Algebraic Representations of Beliefs and Attitudes Ii: Microbelief Models For Dichotomous Belief Data

It may often be the case that the beliefs about which survey researchers query respondents are composed of discrete components, such that holding all of the components isnecessary to give a "yes" response. Simple logical relations, which some researchers have proposed may structure belief data, may obtain between these components, and not between the beliefs that are actually measured. This paper demonstrates that an algebraic inversion of a data matrix, first used in test theory by Haertel and Wiley (1993), can be seen as a unique and interpretable decomposition that can recover information regarding the compositional formulas of the measured beliefs as well as the logical relations between the unobserved components. The inversion is illustrated with a set of data from the GSS. Finally, the conditions under which related techniques are then helpful or not helpful for analyzing survey data are discussed.