30 Vertical Scaling: Statistical Models for Measuring Growth and Achievement

This chapter examines psychometric modeling approaches to vertically scaled educational assessments. We discuss historical approaches, and we examine model assumptions widely used in current practice. We identify a weakness in existing approaches that leaves users overly vulnerable to implausible (e.g., disordinal) grade-by-grade results, and we develop and demonstrate the general feasibility of an alternative approach grounded in hierarchial modeling. The approach we introduce and explore is a hierarchical multi-group item response theory model that allows explicit estimation of the functional form of the grade-to-grade growth patterns. The modeling framework we introduce is very general and incorporates widely used item response theory approaches to vertical scaling as special cases. We explore properties of the model and the estimation algorithms using simulated and real data from achievement tests.