Institut für Politikwissenschaft
Schwerpunkt qualitative empirische Sozialforschung
Goethe-Universität Frankfurt
Fachbereich 03
Prof. Dr. Claudius Wagemann
Qualitative Comparative Analysis (QCA)
and Fuzzy Sets
22 April 2013:
Sets, Set Membership Calibration, Set
Operations
Pros and Cons of Crisp-Sets
Cons
Many real-world phenomena are not simply
dichotomous
The need for dichotomization
Loss of empirical information
Reduced robustness, since analysis becomes more
sensitive
Where to put the threshold?
24.04.2013
Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
2
Pros and Cons of Crisp-Sets
Pros
Some phenomena are truly dichotomous
Also in statistics, dichotomies enjoy a recent revival
Dichotomies correspond better to our everyday
thinking
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
3
Fuzzy Sets
Differences in kind
Differences in degree
Fuzzy sets versus crisp sets
Fuzzy sets versus interval scales
Fuzzy sets versus probabilities
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Basics of Fuzzy Sets – Calibration
• Calibration
– “Application of known and explicit external standards’
(Ragin)
– Common in natural sciences
• Thermometer
– Uncommon in social sciences
• Exceptions exist, such as “poverty” ($1 per day)
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Basics of Fuzzy Sets –
How to Calibrate Fuzzy Sets?
• Theory knowledge
– Meaning of concept
• Case knowledge
– Multiple data sources
• Back and forth between ideas and evidence
– Adjustment of fuzzy set definitions and scores
• Strong link between theory and data (validity)
• Qualitative and quantitative information
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Basics of Fuzzy Sets – Lingusitic Qualifiers
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Direct Method of Calibration
• Fix the qualitative anchors at 0.95, 0.5 and 0.05
• Logit function
• Indirect method of calibration
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
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A*B
A+B
Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
~A
~B
9
Set Operations
A
B
A*B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
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A+B
Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
~A
~B
10
Set Operations
A
B
A*B
A+B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
~A
~B
11
Set Operations
A
B
A*B
A+B
~A
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
~B
12
Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
1
1
1
0
0
0
1
1
1
0
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
0.55???
1
1
1
0
0
0
1
1
1
0
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
0.2
1
1
1
0
0
0
1
1
1
0
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
0.2
1
1
1
0
0.9
0
0
1
1
1
0
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
0.2
1
1
1
0
0.9
0
0
1
1
0.1
1
0
0
1
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
0.2
1
1
1
0
0.9
0
0
1
1
0.1
1
0
0
1
0.8
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
A*B
A+B
~A
~B
1
1
0
0
0.9
0.3
0
0.4
0
1
1
0
0.2
0.4
0.7
0.4
0
1
0
0
0.2
0.3
0
0.4
1
1
1
0
0.9
0.4
0.7
0.4
0
0
1
1
0.1
0.7
1
0.6
1
0
0
1
0.8
0.6
0.3
0.6
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
1
0
1
0
0
0
1
1
0.7
0.2
0.9
0.4
0.1
0.2
0.8
0.7
24.04.2013
A + B*(~C + ~D)
Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
~AB + B~C
21
Set Operations
A
B
C
D
A + B*(~C + ~D)
1
0
1
0
1
0
0
1
1
0.7
0.2
0.9
0.4
0.1
0.2
0.8
0.7
24.04.2013
Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
~AB + B~C
22
Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0.7
0.2
0.9
0.4
0.1
0.2
0.8
0.7
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0
0.7
0.2
0.9
0.4
0.1
0.2
0.8
0.7
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0
0
0.7
0.2
0.9
0.4
0.1
0.2
0.8
0.7
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0
0
0.7
0.2
0.9
0.4
0.7
0.1
0.2
0.8
0.7
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0
0
0.7
0.2
0.9
0.4
0.7
0.2
0.1
0.2
0.8
0.7
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0
0
0.7
0.2
0.9
0.4
0.7
0.2
0.1
0.2
0.8
0.7
0.2
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Set Operations
A
B
C
D
A + B*(~C + ~D)
~AB + B~C
1
0
1
0
1
0
0
0
1
1
0
0
0.7
0.2
0.9
0.4
0.7
0.2
0.1
0.2
0.8
0.7
0.2
0.2
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Notational Systems
“AND”
Logic of propositions:
Conjunction 
Boolean algebra:
Multiplication *
Set theory:
Intersection 
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Notational Systems
“OR”
Logic of propositions:
Disjunction 
Boolean algebra:
Addition +
Set theory:
Union 
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Notational Systems
“NOT”
Logic of propositions:
Complement  
Boolean algebra:
Negation 1 - A
Set theory:
Negative set
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Notational Systems
Inclusion
Logic of propositions:
If-then relation  
Boolean algebra:
./.
Set theory:
Subset 
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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Negation of Expressions
~(A*B) = ~A + ~B
~(A + B) = ~A~B
~(AB + ~CD) =
(~A+~B) * (C + ~D)
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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More Rules
A*~A = 0
A + ~A = 1
A + AB = A
A*(B + C) = AB + AC
A*(AB + ~A~B + A~B) =
= AAB + A~A~B + A~B =
= AB + A~B =
= A (B + ~B) =
=A
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Qualitative Comparative Analysis (QCA) and
Fuzzy Sets
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