Boolean analysis

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Boolean analysis was introduced by Flament (1976). [1] The goal of a Boolean analysis is to detect deterministic dependencies between the items of a questionnaire or similar data-structures 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.

Determinism is the philosophical idea that all events, including moral choices, are determined completely by previously existing causes. Determinism is at times understood to preclude free will because it entails that humans cannot act otherwise than they do. It can also be called hard determinism from this point of view. Hard determinism is a position on the relationship of determinism to free will. The theory holds that the universe is utterly rational because complete knowledge of any given situation assures that unerring knowledge of its future is also possible. Some philosophers suggest variants around this basic definition. Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and considerations. The opposite of determinism is some kind of indeterminism. Determinism is often contrasted with free will.

Questionnaire research instrument consisting of a series of questions and other prompts for the purpose of gathering information from respondents

A questionnaire is a research instrument consisting of a series of questions for the purpose of gathering information from respondents. The questionnaire was invented by the Statistical Society of London in 1838.

Contents

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), item tree analysis 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.

Item tree analysis

Item tree analysis (ITA) is a data analytical method which allows constructing a hierarchical structure on the items of a questionnaire or test from observed response patterns.
Assume that we have a questionnaire with m items and that subjects can answer positive (1) or negative (0) to each of these items, i.e. the items are dichotomous. If n subjects answer the items this results in a binary data matrix D with m columns and n rows. Typical examples of this data format are test items which can be solved (1) or failed (0) by subjects. Other typical examples are questionnaires where the items are statements to which subjects can agree (1) or disagree (0).
Depending on the content of the items it is possible that the response of a subject to an item j determines her or his responses to other items. It is, for example, possible that each subject who agrees to item j will also agree to item i. In this case we say that item j implies item i. The goal of an ITA is to uncover such deterministic implications from the data set D.

Boolean analysis is an explorative method to detect deterministic dependencies between items. The detected dependencies must be confirmed in subsequent research. 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.

In statistics, exploratory data analysis (EDA) is an approach to analyzing data sets to summarize their main characteristics, often with visual methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task. Exploratory data analysis was promoted by John Tukey to encourage statisticians to explore the data, and possibly formulate hypotheses that could lead to new data collection and experiments. EDA is different from initial data analysis (IDA), which focuses more narrowly on checking assumptions required for model fitting and hypothesis testing, and handling missing values and making transformations of variables as needed. EDA encompasses IDA.

In statistics, a latent class model (LCM) relates a set of observed multivariate variables to a set of latent variables. It is a type of latent variable model. It is called a latent class model because the latent variable is discrete. A class is characterized by a pattern of conditional probabilities that indicate the chance that variables take on certain values.

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).

Educational psychology is the branch of psychology concerned with the scientific study of human learning. The study of learning processes, from both cognitive and behavioral perspectives, allows researchers to understand individual differences in intelligence, cognitive development, affect, motivation, self-regulation, and self-concept, as well as their role in learning. The field of educational psychology relies heavily on quantitative methods, including testing and measurement, to enhance educational activities related to instructional design, classroom management, and assessment, which serve to facilitate learning processes in various educational settings across the lifespan.

There are several attempts to use boolean analysis, especially item tree analysis to construct knowledge spaces from data. Examples can be found in Held and Korossy (1998), or Schrepp (2002).

In mathematical psychology, a knowledge space is a combinatorial structure describing the possible states of knowledge of a human learner. To form a knowledge space, one models a domain of knowledge as a set of concepts, and a feasible state of knowledge as a subset of that set containing the concepts known or knowable by some individual. Typically, not all subsets are feasible, due to prerequisite relations among the concepts. The knowledge space is the family of all the feasible subsets.

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. Romme (1995a) introduced Boolean comparative analysis to the management sciences, and applied it in a study of self-organizing processes in management teams (Romme 1995b).

Social science is a category of academic disciplines, concerned with society and the relationships among individuals within a society. Social science as a whole has many branches. These social sciences include, but are not limited to: anthropology, archaeology, communication studies, economics, history, human geography, jurisprudence, linguistics, political science, psychology, public health, and sociology. The term is also sometimes used to refer specifically to the field of sociology, the original "science of society", established in the 19th century. For a more detailed list of sub-disciplines within the social sciences see: Outline of social science.

A. Georges L. (Sjoerd) Romme is a Dutch organizational theorist and professor of entrepreneurship and innovation at the Eindhoven University of Technology.

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 (for example, 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 (Ganter and Wille, 1996). Similar to knowledge space theory this approach concentrates on the formal description and visualization of existing dependencies. Formal concept analysis analysis offers very effective ways to construct such dependencies from data, with a focus on if-then expressions ("implications"). There is even a method, called attribute exploration [2] , for extracting all implications from hard-to-access data.

Another related field is data mining. Data mining deals with the extraction of knowledge from large databases. Several data mining algorithms 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.

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In formal concept analysis (FCA) implications relate sets of properties. An implication A→B holds in a given domain when every object having all attributes in A also has all attributes in B. Such implications characterize the concept hierarchy in an intuitive manner. Moreover, they are "well-behaved" with respect to algorithms. The knowledge acquisition method called attribute exploration uses implications.

References

  1. ↑ Flament, C. (1976). "L'analyse booleenne de questionnaire", Paris: Mouton.
  2. ↑ Ganter, Bernhard and Obiedkov, Sergei (2016) Conceptual Exploration. Springer, ISBN   978-3-662-49290-1
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