Abstract:

CART is a family of advanced mathematical (statistical) techniques that cluster individuals into a number of mutually exclusive and exhaustive groups with markedly different average outcome measures, based on the interaction effects of the explanatory variables. CART has some attractive advantages. First, most traditional mathematical techniques rely on the development of a mathematical model to describe the relationship between outcome and explanatory measures. Complex interactive relations among explanatory measures are often difficult to pinpoint. Correctly identifying and correctly modeling interaction effects are not necessary in CART. Second, CART does not produce complex mathematical equations. Its results therefore are easy to understand and interpret. CART has a great potential to become an advanced mathematical modeling tool for conducting applied research in many fields.

Ma (2005) pursued an integration of hierarchical linear model (HLM) with CART via a latent variable approach. This “hybrid” model of CART with HLM can be considered the initial response of researchers to the challenge for CART to extend its analytical power. In such an integration, HLM functions to set up a multilevel growth model structure with repeated measures nested within students to estimate a rate of growth in academic achievement over time for every student. CART is then used to segregate the rate of growth in academic achievement according to various individual characteristics of students.

In Ma (2005), HLM is applied to estimate a growth model with repeated measures nested within students. For the CART analysis, student characteristics alone are used to describe and predict growth in mathematics achievement of students forming different groups. A further advancement is to build school characteristics into the CART structure so that both student and school characteristics can be used to describe growth in mathematics achievement of students forming different groups. This present paper aims to achieve this important advancement. One approach to integrate school characteristics into a CART analysis which focuses on student characteristics to set up the tree structure is to find out what school characteristics predict membership for a particular group (CART terminal). This involves analyzing students within each CART terminal in terms of their school characteristics. Specifically, within a particular terminal, count is first conducted to find out how many students from a certain school end up in this particular terminal. As a result, each school represented in this particular terminal has a frequency (count) as the outcome measure following a Poisson distribution. A Poisson analysis is appropriate involving a certain number of school-level variables (school characteristics). The results indicate important school-level variables that significantly characterize this particular terminal. The interpretation of both student characteristics and school characteristics reflects the multilevel nature of this CART model. Overall, the multilevel CART model at this early stage demonstrates some good fidelity of the model, appearing to be a valid mathematical (statistical) approach to describe and predict growth in mathematics achievement from characteristics of both students and schools.

Ma, X. (2005). Growth in mathematics achievement during middle and high school: Analysis with classification and regression trees. Journal of Educational Research, 99, 78-86.