Don’t conflate fuzzy set membership with cases in QCA

In 2000, Ragin added the idea of fuzzy sets to Qualitative Comparative Analysis (QCA) and they have been widely used since then. However, one sometimes still finds misunderstandings about what fuzzy sets are, in particular when it comes to the interpretation of consistency and coverage scores.

In a criticism of QCA by Stockemer (which is criticized for a variety of reasons and to which Stockemer again replied), he runs a fuzzy-set analysis. The outcome is ‘high levels of representation of women in a parliament’ which is not important here. The solution coverage is 0.23, which leads Stockemer to conclude that “the combined solution only covers 23 per cent of all cases” (p. 94).

This statement is not true for fuzzy-set QCA (and would neither be true for crisp-set QCA). For fuzzy sets, solution coverage measures the percentage of the sum of the outcome membership values that are covered by the consistent cases’ membership in the solution. As most cases have a membership of less than 1 in the outcome and the solution, this usually is unrelated to any case-related percentages. If we look at the data, we see that the analysis includes 54 cases in total. Seven of these 54 cases are members of the solution. Five cases are typical, one is a deviant case consistency in degree and one is deviant in kind. If these cases are meant by “combined solution only covers 23 per of all cases”, the highest percentage we get is 13 (7/54). If we limit the perspective to the members of the outcome, that is, we speak of coverage for crisp-set QCA now, we have a coverage value of 0.3 (6/20). (I leave aside here that one should not simply switch to a crisp-set perspective based on fuzzy-set results.) If we additionally limit it to the typical cases, we have a coverage value of 0.26 (5/19, because one member of the solution is not a typical case). Regardless of how coverage is calculated if not in the proper way, we do not get a coverage value of 0.23.

It is relatively easy to be on the safe side when interpreting consistency and coverage scores. Just interpret them without any metric, for example, by saying “Solution consistency is 0.34” or “Raw coverage of the first term in the solution is 0.23”. This might be less intuitive than speaking of percentages of cases or percentages more generally, but it is the proper reading of the scores and avoids misunderstandings.


About ingorohlfing

I am Professor for Methods of Comparative Political Research at the Cologne Center for Comparative Politics at the University of Cologne ( My research interests are social science methods with an emphasis on case studies, multi-method research, and philosophy of science concerned with causation and causal inference. Substantively, I am working on party competition and parties as organizations.
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