The IRT Procedure

Marginal Likelihood

Based on the model that is specified in the section Notation for the Item Response Theory Model, the marginal likelihood is

\[  L(\btheta |U) = \prod _{i=1}^ N \int \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \phi (\bm {\eta };\bmu ,\bSigma )d\bm {\eta } = \prod _{i=1}^ N \int f(u_ i|\bm {\eta }) \phi (\bm {\eta };\bmu ,\bSigma )d\bm {\eta }  \]

where $v_{ijk}=I(u_{ij}=k)$, $\phi (\bm {\eta })$ is the multivariate normal density function for the latent factor $\bm {\eta }$, and $\btheta $ is a set of all the model parameters. The corresponding log likelihood is

\[  \log L(\btheta |U) = \sum _{i=1}^ N \log \int \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \phi (\bm {\eta };\bmu ,\bSigma )d\bm {\eta }  \]

Integrations in the preceding equation cannot be solved analytically and need to be approximated by using numerical integration,

\[  \log \tilde{L}(\btheta |U) = \sum _{i=1}^ N \log \left[ \sum _{g=1}^{G^ d}\left[\prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk}(\mb{x}_ g))^{v_{ijk}} \frac{\phi (\mb{x}_ g;\bmu ,\bSigma )}{\phi (\mb{x}_ g;0,I)}\right]w_ g\right]  \]

where d is the number of factors, G is the number of quadrature points per dimension, and $\mb{x}_ g$ and $w_ g$ are the quadrature points and weights, respectively.