Exploring Alternate Latent Trait Metrics with the Filtered Monotonic Polynomial IRT Model

Abstract

Item response theory (IRT) is a broad modeling framework that makes precise predictions about item response behavior given individuals’ locations on a latent (unobserved) variable. If the item-trait regressions, also known as item response functions (IRFs), are monotonically increasing and if assumptions about unidimensionality and local independence are satisfied, then examinees can be ordered uniquely on the latent trait. Scales that satisfy these three assumptions can be transformed monotonically without altering scale properties—that is, they define an ordinal-level scale (Stevens, 1946). When fitting an IRT model, however, the scale of the latent variable—that is, its location and interval spacing—must be identified by introducing extra assumptions. In practice, the scale is identified by specifying either the parametric form of the IRF (parametric IRT) or the distribution of the latent trait (nonparametric IRT). Filtered monotonic polynomial IRT (FMP) has been proposed as a type of nonparametric IRT method (Liang & Browne, 2015), but shares important properties with parametric methods. In this dissertation, it is demonstrated that any IRF defined within the FMP framework can be re-expressed as another FMP IRF by taking linear or nonlinear transformations of the latent trait. A general form for these transformations is presented in terms of matrix algebra. Finally, I propose a composite FMP IRT model in which nonlinear transformations of the latent trait are modeled explicitly by a monotonic composite function.I argue that the composite model offers many advantages over existing methods. First, the composite FMP model narrows the methodological gap between para- metric and nonparametric item response models, allowing for item banking and adaptive testing within a flexible modeling framework. Second, this composite model suggests a sequential NIRT curve-fitting method that allows users to explore both alternate (e.g., non-normal) latent densities and flexible IRF shapes. Finally, the composite FMP model allows users to explore and employ alternate scalings of the latent trait without sacrificing the methodological advantages of parametric models.