scholarly journals Using Eight-Variable Karnaugh Maps to Unravel Hidden Technicalities in Qualitative Comparative Analysis

2021 ◽  
pp. 26-42
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Comparative Analysis ◽  
Qualitative Comparative Analysis ◽  
Digital Design ◽  
Boolean Formula ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Boolean Minimization ◽  
Minimization Technique ◽  
Context Specific ◽  
Karnaugh Map

We use a regular and modular eight-variable Karnaugh map to reveal some technical details of Boolean minimization usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, that looks better than the published solution in two aspects, namely: (a) it is based on a smaller minimal set of supporting variables, and (b) it provides a more compact Boolean formula. However, we refrain from labelling our solution as a better one, but instead we stress that it is simply un-comparable with the published solution. The comparison between any two rival solutions should be context-specific and not tool-specific. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as is in QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. Moreover, each of them is unate as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.

scholarly journals Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

2021 ◽  
pp. 117-137
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Systematic Investigation ◽  
Exact Matching ◽  
Minimal Set ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Switching Functions ◽  
Programming Techniques ◽  
Karnaugh Map

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

scholarly journals Utilization of Eight-Variable Karnaugh Maps in the Exploration of Problems of Qualitative Comparative Analysis

2021 ◽  
pp. 57-84
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Comparative Analysis ◽  
Boolean Function ◽  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Programming Approach ◽  
Minimal Set ◽  
First Case ◽  
Desirable Feature ◽  
Karnaugh Map ◽  
Variable Problem

Qualitative Comparative Analysis (QCA) is an emergent methodology of diverse applications in many disciplines. However, its premises and techniques are continuously subject to discussion, debate, and (even) dispute. We use a regular and modular Karnaugh map to explore a prominent recently-posed eight-variable QCA problem. This problem involves a partially-defined Boolean function (PDBF), that is dominantly unspecified. Without using the algorithmic integer-programming approach, we devise a simple heuristic map procedure to discover minimal sets of supporting variables. The eight-variable problem studied herein is shown to have at least two distinct such sets, with cardinalities of 4 and 3, respectively. For these two sets, the pertinent function is still a partially-defined Boolean function (PDBF), equivalent to 210 = 1024 completely-specified Boolean functions (CSBFs) in the first case, and to four CSBFs only in the second case. We obtained formulas for the four functions of the second case, and a formula for a sample fifth function in the first case. Although only this fifth function is unate, each of the five functions studied does not have any non-essential prime implicant, and hence each of them enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. According  to our scheme of first identifying a minimal set of supporting variables, we avoided the task of drawing prime-implicant loops on the initial eight-variable map, and  postponed this task till the map became dramatically reduced in size. Our map techniques and results are hopefully of significant utility in future QCA applications.

Aggregation Bias and Ambivalent Cases: A New Parameter of Consistency to Understand the Significance of Set-theoretic Sufficiency in fsQCA

2019 ◽  
Vol 18 (2) ◽  
pp. 229-255
Author(s):  
Francesco Veri
Keyword(s):  
Comparative Analysis ◽  
Fuzzy Set ◽  
Negative Impact ◽  
Qualitative Comparative Analysis ◽  
Aggregation Bias ◽  
Random Data ◽  
Additional Tool ◽  
Boolean Minimization ◽  
Per Se ◽  
The Difference

Abstract The Boolean minimization, used in fuzzy-set qualitative comparative analysis (fsQCA) to establish sufficient relationships between conditions and outcome, automatically produces false positive subset relationships in the presence of random data. However, because this type of aggregation bias mainly produces ambivalent subset relationships between the condition(s) and the outcome, such false positives do not pose a problem for the fsQCA results per se. The aggregation bias has a negative impact on fsQCA analysis only because the consistency score is not able to detect set-theoretic subset relationships. Indeed, the existent parameter of consistency does not distinguish whether the subset relationship between conditions and outcome is the result of the mere Boolean minimization or whether it has set-theoretic significance. This article proposes a new consistency formula that provides information about subset relationships between conditions and outcome and detects the difference between randomly-generated subsets and meaningful subset relationships. The new parameter of consistency proposed here can be considered as an additional tool to test the significance of a meaningful sufficient relationship without being subject to the aggregation bias.

scholarly journals Cultural and Religious Diversity in Hospice and Palliative Care: A Qualitative Cross-Country Comparative Analysis of the Challenges of Health-Care Professionals

2018 ◽  
Vol 81 (4) ◽  
pp. 648-669 ◽  
Author(s):  
Panagiotis Pentaris ◽  
Louise L. Thomsen
Keyword(s):  
Palliative Care ◽  
Comparative Analysis ◽  
Health Care Professionals ◽  
Hospice Care ◽  
Current Knowledge ◽  
Qualitative Comparative Analysis ◽  
Strong Relationship ◽  
Important Change ◽  
Context Specific ◽  
Hospice And Palliative Care

