There are already many explanatory books on QCA, including fsQCA (B. Rihoux, C.C. Ragin (2009), I.O Pappas and A.C Woodside (2021)), and explanatory books and translations written in Japanese have also been published (Ishida et al., 2016).

In QCA, data is viewed as conditions that lead to certain outcomes. This is a significant difference from quantitative analysis methods commonly used, such as regression analysis. In quantitative data analysis, the focus is on the correlation between explanatory variables and the dependent variable (outcome), and the relationship is explained by a single equation. If there is a positive correlation between the explanatory variable and the dependent variable, the high value of the dependent variable is due to the high value of the explanatory variable, and the low value of the dependent variable is due to the low value of the explanatory variable. If there is a negative correlation, the high value of the dependent variable is due to the low value of the explanatory variable, and the low value of the dependent variable is due to the high value of the explanatory variable. In any case, the occurrence and non-occurrence of an event are symmetrically explained by whether something is high or low.

In QCA, the issue is what combination of conditions leads to the outcome. Conversely, it is also possible to discuss what combination of conditions does not lead to the outcome. Sometimes, the conditions that lead to the outcome and the conditions that do not lead to the outcome are asymmetrical. Such phenomena often occur in the real world. For example, even if one boy’s good grades are due to studying hard, another boy’s poor grades may not be due to not studying but may be due to a learning disability, having to work at home, or not being able to buy textbooks. Trying to explain everything with a single equation often obscures the essence. Such analysis is important in social sciences. In such cases, instead of aggregating multidimensional phenomena into multiple regression analysis, analyzing what combinations of conditions lead to what outcomes may better reveal the essence.

In this sense, QCA, including fsQCA is expected to capture phenomena that cannot be captured by traditional quantitative analysis centered on multiple regression analysis. This is probably why there are many explanatory books. However, many of these explanatory books are extremely unreliable. They only continue with historical explanations and lack essential discussions and foundational mathematical explanations. As a result, outsiders have no idea what the analysis method means.

For example, some explanatory books mention that one of the advantages of QCA is that it can produce results even when the sample size is small relative to the data items. Indeed, when the sample size is small relative to the data items, multiple regression analysis cannot be performed. This is a mathematical issue. However, there is naturally a question of whether the results of QCA are reliable when the sample size is small. Even if a specific combination of conditions explains the outcome when the sample size is small, it may just include unrelated items, and even if there are many results corresponding to that combination in a small sample size, it is questionable whether it can be generalized. In other words, QCA does not have a general method for examining statistical significance, such as variance ratio tests.

QCA also considers statistical reliability and shows coverage as an analysis result. This is conscious of the reliability of the conclusion, but statistically speaking, it is common sense that the meaning of coverage of 1/1=1.00 and 10/10=1.00 is different. Looking at such discussions, traditional tutorials are extremely unreliable. However, fsQCA has the potential to bring new perspectives to our analysis. Rather, we need to recognize the limitations and effectiveness of fsQCA while mastering it, and combine it with traditional quantitative analysis methods (especially correlation analysis and principal component analysis) to complete fsQCA as a new analytical method.

fsQCA has the potential to bring perspectives to our analysis that cannot be captured by traditional quantitative analysis. However, there are several unresolved issues. In particular, the operation of converting numerical data into “membership scores” for conditions (Calibration) poses significant problems. It is essential to understand these issues and possibilities when learning the fsQCA method. This explanation will first discuss what kind of arguments can be developed with traditional quantitative analysis methods, including statistical evaluations. Then, it will explain the mathematical foundations of QCA methods (csQCA and fsQCA) and attempt practical applications of csQCA and fsQCA with specific examples. Finally, it will point out the issues that QCA needs to address in the future.

**Table of Contents**

- Introduction

I-1. Background and Content

I-2. Why fsQCA?

II. Attempts of Quantitative Analysis

II-1. Materials

II-2. Comprehensive Correlation Analysis

II-3. Distance Matrix and Spatial Relationships of Data (MDS)

II-4. Principal Component Analysis

II-5. Regression Analysis

II-6. Factor Analysis

II-7. Organization of Numerical Analysis Results

**III. Organization **of Binary Logic (Set Theory) and Trial of csQCA

III-1. Content of This Chapter

III-2. Set Theory, Boolean Operations, and Truth Tables

III-3. How to Use Truth Tables

III-4. Trial of csQCA Using Lipset’s Theory Verification as a Subjec

IV. Concepts and Methods of fsQCA

IV-1. Content of This Chapter

IV-2-1. Fuzzy Operations and Membership Functions

IV-2-2. Sufficient Conditions and Consistency in Fuzzy Sets

IV-3. Attempting og fsQCA

IV-3-1. Analysis of Interwar Europe

IV-3-2. Verification of Lipset’s Theory

IV-4. Summary and Additional Remarks on fsQCA

V. Considerations on Membership Functions

V-1. Remaining Issues

V-2-1. Membership Scores Using Symmetric Probability Distributions (Normal Distribution)

V-2-2. Summary of fsQCA

key word

Calibration, Consistency, Coverage, Minimization

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