Fuzzy-set Case Studies

Abstract

Contemporary case studies rely on verbal arguments and set theory to build or evaluate theoretical claims. While existing procedures excel in the use of qualitative information (information about kind), they ignore quantitative information (information about degree) at central points of the analysis. Effectively, contemporary case studies rely on crisp sets. In this article, I make the case for fuzzy-set case studies. I argue that the mechanisms that are the focal points of contemporary case study methods can be modeled as set-theoretic causal structures. I show how case study claims translate into sufficiency statements. And I show how these statements can be evaluated using fuzzy-set tools. This procedure permits the use of both qualitative and quantitative information throughout a case study. As a consequence, the analysis can determine whether one or more cases are both qualitatively and quantitatively consistent with its claims. Or whether some or all cases are consistent by kind but not by degree.

Keywords

case studies causal mechanisms fuzzy sets set-theoretic consistency equifinality

Introduction

Case studies are all about qualitative distinctions. Surely, most case studies deal with large amounts of information backed by extensive substantive knowledge about the case in question. But at the heart of the analysis are verbal arguments. And verbal arguments can be expressed as logical connections between sets fundamentally based in qualitative distinctions between different types of cases.

In a string of important books, Ragin, Goertz, and Mahoney have alerted the qualitative literature to the set-theoretic nature of its claims (Goertz 2006b; Goertz and Mahoney 2012; Ragin 1987, 2000, 2008). By now, it is widely realized in the methodological literature that case studies, and the study of causal mechanisms in case studies using process tracing or congruence analysis, have evaluations of set membership and relations at their core (Beach and Pedersen 2013; Blattter and Haverland 2012; see also Rohlfing 2012). Research practise may not fully or explicitly align with the set-theoretic framework. But, as Elman (2013) shows, the prescriptive note in parts of the literature is that it ought to. I concur with this prescription. And I will show how it can become an information preserving device for case study research.

Case studies excel in the use of detailed and subtle information gained from the researchers’ knowledge of the case(s) they analyze. Detailed knowledge of the context helps turn observations into evidence for or against propositions. But, in the end, the analysis bottoms out in an evaluation of whether a cause, an outcome, or a mechanism element is present in the case or not (Beach and Pedersen 2013; see also, Mahoney 2010). In the language of set theory, the detailed knowledge of cases is used for purposes of calibration; for scoring cases in the relevant causes, mechanisms, and outcomes (see Ragin 2008). But it is not used at the key analytic moment where relations between the presence or absence of causes, outcomes, and mechanism element are evaluated. Here verbal arguments are evaluated. And without the right tools, verbal arguments do not appreciate all the available information.

Thus, contemporary case studies do not have the tools to utilize all the nuance and subtlety of evidence beyond calibration or scoring of cases. In the analysis, the sets of which mechanisms are comprised are essentially crisp. I will argue that we can—and should—go further.

The consequence of current practice is that we can evaluate only whether cases conform to, are consistent with, expectations in terms of kind. The analysis can only spot if a case is not of the type expected by theory. In the terms I will use, the analysis can use qualitative information—information about kind—to detect if a case is qualitatively inconsistent (Schneider and Rohlfing 2013 refer to this as deviance in kind). But the analysis does not use quantitative information—information about degree—and hence does not have the tools to evaluate the extent to which a case is quantitatively inconsistent (Schneider and Rohlfing 2013 refer to this as deviance in degree). Qualitative inconsistency arises in set relations when a case does not belong to the predicted type. Quantitative inconsistency, by contrast, occurs when the case is not a member of this type to the predicted degree.

A few examples will clarify my point. Consider first a well-known example, that is, Tannenwald’s (1999) study of the nonuse of nuclear weapons after World War II. She argues that, when the decision was made to bomb Hiroshima and Nagasaki, a “nuclear taboo” had not yet emerged and that, therefore, decision makers were disinclined to consider normative arguments against the use of nuclear weapons. But there were exceptions, “[S]ome scientists and even a few officials expressed concerns about the moral implications of using the bomb without warning” (Tannenwald 1999:443). Although these arguments “did not find a receptive audience” (Tannenwald 1999:443), it appears from this that the Hiroshima and Nagasaki case did not fully conform to expectations. The case might have been qualitatively consistent but quantitatively inconsistent. But Tannenwald’s approach forces her to disregard the quantitative information and to argue simply that the case did not represent an instance of the nuclear taboo in action.

Another example from a different field is Grzymała-Busse’s (2007) study of state exploitation in East Central Europe in her Rebuilding Leviathan. Her argument is that robust political competition led policy makers in East Central Europe to abandon state exploitation. A central mechanism is that robust competition forces policy makers to adopt formal institutions that constrain their capacity for state exploitation.

In the Polish case, competition was robust and the relevant institutions were in place relatively early. But the “appearance and disappearance of parties from the former anticommunist opposition took their toll on the opposition’s ability to serve as a critical and plausible alternative” (Grzymała-Busse 2007:68). And the implementation of the Polish institutions was deficient, stalled, distorted, and circumvented leading to qualifications, “That is not to say that the Polish state had the reputation for integrity that the Estonian, Hungarian or Slovenian states did” (Grzymała-Busse 2007:104). Additionally, Grzymała-Busse cites critical reports stating that as many as 200,000 state jobs could be filled by Polish political parties even after reforms of formal institutions (Grzymała-Busse 2007:68; see also Heywood and Meyer-Sahling 2013; O’Dwyer 2006).

Like Tannenwald, Grzymała-Busse’s approach forces her to disregard this information and conclude that Polish policy makers did face robust political competition and adopted formal institutions preventing themselves from state exploitation because of it.

Tannenwald’s and Grzymała-Busse’s studies are not necessarily in error. But their approach does push them to eventually disregard quantitative information that could potentially make one or several of their cases quantitatively inconsistent. If this is the case, I will argue, it may give reason to question, refine, or even reject their propositions. For studies of a single case, this point is even more pronounced as quantitative inconsistency in such studies will leave no case fully consistent. But so long as quantitative information is not leveraged analytically, we cannot know whether any cases are fully consistent. We do not know whether cases conform to our expectations beyond belonging to the right type. Providing a solution to this problem is the purpose of the present article.

I propose a straightforward procedure that will permit case studies to evaluate their cases’ consistency with expectations using both qualitative and quantitative information. I base my procedure on the use of fuzzy sets. Fuzzy sets simultaneously use qualitative information to establish differences in kind and quantitative information to establish differences in degree (Ragin 2000, 2008). Using this information, case studies can retain all details and all subtlety beyond calibration by modeling all parts of their claims—causes, outcomes, and mechanisms—as fuzzy sets. My procedure will show how this can be done.

Case studies will still be acutely concerned with qualitative distinctions as these take priority in fuzzy-set analyses in the tradition following Ragin. But using fuzzy sets, we can retain all information in all steps of the analysis. In essence, what I am proposing is fuzzy-set case studies. Briefly, the procedure involves analyzing claims about mechanisms graphically to translate them into sufficiency statements, translating these statements into consistency evaluations, compiling these evaluations, and inputting qualitative and quantitative information from the case or cases in question.

The procedure can be employed after fuzzy-set qualitative comparative analysis (QCA) and thus extend existing mixed-methods designs (Schneider and Rohlfing 2013, 2016) to the study of fuzzy-set mechanisms. But the use of QCA is not necessary. As in fuzzy-set QCA, sets will have to contain both qualitative and quantitative information. But the use of QCA procedures and algorithms is entirely optional. Given enough information, any set can be a fuzzy set. This does not hinge on employing QCA (see also Goertz 2006b; Goertz and Mahoney 2005; Ragin 2000, 2006).