Research has abundantly demonstrated a strong relationship between culture, religion, and the experiences of death, dying, and bereavement. Consequently, cultural competence and a religiously sensitive practice have become highly relevant to social policies and professional practice. However, our current knowledge of culturally competent and religiously sensitive end-of-life care is primarily context specific, with little space for generalizability. This article reports on findings from a qualitative comparative analysis of two nation-specific studies that examined religious literacy and cultural competency, respectively, among palliative-care professionals, drawing on similarities and attempting to identify further applicability of nation-centered knowledge. The study recognized six practice-based approaches in palliative and hospice care, when responding to cultural and religious or nonreligious identities of services users. These approaches intersect with each other via three organizational layers identified in the study: foundations, culture, and professionals. Each identified practice-based approach seems to be incomplete when working with individuals for whom religion, belief, and cultural identities are important. Change in practice is possible if all three organizational layers are considered simultaneously, while further research will shed more light about the benefits and challenges of each approach.

scholarly journals Qualitative Comparative Analysis as a Tool for Concept Clarification, Typology Building, and Contextualized Comparisons in Gender and Feminist Research

2016 ◽  
Vol 12 (03) ◽  
Author(s):  
Rossella Ciccia
Keyword(s):  
Comparative Analysis ◽  
Political Representation ◽  
Qualitative Comparative Analysis ◽  
Feminist Research ◽  
Feminist Studies ◽  
Reproductive Politics ◽  
Systematic Analysis ◽  
Holistic View ◽  
Representation Of Women ◽  
Context Specific

Qualitative Comparative Analysis (QCA) is a method for the systematic analysis of cases. A holistic view of cases and an approach to causality emphasizing complexity are some of its core features. Over the last decades, QCA has found application in many fields of the social sciences. In spite of this, its use in feminist research has been slower, and only recently QCA has been applied to topics related to social care, the political representation of women, and reproductive politics. In spite of the comparative turn in feminist studies, researchers still privilege qualitative methods, in particular case studies, and are often skeptical of quantitative techniques (Spierings 2012). These studies show that the meaning and measurement of many gender concepts differ across countries and that the factors leading to feminist success and failure are context specific. However, case study analyses struggle to systematically account for the ways in which these forces operate in different locations.

Boolean Algebra

2021 ◽  
pp. 30-32
Author(s):  
Jasmin Hasić
Keyword(s):  
Comparative Analysis ◽  
Boolean Algebra ◽  
Binary Data ◽  
Qualitative Comparative Analysis ◽  
Boolean Logic ◽  
Boolean Minimization ◽  
The Social ◽  
Boolean Equation ◽  
Basic Features ◽  
Causal Complexity

This chapter addresses Boolean algebra, which is based on Boolean logic. In the social sciences, Boolean algebra comes under different labels. It is often used in set-theoretic and qualitative comparative analysis to assess complex causation that leads to particular outcomes involving different combinations of conditions. The basic features of Boolean algebra are the use of binary data, combinatorial logic, and Boolean minimization to reduce the expressions of causal complexity. By calculating the intersection between the final Boolean equation and the hypotheses formulated in Boolean terms, three subsets of causal combinations emerge: hypothesized and empirically confirmed; hypothesized, but not detected within the empirical evidence; and causal configurations found empirically, but not hypothesized. This approach is both holistic and analytic because it examines cases as a whole and in parts.

scholarly journals Qualitative comparative analysis in educational policy research: Procedures, processes, and possibilities

2019 ◽  
Vol 12 (2) ◽  
pp. 205979911984098 ◽  
Author(s):  
Andrea J Bingham ◽  
Sean Dean ◽  
Jessica Castillo
Keyword(s):  
Comparative Analysis ◽  
Educational Policy ◽  
Research Method ◽  
Quantitative Methods ◽  
Qualitative Data ◽  
Qualitative Comparative Analysis ◽  
Policy Research ◽  
Specific Nature ◽  
Context Specific ◽  
Oriented Research

In educational policy research, linking specific practices to specific outcomes is an important (though not the only) goal, which can bias researchers (and funders) toward employing purely quantitative methods. Given the context-specific nature of policy implementation in education, however, we argue that understanding how specific practices lead to specific outcomes in specific conditions or contexts is critical to improving education. Qualitative comparative analysis is a method of qualitative research that we argue can help to answer these kinds of questions in studies of educational policies and reforms. Qualitative comparative analysis is a case-oriented research method designed to identify causal relationships between variables and a particular outcome. Distinct from quantitative causal methods, qualitative comparative analysis requires qualitative data to identify conditions (and combinations of conditions) that lead to a particular result; it is context driven, just as many educational reforms must necessarily be. We contend that qualitative comparative analysis has the potential to be of use to educational researchers in investigating complex problems of cause and effect using qualitative data. As such, our aim here is to provide a general overview of the characteristics, processes, and outcomes of qualitative comparative analysis. In so doing, we hope to offer guidance to educational researchers around how and when to use qualitative comparative analysis, as well as recommendations for current educational issues that could be investigated with qualitative comparative analysis.

