In Modern Test Theory, response variables are a function of a common latent variable that represents the measured attribute, and error variables that are unique to the response variables. While considerable thought goes into the interpretation of latent variables in these models (e.g., validity research), the interpretation of error variables is typically left implicit (e.g., describing error variables as residuals, i.e., as `whatever is unexplained by the model'). Yet, many psychometric assumptions are essentially assumptions about error and thus being able to reason about psychometric models requires the ability to reason about errors. We propose the causal theory of error as a framework that helps reasoning about errors in terms of the data-generating mechanism. In this framework, the error variable reflects all unique causes of the response variable that together with the latent variable determine the item responses. We show that different assumptions about error scores (1) imply different psychometric models, (2) have different implications for the chance experiment that underlies the notion of randomness in the model, and (3) have different implications for item bias and local homogeneity. In the causal theory of error, these assumptions concern the unique causes of the item response that may be person characteristics that have the sampling of people as their source of variability, or properties of the measurement circumstances that have the sampling of measurement occasions as their source of variability.
Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints
