2. 1. Qualitative comparative analysis QCA
Purpose of QCA:
Comparison of binary conditions X1, X2, ...
with regard to a binary outcome Y
Tab. 1: An exemplary dataset:
Deference to persons, by gender and status:
________________________________
X1 X2 Y
Person Woman Chief Deference
________________________________
1 1 1 1
2 0 1 1
3 1 0 0
4 0 0 0
________________________________
Legend: Woman: 1 = yes, 0 = no. Chief: 1 = yes,
0 = no. Deference: 1 = deference to person, 0 =
deference to others. Assumption: Y = X2.
The three steps of „classical“ QCA:
1) Translation of data with Y=1 into a Boolean
expression in disjunctive normal form:
1 case = 1 set of conjunctions
E.g., from tab. 1 follows formula
Y = (X1 AND X2) OR (NOT X1 AND X2)
2) Simplification of this Boolean expression
with the Quine-McCluskey algorithm.1)
E.g., from above follows: Y <==> X2
3) Exploration of the simplified Boolean formula
E.g., from simplified formula follows:
Gender X1 has no influence on deference Y.
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3. Problems of „classical“ QCA:
1) Missing instantiations (cases) for certain
sets of preconditions:
E.g. case 1 in tab. 2.
2) Contradictory outcomes for certain sets
of preconditions:
E.g. variable Y of cases 3a,b,c in tab. 2
Tab. 2: A modified exemplary dataset, based on tab. 1:
________________________________________
X1 X2 Y Y*=
Case Woman Chief Deference Rec. Y
________________________________________
1 1 1 ? --
2 0 1 1 1
3a 1 0 0
3b 1 0 0
3c 1 0 1 0
4 0 0 0 0
________________________________________
Legend: Y*: Recoding of Y by „classical“ QCA-methodology.
Other definitions: see tab. 1.
„Classical“ solutions to the problems of QCA:
1) Elimination of inconsistent cases
2) Missing values for missing cases
3) Quantification by fuzzy-set QCA
General critique of the „classical“ solutions:
a) Unwarranted simplifications/omissions of data
b) Quantitative answers in qualitative research.
Alternative solutions to the problems of QCA:
Three-valued modal logic.
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4. 2. An overview of three-valued modal logic
Basic feature 1:
Third truth-value
i = indeterminate whether true or false.
Examples with truth-value i:
Propositions about events in the future.
Propositions with missing instantiations.
Boolean operators:
Extension from 2 to 3 truth-values possible
but not needed for this article:
See Lukasiewicz (1970) and others.2)
Basic feature 2:
Two modal operators:
POS = Possibility of a proposition
NEC = Necessity of a proposition
Tab. 3: The definition of the modal operators:
_________________________________________________________________
Y NOT Y POS Y NEC Y POS NOT Y NEC NOT Y
_________________________________________________________________
0 1 0 0 1 1
i i 1 0 1 0
1 0 1 1 0 0
_________________________________________________________________
Legend: NOT: Negation; POS: Possibility; NEC: Necessity; 0 = false; 1 = true; i = indeterminate.
Interpretation of modal operators:
X => NEC Y: „X is a strict trigger of Y“
X => POS Y: „X is a possible trigger of Y“
X => NEC NOT Y: „X is a strict inhibitor of Y“
X => POS NOT Y: „X is a possible inhibitor of Y“
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5. 3. QCA with three-valued modal logic
Step 1:
Make missing knowledge more visible:
a) Replace missing instantiations of Y by i
b) Replace contradictory outcomes of Y by i
Results of step 1:
New three-valued dependent variable Y‘,
difficult to treat with conventional QCA-software.
Step 2:
Creation of four new variables derived from Y‘:
NEC Y‘, NEC NOT Y‘, POS Y‘, POS NOT Y‘ (see tab. 3)
Tab. 4: Results of the application of steps 1 and 2 to tab 2:
_______________________________________________________________________
X1 X2 Y Y‘ = NEC NEC POS POS
Case Woman Chief Deference Rec. Y Y‘ NOT Y‘ Y‘ NOT Y‘
_______________________________________________________________________
1 1 1 -- i 0 0 1 1
2 0 1 1 1 1 0 1 0
3a 1 0 0
3b 1 0 0
3c 1 0 1 i 0 0 1 1
4 0 0 0 0 0 1 0 1
______________________________________________________________________
Legend: Y ‘: Recoding of Y by methodology described in step 1. Other cols.: see previous tables.
Step 3:
Application of standard QCA to each of the variables
NEC Y‘, NEC NOT Y‘, POS Y‘, POS NOT Y‘,
which are all binary Boolean =>
Possibility of using fs/QCA-software.3)
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6. Results of step 3:
Four simplified Boolean expressions, which explain
NEC Y‘, NEC NOT Y‘, POS Y‘, POS NOT Y‘
Example:
X1 OR X2 => POS Y‘
Step 4:
Unification of the results of step 3
by the use of four new Boolean operators:
(1) Strict implication X ––> Y‘
means X => NEC Y‘
(2) Strict inhibition X –//–> Y‘
means X => NEC NOT Y‘
(3) Possible implication X ----> Y‘
means X => POS Y‘
(4) Possible inhibition X --//--> Y‘
means X => POS NOT Y‘
Illustrative example of step 4:
X1 OR X2 => POS Y‘
is replaced by
X1 OR X2 ----> Y‘
Step 5:
Exploration of the results of step 4 by
drawing logical inferences.
