Smoothing the joint and marginal distributions of scored two-way contingency tables in test equating


Department of Statistics, 3000 Steinberg Hall–Dietrich Hall, Wharton School of the University of Pennsylvania, Philadelphia, PA 19104–6302, USA.

Research Statistics Group, Educational Testing Service.


If the row and column variables of a two-way contingency table have numerical scores, then the table is said to be scored. Scored contingency tables play an important role in equating two exams containing common items: the tables are typically quite large, and often relatively sparse, and exhibit strong positive dependence. For equating, stable, monotone estimates are required for the conditional distribution of the score on the new items given the score on the common items. We obtain such-estimates by using generalized log-linear models in a new way: we smooth both the interior and the margins of the two-way table, yielding the smoothest joint distribution on the table having the same low-order moments as the observed sample.