The IRT Procedure
Prior Distributions for Parameters
Technical issues often arise when the marginal likelihood is maximized for three- and four-parameter models. For example, the optimization algorithm might not converge, or the estimates for one or both of the guessing or ceiling parameters could reach their boundaries of 0 or 1, respectively, resulting in the standard errors for the guessing and ceiling parameters not being computable. To solve these technical issues, prior distributions for the guessing, ceiling, and slope parameters are often incorporated into the marginal likelihood to produce the joint posterior distribution,
where 
is the likelihood function and 
, 
, and 
are the prior distributions of the guessing, ceiling, and slope parameters, respectively. Parameter estimates are then obtained by maximizing the joint posterior distribution. In Bayesian statistics, these parameter estimates are called maximum a posteriori probability (MAP) estimates. For more information about prior distributions and Bayesian analysis of IRT models, see Baker and Kim (2004).
The following subsections describe the prior distributions that PROC IRT incorporates for three- and four- parameter models.
Prior Distribution for the Guessing and Ceiling Parameters
The following beta prior distribution is used for the guessing and ceiling parameters:
where 
is either the guessing or the ceiling parameter and 
is the beta function. The mean of the beta distribution is 
. Because the 
and 
parameters are not practically meaningful by themselves, PROC IRT does not use these two parameters to specify this prior distribution. Instead, a mean and a weight are used as the parameters for the distribution. The mean parameter, 
, corresponds to the mean of the beta distribution, and it represents the probability of getting a correct response. The weight parameter, 
, represents the confidence of the prior information. The more confident you are that the parameter value is near the mean parameter value, the larger the weight parameter you specify.
By default, the mean parameter is 0.2 for the guessing prior and 0.9 for the ceiling prior. The default weight is 20 for both the guessing and the ceiling priors. You can change the default settings by using the GUESSPRIOR= and CEILPRIOR= options in either the PROC IRT statement or the MODEL statement.
Prior Distribution for the Slope Parameter
The normal prior distribution is used for the slope parameters for three- and four-parameter models:
Although the slope parameters exist in all IRT models, no prior distribution for the slopes would be incorporated into models other than the three- and four-parameter models. In PROC IRT, you uses the 
and 
parameters directly to specify this normal prior for the slope parameter. The mean parameter, , represents prior information about the value of the slope parameter. The inverse of the variance parameter, 
, represents the precision of the prior information. A smaller variance parameter means that the prior information is more precise and has a bigger impact on the posterior distribution.
By default, the mean and variance parameters are 0 and 1. To change the default, you can use the SLOPEPRIOR= option in either the PROC IRT statement or the MODEL statement.