An examination of regionality in a configuration of smallest space analysis using Loevinger’s homogeneity coefficient

Abstract

When applying smallest space analysis to behavioural science data, we may observe a radex structure in the two-dimensional output. This study proposes a method to support the investigation of the radex. A radex structure includes both polar and modular facets as defined in facet theory, where the modular facet of the concentric circles represents the frequency of variables. The polar facet of the pie shape, intersected by lines emanating from the centre, represents the semantic theme, grouped by variable clusters. When a dataset of dichotomous variables is analysed in this type of structure, variables with a higher proportion of positive answers often fall near the centre, whereas variables with a lower proportion of positive answers tend to fall near the outer area. When a few variable points fall along the line emanating from the centre to the outer rim, that series of variables may form a Guttman scale. The proposed method first determines the centre of the concentric circles, based on variable frequencies, using the quasi-Newton method of optimisation. Next, weighted Loevinger’s homogeneity coefficients ($H_{w}$) are used to measure the degree of Guttman scalability. The author examined possible borders among the regions of the polar facet by investigating the peaks and troughs of $H_{w}$ values, using different ways of weighting homogeneity coefficients. Applications of the method to artificial and survey data showed that the proposed method was adequately effective for detecting the centre of concentric circles and radial lines to suggest regionality for the polar facet in a radex.