Introduction to Fuzzy-Set Analysis
Donal Crilly
London Business School
Presentation for QCA PDW, Orlando, 2013
Rapid increase in number of articles using
fuzzy set analysis (fsQCA)
Source: Marx et al. (2012)
Not all Democracies1 are Equal.
1 Or most other things social scientists care about
Social phenomena differ in kind and degree Difference in kind: democracy versus
authoritarian regime Difference in degree: Norway vs. Italy (equally
democratic?)
fsQCA combines set-theoretic analysis with gradations in set membership
Crisp sets (0 or 1): differences in kind Fuzzy sets (between 0 and 1): differences in kind and
degree
Why Fuzzy Sets (fsQCA)?
Calibration
Membership has to be “purposefully calibrated” (Ragin, 2008: 30)
Calibration =/= measurement
Source: Economist Intelligence Unit Democracy Index, 2010
Norway outscores its neighbors on most dimensions, but these indicators don’t tell us whether Norway is democratic.
Is the UK (score 8.16) a democracy or a dictatorship?
Cannot consider a country a democracy simply because its score is above the sample mean
Ultimately, must depend on some qualitative assessment of what warrants being considered a democracy
Measurement vs. Calibration
Measurement Calibration
Aims for fine gradations between cases
Shows relative positions of cases All variation treated as important
Aims to make position of a case interpretable
Considers how well cases meet requirements for inclusion in a category
Not all variation treated as important
Calibration Approaches
Crisp set Three-value fuzzy set
Four-value fuzzy set
Six-value fuzzy set
“Continuous” fuzzy set
1 = fully in 0 = fully out
1 = fully in 0.5 = neither fully in nor fully out 0 = fully out
1 = fully in 0.67 = more in than out 0.33 = more in than out 0 = fully out
1 = fully out 0.8 = mostly (not fully) in 0.6 = more or less in 0.4 = more or less out 0.2 = mostly (not fully) out 0 = fully out
1 = fully in More in than out 0.5 = cross over: neither in nor out More out than in 0 = fully out
Based on Ragin (2008)
Calibration: Examples Identify distinct qualitative groupings based on
substantive knowledge Not simply ordinal scales!
Use external standards wherever possible For example, democracy classification (EIU)
Full democracies (1), Flawed democracies (0.66), Hybrid regimes (0.33), and Authoritarian regimes (0)
Country development (based on UNDP HDI cf. Crilly, 2011) Very high (1), high( (0.66), Medium (0.33), and Low (0)
Democracy Index
Continuous Fuzzy Set: Direct Calibration Method
Used to transform interval-scale variables into membership scores between 0 and 1
Three ‘qualitative’ anchors 1. Threshold for full membership 2. Threshold for full non-membership 3. Cross-over point (maximum ambiguity)
E.g. Firm size based on European Union enterprise size classes (Fiss, 2011)
1. Full membership: 250 + employees 2. Full non-membership: < 10 employees 3. Cross-over point: 50 employees
This calibration can be performed using fsQCA software
EMPIRICAL EXAMPLE
Aim: To understand why some firms implement CSR policy consistently whereas others decouple (adopt without implementing)
Data: 359 interviews across 17 firms and their stakeholders, social performance data
Why QCA? Useful for identifying how characteristics of firms and their environments combine to shape firms’ responses
Why fsQCA? Distinction between implementation/decoupling is not binary
Faking it or Muddling Through? Decoupling in
Response to Stakeholder Pressures (Crilly, Zollo & Hansen)
Steps
Calibrating set membership
Constructing truth table
Reducing number of truth table rows based on Minimum acceptable solution frequency
Minimum acceptable consistency
Generating simplified combinations from the truth table rows
(Optional: Identifying cases that are members in each configuration)
Condition Condition Calibration
Outcome Implementation 0, 0.5, 1
Causal conditions
Information asymmetry
0, 0.33, 0.66, 1
Stakeholder consensus
0, 0.5, 1
Organizational interest
0, 0.33, 0,66, 1
Managerial consensus
0, 0.5, 1
Conditions
Example: Calibrating Managerial Consensus (based on measure of variance
of responses)
Membership Threshold Evidence from firm at threshold
1 Variance below 0.30 “Agreeing what’s important can’t be decentralized. You have to do it centrally and then roll it out.”
0.5 Variance 0.30 – 0.60 Consensus/dissension not a main theme
0 Variance above 0.60 “All our units are very decentralized. We realize we have to be in greater harmony because the world doesn’t view us as these separate functions.”
Calibration Table
Note: Cases with values of 0.5 are dropped from the fuzzy-set analysis in the
fsQCA software program. Transform them by subtracting 0.001 (or adding 0.001).
Sample Truth Table
Configurations Associated with Implementation and Decoupling
Identifying Case Membership in Configurations
Assign cases to configurations on the basis of their membership of
at least 0.5 in the configuration
In Closing…
fsQCA enables you to capture differences in kind and degree in the phenomena you study
Middle way between qualitative and quantitative measurement
Calibration: simultaneously quantitative and qualitative
Fuzzy sets: advantages over conventional variables
And potential applications beyond social sciences (e.g. athlete selection
for triathlon… but wait until Rio 2016)