The procedure is compatible with a range of existing and ongoing developments in case study methodology, including Bayesianism (Abell 2009; Beach and Pedersen 2013; Bennett 2008), process tracing tests (Collier 2011; Mahoney 2012; Rohlfing 2014), and congruence analysis (Blattter and Haverland 2012). But it extends them by permitting the use of quantitative information in the evaluation of causal claims about mechanisms. In particular, the Bayesian approach to process tracing focuses on epistemological uncertainty, informally assigning probabilities to mechanism elements belonging to the predicted type (Beach and Pedersen 2013). But it does not appreciate quantitative information ontologically. Fuzzy sets do, and hence my procedure extends the Bayesian perspective as well.

I write that the procedure permits the use of quantitative information in case studies. This does not mean that quantitative information is always available to case study researchers. When no quantitative information is available, my procedure cannot be of any assistance.

I will give two replies to this seeming limitation. First, unavailability of quantitative information is a limitation in data, not in my procedure. And second, that quantitative information often is available to case study researchers as evident, for instance, from my discussion of Tannenwald and Grzymała-Busse. When quantitative information is available, I propose my procedure as a viable way of levering it.

The article is structured as follows. In the second section, I discuss tempting or possible options for fuzzy-set case studies already existing in the literature. In the third section, I show how we can draw set-theoretic statements from graphical representations of causal structures (henceforth simply structures). By structures, I mean the overarching systems linking start-up conditions (or causes or independent variables), outcomes (dependent variables), and mechanism elements (the steps in a mechanism linking start-up conditions to the outcome). In gist, I lay out what a mechanism looks like from a set-theoretic point of view. In the fourth section, I detail the procedure for fuzzy-set case studies I suggest in its most basic form. In this section, I assume that one case is selected and that no elements in the structure are equifinal. In fifth and sixth sections, I loosen these assumptions and extend my procedure into situations where structures are equifinal, where multiple cases are selected, and where both complications coincide. At this point, software might become necessary. When it is, available packages like “fs/QCA” (Ragin and Davey 2014) or the “QCA” package for R (Duşa and Thiem 2014; Thiem and Duşa 2013) will suffice. I conclude in the seventh section.

Existing Literature

In this section, I will discuss existing approaches similar to my own. The use of fuzzy sets in case studies is a novelty so instead of reviewing alternatives as such, I discuss plausible candidate approaches that could be molded to support using quantitative information in case studies. I examine whether each approach is suited for analyzing and testing mechanisms conceived as causal chains in one or a few cases.

First, because fuzzy sets are associated with Ragin’s (2008) QCA, it would be tempting to use this procedure to examine structures. But QCA is not immediately useful for analyzing sufficiency chains. In fact, QCA does not detect chains but treats the resulting dependencies in the data as logical remainders (see Baumgartner 2009:77-79, 92, 2013:11). Additionally, Baumgartner and Epple (2014) have argued that modeling causal chains using QCA sequentially leads to logical contradictions. Since I am concerned with causal chains, fuzzy-set QCA does not provide a viable way forward.

Second, it would seem possible that mechanisms could be modeled using temporal QCA (Caren and Panofsky 2005; Ragin and Strand 2008). Temporal QCA introduces a “then” operator, permitting the study of causal order. But it does not study causal chains. Consequently, it is not an appropriate tool for studying mechanisms (its developers never claimed that it was), although it can be a useful tool for when sequencing matters. Thus, temporal QCA can be used, for instance, to evaluate Shefter’s (1994) argument in his classic Political Parties and the State that bureaucracies are more stable and professional when they are established before democratization than if they are established after. But it is not useful for the study of mechanisms, including the very detailed mechanisms uncovered in Shefter’s book. Temporal QCA is no more an alternative than fuzzy-set QCA.

Third, my approach is related to Goertz and Mahoney’s “two-level theories” and their proposal to test these using fuzzy sets (Goertz and Mahoney 2005:522-33). Two-level theories consider fuzzy-set relationships between “primary-level variables” and “secondary-level variables.” These could potentially be interpreted as “start-up conditions and outcome” and “mechanism elements,” respectively. Two-level theories are not primarily concerned with mechanisms, but they could be useful. I depart from Goertz and Mohney (2005) by showing how the entire structures of sufficiency chains can be evaluated simultaneously, by using evaluation criteria which permit slight quantitative deviations from predictions, and by clarifying under what conditions hypothesised fuzzy-set relationships are supported both qualitatively and quantitatively. Additionally, two-level theories consider relationships between levels as causal, ontological, or substitutable (Goertz and Mohney 2005). By contrast, my focus on mechanisms prompts a focus on causal relations only. Overall, my approach complements Goertz and Mahoney’s (2005) but extends on their discussion of causal relationships considerably.

Fourth, Mahoney, Kimball, and Koivu (2009) discuss “sequence elaboration” as a tool to study multiple connected set-theoretic relationships. Sequence elaboration looks at structures linking a set, interpretable as a cause, to another set, interpretable as an outcome, both directly and indirectly through a third set, interpretable as a mechanism. Thus, the analysis looks at a small structure with three elements. And permits each element to be linked to others through either sufficiency or necessity relations. Mahoney et al. (2009) subsequently show the consequences of each combination of these relations. They do not propose a fuzzy-set version of sequence elaboration. However, doing so would be natural following procedures similar to those I will propose.

But contrary to sequence elaboration, my approach is specifically designed to handle mechanisms. This means that I depart from sequence elaboration in a number of ways. First I do not consider direct links between causes and outcomes but link them only through mechanisms. Second, as I will argue, causal mechanisms result in structures that comprise sufficiency statements rather than combinations of sufficiency and necessity relations. Third, my exclusive focus on sufficiency statements allows the structures I analyze to become much more complex than those analyzed by sequence elaboration and to do so without creating logically contradictory combinations of statements (see Mahoney et al. 2009:134-41). This is advantageous to researchers who approach their case(s) with complex causal models.

Finally, Baumgartner (2013, 2009; Baumgartner and Epple 2014) has proposed coincidence analysis (or CNA) as an alternative to QCA, permitting the study of causal structures proper. Briefly, in CNA, structures are uncovered in an analysis relying on dependencies in a data matrix. Basically the analysis determines sufficiency chains by cross-case analysis, searching for patterns of minimally sufficient conditions to form a structure. The output of CNA is one or several structures that are consistent with the data. This is not the place to go through its procedures (interested readers can consult Baumgartner 2009).¹

CNA outperforms QCA when data form causal chains (for direct comparisons, see Baumgartner 2013; Baumgartner and Epple 2014). But CNA is not appropriate for single case studies or studies where only a subset of the cases in a population of interest are selected. CNA is tailored to determine possible structures across cases and explicitly assumes that information is perfectly available on all start-up conditions, mechanism elements, and outcomes in all cases (Baumgartner 2009:78-79). But this “examine-all-cases” possibility is not always achievable. Data will often be missing on some of the relevant elements of the mechanism, if not on start-up conditions or outcomes. In fact, much of what the study of mechanisms in case studies does is gathering within-case data which is used, along with contextual knowledge, to assess whether the case is a member of the sets in the mechanism or not (Beach and Pedersen 2013; Collier 2011). My procedure is viable even when all the necessary data are not available for all cases.

This section has already signaled some of the characteristics of my approach. But the main point has been to argue that I propose to fill a gap in the literature by showing how hypothesized structures can be examined using fuzzy sets. In the next two sections, I detail exactly how I propose to do so.

Graphical Analysis of Structures

Fuzzy-set case studies require more precision than traditional case studies on three accounts. First, measurement of start-up conditions, outcomes, and mechanism elements must be at least marginally more precise than traditional case studies. Otherwise the approach will not leverage any of the potential informational superiority of fuzzy sets. I assume this requirement satisfied throughout. As mentioned, this assumption will not be fulfilled for all applied research. But it is fulfilled whenever information is available to make qualifications such as those Tannenwald (1999) makes for Hiroshima and Nagasaki or Grzymała-Busse (2007) makes for Poland.

Second, fuzzy-set case studies demand a specific idea about the causal structure of the mechanism between start-up conditions and outcomes (cf. Goertz and Mahoney 2005, 500). Essentially, I propose to treat structures as causal models. Fuzzy-set case studies as I propose them require the researcher to distil and specify the structure to be subjected to testing on the basis of theory or existing knowledge. Fortunately, this is entirely analogous to accepted practise, that is, researchers specify the observable implications of a particular theoretical account—whether they or someone else built it—and match these implications with data to see if the account is, or at least plausibly is, correct (see e.g., Collier 2011; George and Bennett 2005:115-20, 181-232; Hall 2008).²

Third, fuzzy-set case studies require a precise idea about how a mechanism’s constituent parts relate to each other. I will be discussing only sufficiency claims (see also Collier 2011:825; Goertz and Mahoney 2005:501-2). The reason for this choice is conceptual. Let us define mechanisms broadly as “the intervening process through which causes exert their effects” (Goertz and Mahoney 2012:100) and adopt the idea that mechanisms produce their associated outcomes (e.g., Beach and Pedersen 2013). If we accept these premises and the set-theoretic framework, I will argue, start-up conditions must be individually or jointly sufficient for mechanism elements which are either individually or jointly sufficient for outcomes (see also the Online Appendix to Abell 2009).

To see why, consider the obvious alternative, that is, necessity relations between either start-up conditions and mechanism or mechanism and outcome. If a mechanism is necessary for an outcome and is present in a case, it does not produce the outcome. Instead, its presence permits that any outcome can occur in the case. Similarly, if a start-up condition is necessary but insufficient for a mechanism, anything can happen when the start-up condition is present. These conclusions are not conceptually attractive.

If start-up conditions are to produce the outcome through a mechanism, each mechanism element in the structure must have at least one sufficient condition or set of conditions contained either in start-up conditions or in prior mechanism elements of the structure. And eventually, at least one mechanism element (possibly in conjuncture with start-up conditions or other mechanism elements) must be sufficient for the outcome. If this is not the case, the productive relation between start-up conditions and outcome can break down as insufficiency permits that anything can happen in the presence of the insufficient condition.

This does not mean that all mechanism elements and the outcome must have a necessary and sufficient condition prior to them. Nor does it mean that the structure cannot contain necessary and sufficient conditions or individually insufficient conditions within it. But the latter must be jointly sufficient in conjuncture with other included start-up conditions or mechanism elements. In essence, productive mechanisms require causal structures that are made up of sufficiency chains.

Next, I provide guidelines for turning structures of sufficiency chains into sufficiency statements that can be tested against data in a fuzzy-set case study. To do this, I depict hypothesized structures in graphs. The graphs I will present may look familiar to readers acquainted with structural equation models or directed acyclic graphs for causal analysis (see Morgan and Winship 2007). But, as will become very clear, my approach is different in important ways. It shares much more with the Boolean structures discussed by Baumgartner (2009), the time-ordered diagraphs discussed by Abell (2009), or the set diagrams discussed by Mahoney and Vanderpoel (2015).

Just to be clear, there is nothing in this article that cannot be handled without graphs. But using graphs is helpful, both for presentational purposes and insofar as researchers themselves graphically represent their model’s causal structure for clarity and to avoid making mistakes. Figure 1 provides a few examples of hypothetical structures. For descriptive purposes, it is useful to introduce a little vocabulary from graph theory.

Figure 1.

Exemplary causal structures.

In the figure, vertices (the boxes) are start-up conditions (I denote these by ι), mechanism elements (denoted by μ), and outcome (denoted by ω). Edges (the arrows) are “sufficiency links” between vertices directed to indicate that the initial vertex or vertices (at the arrow’s tail or tails) are sufficient for the terminal vertex (at the arrow’s head). I illustrate hyper-edges joining more than two vertices using • (similar to Goertz and Mahoney 2005). Graphs can be used to distil sufficiency statements in the context of case studies using two simple rules (both of which are quite similar to Abell 2009:45-50).

If an edge in a hypothesized structure connects exactly one initial vertex and one terminal vertex, the initial vertex is hypthesised to be sufficient for the terminal vertex. Put differently, if an arrow connects exactly two parts of a structure, the part at its tail is claimed to be sufficient for the part at its head.

If a hyper-edge connects multiple initial vertices to a single terminal vertex, the initial vertices are hypothesized to be jointly sufficient for the terminal vertex. Put differently, if an arrow connects parts of a structure at several tails to a part at its head, the parts at its tails are jointly sufficient for the part at its head.

Notice that these rules say nothing about necessity. It is possible that an initial vertex is necessary for its terminal vertex. Similarly, it is possible that multiple initial vertices in hyper-edges are individually necessary for their common terminal vertex. But they do not have to be. Notice also that the second rule does not result in equifinality. Instead, the first rule allows equifinality if multiple edges connect multiple initial vertices to a single terminal vertex (and are not joined in one hyper-edge). I discuss this issue further in the fifth section.

A few examples will illustrate the point. Consider (a) in Figure 1. Here a single start-up condition ι _a ₁ initiates a simple causal chain though two steps μ _a ₁ and μ _a ₂ ultimately resulting in an outcome ω _a . Thus, three vertices μ _a ₁, μ _a ₂, and ω _a are terminal vertices for three edges with exactly one initial vertex. Consequently, following the first rule, if ι _a ₁ is to be sufficient for ω _a , the following three sufficiency statements should hold. If at least one of them do not hold, at least a subgraph of (a) is not supported by the data. This ought to be enough for researchers to reconsider their structure (George and Bennett 2005:29-30, 207).

1a. ι _a ₁ is sufficient for μ _a ₁,

2a. μ _a ₁ is sufficient for μ _a ₂, and

3a. μ _a ₂ is sufficient for ω _a .

(b) in Figure 1 is somewhat more complex. Here, two start-up conditions ι _b ₁ and ι _b ₂ collaborate to produce the first element of the mechanism μ _b ₂. Notice that (b) does not involve equifinality. Both ι _b ₁ and ι _b ₂ are needed to produce μ _b ₁. This element, in turn, sets off two other mechanism elements μ _b ₂ and μ _b ₃. Finally, these two conditions produce ω _b in conjunction. Thus, μ _b ₂ and μ _b ₃ are terminal vertices for edges with exactly one initial vertex, whereas μ _b ₁ and ω _b are terminal vertices for hyper-edges with two initial vertices each. Consequently, following the first and second rule, the following sufficiency statements should hold for ι _b ₁ and ι _b ₂ to be sufficient for ω _b :

1b. ι _b ₁ and ι _b ₂ are jointly sufficient for μ _b ₁,

2b. μ _b ₁ is sufficient for μ _b ₂,

3b. μ _b ₁ is sufficient for μ _b ₃, and

4b. μ _b ₂ and μ _b ₃ are jointly sufficient for ω _b .

So far I have discussed only instances where start-up conditions play no other role than starting up the mechanism. This is not a necessary limitation. Start-up conditions can be relevant because they collaborate with one or several mechanism elements to produce another mechanism element. (c) in Figure 1 depicts such a situation (which, again, does not involve equifinality). ι _c ₂ produces μ _c ₁ which then conjoins with ι _c ₁ to produce μ _c ₂. This condition subsequently produces the outcome ω _c . Here μ _c ₁ and ω _c are terminal vertices for edges with exactly one initial vertex, whereas μ _c ₂ is the terminal vertex for a hyper-edge connecting it to two initial vertices. Following the first and second rule, that the following sufficiency statements should hold:

1c. ι _c ₂ is sufficient for μ _c ₁,

2c. ι _c ₁ and μ _c ₁ are jointly sufficient for μ _c ₂, and

3c. μ _c ₂ is sufficient for ω _c .

