This paper discusses new psychometric analyses that improve capabilities for relating performance on achievement test items to instruction received by examinees. The modeling discussion is closely tied to data for eighth graders in the United States provided by the Second International Mathematics Study (SIMS), comprising not only responses to a set of achievement items at the beginning and end of the eighth grade but also a relatively rich set of student background information, including opportunity-to-learn information specific to each item. Taking item response theory (IRT) as a starting point, this paper summarizes B. Muthen's work on a set of new psychometric analysis techniques that give a richer description of achievement-instruction relations. Six topics that expand standard IRT and deal specifically with the effects of varying instructional opportunities are discussed: (1) variation in IRT measurement characteristics; (2) multidimensional modeling; (3) modeling with heterogeneity levels; (4) estimation of trait scores; (5) predicting achievement by relating trait to student background; and (6) relationship of change in general and specific traits to instructional opportunities. The core test of 40 items used in the SIMS, four tables, and 38 graphs are provided. (TJH)