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Parameters of fit and intermediate solutions in multi-value Qualitative Comparative Analysis

Quality & Quantity volume 49, pages 657–674 (2015)Cite this article

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. Note that the recent exchange between Denk (2010) and Rohlfing (2012) on “multi-level” QCA is unrelated to multi-value QCA.

  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. Some authors have proposed alternative minimization procedures that do not explicitly use this core mechanism (e.g. Baumgartner 2009; Eliason and Stryker 2009; Thiem and Duşa forthcoming). I do not discuss them here.

  6. The QCA package to be introduced later in Sect. 5.1 is the only software so far than can process outcome factors with multiple levels directly. In Tosmana, such factors have to be dichotomized before the analysis (Cronqvist and Berg-Schlosser 2009, p. 84).

  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. Consistency has been traditionally referred to as inclusion in the literature (Smithson 2005; Smithson and Verkuilen 2006).

  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. Raw coverage scores correspond to the coverage statistic introduced in Sect. 3. Unique coverage scores take into account the overlap in coverage between different prime implicants. See Ragin (2006) for more details.

  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 \(\mathbf {H}^{\{1\}}\mathbf {G}^{\{1\}}\mathbf {F}^{\{1\}}\).

References

  • Ackrén, M., Olausson, P.M.: Condition(s) for island autonomy. Int. J. Minor. Gr. Rights 15(2–3), 227–258 (2008)

    Article  Google Scholar 

  • Balthasar, A.: The effects of institutional design on the utilization of evaluation. Evaluation 12(3), 353–371 (2006)

    Article  Google Scholar 

  • Baumgartner, M.: Inferring causal complexity. Sociol. Methods Res. 38(1), 71–101 (2009)

    Article  Google Scholar 

  • Berg-Schlosser, D.: Determinants of democratic successes and failures in Africa. Eur. J. Polit. Res. 47(3), 269–306 (2008)

    Article  Google Scholar 

  • Cronqvist, L.: Tosmana: Tool for small-n analysis, version 1.3.2.0 [computer programme]. University of Trier, Trier, (2011)

  • Cronqvist, L., Berg-Schlosser, D.: Multi-value QCA (mvQCA). In: Rihoux, B., Ragin, C.C. (eds.) Configurational Comparative Methods: Qualitative Comparative Analysis (QCA) and Related Techniques, pp. 69–86. Sage, London (2009)

    Chapter  Google Scholar 

  • Delhi, V.S.K., Mahalingam, A., Palukuri, S.: Governance issues in BOT based PPP infrastructure projects in India. Built Environ. Proj. Asset. Manag. 2(2), 234–249 (2012)

    Article  Google Scholar 

  • Denk, T.: Comparative multilevel analysis: proposal for a methodology. Int. J. Soc. Res. Method. 13(1), 29–39 (2010)

    Article  Google Scholar 

  • Dubrova, E.: Multiple-valued logic synthesis and optimization. In: Hassoun, S., Sasao, T. (eds.) Logic Synthesis and Verification, pp. 89–114. Springer Science+Business Media, New York (2002)

    Chapter  Google Scholar 

  • Eliason, S.R., Stryker, R.: Goodness-of-fit tests and descriptive measures in fuzzy-set analysis. Sociol. Methods Res. 38(1), 102–146 (2009)

    Article  Google Scholar 

  • Gross, M., Garvin, M.: Structuring PPP toll-road contracts to achieve public pricing objectives. Eng. Proj. Organ. J. 1(2), 143–156 (2011)

    Article  Google Scholar 

  • Hartmann, C., Kemmerzell, J.: Understanding variations in party bans in Africa. Democratization 17(4), 642–665 (2010)

    Article  Google Scholar 

  • Herrmann, A.M., Cronqvist, L.: When dichotomisation becomes a problem for the analysis of middle-sized datasets. Int. J. Soc. Res. Method. 12(1), 33–50 (2009)

    Article  Google Scholar 

  • Hohn, F.E.: Applied Boolean Algebra: An Elementary Introduction, 2nd edn. Macmillan, New York (1966)

    Google Scholar 

  • Huntjens, P., Pahl-Wostl, C., Rihoux, B., Schlüter, M., Flachner, Z., Neto, S., Koskova, R., Dickens, C.: Adaptive water management and policy learning in a changing climate: a formal comparative analysis of eight water management regimes in Europe, Africa and Asia. Environ. Policy Gov. 21(3), 145–163 (2011)

    Article  Google Scholar 

  • Jordan, E., Gross, M.E., Javernick-Will, A.N., Garvin, M.J.: Use and misuse of Qualitative Comparative Analysis. Constr. Manag. Econ. 29(11), 1159–1173 (2011)

    Article  Google Scholar 

  • Klüver, H.: Europeanization of lobbying activities: when national interest groups spill over to the European level. J. Eur. Integr. 32(2), 175–191 (2010)

