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

upper L left-parenthesis bold-italic theta vertical-bar upper U right-parenthesis equals product Underscript i equals 1 Overscript upper N Endscripts integral product Underscript j equals 1 Overscript upper J Endscripts product Underscript k equals 1 Overscript upper K Endscripts left-parenthesis upper P Subscript i j k Baseline right-parenthesis Superscript v Super Subscript i j k Baseline phi left-parenthesis bold-italic eta semicolon bold-italic mu comma bold upper Sigma right-parenthesis d bold-italic eta equals product Underscript i equals 1 Overscript upper N Endscripts integral f left-parenthesis u Subscript i Baseline vertical-bar bold-italic eta right-parenthesis phi left-parenthesis bold-italic eta semicolon bold-italic mu comma bold upper Sigma right-parenthesis d bold-italic eta

where v Subscript i j k Baseline equals upper I left-parenthesis u Subscript i j Baseline equals k right-parenthesis, phi left-parenthesis bold-italic eta right-parenthesis is the multivariate normal density function for the latent factor bold-italic eta, and bold-italic theta is a set of all the model parameters. The corresponding log likelihood is

log upper L left-parenthesis bold-italic theta vertical-bar upper U right-parenthesis equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts log integral product Underscript j equals 1 Overscript upper J Endscripts product Underscript k equals 1 Overscript upper K Endscripts left-parenthesis upper P Subscript i j k Baseline right-parenthesis Superscript v Super Subscript i j k Baseline phi left-parenthesis bold-italic eta semicolon bold-italic mu comma bold upper Sigma right-parenthesis d bold-italic eta

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

log ModifyingAbove upper L With tilde left-parenthesis bold-italic theta vertical-bar upper U right-parenthesis equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts log left-bracket sigma-summation Underscript g equals 1 Overscript upper G Superscript d Baseline Endscripts left-bracket product Underscript j equals 1 Overscript upper J Endscripts product Underscript k equals 1 Overscript upper K Endscripts left-parenthesis upper P Subscript i j k Baseline left-parenthesis bold x Subscript g Baseline right-parenthesis right-parenthesis Superscript v Super Subscript i j k Superscript Baseline StartFraction phi left-parenthesis bold x Subscript g Baseline semicolon bold-italic mu comma bold upper Sigma right-parenthesis Over phi left-parenthesis bold x Subscript g Baseline semicolon 0 comma upper I right-parenthesis EndFraction right-bracket w Subscript g Baseline right-bracket

where d is the number of factors, G is the number of quadrature points per dimension, and bold x Subscript g and w Subscript g are the quadrature points and weights, respectively.

Last updated: December 09, 2022