This dissertation is a collection of two essays that address issues of class size effects on student achievement and power analysis methods for model of changes.Class size reduction policies have been widely implemented around the world in the past decades. However, findings about the effects of class size on student achievement have been mixed. In addition, most of the studies about class size effects have focused on the effects on the average achievement for all students. Only a few studies have focused on the differential class size effects across the student achievement distribution, and their findings have been mixed. The first essay (Chapter 1 and Chapter 2) was designed to evaluate class size effects on student achievement. In particular, Chapter 1 employed instrumental variables (IV) methods to examine the causal effects of class size on fourth grade mathematics achievement using data from TIMSS (Trends in International Mathematics and Science Study). While I found some evidence of class size effects in Romania and the Slovak Republic, overall there were no systematic patterns of class size effects. The results indicate that in most European and Asian countries class size reduction may not improve mathematics achievement in fourth grade. The first essay also evaluated the differential class size effects across mathematics achievement distribution. In particular, Chapter 2 employed quantile regression analysis, coupled with instrumental variables methods, to examine the causal effects of class size on fourth grade mathematics achievement. While I found some evidence of quantile-specific class size effects in Romania and the Slovak Republic, overall there were no systematic patterns of class size effects. What is more, there was no evidence to show that high- or low-achievers benefited more from smaller classes. The results indicate that in most European and Asian countries class size reduction may not increase or reduce the achievement gap between low- and high-achieving students in fourth grade.The second essay of this dissertation (Chapter 3) was designed to provide methods for three-level models in studies of polynomial change. Experiments that involve nested structures often assign entire groups to treatment conditions and follow them over time to assess group differences in the average of change, rate of acceleration, or higher degree polynomial effect. Chapter 3 provide methods for power analysis in three-level polynomial change models for cluster randomized designs (i.e., treatment at the third level) and block randomized designs (i.e., treatment at the second level). Both unconditional models and conditional models that include covariates at the second (e.g., student) and the third (e.g., school) levels are discussed. The power computations take into account nesting effects at the second and at the third level, the duration of study, sample size effects (e.g., the numbers of schools and students), and covariates effects. Chapter 3 also provided illustrative examples to show how powers are influenced by the study duration, sample sizes and covariates at the second and the third level.
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