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)