    Google Scholar 

  • Mannewitz, T.: Alte Bekannte im neuen System: Gründe für die Aufstiege und Niedergänge der mittelosteuropäischen Postkommunisten seit 1990. Zeitschrift für Vergleichende Politikwissenschaft 4(2), 261–293 (2010)

  • McCluskey, E.J.: Introduction to the Theory of Switching Circuits. Princeton University Press, Princeton (1965)

    Google Scholar 

  • R Development Core Team.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2013)

  • Ragin, C.C.: The Comparative Method: Moving Beyond Qualitative and Quantitative Strategies. University of California Press, Berkeley (1987)

    Google Scholar 

  • Ragin, C.C.: Fuzzy-Set Social Science. University of Chicago Press, Chicago (2000)

    Google Scholar 

  • Ragin, C.C.: Set relations in social research: evaluating their consistency and coverage. Polit. Anal. 14(3), 291–310 (2006)

    Article  Google Scholar 

  • Ragin, C.C.: Redesigning Social Inquiry: Fuzzy Sets and Beyond. University of Chicago Press, Chicago (2008)

    Book  Google Scholar 

  • Ragin, C.C., Davey, S.: fs/QCA: Fuzzy-set/Qualitative Comparative Analysis, version 2.5 [computer programme]. Department of Sociology, University of Arizona, Tucson, AZ (2009)

  • Ragin, C.C., Sonnett, J.: Between complexity and parsimony: limited diversity, counterfactual cases and comparative analysis. In: Kropp, S., Minkenberg, M. (eds.) Vergleichen in der Politikwissenschaft, pp. 180–197. VS Verlag für Sozialwissenschaften, Wiesbaden (2005)

    Chapter  Google Scholar 

  • Rihoux, B.: Qualitative Comparative Analysis (QCA) and related systematic comparative methods: recent advances and remaining challenges for social science research. Int. Sociol. 21(5), 679–706 (2006)

    Article  Google Scholar 

  • Rihoux, B., Álamos Concha, P.: From niche to mainstream method? A comprehensive mapping of QCA applications in journal articles from 1984 to 2011. Polit. Res. Quart. 66(1), 175–184 (2013)

    Article  Google Scholar 

  • Rohlfing, I.: Analyzing multilevel data with QCA: a straightforward procedure. Int. J. Soc. Res. Method. 15(6), 497–506 (2012)

    Article  Google Scholar 

  • Sager, F., Andereggen, C.: Dealing with complex causality in realist synthesis: the promise of Qualitative Comparative Analysis. Am. J. Eval. 33(1), 60–78 (2012)

    Article  Google Scholar 

  • Schneider, C.Q., Wagemann, C.: Set-theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis (QCA). Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  • Schneider, C.Q., Wagemann, C.: Doing justice to logical remainders in QCA: moving beyond the standard analysis. Polit. Res. Q. 66(1), 211–220 (2013)

    Google Scholar 

  • Smithson, M.: Fuzzy set inclusion: linking fuzzy set methods with mainstream techniques. Sociol. Methods Res. 33(4), 431–461 (2005)

    Article  Google Scholar 

  • Smithson, M., Verkuilen, J.: Fuzzy Set Theory: Applications in the Social Sciences. SAGE, London (2006)

    Google Scholar 

  • Thiem, A.: Clearly crisp, and not fuzzy: a reassessment of the (putative) pitfalls of multi-value QCA. Field Methods 25(2), 197–207 (2013a)

    Article  Google Scholar 

  • Thiem, A.: Unifying configurational comparative methods: Generalized-set Qualitative Comparative Analysis. Sociol. Methods Res. (2013b). doi:10.1177/0049124113500,481

  • Thiem, A., Duşa, A.: Boolean minimization in social science research: a review of current software for Qualitative Comparative Analysis (QCA). Soc. Sci. Comput. Rev. 31(4), 505–521 (2013a)

    Article  Google Scholar 

  • Thiem, A., Duşa, A.: QCA: a package for Qualitative Comparative Analysis. R J. 5(1), 87–97 (2013b)

    Google Scholar 

  • Thiem, A., Duşa, A.: Qualitative Comparative Analysis with R: A User’s Guide. Springer, New York (2013c)

    Book  Google Scholar 

  • Thiem A, Duşa A.: When more than time is of the essence: Enhancing the minimization of Boolean output functions with eQMC. J. Math. Sociol. (forthcoming)

  • Vink, M.P., van Vliet, O: Not quite crisp, not yet fuzzy? Assessing the potentials and pitfalls of multi-value QCA. Field Methods 21(3), 265–289 (2009)

    Google Scholar 

  • Vink, M.P., van Vliet, O: Potentials and pitfalls of multi-value QCA: response to Thiem. Field Methods 25(2), 208–213 (2013)

    Google Scholar 

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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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  1. Department of Philosophy, University of Geneva, Rue de Candolle 2 / Bât. Landolt, 1211 , Geneva, Switzerland

    Alrik Thiem

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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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