To illustrate a live example let me return to Grzymała-Busse (2007). As mentioned, she is interested in Rebuilding Leviathan in how robust political competition forces elites to avoid state exploitation. In her book, she outlines three mechanisms through which this happens. First, governments facing robust competition moderate their behavior. They avoid discretionary hiring in the administration, the establishment of extra-budgetary funds, and the worst excesses in privatization. Second, as I mentioned in the introduction, incumbents facing robust competition build formal state institutions, including civil service laws and oversight organs. They do so in anticipation that they will lose office to their robust competitors and do not want to see these have unrestrained and unmonitored access to state resources. Third, governments attempt to co-opt their robust competitors by giving them central roles in parliamentary committees and by instating inclusive party financing laws. This makes the opposition better capable of monitoring and contesting government behavior. Where all of this happens, changing incumbents are eventually unable to exploit state resources for the benefit of their organization.

I translate these arguments into a graph in Figure 2. Using the two rules, I have outlined the central claim in Rebuilding Leviathan becomes the following: Robust political competition (ι_competition) is sufficient for governments to moderate their behavior (μ_moderation), to build formal institutions in anticipation of their own loss of power (μ_anticipation), and to co-opt their parliamentary opposition (μ_co-optation). These three mechanism elements are then jointly sufficient for governments to abandon state exploitation (denoted ω_{∼exploitation} where ∼ indicates negation or absence). Thus, activating the first rule three times and the second once, the following sufficiency statements should hold:

1GB. ι_competition is sufficient for μ_moderation,

2GB. ι_competition is sufficient for μ_anticipation,

3GB. ι_competition is sufficient for μ_co-optation, and

4GB. μ_moderation, μ_anticipation, and μ_co-optation are jointly sufficient for ω_{∼exploitation}.

Figure 2.

Grzymała-Busse’s structure.

As these examples should make clear, the two rules I have employed can examine quite complex causal structures easily. A simple graphical analysis can highlight all the sufficiency statements that a given structure requires. But these considerations are all preliminaries to the main event of this article. They are necessary to show how fuzzy-set case studies are feasible for sufficiency relations. But none of what I have said in this section directly addresses the loss of quantitative information in case studies. In the next section, I introduce my solution.

A Straightforward Procedure

The primary task of fuzzy-set case studies is to use both qualitative and quantitative information when evaluating statements like those I derived from the graphs in the previous section. In this section, I will show how this end can be achieved using standard fuzzy algebra.

In fuzzy algebra sets are assigned fuzzy-set membership scores using a membership function denoted m(·). The membership function assigns membership scores to each case. Cases where m(·) > .5 are qualitative members of the set whereas cases where m(·) < .5 are qualitatively out of the set. If two sets are connected by logical OR the fuzzy-set operator takes the maximum of their fuzzy-set membership. If two sets are connected by logical AND the fuzzy-set operator takes the minimum of their fuzzy-set membership (Ragin 2000).

The key question when examining sufficient set relations is whether the data are consistent with the claim that membership of some set σ is sufficient for membership of another set υ. Consistency here means the degree to which data agree with the sufficiency claim.

To calculate the consistency of sufficiency relations for n cases in the QCA context, Ragin (2006) has proposed the following quantity:

C o n s i s t e n c y = \frac{\sum m i n [m (σ_{k}), m (υ_{k})]}{\sum m (σ_{k})},

for $k \in {1, 2, . . . n}$ . Where m(σ _k ) indicates the fuzzy-set membership of case k in condition σ. This quantity is bounded between 0 (fully inconsistent) and 1 (fully consistent). Both σ and υ can be arbitrarily complex expressions without altering the consistency measure.

I will not extensively discuss the issue of coverage (Ragin 2006) or relevance (Goertz 2006a) as means of evaluating the extent to which a structure “accounts for” examined cases. But I will note that extending my framework to include a coverage evaluation as well as a consistency evaluation is possible. Evaluations would include both equation (1) and $C o v e r a g e = \sum m i n [m (σ_{k}), m (υ_{k})] / \sum m (υ_{k})$ . Coverage evaluations of sufficiency statements can be treated just as I propose to treat consistency evaluations.

The advantage of including a coverage evaluation would be that it provides a measure of how close a sufficiency statement is to also being necessary. Incorporating coverage evaluations into consistent structures is worth doing. But consistency takes priority. If we do not consider set relations consistent, their coverage is not substantially meaningful. Consequently, I focus my attention on consistency in the following presentation.

Returning to consistency and equation (1) mentioned previously, cases where m(σ _k ) > .5 and m(υ _k ) < .5 are qualitatively inconsistent with the claim that σ is sufficient for υ. But this we do not need the consistency measure or fuzzy-set case studies to realize. Instead, the advantage of using a consistency measure is that we can evaluate which cases are quantitatively consistent. If the consistency score is at a high enough level (I will call this level ρ), the level of quantitative consistency is acceptable. If it is not, the researcher ought to reconsider, refine, or reject the hypothesised structure as it does not produce consistent predictions.

But to ensure qualitative consistency, we may demand that cases have a fuzzy-set membership higher than .5 in all vertices in the structure. I will refer to this requirement as the qualitative consistency constraint (QCC).

As I will discuss it, the QCC follows the recommendation in the literature to select cases for in-depth study where both input conditions, mechanism elements, and outcome are qualitatively present (e.g., Beach and Pedersen 2013). But it overlooks cases where neither input conditions nor outcome is present. These cases are also accounted for by a given structure and may be consistent with it. An extended QCC could take account of these cases too and permit both the outcome and its absence to be studied.³ But for simplicity I will discuss only the simple QCC in the main text (note 3 explains the extended QCC).

If a QCC is not applied, cases may—in admittedly rare instances—be quantitatively consistent but qualitatively inconsistent with the hypothesised structure. Theoretically, if ρ < 1, it is possible that ρ is high but that a case is of the wrong type. If the case is close to a .5 membership in both initial and terminal vertices for a sufficiency statement it may have an above .5 membership in the initial vertex or vertices but a below .5 membership in the terminal vertex and still produce a high ρ in the evaluation of the relevant statement. But this situation is unattractive as the case does not qualitatively conform to the structure. I introduce QCC as a solution to this problem.

Satisfying a QCC is sufficient for traditional case studies to accept a structure as consistent. But it is insufficient for fuzzy-set case studies. Here we require an additional evaluation of consistency at least at a level of ρ.

Suppose for now that exactly one case is analyzed. With just one case, the calculation of consistency in equation (1) reduces to (suppressing case identifiers):

C o n s i s t e n c y = \frac{m i n [m (σ), m (υ)]}{m (σ)} .

Which is Rubinson’s (2013) measure of “observation consistency.” To fix ideas, let us focus on a particular structure. Take (c) in Figure 1. As I showed previously, for this structure to be supported, we require the following three sufficiency statements to simultaneously hold: (1c) requiring that ι _c ₂ is sufficient for μ _c ₁, (2c) requiring that ι _c ₁ and μ _c ₁ are jointly sufficient for μ _c ₂, and (3c) which requires that μ _c ₂ is sufficient for ω _c . Translated into consistency evaluations, these statements become three expressions. Each applying equation (2) to one of c1 – c3. Thus, for (c), the following must hold for the selected case in order for the proposed mechanism to find support in the data (I omit m(·) and case identifiers but note that all elements in the expressions are fuzzy-set membership scores).

\begin{aligned} \frac{m i n (ι_{c 2}, μ_{c 1})}{ι_{c 2}}, \\ \frac{m i n (ι_{c 1}, μ_{c 1}, μ_{c 2})}{m i n (ι_{c 1}, μ_{c 1})}, \\ \frac{m i n (μ_{c 2}, ω_{c})}{μ_{c 2}} . \end{aligned}

All subject to the QCC for (c), QCC _c , which holds (in its simple version) that $m i n [m (ι_{c 1}), m (ι_{c 2}), m (μ_{c 1}), m (μ_{c 2}), m (ω_{c})] > .5$ . The expressions in equation (3) can be used to evaluate c1 – c3.

