Using Heteroskedastic Ordered Probit Models to Recover Moments of Continuous Test Score Distributions from Coarsened Data

Test score distributions of schools or demographic groups are often summarized by frequencies of students scoring in a small number of ordered proficiency categories. We show that heteroskedastic ordered probit (HETOP) models can be used to estimate means and standard deviations of multiple groups' test score distributions from such data. Because the scale of HETOP estimates is indeterminate up to a linear transformation, we develop formulas for converting the HETOP parameter estimates and their standard errors to a scale in which the population distribution of scores is standardized. We demonstrate and evaluate this novel application of the HETOP model with a simulation study and using real test score data from two sources. We find that the HETOP model produces unbiased estimates of group means and standard deviations, except when group sample sizes are small. In such cases, we demonstrate that a "partially heteroskedastic" ordered probit (PHOP) model can produce estimates with a smaller root mean squared error than the fully heteroskedastic model.