The IRT Procedure
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Overview
- Getting Started
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Syntax
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DetailsNotation for the Item Response Theory ModelAssumptionsPROC IRT Contrasted with Other SAS ProceduresResponse ModelsMarginal LikelihoodApproximating the Marginal LikelihoodMaximizing the Marginal LikelihoodFactor Score EstimationModel and Item FitItem and Test informationMissing ValuesOutput Data SetsODS Table NamesODS Graphics
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Examples
- References
Marginal Likelihood
Based on the model that is specified in the section Notation for the Item Response Theory Model, the marginal likelihood is
![\[ L(\btheta |U) = \prod _{i=1}^ N \int \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \phi (\bm {\eta };\bmu ,\bSigma )\, d\bm {\eta } = \prod _{i=1}^ N \int f(u_ i|\bm {\eta }) \phi (\bm {\eta };\bmu ,\bSigma )\, d\bm {\eta } \]](images/statug_irt0061.png)
where 
, 
is the multivariate normal density function for the latent factor 
, and 
is a set of all the model parameters. The corresponding log likelihood is
![\[ \log L(\btheta |U) = \sum _{i=1}^ N \log \int \prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk})^{v_{ijk}} \phi (\bm {\eta };\bmu ,\bSigma )\, d\bm {\eta } \]](images/statug_irt0066.png)
Integrations in the preceding equation cannot be solved analytically and need to be approximated by using numerical integration,
![\[ \log \tilde{L}(\btheta |U) = \sum _{i=1}^ N \log \left[ \sum _{g=1}^{G^ d}\left[\prod _{j=1}^ J \prod _{k=1}^ K(P_{ijk}(\mb{x}_ g))^{v_{ijk}} \frac{\phi (\mb{x}_ g;\bmu ,\bSigma )}{\phi (\mb{x}_ g;0,I)}\right]w_ g\right] \]](images/statug_irt0067.png)
where d is the number of factors, G is the number of quadrature points per dimension, and 
and 
are the quadrature points and weights, respectively.