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Importing set-theoretic tools into quantitative research: the case of necessary and sufficient conditions

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Abstract

In this article, we import set-theoretic methods from the qualitative research tradition into the quantitative research tradition. We focus specifically on set-theoretic methods designed to analyze the extent to which a condition is necessary and is sufficient for an outcome of interest. We use these methods to reanalyze four major studies from the quantitative tradition. We find that set-theoretic methods call attention to asymmetrical patterns in the data that otherwise go unnoticed and unanalyzed. We develop a general set-theoretic framework for the study of necessity and sufficiency. We conclude that the use of this framework can enrich existing and future quantitative research in the social sciences.

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Notes

  1. For basic statistics on smoking and lung cancer, see Godtfredsen et al. (2005) and Islami et al. (2015). For a short summary of the multimethod research program that established smoking to be a cause of lung cancer, see Freedman (1999).

  2. See Rubinson (2019) for discussion of different types of diagrams that can be used with QCA.

  3. Non-QCA approaches for analyzing necessary conditions and sufficient conditions do exist (e.g., Clark, Gilligan, and Golder 2006; Eliason and Stryker 2009; Goertz 2006; Seawright 2015), but they lack important aspects of the QCA apparatus, including its set-theoretic foundations.

  4. In the QCA literature, this consideration is addressed via the study of “unique coverage” (Ragin 2008).

  5. More generally, the negation of a condition is not equivalent to the opposite of a condition. For example, consider the condition of development and its opposite, underdevelopment. The condition of not-development is not equivalent to the condition of underdevelopment. Not all cases of not-development qualify as cases of underdevelopment.

  6. However, more than half of the subjects were willing to administer at least a “slight shock” on the basis of written instructions and in the absence of an authority figure (Milgram 1973: Table 2 on pp. 60–61).

  7. Conformity theory proposes that individuals in a group relinquish decision-making authority to experts, especially in crises. Agentic state theory proposes that subjects view themselves as agents carrying out the wishes of a legitimate authority figure and thus believe that they are not accountable for their actions.

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Acknowledgements

Earlier versions of the this paper were presented at the 2020 Annual Meetings of the American Political Science Association and the 2020 Annual Meetings of the American Sociological Association. For helpful comments, we thank Gary Goertz, Ingo Rohlfing, and the anonymous reviewers at Quality and Quantity.

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  1. Department of Sociology, Northwestern University, 1810 Chicago Ave., Evanston, IL, 60208, USA

    James Mahoney & Andrew Owen

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Appendix

Appendix

1.1 Calculating necessity and sufficiency with graded conditions

Procedures for assessing degrees of necessity and degrees of sufficiency with continuously coded graded conditions have been around since Zadeh (1965). Recent formulations for set-theoretic analysis include Ragin (2008: chap. 3) and Schneider and Wagemann (2012: chap. 3). With graded conditions, some cases have partial degrees of membership. For example, if the condition is developed country, some countries are only partly in the condition. With continuous-set measurement, analysts code cases continuously from 1.0 to 0.0 based on their degree of membership in the condition. Cases with a 1.0 membership value are full members; cases with a 0.0 membership value are full non-members. Cases with 0.5 membership are true crossover cases, half in and half out of the condition. Cases with membership values > 0.5 < 1.0 are more in a condition than out; cases with membership values > 0.0 < 0.5 are more out of a condition than in.

When cases are coded continuously in this way, the formulas for calculating the degree of necessity and degree of sufficiency are simply:

$$\% {\text{ }}\;{\text{Necessary}}\;{\text{ }}\left( {X \to Y} \right){\text{ }} = \;\Sigma \left[ {{\text{min}}X_{i} ,Y_{i} } \right]/{\text{ }}\Sigma \left( {X_{i} } \right);{\text{ and}}$$
$$\% {\text{ }}\;{\text{Sufficient }}\;\left( {X \to Y} \right){\text{ }} = \;\Sigma \left[ {{\text{min}}X_{i} ,Y_{i} } \right]/{\text{ }}\Sigma \left( {Y_{i} } \right).$$

It also possible to assess degrees of necessity and sufficiency with interval variables. The relevant techniques are developed in Goertz (2006) and Eliason and Stryker (2009).

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Mahoney, J., Owen, A. Importing set-theoretic tools into quantitative research: the case of necessary and sufficient conditions. Qual Quant 56, 2001–2022 (2022). https://doi.org/10.1007/s11135-021-01188-6

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