#### Author ORCID Identifier

#### Year of Publication

2019

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Education

#### Department/School/Program

Educational, School, and Counseling Psychology

#### First Advisor

Dr. Xin Ma

#### Abstract

The significance of this dissertation research is twofold with both methodological advancement and empirical update. In this dissertation research, quantile regression (QR) was introduced to social sciences researchers as a response to the weaknesses of the traditional mean-based regression often referred to as multiple regression. General advantages of QR includes being more flexible for modeling data with heterogeneous conditional distributions, more robust to outliers, and having richer characterization and description of the data. Results of QR allow researchers to not only describe a general trend of changes in the effects of the independent variables across a continuous distribution of the dependent variable but also provide information on characteristics of any shift in the distribution caused by the independent variables. These shifts pertain to location, scale, and shape shifts. This dissertation research reviewed graphical ways to examine location, scale, and shape shifts, and more importantly, developed statistical ways to quantify location, scale, and shape shifts (i.e., test for statistical significance of location, scale, and shape shifts).

Overall, this dissertation demonstrated that the introduction of QR as an advanced statistical procedure will advance the quantitative landscape of social sciences research. The results of this dissertation showed that QR can detect the differential effects of independent variables on the dependent variables that mean-based regression cannot detect and therefore uncovers more detailed relationships. This quality of QR enables more in-depth research than mean-based regression in many fields. The results of this dissertation also showed that QR allows for the understanding of relationships between variables outside the mean of the data, making it useful in understanding outcomes that are non-normally distributed and that have non-linear relationships with the independent variables. Finally, this dissertation introduced ways to detect and describe distributional shifts caused by the independent variables. The median regression line describes the (central) location shift. In addition to the estimated location shifts, the other QR lines provide information about the scale and shape shifts. This dissertation developed the bootstrapping approach to test for statistical significance when comparing location, scale, and shape shifts between parameters within and between samples (i.e., studies).

This dissertation research applied QR to the examination of individual differences in mathematics achievement and mathematics self-efficacy, using the 2003 and 2012 Programme for International Student Assessment (PISA) data. The QR results showed that the effects of many student characteristics were not constant across the mathematics outcomes distributions (i.e., mathematics achievement and mathematics self-efficacy). This suggested that individual differences were valued heterogeneously across the mathematics outcomes distributions. There was only one statistically significant location shift in terms of individual differences associated with family structure in both mathematics achievement and mathematics self-efficacy between 2003 and 2012. There was only one statistically significant scale shift in terms of individual differences associated with father SES in mathematics achievement for the middle 40 percent of the students between 2003 and 2012. There was only one statistically significant scale shift in terms of individual differences associated with gender in mathematics self-efficacy for the middle 40 percent of the students between 2003 and 2012. There was only one statistically significant shape shift in terms of individual differences associated with gender in mathematics self-efficacy between 2003 and 2012. Even though QR and LMR results can be similar in terms of statistical significance, they can differ dramatically in magnitude. Studentsâ€™ age, gender, and socioeconomic status were typical examples in this study. The effect of student age generally became more positive as student mathematics achievement increased in 2003. This suggests that age had a stronger effect on better-performing students than lower-performing students in 2003. It also means that there are more age differences in the upper tail of student mathematics achievement distribution than in the lower tail.

#### Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2019.049

#### Recommended Citation

Yuan, Jing, "FROM MEAN TO QUANTILES: RETHINKING INDIVIDUAL DIFFERENCES IN MATHEMATICS ACHIEVEMENT AND MATHEMATICS SELF-EFFICACY" (2019). *Theses and Dissertations--Educational, School, and Counseling Psychology*. 82.

https://uknowledge.uky.edu/edp_etds/82

#### Included in

Curriculum and Instruction Commons, Educational Assessment, Evaluation, and Research Commons