Scaling

General Objectives

Scaling, in contrast, is used to quantify phenomena that cannot be counted directly. For example, scaling techniques have been employed to measure the attitudes of survey respondents, the severity of crimes, the prestige of occupations, and the ideology of legislators. In each of these cases, we believe that the phenomenon in question (i.e., attitudes, severity, prestige, ideology) actually exists. But none of these things can be “counted” in a manner similar to the way one would assess the calibrations on a ruler or a scale. It is for such phenomena that scaling methods exist.

There is no single technique called “scaling.” Instead, there is a very wide variety of scaling models. The common characteristic is that they all seek to recover systematic structure in real-world observations by representing elements of the input data matrix (i.e., observations, stimuli, variables, etc.) as objects (usually, points or vectors) within a geometric space. Thus, a scale is an abstract model of data that are, themselves, culled from empirical observations. The way that one proceeds from observations, to data, to the geometric model depends upon the specific scaling strategy.

In any case, performing a scaling analysis is similar to putting together a jigsaw puzzle. Like the pieces of a puzzle, individual data points provide only fragmentary, limited information. When puzzle pieces are put together, the result is a picture that an observer can comprehend and understand. Similarly, when observations are combined into some geometric structure, the resultant “picture” shows the relative positions of the objects in the data matrix and thereby helps the analyst understand the contents and implications of the data themselves.

Data and Scaling Methods

How does one choose a specific scaling model for a particular data analysis situation? The answer depends upon the researcher's interpretation of information contained within the input data matrix and the type of model that he or she wants to construct to represent that information. The two most common types of data matrices are multivariate and similarities. Multivariate data refer to a matrix in which the rows and columns represent different objects (usually, observations and variables), and the cell entries provide scores for one set of objects with respect to the other (observations' variable values). For example, survey respondents' ratings of the incumbent president on each of 10 characteristics would be an example of multivariate data.

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