Modeling Nonlinear Effects of Person‐by‐Item Covariates in Explanatory Item Response Models: Exploratory Plots and Modeling Using Smooth Functions

Abstract

Explanatory item response models (EIRMs) have been applied to investigate the effects of person covariates, item covariates, and their interactions in the fields of reading education and psycholinguistics. In practice, it is often assumed that the relationships between the covariates and the logit transformation of item response probability are linear. However, this linearity assumption obscures the differential effects of covariates over their range in the presence of nonlinearity. Therefore, this paper presents exploratory plots that describe the potential nonlinear effects of person and item covariates on binary outcome variables. This paper also illustrates the use of EIRMs with smooth functions to model these nonlinear effects. The smooth functions examined in this study include univariate smooths of continuous person or item covariates, tensor product smooths of continuous person and item covariates, and by‐variable smooths between a continuous person covariate and a binary item covariate. Parameter estimation was performed using the mgcv R package through the maximum penalized likelihood estimation method. In the empirical study, we identified a nonlinear effect of the person‐by‐item covariate interaction and discussed its practical implications. Furthermore, the parameter recovery and the model comparison method and hypothesis testing procedures presented were evaluated via simulation studies under the same conditions observed in the empirical study.