A discretization model of measurement errors

Classical (random) errors cause measurements to differ when they should truly be the same. Here we present a model of the inverse type of errors: that cause measurements to be the same when they should truly differ. We term these errors discretization errors, develop their mathematical model, and show how they can be corrected for when calculating slopes and correlations in linear models using a correction similar to the Spearman one for classical errors, and correction factors similar to classical reliabilities. This allows a general way to model and correct for these types of errors. Any situation where multiple true scores get collapsed can be understood with this model. We discuss how existing approaches fit into this general framework (e.g., scoring binary responses using Item Response Theory), and sketch the broader range of situations where the model can be applied. Discretization errors have the capacity of organizing, guide modelling, and furthering our understanding about an extensive class of errors beyond classical ones, that psychological researchers frequently encounter.