Coverage in Fuzzy Subset Relations

crossover anchor

τ

c

2.19979

polynomial

μ

quad

2

root

μ

root

0.5

seed

Show

,Plot[0.5,{x,0,Max[x]},PlotStyle{Gray,Dashed}],ImageSize{225,225}

μ	ω	ζ
lin	0.5181	1.0000
quad	0.5010	1.0000
root	0.5397	1.0000
log	0.5129	1.0000

Fuzzy-set qualitative comparative analysis (fsQCA) is a relatively new method used in the social sciences to analyze whether a set of causal conditions (

C

) is necessary (

C⊃O

), sufficient (

C⊂O

), or both necessary and sufficient (

C=O

) for an outcome (

O

) to occur. These subset-theoretic relations are assessed on the basis of consistency (

ζ

) and coverage (

ω

). The left graphic shows how four popular membership functions assign fuzzy condition set membership scores to 30 cases from a normally distributed base variable (

X

). This assignment is based on the location of the crossover anchor (

τ

c

), which defines the point of maximum set membership ambiguity at 0.5. Functions include the linear function

μ

lin

, the quadratic function

μ

quad

, the root function

μ

root

, and the logistic function

μ

log

. The right graphic in the middle visualizes the resulting subset-theoretic relation for the case of sufficiency. The table on the right shows how

ω

and

ζ

change as a result of altering

τ

c

,

μ

quad

, or

μ

root

. Different outcome set scores can be generated by changing the seed.