Bootstrap in CATPCA: announcement

[Linting, Marielle]

Dear listers,
Now available on DevCentral is a nonparametric bootstrap procedure for
the CATPCA program from the SPSS Categories module. This procedure is
used to perform a bootstrap study with 1000 bootstrap samples (by
default) on a data set specified by the user, on which a two-dimensional
PCA is performed. Although this procedure has been developed for
nonlinear PCA (CATPCA) which offers provisions for analyzing (possibly)
nonlinearly related variables of all types of measurement level, it can
be applied to data sets with only linearly related numeric variables as
well (by specifying a numeric analysis level for all variables in the
data set and using the discretization option "multiplying"). A user's
guide is provided along with the macro files to run the procedure.
Below, a somewhat more elaborate description of the general purpose of
the bootstrap in CATPCA is given.

In the daily research practice, data often do not meet the assumptions
made by the traditional analysis methods, such as (multivariate)
normality or numeric measurement levels. In such situations,
multivariate categorical analysis techniques might be more effective.
Nonlinear principal components analysis (nonlinear PCA), performed by
CATPCA in the Categories module, is such a technique that has been
developed as a nonlinear alternative to standard linear PCA, and can be
used to explore the correlational structure between different types of
variables (nominal, ordinal, and numeric) that may be nonlinearly
related to each other. As PCA does not make particular assumptions about
the distributions of the variables at hand, it does not seem
theoretically sensible to apply standard (asymptotic) formulas for
statistical inference. Therefore, in order to bring nonlinear PCA beyond
its exploratory reputation, there is a need for easily applicable
methods for performing inference on the elements of the nonlinear PCA
solution without making prior assumptions about the data. For this
purpose, the nonparametric bootstrap procedure mentioned above has been
developed that establishes the stability of the nonlinear PCA solution,
paying special attention to the graphical representation of the results.
Bootstrap results for the eigenvalues, component loadings, and component
scores are displayed by confidence ellipses, and results for the
quantified variables are displayed by confidence intervals.

More information on the CATPCA procedure, along with an illustrative
example, can be found in: Linting, M., Meulman, J. J., Groenen, P. J.
F., & Van der Kooij, A. J. (2007). Nonlinear Principal Components
Analysis: Introduction and application. Psychological Methods, 12,
336-358.  

The exact procedure used to obtain, represent, and interpret the
bootstrap results is described in an accompanying paper: Linting, M.,
Meulman, J. J., Groenen, P. J. F., & Van der Kooij, A. J. (2007).
Stability of Nonlinear Principal Components Analysis: An empirical study
using the balanced bootstrap. Psychological Methods, 12, 359-379.



***********************************************************
M. Linting
Data Theory Group
Faculty of Social and Behavioral Sciences
Leiden University
The Netherlands
email: [hidden email]
homepage: http://www.datatheory.nl/pages/linting.html
***********************************************************


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