In QCA, consistency levels are conventionally required to be at least .75 (applications sometimes demand higher values). But if just one case is selected for a fuzzy-set case study, I recommend demanding ρ = 1. This is conceptually attractive as it does not permit the case to be qualitatively or quantitatively inconsistent. But it is possible to permit imperfect quantitative consistency at thus set ρ < 1.

Generally, if the three sufficiency statements hold, the smallest of the expressions must be larger than or equal to the acceptable consistency threshold ρ. Thus, we can compile equation (3) to get the following inequality (again omitting m(·) and case identifiers):

m i n (\frac{m i n (ι_{c 2}, μ_{c 1})}{ι_{c 2}}, \frac{m i n (ι_{c 1}, μ_{c 1}, μ_{c 2})}{m i n (ι_{c 1}, μ_{c 1})}, \frac{m i n (μ_{c 2}, ω_{c})}{μ_{c 2}}) \geq ρ .

That is, the selected case should be consistent with each statement c1 – c3 at least at a level of ρ, subject to QCC _c . I will discuss general expressions like this for the remainder of this article. I will emphasize, however, that quantitative inconsistencies are a cause for concern. Frequently, setting ρ to 1 is advisable. Not least in single-case studies where quantitative inconsistency would leave no case fully consistent with the hypothesized structure.

Let me return one last time to Grzymała-Busse (2007). The four statements I distilled from Rebuilding Leviathan can be translated into consistency expressions and compiled into the following inequality that should hold for her argument to be supported (as usually omitting m(·) and case identifiers).

\begin{aligned} m i n (\frac{m i n (ι_{c o m p e t i t i o n}, μ_{m o d e r a t i o n})}{ι_{c o m p e t i t i o n}}, \frac{m i n (ι_{c o m p e t i t i o n}, μ_{a n t i c i p a t i o n})}{ι_{c o m p e t i t i o n}}, \\ \frac{m i n (ι_{c o m p e t i t i o n}, μ_{c o - o p t a t i o n})}{ι_{c o m p e t i t i o n}}, \\ \frac{m i n (μ_{m o d e r a t i o n}, μ_{a n t i c i p a t i o n}, μ_{c o - o p t a t i o n}, ω_{: e x p l o i t a t i o n})}{m i n (μ_{m o d e r a t i o n}, μ_{a n t i c i p a t i o n}, μ_{c o - o p t a t i o n})}) \geq ρ . \end{aligned}

Subject to a (once again simple) QCC holding that the case studied should have a membership in ι_competition, all three of the mechanisms it sets off, and the outcome of above .5.

If equation (5) and the relevant QCC are satisfied, the case under consideration is both qualitatively and quantitatively consistent with the hypothesised structure. If the QCC is not satisfied, the case is qualitatively inconsistent with the structure. And if equation (5) is not satisfied, the case is quantitatively inconsistent with the structure. In the two latter instances, the researcher may want to examine which set in the structure violates the QCC. Or unpack equation (5) into individual expressions, as in equation (3), and set each in an inequality with ρ in order to find which sufficiency statement is quantitatively inconsistent. Irrespectively, a violated QCC or compiled inequality such as equation (5) entails that the structure is not fully consistent and should be reconsidered.

Using the procedure I have outlined, we can discover the inconsistencies that are not discovered by analyses of start-up conditions and outcomes alone. And which are not discovered by traditional case studies if QCC is satisfied. Furthermore, the procedure is straightforward. The math needed to evaluate equation (4) or equation (5) involves nothing more than finding the smallest of a set of numbers and simple division. Next, I turn to some extensions.

Equifinality

So far I have only dealt with structures that are not equifinal. This might be plausible in some cases. At least, a number of case studies emphasize only one set of conditions that generate the outcome they are interested in. Grzymała-Busse’s focus on robust political competition in Rebuilding Leviathan (2007) is a case in point. But there are exceptions.

For instance, in her Europe Undivided, Vachudova (2005) argues that two “patterns of political change” (2005:5) emerged in East Central Europe after communism fell. Initially, countries were divided into “liberal” and “illiberal” democracies. But influenced by the process of accession to the European Union (EU), the region eventually saw “a significant—but far from complete—convergence” (Vachudova 2005:257) to liberal democracy. I will outline Vachudova’s structure in more detail subsequently. Suffice it here to say that, if we permit the convergence to be complete enough for qualitative similarity, Vachudova presents us with an equifinal argument.

In equifinal structures, graphical analysis can help determine statements that should hold just as I showed in the last section. The procedure becomes slightly more complicated but not much.⁴

Figure 3 depicts some structures to guide the discussion. In panels (d), (e), and (f) the claim is that either ι₃ or the conjuncture of ι₁ and ι₂ are sufficient for the outcome ω. This is denoted “+” in the figure for clarity. Generally, this notation is unnecessary. The structures themselves are equifinal, with one exception I will discuss shortly (the reader can verify this by following edges against their direction backward from the outcomes and confirm that this leads to multiple and separate start-up conditions). In (d), the outcome is produced in either one or both of two ways. Either ι _d ₃ produces μ _d ₃ which then produces ω _d . Or ι _d ₁ produces μ _d ₁, while ι _d ₂ produces μ _d ₂. Subsequently μ _d ₁ and μ _d ₂ produce μ _d ₄ in conjuncture. μ _d ₄ in turn produces the outcome ω _d . This structure can be treated like (a)–(c) in Figure 1. The only difference is that ω _d is a terminal vertex for two edges, each with one initial vertex. This activates the first of my two rules twice and results in two sufficiency statements (one for each edge). Thus, the following statements should hold:

1d. ι _d ₁ is sufficient for μ _d ₁,

2d. ι _d ₂ is sufficient for μ _d ₂,

3d. ι _d ₃ is sufficient for μ _d ₃,

4d. μ _d ₁ and μ _d ₂ are jointly sufficient for μ _d ₄,

5d. μ _d ₃ is sufficient for ω _d , and

6d. μ _d ₄ is sufficient for ω _d .

Figure 3.

Equifinal structures.

I have included (e) in Figure 3 to illustrate a potential pitfall. This structure involves a contradiction. The graph demands that either ι _e ₃ or the conjuncture of ι _e ₁ and ι _e ₂ are sufficient for the outcome ω _e . At the same time, it demands that μ _e ₃ and μ _e ₄ are jointly sufficient to produce ω _e . These two demands are contradicting. If μ _e ₃ needs μ _e ₄ to produce ω _e the two equifinal terms connected by the + cannot actually be equifinal. After all μ _e ₄ is ultimately a product of ι _e ₁ and ι _e ₂, but the equifinal claim states that ω _e can be achieved without either of these two input conditions so long as ι _e ₃ is present. By contrast, the structure depicted in (f) is not contradictory. It simply demands that there are two equifinal ways to produce μ _f ₄, which will then produce ω _f .

When sufficiency statements have been derived for equifinal structures, these need to be translated into consistency expressions and inequalities with ρ as in the previous section. I am still assuming that just one case is selected. This may seem somewhat counterintuitive for equifinal structures. But it will be useful subsequently. And it will help show the disadvantages of selecting a case which is covered by multiple equifinal start-up conditions (known as cases with joint coverage in the QCA literature).

