## Abstract

Multi-value Qualitative Comparative Analysis (mvQCA) is a variant of QCA that continues to exist under the shadow of crisp and fuzzy-set QCA. The lack of support for parameters of fit and intermediate solutions has contributed to this undeserved status. This article introduces two innovations that put mvQCA on a par with its two sister variants. First, consistency and coverage as the two most important parameters of fit are generalized. Second, the concepts of easy and difficult counterfactuals for deriving intermediate solutions are imported. I demonstrate how to leverage these features in the QCA software package for the R environment. For researchers who do not use QCA, I explain how to exploit Veitch–Karnaugh maps instead for solving set-theoretic minimization problems of low to moderate complexity.

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## Notes

- 1.
See http://www.compasss.org/bibdata.htm. Accessed 15 Feb 2014. The application share of fsQCA is about 36.9 %, that of csQCA about 60.2 %.

- 2.
- 3.
A sans-serif type face indicates a piece of software and a typewriter font face commands, functions and objects that are part of a software.

- 4.
Multivalent logic requires a significant generalization of Boolean algebra, particularly if both condition and outcome factors are allowed to assume multivalent structures. See Dubrova (2002) for a comprehensive but technical introduction.

- 5.
- 6.
- 7.
Note that not all minimization algorithms in QCA follow this pairwise elimination procedure since it is computationally highly demanding and thus not very efficient. For example, both the QCA package and Tosmana use different algorithms.

- 8.
As Sager and Andereggen (2012, p. 70) in fact present incorrect solutions in their original article, I will also correct these along the way.

- 9.
For reasons of space and layout, I use acronyms other than those in the original study.

- 10.
The terms

*bivalent*,*trivalent*, etc. specify the number of levels a factor is comprised of. - 11.
Although Ragin has introduced these statistics in relation to fsQCA, they apply equally to csQCA in their set membership score form, but not their level indicator form. However, since mvQCA generalizes csQCA, not fsQCA, I use the simple level indicator form here.

- 12.
So far, only simple outcomes have been analysed in QCA, but there exists no reason why consistency could not be extended to complex outcomes.

- 13.
Similarly, the number of configurations \(d\) that can be formed from \(k\) conditions factors with \(p_{j}\) levels is given by \(d = \prod _{j=1}^{k}{p_{j}}\), which reduces to \(d = 2^{k}\) for csQCA as all condition factors are bivalent.

- 14.
Configurations \(\mathcal {C}_{8}\) and \(\mathcal {C}_{11}\) could be coded as so-called

*contradictions*(not to be confused with*contradiction*in the logical sense of the word) since their cases display the outcome as often as its negation. The possibility for contradictions has been excluded for reasons of simplicity. The Tosmana software would have automatically coded configurations \(\mathcal {C}_{6}, \mathcal {C}_{8}\) and \(\mathcal {C}_{11}\) as contradictions. - 15.
I will use the term

*conservative*rather than*complex*because it is closer to the definition of this solution type. - 16.
In the Quine-McCluskey algorithm, for example, this procedure works in two steps. First, all logical remainder configurations are added to the output function. Second, after the prime implicants have been derived, these configurations are removed again from the prime implicant chart before the solution is finalized (Ragin 1987, p. 110).

- 17.
Schneider and Wagemann (2013) propose TESA—theory-enhanced standard analysis—which consists in breaking the algorithmic link between simplifying assumptions, easy and difficult counterfactuals in the derivation of parsimonious and intermediate solutions.

- 18.
From a formal point of view, only solutions with more than one prime implicant are causally interpretable (Baumgartner 2009). I disregard this requirement here for reasons of simplicity.

- 19.
Note that this solution corrects the one presented by (Sager and Andereggen 2012, 70).

- 20.
At the time of writing, 5065 extension packages were available.

- 21.
CSV files are very convenient, not least because they can also be read by the fs/QCA and Tosmana software.

- 22.
The object cond is a user-defined character vector containing the names of the conditions to be included. It is subset to the first five names since sa contains both proximate and remote condition factors.

- 23.
- 24.
The full truth table including logical remainders can be obtained by passing the optional argument complete = TRUE to the truthTable() function.

- 25.
The parsimonious solution will always be printed. However, as there is only one conservative solution in this example, it is not printed again.

- 26.
- 27.
Whether or not such counterfactuals should then be included as part of the solution is a question to be discussed in future research. The default behaviour of the QCA package is to omit and record them as non-simplifying counterfactuals.

- 28.
Unique coverage scores for a prime implicant can be calculated as the number of all covered positive cases minus the cardinality of the union of all other prime implicants divided by the total number of positive cases. For example, \(\mathbf {L}^{\{0\}}\mathbf {H}^{\{1\}}\mathbf {F}^{\{1\}}\) has a unique coverage of \((9 - 8)/12 \approx 0.083\) because two of its cases are also covered by \(\mathbf {H}^{\{1\}}\mathbf {G}^{\{1\}}\mathbf {F}^{\{1\}}\).

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## Acknowledgments

I am grateful to Michael Baumgartner, Tim Haesebrouck and the two anonymous reviewers for very helpful comments and suggestions.

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Thiem, A. Parameters of fit and intermediate solutions in multi-value Qualitative Comparative Analysis.
*Qual Quant* **49, **657–674 (2015). https://doi.org/10.1007/s11135-014-0015-x

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### Keywords

- Configurational comparative methods
- Consistency
- Coverage
- csQCA
- fsQCA
- mvQCA
- QCA
- Qualitative Comparative Analysis