On the fractal dimension of a social measurement: I

Many scales in psychology, education and social measurement in general, constructed to measure a single variable, are nevertheless designed to measure different aspects of the variable. The scales are often composed of subscales which measure these various aspects. These different aspects are conceptually related to the variable and are employed because they make the scale more valid than if only one of the aspects were measured. The aspects are expected to be reasonably highly correlated, but most unlikely to be perfectly correlated. To the degree that the aspects are not perfectly correlated, to that degree the scale could be considered not to measure a unidimensional variable. This paper suggests that the concepts of fractal geometry may be useful in characterizing this non-unidimensionality. Fractal geometry of physical entities is concerned with characterizing roughness, and the approach taken in this paper is that the presence of dimensions other than the dominant dimension adds roughness to the measurement. However, because of some special features of social measurement when compared to physical measurement, the term thickness of a measurement is also introduced to complement the concept of roughness in social measurement. The distinctive feature of the approach, compared with those that attempt to characterize the different dimensions explicitly, is that the focus remains on the main variable to be measured. The calculation and interpretation of the fractal dimension is made possible by the application of the unidimensional Rasch model. The paper illustrates the calculation of the fractal dimension of a social measurement by analyzing an Australian Scholastic Aptitude Test and considers some implications of such a characterization for social measurement.