In case of equifinality, it is useful to compile all statements relating to vertices in each possible path from a subset of start-up conditions to the outcome. By path, I mean simply a possible way to get from at least one start-up condition to the outcome by following edges and hyper-edges from initial to terminal vertices. In multiple initial vertices of hyper-edges marked by •, all initial vertices are jointly needed to form a path. Equifinality, as I showed in the discussion of structure (e) mentioned previously, must entail a minimum of two paths. But there could be more.

To fix ideas Figure 4 shows the two possible paths in structure (d) from Figure 3. Thick lines indicate one path. In the upper panel, I highlight the more complex path from ι _d ₁ and ι _d ₂ through μ _d ₁ and μ _d ₂ to μ _d ₄ to ω _d (I refer to this as path 1). In the lower panel, I highlight the simpler path from ι _d ₃ through μ _d ₃ also arriving at ω _d (path 2). There are no more possible paths in (d). The first step for the equifinal structure is thus to transform the sufficiency statements in each of these two paths into consistency evaluations and compile each path under its own minimum operator. Subsequently, the compiled expressions can be connected by the fuzzy-set equivalent of logical OR (the maximum operator) if this is desirable.

Figure 4.

Paths in structures.

But when is it desirable? This turns out to be as much a practical as a methodological question. It would be desirable for cases with joint coverage. But these are themselves undesirable for reasons I will explain. Once joint coverage is ruled out, desirability becomes mostly a matter of the amount of available information.

Let us stick with structure (d). From the graphical analysis mentioned previously followed six sufficiency statements that should hold if the mechanism is correct. But when just one case is selected, these six do not all have to hold. Equifinality lightens the test. Within one case either (rearranging the statements somewhat).

1d. ι _d ₁ is sufficient for μ _d ₁,

2d. ι _d ₂ is sufficient for μ _d ₂,

4d. μ _d ₁ and μ _d ₂ are jointly sufficient for μ _d ₄,

6d. μ _d ₄ is sufficient for ω _d ,

3d. ι _d ₃ is sufficient for μ _d ₃,

5d. μ _d ₃ is sufficient for ω _d ,

or both.

Each of these two lists of statements can be individually treated exactly like in the previous section. 1d, 2d, 4d, and 6d can be translated into expressions as in equation (3) and compiled under a single minimum operator as in equation (4), so can 3d and 5d in parallel. The two expressions can then be further compiled under a maximum operator and placed in an inequality with ρ. The resulting inequality is (once again omitting m(·) and case identifiers):

\begin{aligned} m a x (m i n [\frac{m i n (ι_{e 1}, μ_{e 1})}{ι_{e 1}}, \frac{m i n (ι_{e 2}, μ_{e 2})}{ι_{e 2}}, \\ \frac{m i n (μ_{e 1}, μ_{e 2}, μ_{e 4})}{m i n (μ_{e 1}, μ_{e 2})}, \frac{m i n (μ_{e 4}, ω_{e})}{μ_{e 4}}], \\ m i n [\frac{m i n (ι_{e 3}, μ_{e 3})}{ι_{e 3}}, \frac{m i n (μ_{e 3}, ω_{e})}{μ_{e 3}}]) \geq ρ . \end{aligned}

Each path requires its own QCC, which I will label its path QCC (or PQCC for short). In (d), we have two such constraints (again limiting attention to simple QCCs), one for path 1 which I will label P ₁QCC _d and which holds that:

m i n [m (ι_{d 1}), m (ι_{d 2}), m (μ_{d 1}), m (μ_{d 2}), m (μ_{d 4}), m (ω_{d})] > .5 .

And another for path 2 which I will label P ₂QCC _d and which holds that:

m i n [m (ι_{d 3}), m (μ_{d 3}), m (ω_{c})] > .5 .

In equation (6) when both paths are compiled in one expression, we can satisfy an overall constraint QCC _d by requiring that either P ₁QCC _d or P ₂QCC _d holds.

This may sound more complicated than it is. Equation (6) states that, just following the arrows of the graph, ι _d ₁ has to be consistently sufficient for μ _d ₁, ι _d ₂ consistently sufficient for μ _d ₂, the minimum of μ _d ₁ and μ _d ₂ consistently sufficient for μ _d ₄, and μ _d ₄ consistently sufficient for ω _d (let us call the situation where all these statements are true α). Or ι _d ₃ has to be consistently sufficient for μ _d ₃ and μ _d ₃ consistently sufficient for ω _d (let us call the situation where these two statements are true ζ). All subject to the constraint that the selected case must have a membership above .5 in all parts of path 1 for α, all parts of path 2 for ζ, or both. The only addition from the last section is the maximum operator. Calculation still only involves simple division and finding the smallest, and now also the largest, number in a set of numbers.

Next let me briefly consider why joint coverage is a problem. In our (d) example, suppose an analysis of a jointly covered case finds that α is true but ζ is not. Does this mean that one part of the structure is incorrect? It could. α is supported and joint coverage will not undermine this support. But the problem is that it becomes difficult to know whether ζ is not operating because α already caused the outcome to occur. The analysis cannot determine whether or not ζ is correct. For this reason, selecting cases jointly covered by several paths in a structure should be avoided (Schneider and Rohlfing 2013).

Once we realize this, using the maximum operator becomes a more practical issue. For practical purposes equation (6) might be demanding too much. After all, if we know that a case is uniquely covered by a specific path we might not wish—for reasons of time, manpower, or budget—to examine the mechanism associated with another path in that case. If the selected case is not jointly covered, we may test a part of the hypothesized structure only. Specifically, we can test whether 1d, 2d, 4d, and 6d are true if the case is uniquely covered by ι _d ₁ and ι _d ₂. If this is the case, we need to only examine (again omitting m(·) and case identifiers)

\begin{aligned} m i n (\frac{m i n (ι_{d 1}, μ_{d 1})}{ι_{d 1}}, \frac{m i n (ι_{d 2}, μ_{d 2})}{ι_{d 2}}, \\ \frac{m i n (μ_{d 1}, μ_{d 2}, μ_{d 4})}{m i n (μ_{d 1}, μ_{d 2})}, \frac{m i n (μ_{d 4}, ω_{d})}{μ_{d 4}}) \geq ρ . \end{aligned}

Subject to P ₁QCC _d . Or we can test whether 3d and 6d are true if the case is uniquely covered by ι _d ₃. In this case, we need only examine:

m i n (\frac{m i n (ι_{d 3}, μ_{d 3})}{ι_{d 3}}, \frac{m i n (μ_{d 3}, ω_{d})}{μ_{d 3}}) \geq ρ .

Subject to P ₂QCC _d . Either way, the whole of the hypthesized structure is not examined. But only one path. When practical limitations apply and joint coverage is avoided, joining expressions for several paths under a maximum operator is not generally desirable.

Vachudova’s (2005) study points to another situation where joining expressions is not useful, that is, when joint coverage is impossible. In particular, when paths embed start-up conditions or mechanism elements that are the negation (denoted ∼) of each other. I present a stylized and somewhat simplified graphical representation of Vachudova’s argument in Figure 5. Briefly, her argument is that two patterns emerged in East Central Europe. High-quality political competition after the fall of communism (ι_competition) reinforced by the attractiveness of EU membership (the EU’s “passive leverage” ι_EUpassive) pushed some countries’ governments toward reform of their political institutions and their economies (μ_reform), which led to liberal democracy (ω_liberal.dem).

Figure 5.

Vachudova’s structure.

In states where high-quality competition was absent after communism fell, passive leverage did not have an impact. By contrast lack of high-quality competition (ι_{∼competition}) led to a lack of reform (μ_∼reform). Combined with rising active pressure from EU institutions (the EU’s active leverage μ_EUactive), the lack of reform generated new political competitors endogenously (μ_{late.competition}) which then led to reform and subsequently liberal democracy.

