Estimating quasi-loglinear models for a Rasch table if the numbers of items is large

The Rasch Model and various extensions of this model can be formulated as a quasi loglinear model for the incomplete subgroup x score x item response 1 x ... x item response k contingency table. By comparing various loglinear models, specific deviations of the Rasch model can be tested. Parameter estimates can be computed using programs such as GLIM, ECTA, and MULTIQUAL, but this becomes impractical if the number of items is large. In that case, the tables of observed and expected counts become too large for computer storage. In this paper, a method of parameter estimation is described that does not require the internal representation of all observed and expected counts, but rather uses only the observed and expected sufficient statistics of the parameter estimates, which are the marginal tables corresponding to the model terms only. The computational problem boils down to computation of the expected sufficient statistics which, in its raw form, amounts to summation of a very large number of expected counts. However, it is shown that, depending on the structure of the model, the number of computations can be reduced considerably by making use of the distributive law. As a result, simpler models may be computed much more efficiently in terms of both storage and processing times.