Slides of talk (PDF, 1595 Kb)
The purpose of this paper is to introduce the logical and arithmetic
operators and simple matrix functions of Minitab® –a well-known software
package for teaching statistics- as a computer-aid to teach Principal
Components Analysis (PCA) to graduate students in the field of
Education.
PCA, originally proposed by Pearson (1901) is a mathematical
technique –a vector space transform- that has its roots in linear
algebra and in statistics. Its main purpose is to reduce a correlated
multidimensional data set to an uncorrelated lower dimensional space
with maximum variance. PCA concepts can be a roadblock for
non-mathematical oriented students, since statistical definitions (i.e.,
variance-covariance, correlation) need to be connected to matrix
algebra (eigenvectors of a variance-covariance matrix) and to graphical
vector representation (including matrix rotation).
Effective teaching of PCA requires students to develop a “feeling”
for the intuitive sense of eigenvalues and eigenvectors and Minitab®
provides a flexible vector-based environment and tools which students
can use for meaningful learning of such concepts.
Though MINITAB by no means falls under the definition of a computer
algebra system (CAS), it is well suited –among other functions- to
manipulate, calculate, transform, and save data or matrices (transpose,
inverse, eigen-analysis and matrix arithmetic) and has been reported to
be successful teaching linear algebra (Greenwell, 1985). Following
Güyer’s general taxonomy of computer aided mathematics education
software (Güyer, 2008), Minitab® is here considered as a general purpose
software that can efficiently be used as a teaching tool and may
interact with other CAS. It is not as sophisticated as Maple® or
Mathematica®, but the purpose here is to illustrate how to teach PCA by
manipulating data sets and taking advantage of the software editing
facilities (Bassett, Brooks, & Morgan, 1995).
This presentation will also show how to use the graphical
capabilities of Minitab® and illustrate the bi-plot graphic display of
matrices with application to PCA (Gabriel, 1971). Bi-plots can be used
to support students’ conceptual and cognitive difficulties with the
geometrical interpretation of eigenvectors and eigenvalues. Earlier,
Gould (1967) suggested to consider a matrix of coordinates of points in
space and interpret the eigenvalues and associated functions as
geometric properties of the arrangements of these points.
Finally this paper wants to illustrate Minitab® as a learning tool to
support graduate students in the field of Education in their conceptual
understanding of reducing data dimensionality and other multivariate
methods. These students are commonly introduced to PCA as one of the
most commonly used exploratory data reduction procedure used in
educational research at the same time they experience the analysis and
computation of large number of variables. For such purpose ample
examples of PCA guides to educational research exist (ex., Osborne &
Costello, 2004; Pohlmann, 2004) and its application is evidenced by the
wide range of educational research studies as exemplified by numerous
perceived self-efficacy beliefs studies (e.g. Curts, Tanguma, &
Peña, 2008; Dellinger, Bobbett, Oliver, & Ellet, 2008). |