Clearly there are two paths in Vachudova’s structure. Formulated as sufficiency statements either:

1VA. ι_EUpassive and ι_competition are jointly sufficient for μ_reform and

2VA. μ_reform is sufficient for ω_liberal.dem.

3VA. ι_{∼competition} is sufficient for μ_∼reform,

4VA. ι_EUactive and μ_∼reform are jointly sufficient for μ_{late.competition},

5VA. μ_{late.competition} is sufficient for μ_reform, and

6VA. μ_reform is sufficient for ω_liberal.dem.

Notice that in this case joint coverage is impossible as one path relies on a start-up condition that is a negation of a start-up condition in the other path. Thus, for the early competitive states, we need to only evaluate the following:

\begin{aligned} m i n (\frac{m i n (ι_{E U p a s s i v e}, ι_{c o m p e t i t i o n}, μ_{r e f o r m})}{m i n (ι_{E U p a s s i v e}, ι_{c o m p e t i t i o n})}, \\ \frac{m i n (μ_{r e f o r m}, ω_{l i b e r a l . d e m})}{μ_{r e f o r m}}) \geq ρ . \end{aligned}

Subject to the relevant PQCC. Whereas for states where high-quality competition was absent after communism fell, we need to evaluate:

\begin{aligned} m i n (\frac{m i n (ι_{~ c o m p e t i t i o n}, μ_{~ r e f o r m})}{ι_{~ c o m p e t i t i o n}}, . \\ \frac{m i n (ι_{E U a c t i v e}, μ_{~ r e f o r m}, μ_{l a t e . c o m p e t i t i o n})}{m i n (ι_{E U a c t i v e}, μ_{~ r e f o r m})}, \\ . \frac{m i n (μ_{l a t e . c o m p e t i t i o n}, μ_{r e f o r m})}{μ_{l a t e . c o m p e t i t i o n}}, \frac{m i n (μ_{r e f o r m}, ω_{l i b e r a l . d e m})}{μ_{r e f o r m}}) \geq ρ . \end{aligned}

Subject to the PQCC relevant to this path. Joining these two expressions under a maximum operator serves little purpose if only one case is analyzed. The structure prevents joint coverage.

Thus, while I have shown how equifinal structures can be evaluated as wholes in fuzzy-set case studies, there are two reasons for examining only parts of an equifinal structure in a case. Either because joint coverage is impossible. Or because joint coverage is avoided and practical limitations make a more limited study of a uniquely covered case attractive. For multiple cases, however, considering equifinal causal structures in full might be helpful as I will discuss in the next section.

Multiple Cases

Similar to my initial assumption that structures are not equifinal my assumption that just one case is studied does not have to be justified in empirical research. Indeed, the three empirical case study works I have discussed all study multiple cases (Grzymała-Busse 2007; Shefter 1994; Vachudova 2005). In this section, I show how my approach handles multiple cases. To prepare the reader, the major advantage of analyzing multiple cases is that the QCC may be lifted.

Recall that Ragin’s (2006) consistency measure in equation (1) by means of Rubinson’s (2013) measure of observation consistency in equation (2) formed the basis for my procedure for one case. All that is needed for the inclusion of multiple cases is to go straight from Ragin’s formula to presenting expressions of statements. This permits taking account of all examined cases when determining the overall consistency of the structure. If, for instance, k cases are selected to examine structure (c) in Figure 1, the overall consistency of the structure is evaluated using a slight alteration of equation (4):

\begin{aligned} m i n (\frac{\sum m i n (ι_{c 2 (i)}, μ_{c 1 (i)})}{\sum ι_{c 2 (i)}}, \frac{\sum m i n (ι_{c 1 (i)}, μ_{c 1 (i)}, μ_{c 2 (i)})}{\sum m i n (ι_{c 1 (i)}, μ_{c 1 (i)})}, \\ \frac{\sum m i n (μ_{c 2 (i)}, ω_{c (i)})}{\sum μ_{c 2 (i)}}) \geq ρ, \end{aligned}

for cases in $i \in {1, 2, . . . k}$ . The three fractions contained within the minimum operator in equation (13) are standard consistency calculations as implemented in existing QCA software such as fs/QCA and the QCA package for R. They can be easily calculated in either software. And equation (13) gives the overall consistency of the entire structure for multiple cases.

This has the advantage that it tests the conformity of all selected cases with the (c) structure simultaneously. Consequently, a case with a poor fit can potentially be tolerated by the analysis if sufficiently many other cases are consistent. If a researcher is satisfied with having one or a few qualitatively inconsistent cases, equation (13) does not need to be subject to a QCC. If qualitative inconsistency is not tolerated, a QCC can be applied. For the (c) structure, this restriction would hold (again focussing on the simple QCC):

m i n [m (ι_{c 1 (i)}), m (ι_{c 2 (i)}), m (μ_{c 1 (i)}), m (μ_{c 2 (i)}), m (ω_{c (i)})] > .5,

for all $i \in {1, 2, . . . k}$ . If a QCC is not applied, individual cases may become qualitatively inconsistent with the hypothesized structure even when ρ is at acceptable levels. This may well highlight that something is missing from, or wrong with, the hypothesized structure (see, e.g., Rubinson 2013). Qualitatively inconsistent cases should at least be discussed.

A complication arises again for equifinal structures. Reconsider (d) in Figure 3. If we are to learn about both paths in (d) from start-up conditions to outcome as illustrated in Figure 4, we should analyse at least one case from each (Goertz 2008; Schneider and Rohlfing 2013). All cases can be analyzed simultaneously by relying on Ragin’s (2006) consistency measure, resulting in an expression compiled under a maximum operator as in equation (6).

But as before, this would mean gathering information on mechanism elements in cases where we do not expect these to be relevant, and now in multiple cases. Consequently, it may for practical reasons be worth examining each equifinal path from inputs to outcome in separate inequalities.⁵ Analyzing multiple cases from each path means following the rational I have laid out previously in this subsection. If, for instance, multiple cases n (without joint coverage) are selected from the first path in (d) and multiple cases m (also without joint coverage) are selected from the second path in (d), the following inequalities should hold. For cases covered by path 1 (beginning with ι _d ₁ and ι _d ₂), the following should hold (once again omitting m(·)):

\begin{aligned} m i n (\frac{\sum m i n (ι_{d 1 (i)}, μ_{d 1 (i)})}{\sum ι_{d 1 (i)}}, \frac{\sum m i n (ι_{d 2 (i)}, μ_{d 2 (i)})}{\sum ι_{d 2 (i)}}, \\ \frac{\sum m i n (μ_{d 1 (i)}, μ_{d 2 (i)}, μ_{d 4 (i)})}{\sum m i n (μ_{d 1 (i)}, μ_{d 2 (i)})}, \frac{\sum m i n (μ_{d 4 (i)}, ω_{d (i)})}{\sum μ_{d 4 (i)}}) \geq ρ . \end{aligned}

For the n cases. This may or may not be subject to P ₁QCC _d . By parallel, in cases covered by ι _d ₃, the following should hold:

m i n (\frac{\sum m i n (ι_{e 3 (i)}, μ_{e 3 (i)})}{\sum ι_{e 3 (i)}}, \frac{\sum m i n (μ_{e 3 (i)}, ω_{e (i)})}{\sum μ_{e 3 (i)}}) \geq ρ .

For the m cases and optionally subject to P ₂QCC _d .

These two inequalities are separated for practical reasons only. If information is available on all relevant start-up conditions, mechanisms elements, and outcome for all selected cases, a maximum operator can connect the two inequalities mentioned previously. If so, this expression can be optionally subject to QCC _d for all cases $i \in {1, 2, . . . n + m}$ .

