Principles of quantile regression and an application

Abstract

Newer statistical procedures are typically introduced to help address the limitations of those already in practice or to deal with emerging research needs. Quantile regression (QR) is introduced in this paper as a relatively new methodology, which is intended to overcome some of the limitations of least squares mean regression (LMR). QR is more appropriate when assumptions of normality and homoscedasticity are violated. Also QR has been recommended as a good alternative when the research literature suggests that explorations of the relationship between variables need to move from a focus on average performance, that is, the central tendency, to exploring various locations along the entire distribution. Although QR has long been used in other fields, it has only recently gained popularity in educational statistics. For example, in the ongoing push for accountability and the need to document student improvement, the calculation of student growth percentiles (SGP) utilizes QR to document the amount of growth a student has made. Despite its proven advantages and its utility, QR has not been utilized in areas such as language testing research. This paper seeks to introduce the field to basic QR concepts, procedures, and interpretations. Researchers familiar with LMR will find the comparisons made between the two methodologies helpful to anchor the new information. Finally, an application with real data is employed to demonstrate the various analyses (the code is also appended) and to explicate the interpretations of results.