A set of linear conditions on item response functions is derived that guarantees identical observed-score distributions on two test forms. The conditions can be added as constraints to a linear programming model for test assembly that assembles a new test form to have an observed-score distribution optimally equated to the distribution on an old form. For a well-designed item pool and items fitting the IRT model, use of the model results into observed-score pre-equating and prevents the necessity of post hoc equating by a conventional observed-score equating method. An empirical example illustrates the use of the model for an item pool from the Law School Admission Test.

Posterior odds of cheating on achievement tests are presented as an alternative to $p$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document} values reported for statistical hypothesis testing for several of the probabilistic models in the literature on the detection of cheating. It is shown how to calculate their combinatorial expressions with the help of a reformulation of the simple recursive algorithm for the calculation of number-correct score distributions used throughout the testing industry. Using the odds avoids the arbitrary choice between statistical tests of answer copying that do and do not condition on the responses the test taker is suspected to have copied and allows the testing agency to account for existing circumstantial evidence of cheating through the specification of prior odds.