scholarly journals Utilization of Karnaugh Maps in Multi-Value Qualitative Comparative Analysis

2018 ◽  
Vol 3 (1) ◽  
pp. 28-46 ◽  
Author(s):  
Ali Muhammad Ali Rushdi
Keyword(s):  
Comparative Analysis ◽  
Wide Spectrum ◽  
Qualitative Comparative Analysis ◽  
Software Tools ◽  
Sub Saharan Africa ◽  
Common Belief ◽  
Recent Debate ◽  
Input Domain ◽  
Sub Saharan ◽  
Karnaugh Map

A recent debate in the literature of Qualitative Comparative Analysis (QCA) concerns the potentials and pitfalls of the multi-value variant (mvQCA) in comparison with the more established crisp-set QCA (csQCA) and fuzzy-set QCA (fsQCA) variants. So far, the mvQCA methodology has been implemented either algebraically or via specific software tools such as TOSMANA. The main goal of this paper is to enhance the mvQCA methodology through the utilization of several varieties of Karnaugh maps including (a) the Conventional Karnaugh Map (CKM), (b) the Multi-Valued Karnaugh Map (MVKM), and (c) the Variable-Entered Karnaugh Map (VEKM). The paper offers a tutorial exposition of each of these maps in terms of two recently-published problems concerning the legal provision (introduction) and implementation of party bans in sub-Saharan Africa. Results obtained via various map techniques agree exactly among themselves, and are generally more compact than those obtained earlier via elementary algebraic manipulations, or even via software tools. We show, by way of example, that coding multi-valued variables by binary ones has a harmful primary effect of increasing the input domain. This effect is partially counterbalanced by a (contrarily to common belief) beneficial secondary effect of introducing genuine don’t-care configurations. We also address the issue of unresolved contradictory configurations, and propose two strategies to cope with them. The maps used tackle seven binary variables (or their equivalent), a number beyond the typical map limit of six variables. They are used to produce not only the minimal sum of a Boolean function but the complete sum as well. Though this paper is basically intended as a contribution to mvQCA methodology, it is also of significant utility in any field that demands the use of the Karnaugh map. It serves as a unification/exposition of three fundamental variants of the map, and has a definite pedagogical advantage for the wide spectrum of map users.

Standards of Good Practice and the Methodology of Necessary Conditions in Qualitative Comparative Analysis

2016 ◽  
Vol 24 (4) ◽  
pp. 478-484 ◽  
Author(s):  
Alrik Thiem
Keyword(s):  
Social Sciences ◽  
Comparative Analysis ◽  
Necessary Conditions ◽  
Meta Analysis ◽  
Good Practice ◽  
Qualitative Comparative Analysis ◽  
Boolean Minimization ◽  
The Social ◽  
Truth Tables

The analysis of necessary conditions for some outcome of interest has long been one of the main preoccupations of scholars in all disciplines of the social sciences. In this connection, the introduction of Qualitative Comparative Analysis (QCA) in the late 1980s has revolutionized the way research on necessary conditions has been carried out. Standards of good practice for QCA have long demanded that the results of preceding tests for necessity constrain QCA's core process of Boolean minimization so as to enhance the quality of parsimonious and intermediate solutions. Schneider and Wagemann's Theory-Guided/Enhanced Standard Analysis (T/ESA) is currently being adopted by applied researchers as the new state-of-the-art procedure in this respect. In drawing on Schneider and Wagemann's own illustrative data example and a meta-analysis of thirty-six truth tables across twenty-one published studies that have adhered to current standards of good practice in QCA, I demonstrate that, once bias against compound conditions in necessity tests is accounted for, T/ESA will produce conservative solutions, and not enhanced parsimonious or intermediate ones.

scholarly journals Computer Engineers Look at Qualitative Comparative Analysis

2019 ◽  
Vol 4 (4) ◽  
pp. 851-860
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Mohammad Salih Badawi
Keyword(s):  
Comparative Analysis ◽  
Statistical Methods ◽  
Qualitative Comparative Analysis ◽  
Digital Design ◽  
Computer Engineering ◽  
Scientific Disciplines ◽  
The One

Qualitative Comparative Analysis is a variant of Boolean Analysis that complements quantitative or statistical methods in many scientific disciplines. Therefore, its technicalities resemble those of other variants of Boolean Analysis, such as the one employed by computer engineers for digital design. This paper offers a brief look at Qualitative Comparative Analysis from a Computer-Engineering perspective. Critical observations on some technicalities of Qualitative Comparative Analysis are presented, with an aim to initiate constructive and fruitful intellectual debate that might subsequently lead to desirable enhancements and improvements.

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