Scale shrinkage in vertical equating

As an alternative to equipercentile equating in the area of multilevel achievement test batteries, item response theory (IRT) vertical equating has produced unexpected results. When expanded standard scores were obtained to link the Comprehensive Test of Basic Skills and the California Achievement Test, the variance of test scores diminished both within particular grade levels from fall to spring, and also from lower to upper grade levels. Equipercentile equating, on the other hand, has resulted in increasing variance both within and across grade levels, although the increases are not linear across grade levels. Three potential causes of scale shrinkage are discussed, and a more comprehensive, model-based approach to establishing vertical scales is described. Test data from the National Assessment of Educational Progress were used to estimate the distribution of ability at grades 4, 8, and 12 for several math achievement subtests. For each subtest, the variance of scores increased from grade 4 to grade 8; however, beyond grade 8 the results were not uniform. Index terms: developmental scores, equating, IRT scaling, maximum likelihood estimation, National Assessment of Educational Progress (NAEP), scale shrinkage, vertical equating