Teresa Calapez

The purpose of this study is to evaluate the similarity of results obtained with linear and nonlinear PCA over Likert-type items.

As part of a wider project about students' perceptions of mathematics, this study analyses two dimensions: the usefulness of mathematics and the importance of understanding the concepts. Each dimension is theoretically structured by six items, three with a positive connotation and three with a negative connotation (adaptations of Fennema and Sherman, 1976 and Kloosterman and Stage, 1992). Each item is a statement that seeks to register the intensity of agreement on a 5-point scale.

Four questionnaires were constructed: in the first two (A and B) the statements are evaluated by marking the response on a line segment (Visual Analogue Scale), labelled in the extreme and at the midpoint (A) or only at the extremes (B). In the third (C), Likert-type items with five labelled points are used (completely disagree, disagree, neither agree nor disagree, agree and totally agree), while in D only the extremes are labelled. This work concerns only questionnaires type C and D.

Since the variables of this type are structurally ordinal variables (Gob et al, 2007), the use of a nonlinear principal components (e.g. CATPCA) is a possibility for its interdependency analysis. However, the use of linear techniques (PCA) has been widespread as a form of preferential treatment of such data. Do both techniques lead to similar results? If not, then the linearity of the treatment of such items can be questioned.

Drawing on the data collected in the study mentioned, and using the balanced bootstrap as proposed in Linting et al (2007), several confidence intervals are built for the objects scores, either obtained through PCA or CATPCA, and for the correlation coefficient between paired dimensions. With these confidence intervals we expect to have a perception of the comparability of the linear and non-linear solutions.

**References:**

Davison, A.C., D.V. Hinkley; E. Schechtman (1986): Efficient bootstrap simulation”, *Biometrika*, 73, 555–566.

Fennema, E. H.; Sherman, J.A. (1976). Fennema-Sherman mathematics attitudes scales: instrument designed to measure attitudes toward mathematics. *Journal for Research in Mathematics Education*, 7(5), 324-326.

Gob, R.; McCollin, C.; Ramalhoto, F. (2007), “Ordinal Methodology in the Analysis of Likert Scales”, *Quality & Quantity*, 41, pp 601-626.

Kloosterman, P.; Stage, F.K.(1992). Measuring beliefs about mathematical problem solving. *School Science and Mathematics*, 92, 109-115.

Linting, M.; Meulman, J.; Groenen, P.; Van der Kooij, A. (2007): “Stability of Nonlinear Principal Components Analysis: An Empirical Study Using Balanced Bootstrap”, *Psychological Methods*, 12(3), pp 359-379.

Timmerman, M.; Kiers, H.; Smilde, A. (2007): “Estimating confidence Intervals for Principal Components Loadings: A Comparison Between the Bootstrap and Asymptotic Results”, *British Journal of Mathematical and Statistical Psychology*, 60, pp 295-314.

**Keywords:** Likert-type items; PCA; CATPCA; Balanced bootstrap

**Biography:** Teresa Calapez completed her MSc in Statistics and Operational Research in 1991 at Lisbon University's Faculty of Science, and obtained her PhD in Quantitative Methods, Multivariate Data Analysis, from ISCTE – Lisbon University Institute in 2004.

She is currently Assistant Professor in the Quantitative Methods Department at ISCTE–IUL, teaching mainly statistics and data analysis.

She is a member of the Portuguese Statistical Society, a Unide-IUL researcher, and cooperates regularly in Dinâmia/CET projects.

Her main interests are in the field of multivariate data analysis for non metric data.