Linear Item Response Theory, Nonlinear Item Response Theory, and Factor Analysis: A Unified Framework

What it is now known as item response modeling [for an overview see van der Linden and Hambleton (1997)] originated as an effort to overcome the limitations of the factor model when applied to test items. Test items are most often categorical in nature, whereas the factor model was designed for continuous data. Unfortunately, over the years item response modeling and factor modeling have developed rather independently from one another. One of the recurring topics in R. P. McDonald’s career has been establishing bridges between these two fields (McDonald, 1967, 1980, 1981, 1982a, 1982b, 1985a, 1985b, 1986, 1999, 2001; McDonald & Mok, 1995). Two approaches can be used to relate the nonlinear models used in item response theory (IRT) to the linear model used in factor analysis. One approach is to use harmonic analysis (e.g., McDonald, 1967, 1982a). The second approach is to use link functions (e.g., McDonald, 1999; Moustaki & Knott, 2000). This chapter focuses on one particular item response model for binary data, the linear IRT model. In this model, the conditional probability of endorsing an item given the latent traits is simply a linear function. McDonald (1999, chap. 12 and 13; see also McDonald, 1969, 1982a) discussed at length the application of