal ogive model. As particular cases have

The last model of a parameter, is knows as well as the Rasch model but it has own interpretations and derivations (see for example Fischer and Molenaar, 1995).

The BayesianModeling program allows implementing the code in WinBUGS for the models 1L, 2L, 3L, 1P, 2P and 3P IRT models. By considering this links, these models are symmetric IRT models. In addition, news IRT models with asymmetric links are considered also in BayesianModeling which are presented in the next section.