I have shown in this section that multiple cases can be analyzed using the information-preserving tests in fuzzy-set case studies. Multiple cases might even entail fewer restrictions than for single-case studies if outliers can be tolerated and the QCC lifted. However, if exactly one case is selected from each path in an equifinal structure I will recommend not lifting the QCC. In this case, as when single cases are selected from structures that are not equifinal, I would recommend tolerating neither qualitative outliers nor quantitative deviations and thus applying a QCC and setting ρ to 1. On that note, let me conclude.

Conclusions

I started this article with an argument that case studies as they are currently done do not use all the information available to case study researchers. Masses of contextual and substantive knowledge go into determining which type a case belongs to. But the quantitative information about whether the case is a full instance of its type or some qualification is needed does not enter the actual evaluation of causal claims. In the end, traditional case studies evaluate whether start-up conditions, outcomes, and mechanism elements are present or not (Beach and Pedersen 2013).

I have argued that this loss of information is unnecessary, and that we ought to develop procedures to rely on both information about kind and information about degree when evaluating causal claims in case studies. I have presented one plausible option for doing so. The procedures I have discussed are useful for any case study for which information is detailed enough and where a causal model is developed or tested. In principle, it can be applied whenever mechanisms between a set of sufficient causes and their associated outcome are studied in a qualitative fashion.

The procedure is different from other recent advances in set-theoretic analysis (Baumgartner 2009, 2013) in that it begins with a causal structure hypothesized by the researcher. Theory development is certainly possible to evaluate using my procedure. But I have not provided tools to automatically develop structures from data. Instead, I have proposed to use graphical analysis and a few simple rules to distil from such structures lists of sufficiency statements which should hold if the structure is correct. Once a list has been determined, I have suggested using consistency formula to determine whether the elements of the list are acceptably consistent with sufficiency by established set-theoretic standards. To summarize, the analysis proceeds in four steps.

First, graphically examine the hypothesized structure, apply the two rules I emphasized in the third section, and construct the according sufficiency statements.

Second, reformulate the constructed sufficiency statements in terms of an appropriate consistency measure.

Third, compile the reformulated statements under a minimum operator and place them in an expression demanding that the compiled is larger than or equal to a satisfactory consistency level ρ subject to a QCC.

Fourth, input the case’s fuzzy-set membership scores and compute whether both the compiled inequality and the QCC are satisfied. If both are satisfied, the data supports the hypothesised structure. If the QCC is violated, the case is qualitatively inconsistent with the hypothesized structure. And if the compiled inequality is violated, that is, if consistency is not at the acceptable level ρ, the case is quantitatively inconsistent with the structure.

Finally, I have probed into some complications that may arise from doing fuzzy-set case studies. In particular, I have discussed how to deal with equifinal causal structures and studies of multiple cases.

Where a structure is equifinal, there are two options. Either follow the four steps mentioned previously for each path in the structure, while subjecting each of the multiple paths to its own PQCC. This will result in multiple compiled expression, subject to each of their PQCC. Only one of these inequalities need to hold for a case to support the structure. This will require some but not all information to be collected for the case. Or follow the three first steps mentioned previously and compile the compiled expressions for each path under a maximum operator, demanding that the total expression is larger than or equal to ρ. Proceed to the fourth step, subjecting the expression to a single QCC demanding that at least one PQCC is true for the case (again using a maximum operator). This requires all information to be collected for the case.

Where multiple cases are analyzed, I propose relying on a consistency measure that can handle multiple cases. Ragin’s (2006) consistency measure designed for QCA is one plausible option. After this adjustment, the procedure can proceed as for single cases. The main difference is that, when multiple cases are analyzed, the QCC (or multiple PQCCs) may be lifted if the researcher is satisfied that some cases might qualitatively contradict the structure so long as the structure as a whole is consistent at least at the level of ρ.

My approach is quite flexible, but it is subjected to some limitations as I have presented it. In particular, it only applies to structures where start-up conditions are sufficient for outcomes. This is intentional. Standard case study methods like process tracing and congruence analysis are concerned with causal mechanisms. And a concern with how causal mechanisms produce their associated outcomes requires an interest in sufficiency. Furthermore, I have restricted my attention mainly to consistency and cases where both input conditions and outcome are present. However, as I have argued, the procedure is amendable to evaluations of coverage and to the study of cases where neither input conditions nor outcomes are present.

I have not touched on the issue of calibration. But I will end on an important point related to it. My procedure is compatible with the use of quantitative indicators—interval and ratio scales—used directly to form mechanism elements (called the direct method of calibration in the QCA literature see Ragin 2008). But this type of information will often not be available for case study research. My procedure is also compatible with more coarse coding of cases according to qualitative indicators such as assessments from texts or interviews. For instance using Ragin’s (2000) linguistic qualifiers to assign a fuzzy membership (called the indirect method of calibration see Ragin 2008).

A particular complication occurs if empirical measures are very coarse, or if they are not equally coarse. In these instances, calibration may create differences between the membership assigned to a case and the case’s real, unobserved membership. As a result, quantitative consistency or inconsistency might be measurement artefacts. Dealing with the artificial consistencies that may be created from the use of crisp sets in traditional case studies, by revealing quantitative inconsistencies, is one way of understanding the advantage of fuzzy-set case studies. But the consequences of coarseness remain an issue.

The unequal coarseness instance may require some explication to be appreciated. For instance, say a five-value fuzzy set assigns a fuzzy score of .75 to a case whose real, unobserved fuzzy score is .85, and a seven-value fuzzy set assigns a fuzzy score of .83 to the case whose real, unobserved fuzzy score again is .85. Further, say the five-value fuzzy set is the terminal vertex for an edge in a hypothesized structure and the seven-value fuzzy set is the sole initial vertex for that same edge. An evaluation of the relationship between the two results in a consistency of ( $m i n [.83, .75] / .83$ ) .90, even though the real consistency is a perfect 1. The inconsistency only results from coarseness.

One good solution to the coarseness problem would be to recalibrate sets involved in individual (not compiled) statements. This would mean one of two things.

First, the seven-value fuzzy set could be coarsened. In the example, this would result in perfect consistency as the case would be assigned a score of .75 in the coarsened fuzzy set (the closest score of a five-value fuzzy set). This would mean disregarding information and does not solve problems related to coarseness as such. But it can avoid the problem of unequal coarseness and can do no worse than contemporary case studies that disregard all quantitative information by default.

Second, coarse sets could be conceptualized as placing cases within appropriate bounds of more fine-grained, but unobserved, sets. This would address the coarseness problem but would necessitate multiple evaluations of each statement corresponding to the bounds. In my example, the five-value fuzzy set assigning .75 would be transformed to assign both .83 and .67 (the closest scores of a seven-value fuzzy set). Consistency evaluations would then be made for both these values, resulting in scores of 1 and .81, respectively. The conclusion would be that perfect consistency is a possibility thought, naturally, we do not have the information to be sure that consistency is above .81.

Beyond these intuitions, a full discussion is better left for a separate paper (see also Maggetti and Levi-Faur 2013, and the references herein). Additionally, note that fuzzy set case studies do not create the coarseness problem. In fact, they have the attractive property of diminishing the risk of accepting artificial consistency. But they do highlight that it is a problem and a fruitful venue for future research.

I will end by strongly emphasizing that my procedure is a complement to, not a substitute for, the efforts involved with doing case studies and making claims on mechanisms convincing. I have said that contextual and substantive knowledge goes into classifying cases in traditional case studies. This process does not become less cumbersome in fuzzy-set case studies. In fact, because the analysis becomes attentive to both qualitative and quantitative information the process is likely to become more cumbersome as more information is required. But the additional work comes with a benefit as my procedure permits case study researchers to use all available information throughout their analyses.

Footnotes

Acknowledgment

I thank Damien Bol, Ingo Rohlfing, Markus Siewert, and two anonymous reviewers for helpful comments and suggestions.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Supplementary Material

The online [appendices/data supplements/etc.] are available at .

Notes

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