The IRT Procedure

Newton-type algorithms maximize the marginal likelihood directly, based on the first and second derivatives. Two of the most widely used estimation algorithms are the Newton-Raphson and Fisher scoring algorithms, which rely on the gradient and Hessian of the log likelihood. However, for latent variable models that contain categorical responses, the Hessian matrix is often expensive to compute. As a result, several quasi-Newton algorithms that require only gradients have been proposed. In the field of IRT, Bock and Lieberman (1970) propose replacing the Hessian with the following information matrix:

To calculate the preceding expectation, you need to sum over not just the observed but all possible response patterns; this becomes computationally very expensive when the number of items is large. Fortunately, other quasi-Newton algorithms that do not have these computational difficulties have been proposed. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is one of the most popular quasi-Newton algorithms that approximate the Hessian matrix with gradient.

For the objective function, , the first derivatives with respect to are

where

and

where, in the preceding two equations, indicate parameters that are associated with item j and represents parameters that are related to latent factors.

The expectation-maximization (EM) algorithm starts from the complete data log likelihood that can be expressed as follows:

The expectation (E) step calculates the expectation of the complete data log likelihood with respect to the conditional distribution of , as follows:

The conditional distribution is

These conditional expectations that are involved in the Q function can be expressed as follows:

Then

where

and

Integrations in the preceding equations can be approximated as follows by using G-H quadrature:

In the maximization (M) step of the EM algorithm, parameters are updated by maximizing . To summarize, the EM algorithm consists of the following two steps: