Boolean analysis
Boolean analysis was introduced by Flament (1976). The goal of a Boolean analysis is to detect deterministic dependencies between the items of a questionnaire in observed response patterns. These deterministic dependencies have the form of logical formulas connecting the items. Assume, for example, that a questionnaire contains items i, j, and k. Examples of such deterministic dependencies are then i → j, i ∧ j → k, and i ∨ j → k. Since the basic work of Flament (1976) a number of different methods for Boolean analysis have been developed. See, for example, Buggenhaut and Degreef (1987), Duquenne (1987), Leeuwe (1974), Schrepp (1999), or Theuns (1998). These methods share the goal to derive deterministic dependencies between the items of a questionnaire from data, but differ in the algorithms to reach this goal.
Application areas
The investigation of deterministic dependencies has some tradition in educational psychology. The items represent in this area usually skills or cognitive abilities of subjects. Bart and Airasian (1974) use Boolean analysis to establish logical implications on a set of Piagetian tasks. Other examples in this tradition are the learning hierarchies of Gagné (1968) or the theory of structural learning of Scandura (1971). Another example for the use of deterministic dependencies in psychology are approaches to formalize the diagnostic process of psychologists. The goal of this approach is to uncover the rules on which the decisions of diagnosticians are based. See Härtner, Mattes and Wottawa (1980) or Wottawa and Echterhoff (1982) for details. A recent application of Boolean analysis can be found in Held and Korossy (1998) who analyze implications on a set of algebra problems. In this paper item tree analysis (Leeuwe, 1974) is used to extract logical implications from observed response patterns. The extracted implications are then compared to implications obtained by querying an expert. Methods of Boolean analysis are used in a number of social science studies to get insight into the structure of dichotomous data. Bart and Krus (1973) use, for example, Boolean analysis to establish a hierarchical order on items that describe socially unaccepted behavior. Janssens (1999) used a method of Boolean analysis to investigate the integration process of minorities into the value system of the dominant culture.
Boolean analysis is an explorative method to detect deterministic dependencies between items. The detected dependencies must be confirmed in subsequent research. Methods to test such deterministic dependencies statistically are described, for example, in von Eye (1991).
Methods of Boolean analysis do not assume that the detected dependencies describe the data completely. There may be other probabilistic dependencies as well. Thus, a Boolean analysis tries to detect interesting deterministic structures in the data, but has not the goal to uncover all structural aspects in the data set. Therefore, it makes sense to use other methods, like for example latent class analysis, together with a Boolean analysis.
Relations to other areas
Boolean analysis has some relations to other research areas. There is a close connection between Boolean analysis and knowledge spaces. The theory of knowledge spaces provides a theoretical framework for the formal description of human knowledge. A knowledge domain is in this approach represented by a set Q of problems. The knowledge of a subject in the domain is then described by the subset of problems from Q he or she is able to solve.
This set is called the knowledge state of the subject. Because of dependencies between the items (for example, if solving item j implies solving item i) not all elements of the power set of Q will, in general, be possible knowledge states. The set of all possible knowledge states is called the knowledge structure.
Methods of Boolean analysis can be used to construct a knowledge structure from
data (see Theuns, 1998 or Schrepp, 1999). The main difference between both research areas is that Boolean analysis concentrates on the extraction of structures from data while knowledge space theory focus on the structural properties of the relation between a knowledge structure and the logical formulas which describe it.
Closely related to knowledge space theory is formal concept analysis. See Ganter and Wille (1996) for an overview of this research area. Similar to knowledge space theory this approach concentrates on the formal description and visualization of existing dependencies. In contrast Boolean analysis offers a way to construct such dependencies from data. Another related field is data mining (for an overview see Nakhaeizadeh, 1998). Data mining deals with the extraction of knowledge from large databases. Several algorithms (see for example Klementinnen, Mannila, Ronkainen, Toivonen & Verkamo, 1994 or Toivonen, 1996) are developed in this area which extract dependencies of the form j → i (called association rules) from the database. The main difference between Boolean analysis and the extraction of association rules in data mining is the interpretation of the extracted implications. The goal of a Boolean analysis is to extract implications from the data which are (with the exception of random errors in the response behavior) true for all rows in the data set. For data mining applications it is sufficient to detect implications which fulfill a predefined level of accuracy. It is, for example in a marketing scenario, of interest to find implications which are true for more than x% of the rows in the data set. An online bookshop may be interested, for example, to search for implications of the form If a customer orders book A he also orders book B if they are fulfilled by more than 10% of the available customer data.