Guttman scale

In the analysis of multivariate observations designed to assess subjects with respect to an attribute, a Guttman scale (named after Louis Guttman) is a single (unidimensional) ordinal scale for the assessment of the attribute, from which the original observations may be reproduced. The discovery of a Guttman scale in data depends on their multivariate distribution's conforming to a particular structure (see below). Hence, a Guttman scale is a hypothesis about the structure of the data, formulated with respect to a specified attribute and a specified population and cannot be constructed for any given set of observations. Contrary to a widespread belief, a Guttman scale is not limited to dichotomous variables and does not necessarily determine an order among the variables. But if variables are all dichotomous, the variables are indeed ordered by their sensitivity in recording the assessed attribute, as illustrated by Example 1. Example 1: Dichotomous variables A Guttman scale may be hypothesized for the following five questions that concern the attribute "acceptance of social contact with immigrants" (based on the Bogardus social distance scale), presented to a suitable population: Would you accept immigrants as residents in your country? (No=0; Yes=1) Would you accept immigrants as residents in your town? (No=0; Yes=1) Would you accept immigrants as residents in your neighborhood? (No=0; Yes=1) Would you accept immigrants as next-door neighbors? (No=0; Yes=1) Would you accept an immigrant as your child's spouse? (No=0; Yes=1) A positive response by a particular respondent to any question in this list, suggests positive responses by that respondent to all preceding questions in this list. Hence one could expect to obtain only the responses listed in the shaded part (columns 1–5) of Table 1. Table 1. Hypothesized responses to the five social distance variables form a Guttman scale (a cumulative scale) Every row in the shaded part of Table 1 (columns 1–5) is the response profile of any number (≥ 0) of respondents.