Quality & Quantity 35: 91–105, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Recovering the Metric Structure in Ordinal Data:
Linear Versus Nonlinear Principal Components
MATH J. J. M. CANDEL
Department of Methodology and Statistics, Faculty of Health Sciences, Maastricht University, P.O.
Box 616, 6200 MD Maastricht, The Netherlands, Tel.: +31.43.3882273. Fax: +31.43.3618388.
Abstract. Two techniques for data reduction as part of the SPSS package are compared in a Monte
Carlo study: principal components analysis (PCA) and nonlinear principal components analysis
(NPCA). The relative performance of these techniques in recovering the component scores underly-
ing subjects’ scores on observed ordinal variables is studied for two-dimensional spaces. The relative
performance is examined as a function of (a) the sample size, (b) the number of categories in the
variables, (c) the amount of measurement error, (d) the type of nonlinearity in the data, and (e) the
degree of heterogeneity of the marginal distributions of the variables. As expected, when the sample
size increases the performance of NPCA improves when compared to PCA. For the range of values
considered, there is no effect of the number of categories on the relative performance of PCA and
NPCA. For the other factors the effects are more complicated: adding error does not affect PCA
as strongly as NPCA, as expected, but not for heterogeneously distributed variables for a particular
form of nonlinearity, in which case NPCA becomes more appropriate. PCA appears to outperform
NPCA for linear data, but also for a substantial number of nonlinear data sets.
Key words: ordinal data, (nonlinear) principal components analysis, questionnaire-based data,
Monte Carlo study
Various analysis techniques aim for reducing a set of variables to a smaller set.
The motivation for employing such a data reduction technique is threefold. First
of all, one may want to simplify the data by obtaining scores on a smaller number
of new variables. These new variables subsequently can be used as input for other
statistical analyses. Secondly, when one has no a priori or strongly held beliefs as
regards the dimensional structure underlying the data, one may want to uncover
the dimensions that may have given rise to the data. Uncovering this structure is
the aim of exploratory multivariate analysis. Thirdly, the newly created variables
resulting from data reduction, may be constrained to be orthogonal, which may
avoid problems relating to multicollinearity in subsequent statistical analyses (Hair
et al., 1998; Sharma, 1996).