On the fractal dimension of a social measurement II

Many scales in psychology, education and social measurement in general, which are constructed to measure a single variable, are nevertheless composed of subscales which measure different aspects of the variable. Although the presence of subscales captures better the complexity of a variable and increases its validity, it compromises its unidimensionality. This paper reconciles the measurement produced by a scale composed of subscales by resolving the measurement into a main variable common among all subscales and a set of mutually orthogonal variables unique to each subscale and orthogonal to the main variable. Then using the formula for Cronbach’s α calculation of the reliability of traditional test theory, it derives a formula which estimates the summary value characterizing the main variable relative to the mutually orthogonal variables. It also derives formulae elaborating the interpretation of α calculated at different levels of scale in which account is taken of the multidimensionality produced by the subscales. A set of simulation studies, generated according to the Rasch model, illustrates the effectiveness in recovering the summary value of the mutually orthogonal variables, using both the raw scores and the Rasch model estimates of the persons. The concept of the roughness, adapted from fractal geometry and introduced in a companion paper (Andrich, 2006), is used as a metaphor for the impact of the mutually orthogonal variables on the main variable and a possible motif to represent this roughness is suggested. The advantage of such an approach to imperfect unidimensionality, inherent in the design structure of an instrument, is that the focus remains on the main variable to be measured. Data from an Australian Scholastic Aptitude Test (ASAT), which are analyzed illustratively in the companion paper, are reanalyzed to illustrate the interpretation of the formulae that are derived in the paper.