Example of step 5:
X1 OR X2 ----> Y‘
implies
X1 ----> Y‘ and X2 ----> Y‘
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7. 4. On the use of fs/QCA software
Purpose of fs/QCA software:
Qualitative comparative analyses:
a) Fuzzy set method of Ch. Ragin
b) Crisp set method of Ch. Ragin
Source a free software copy:
http://www.u.arizona.edu/~cragin/fsQCA/software.shtml
Software runs under Windows XP
Exemplary data:
See tab. 4: Deference to persons,
by gender and status.
Fig. 1a: Definition of data and variables as step 1 in
the use of fs/QCA:
Legend: For definitions of the variables Women, Chief, and POS_Def
see tab. 4, cols. Women, Chief, and POS Y‘.
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8. Fig. 1b: Data gathering as step 2 in the use of fs/QCA:
Legend: Data from tab. 4, cols. Women, Chief, and POS Y‘.
Step 3 in the use of fs/QCA:
Choice of method of analysis from software menu:
Crisp Sets with Truth Table Algorithm.
Fig. 1c: Model specification as step 4 in the use of fs/QCA:
Legend: Outcome and Causal Conditions from the data-pool Variables.
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9. Fig. 1d: Data cleaning of outcome variable as step 5 in the use of fs/QCA:
Legend: Dependent variable POS_Def has same value as before, because of
100% consistency consist of the original data-table.
Fig. 1e: Specification of meaning of values as
step 6 in the use of fs/QCA:
Legend: No Don‘t Care Cases, no Contradictions,
no Remainders due to contradiction-free, complete data.
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10. Fig. 1f: Extraction of Boolean expression as
step 7 in the use of fs/QCA:
Legend: Fourth line from below: (OR) added by the author.
Interpretation of figure 1f:
Woman OR Chief =>
POS_Def = Possibility of Deference
In other words:
Woman OR Chief ----> Deference
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11. 5. An exemplary application to ethno-political conflict
Research question:
What determines the ethno-political mobilization E
of a region in a situation of wealth:
Size S, Linguistic ability L, Economic growth G?
Reference study:
„Classical“ QCA by Ch. Ragin (1989), pp. 133-149,4)
based on data about 36 ethnic regions in Europe.
Tab. 5: Ethno-political conflict in wealthy regions:
_________________________________________________________________
Config. Size Ling. Abil. Growth Ethn. Mob. Ethn. Mob.
Nr. S L G E E*
_________________________________________________________________
1 0 0 0 0 0
2 0 0 1 i -
3 0 1 0 1 1
4 0 1 1 i 1
5 1 0 0 i -
6 1 0 1 i 1
7 1 1 0 1 1
8 1 1 1 i 1
_________________________________________________________________
Legend: S = Size. L = Linguistic ability. G = Economic growth. E* = Ethno-political mobilization,
coded by Ragin, 1989, Tab. 13. E = Ethno-political mobilization, coded by the author: E=1, if
Ragin, 1989, Tab. 12, reports for all cases conflict level 2; E=0, if Ragin, 1989, Tab. 12, reports for
all cases conflict level 0 or 1; E=i, for all other cases. Sample: Wealthy subnations with W=1 (see
Ragin, 1989, Tab. 13). Source: Ch. Ragin. 1989. The Comparative Method. Berkeley: University
of California Press.
Four Boolean expressions representing
the empirical results of QCA with 3-valued logic:
NOT G AND L ––> E
S OR G OR L ---> E
G OR NOT L --//--> E
NOT S AND NOT G AND NOT L –//–> E
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12. Fig. 2: The effects of different Boolean terms on the conflict E:
NOT G AND L –––––> E
L
S --------> E
G
E <----//----
NOT L
E <––//–– NOT L AND
NOT G AND NOT S
Legend: E = Ethno-political mobilization; G = Growth; L = Linguistic
ability; S = Size.
Interpretation of fig. 2:
The presence of G or S or L
may trigger a conflict E.
L must trigger a conflict E if in
addition there is no growth G.
Linguistic ability L as a prerequisite
of ethnic identity.
The presence of G or the absence of L
may inhibit a conflict E.
The absence of L must inhibit a conflict E
if in addition G and S are both absent.
The presence of G may
both trigger or inhibit a conflict E:
Growth G makes a region more important
but threatens its ethnic identity.
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13. 6. Three-valued QCA: What is different?
Fig. 3a,b: 2- versus 3-valued QCA of tab. 5:
Differences and similarities:
Ragin with 2-valued QCA
L AND G S AND G
––––> E
L AND NOT G
E <––––
Mueller with 3-valued QCA
Ragin with 2-valued QCA
NOT L AND NOT L AND
NOT S AND G NOT G
––//––> E
NOT L AND
NOT S AND NOT G
E <––//––
Mueller with 3-valued QCA
Comments on fig. 3a,b:
The strict triggers of E in 3-valued QCA are a
subset of the strict triggers in 2-valued QCA.
The strict inhibitors of E in 3-valued QCA are a
subset of the strict inhibitors in 2-valued QCA.
General summary:
3-valued QCA is a „prudent“ methodology:
It points to the limits of our theories,
which may be hidden by the use 2-valued QCA.
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14. Notes:
1: For Quine-McCluskey algorithm see: Mendelson, Elliot (1970): Boolean Algebra and
Switching Circuits: chap. 4. New York: McGraw-Hill.
2: Lukasiewicz, Jan (1970 [1920]): Selected Works. Ed. by L. Borkowski. Amsterdam:
North-Holland.
3: For fs/QCA-software see: http://www.u.arizona.edu/~cragin/fsQCA/
4: Ragin, Charles (1989): The Comparative Method. Berkeley: University of California
Press.
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