Better biplots

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Abstract

The elements of a biplot are (i) a set of axes representing variables, usually concurrent at the centroid of (ii) a set of points representing samples or cases. The axes are (approximations to) conventional coordinate axes, and therefore may be labelled and calibrated. Especially when there are many points (perhaps several thousand) the whole effect can be very confusing but this may be mitigated by:

1. Giving a density representation of the points.

2. While respecting the calibrations, moving the axes to new positions more remote from the points, and possibly jointly rotating axes and points.

3. The use of colour — when permissible.

4. Choosing more than one centre of concurrency.

The principles are quite general but we illustrate them by examples of the Categorical Principal Component Analysis of the responses to questions concerning migration in Germany. This application introduces the additional interest of representing ordered categorical variables by irregularly calibrated axes.

Introduction

Biplots can be viewed as the multivariate analogue of scatterplots, where samples/cases are plotted as points relative to two variables. With biplots, the multivariate distribution of a set of variables can be approximated in a low dimensional space (usually two dimensions) giving a useful visualization of the structure of the samples relative to the variables. Biplots were introduced by Gabriel (1971) in the field of principal component analysis; the first textbook on general biplot methods is that of Gower and Hand (1996).

Figure 2.5 of Gower and Hand (1996) illustrates many of the things that can be wrong with a biplot. Surprisingly, many of the deficiencies described in their artificial example continue to occur (compare also Fig. 8):

(i) Inelegant, and often needless, horizontal and vertical scales.

(ii) Unequal scaling in the x and y directions.

(iii) Vector representation of variables.

(iv) The inconvenience of evaluating inner-products.

(v) Variables, whether in vector form or not, being inextricably intermingled with sample points, so interfering with clear labelling.

(vi) Not a deficiency, but nevertheless a problem, is how to display very large numbers of samples.

With a little care, (i) and (ii) are easily handled. Items (iii) and (iv) may be circumvented, especially by the proposal that vector representations be replaced by using calibrated axes, so bringing biplots into line with the familiar everyday use of coordinate axes in all fields of study concerned with the relationships between two or more variables. Calibrated biplot axes are used in the same way as conventional calibrated coordinate axes, and differ only in having as many axes as there are variables. However, because few dimensions are used for the representation, these axes must be non-orthogonal. Thus, a sample point is projected orthogonally onto the axis, and the nearest calibrated point is taken, if necessary using a little visual interpolation between adjacent calibrations. The use of projection is a primary reason why it is essential to scale equally, i.e. a circle must look like a circle. The same point applies to calculating inner products which depend on accurate representations of angles. Although Gower and Hand (1996) had emphasised this point, it was particularly galling when a reviewer took the authors to task for not taking into account the excellent work of Cleveland (1993) on aspect ratios. Valuable as is the aspect ratio concept when used with differently scaled variables, it will destroy the metrics underlying biplots and should never be used with biplots or, indeed, with any form of MDS. A thing to watch is that however hard one tries to get scaling right; computer software, and some publishers, concerned with printing aesthetics may readjust diagrams to fit the page; the aspect ratio will suffer similarly.

What was not covered by Gower and Hand (1996) was (v) the placement of axes; something that is readily handled with manually produced plots, and (vi) what to do about large numbers of sample points. In this paper we address both of these matters, largely in the context of linear biplots. However, the simple ideas presented easily generalise to other situations and here, we provide an example, still in the linear context, with variables measured on ordinal category, rather than numerical, scales.

Section snippets

Calibrated axes

We shall not dwell further on the reasons for favouring calibrated axes. They are not an alternative to using inner products but merely a simple way for their evaluation. The details for calculating the positions of calibration markers are straightforward (see Gower and Hand (1996), pp 38–40) and are discussed in the Appendix. When there are points of special interest, such as the length of one standard deviation, there is no reason why this information should not be shown on a calibrated axis,

Density plots

There are several situations where scatterplots are very popular, for example, in the French Analyse des Données (cf. Le Roux and Rouanet (2004)), in microarray data as well as in the whole area of data mining. In microarray studies, scatterplots can contain tens of thousands of dots leading to perception problems. In this section we present the background to displaying (very) many data points as a smooth density image. This issue is important when the number of scattered data points is so

An example with ordered categorical variables

The International Social Survey Program (ISSP) is a continuing annual program of cross-national collaboration covering topics important for social science research, being one of the most recognized and best documented data sets world-wide (Scheuch, 2000). A self-administered questionnaire contains questions relevant to all participating countries. Sample size varies but should exceed 1000 cases in each country. 34 countries participated in the 2003 ISSP when 11 questions referring to attitudes

Discussion

The aim of the paper was to show useful ways of displaying biplots and to develop a program that aids automatic drawing. Although the developed program allows for iterative changes of axes and labels, some final manual improvement is possible and desirable, for example, using the Adobe Illustrator software. Further, we introduced some new ways of presenting these data, some already mentioned by Gower and Hand (1996). As examples we used a small data set on 21 aircraft described by four

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