Tree-Link: Semi-Parametric Item Response Theory Using Decision and Regression Trees as a Link Function

Item response theory models typically assume the form of the link function between manifest variables and the typically discrete outcome variables is known. This
assumption is not always tenable. For example, the class of models known as “ideal point” or “unfolding models” (Stark et al., 2006) serve as a counter-example that suggests the existence of several forms. This proposal develops a method called Tree-Link Item Response Theory, that focuses on the problem of estimating an approximation of that link function when the data source suggests there may be a non-typical mapping between latent and observed variables (e.g. a logistic curve).

Tree-Link uses a technique borrowed from machine learning, decision trees for categorical data and regression trees for continuous data, as the link function between
the observed and manifest variables to approximate this relationship. This approach constitutes a semi-parametric item response theory model. Other non-parametric
(NIRT) and semi-parametric (SIRT) models for item response theory exist. However, these methods may have stringent assumptions such as monotonicity of the link
function, sum score sufficiency, or distributional assumptions. In extreme cases, e.g. when the sum score is a poor approximation of the latent trait score, fitting NIRT
models may produce nonsense results.

This dissertation develops methods for estimating NIRT models in which the link function is a tree. This approach can reduce or remove the assumptions present incurrent methods for NIRT. Tree-Link recursively partitions the latent ability scale into regions with relatively homogeneous responses. Because of this, it naturally handles data from all levels of measurement (Stevens, 1946), including continuous responses. The methods developed here are evaluated with via simulation study and then applied to example data. Results show Tree-Link can accurately recover estimates of the underlying trait with high fidelity even in the presence of highly irregular link functions. Likewise, the estimated item response function (IRF) is demonstrated to closely match the true IRF.