|

# 4.1 Multiple Correspondence Analysis2

• 2 For a complete mathematical revision of this technique please refer to Michailidis, G. & de Leeuw, (...)

1As reported by Abdi and Valentine (2007):

Multiple Correspondence analysis (mca) is an extension of Correspondence analysis that allows one to analyze the pattern of relationships of several categorical variables (…) Technically mca is obtained by using a standard correspondence analysis on an indicator matrix (i. e., a matrix whose entries are 0 or 1). (p. 651)

2In the same way, Starkweather and Herrington (2014) report that: “using a correspondence analysis with categorical variables is analogous to using a correlation analysis for continuous or nearly continuous variables” (para. 1). Moreover “this analysis is nonparametric and does not offer a statistically significant test because it is not based on a probability of distribution.” (para. 2) In this study, mca was used in order to produce a socioeconomic index (ses index), which explained the largest amount of variance underlying seven socioeconomic variables. Specifically, mca was proposed to answer one question: What is the best way to combine some socioeconomic variables in order to produce a single ses index?

Figure 1. Multiple correspondence analysis: conceptual diagram # 4.2 Cluster Analysis and Contingency Tables

Suppose there are two categorical variables, denoted by X and Y. Let I denote the number of categories of X and J the number of categories of Y. A rectangular table having I rows for the categories of X and J columns for the categories of Y has cells that display the IJ possible combinations of outcomes. A table of this form that displays counts of outcomes in the cells is called a contingency table. (Agresti, 2007, p. 22)

3In this study, contingency tables were used with the purpose of establishing a relation between ses index, created by the analysis described in the section 4.1, and the saber 11 and saber pro outcomes. Before that, a 3-means clusters classification was performed, obtaining three groups for each case. Consequently, the clusters obtained for the tests were high performance, middle performance and low performance. For the ses index, the clusters obtained were high condition, middle condition and low condition. Specially, the question answered by these techniques was: What is the relation among the ses index, the saber 11 test outcomes and saber pro test outcomes?

Figure 2. Cluster analysis and contingency tables: conceptual diagram # 4.3. Multiple Discriminant Analysis

4According to Tacq (1997), in a discriminant analysis the intention is to examine whether a set of variables is capable of distinguishing between different groups. As a result, one linear combination of the discriminating variables is obtained in such a way that different groups are maximally distinguished. Such a linear combination is called a discriminant function. In addition, Wuensch (2014) states that:

With more than two groups one can obtain more than one discriminant function (DF). The first DF is that which maximally separates the groups (...) The second DF, orthogonal to the first, maximally separates the groups on variance not yet explained by the first DF. One can find a total of K = 1 (number of groups minus 1) or p (number of predictor variables) orthogonal discriminant functions, whichever is smaller. (p. 1)

5In the case of this study, mda was used to establish how well the socioeconomic variables could classify saber 11 and saber pro outcomes. In short, mda was used to answer the following two questions: a) is it possible to use socioeconomic variables to classify saber 11 and saber pro performance groups’membership? b) Which are the most significant predictors in these classifications?

Figure 3. Multiple discriminant analysis: conceptual diagram # 4.4. manova

6As reported by Mayers (2013):

Multivariate analysis of variance explores the outcomes of many parametric dependent variables, across one or more independent variables, each with at least two distinct groups or conditions. The goal of this analysis is to look for an effect of one or more independent variables on several dependent variables at the same time. (p. 318)

7In consequence, manova was used during this research to answer the following two questions (Tacq, 1997):

1. Are there significant differences in the Critical Reading and Quantitative Reasoning centroids3 between: a) the saber 11 performance groups? And b) the ses groups?
2. Is there a significant effect of interaction, i. e., is the effect of belonging to a saber 11 performance group on saber pro outcomes different for students of high, middle and low ses ?

# 4.5. Multilevel Analysis

8Schools systems represent an example of a hierarchical structure with students grouped within schools. Some research on the field of education has been oriented towards comparing schools in terms of their students’achievement using a multilevel approach (Goldstein, 2011). Indeed, multilevel modeling of hierarchical social data is an advanced form of regression analysis now widely used to provide estimates of school and student level variances (e.g. Raudenbush & Bryk, 2002).

9In the case of this study, the data were hierarchically structured: students (level one) were located within universities (level two). In consequence, multilevel analyses were used with a threefold purpose: first, to investigate the relation between saber 11 and saber pro at university level; second, to determine whether this relation varies across universities, and finally to compare this relation among different ses conditions. For the previous reasons, the questions answered with multilevel analysis were: a) how universities affect the progress of the students in Quantitative Reasoning and Critical Reading? b) Are there different university effects for saber pro students’outcomes according to their ses index?

Figure 5. Multilevel model: conceptual diagram 