The IRT Procedure
PROC IRT provides three methods of estimating factor scores: maximum likelihood (ML), maximum a posteriori (MAP), and expected a posteriori (EAP). You can specify them by using the SCOREMETHOD= option in the PROC IRT statement.
You can obtain the ML factor score by maximizing the likelihood for each observation with respect to the latent factor. You
can also compute the MAP or EAP factor score by maximizing or by taking the expectation of the posterior distribution of latent
factors for each observation. The likelihood and posterior distribution for each observation, 
, can be expressed, respectively, as
![\[ l(\bm {\eta }|u_ i,\hat{\btheta }) = \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \]](images/statug_irt0110.png){kind=small}
and
![\[ p(\bm {\eta }|u_ i,\hat{\btheta }) \propto \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}}\phi (\bm {\eta };\bmu ,\bSigma ) \]](images/statug_irt0111.png){kind=small}
Factor scores are restricted to the range from –99 to 99. For unidimensional models, the ML factor score is not available for subjects whose response to all the items is either the lowest or the highest level. For example, suppose there are five binary items in the model. For subjects whose response is 1 or 0 to all five items, the ML factor score cannot be estimated. For subjects whose response to all items is the lowest level, the ML factor score is set to –99, and for subjects whose response to all items is the highest level, the ML factor score is set to 99.