Wednesday, 7 September 2011

Revision: Factor Analysis!!

Factor Analysis - Geometric Model

An understanding of the patterns defined by factor analysis can be enhanced through a geometric interpretation. Each nation can be thought of as defining a coordinate axis of a geometric space. For example, the US, the UK, and the

USSR can define a three

-dimensional space as given in Figure 3. Imagine that the axis for the UK is projecting at right angles from the paper. Although pictorially constrained to three dimensions, the space can be analytically extended to fourteen dimensions at right angles to each other and thus represent the fourteen nations.

Now, in this space each characteristic can be con

sidered a point located according to its value for each nation. Such a plot is shown in Figure 3 for the GNP per capita and trade values of the US, UK, and USSR. To make the plot explicit, projections for each point are drawn as dotted lines to each axis.

If for each point in Figure 3 we draw a line from the origin to the point and top the line off with an arrowhead as shown in Figure 4, then we have a vector representation of the data. The characteristics of similarly plotted as vectors in an imaginary space of the fourteen nations (dimensions) would describe a vector space. In this space, consider two vectors representing any two of these characteristics for the fourteen nations.

The angle between these vectors measures the relationship between the two characteristics for the fourteen nations. The closer to 90o the angle is, the less the relationship is. If two vectors are at a right angle, the characteristics they represent are uncorrelated: they have no relationship to each other. In other words, some nations will be high on one characteristic, say GNP per capita, and low on the other, say trade; some nations will be low on GNP per capita and high on trade; some nations will be high on both, and some will be low on both. No regularity exists in their covariation.

The closer the angle between the vectors is to zero, the stronger the relationship between the characteristics. An angle of zero means that nations high or low on one characteristic are proportionately high or low on the other. Obtuse angles mean a negative relationship. At the extreme, an angle of 180o between two vectors means that the two characteristics are inversely related: a nation high on one characteristic is proportionately low on the other.

Let the characteristics be projected in the fourteen-dimensional space defined by the fourteen nations as suggested in Figure 5(a). The configuration of vectors will then reflect the data interrelationships. Characteristics that are highly interrelated will cluster together; characteristics that are unrelated will be at right angles to each other. By inspecting the configuration we can discern the distinct clusters of vectors (if such clusters exist), and these clusters index the patterns of relationship in the data: each cluster is a pattern.

Were we dealing with characteristics of two or three nations, patterns could be found by simply plotting the characteristics as vectors. What factor analysis does geometrically is this: it enables the clusters of vectors to be defined when the number of cases (dimensions) exceeds our graphical limit of three. Each factor delineated by factor analysis defines a distinct cluster of vectors.

Consider Figure 5(a) again. Factor analysis would mathematically lay out such a plot and then project an axis through each cluster as shown in Figure 5(b). This is analogous to giving each vector point in a cluster a mass of one and letting the factor axes fall through their center of gravity. The projection of each vector point on the factor axes defines the clusters. These projections are called loadings and the factor axes are often called factors or dimensions.

Figure 5(c) pictures the power and foreign conflict patterns. For simplicity, the configuration of points is shown, rather than vectors, and the two factor axes are indicated (as actually derived from a factor analysis). The loadings of each characteristic (i.e., each point in space) on each axis are also displayed. This figure may clarify how factor loadings as a set of numbers can define

  • a pattern of relationships and
  • the association of each characteristic with each pattern.

Author – Ankit Gupta

Marketing Group 1

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