FRAMING ANALYTIC TRANSPARENCY IN CAUSAL-ORIENTED MIXED-METHODS RESEARCH

Analytic transparency in C-MMR is daunting because it is multilayered and the layers are interdependent. Although we recognize that each project presents its own challenges, we believe that scholars using C-MMR should consider the following three basic questions in thinking about analytic transparency:

1. 1. What is the causal relationship being explored?

2. 2. What is the intended analytic contribution(s) of each method?

3. 3. What is the empirical contribution of each method?

The nature of the causal relationship being studied is highly project-specific. However, the last two questions raise general issues regarding the role of case studies in triangulation- and integration-based C-MMR, and the varying role of case studies in these approaches suggests different concerns for analytic transparency.

CASE STUDIES IN TRIANGULATION-BASED CAUSAL-ORIENTED MIXED-METHODS RESEARCH

Triangulation-based C-MMR uses large-N and small-N data to study independently the same theoretical relationship. ^{Footnote 1} The basic idea is that separate analytic cuts at understanding the same phenomenon will yield more valid results. The goal is convergence among different methods so that using methods with distinct ontological and epistemological assumptions is a strength; indeed, the more divergent the methods, the better. Perhaps the most common example uses large-N analysis to establish a robust relationship between X and Y and detailed process-tracing case studies to probe the X/Y relationship in specific settings. Here, the quantitative and qualitative analyses typically focus on different types of observable implications of the X/Y relationship. The large-N work centers on outcomes and the small-N work focuses on processes, but the goal in both types of research is to make claims about X as a cause of Y.

The extensive literature on process tracing clarifies the role of case studies in establishing a causal relationship (Bennett Reference Bennett, Box-Steffensmeiser, Brady and Collier2008; Reference Bennett, Brady and Collier2010; Bennett and Checkel Reference Bennett and Checkel2015; Collier Reference Collier2011; Mahoney Reference Mahoney2010; Reference Mahoney2012). Table 1 re-creates a well-known typology of four empirical tests used in process tracing (Bennett Reference Bennett, Brady and Collier2010; Collier Reference Collier2011; Van Evera Reference Van Evera1997). Doubly decisive tests provide the strongest evidence, and passing them confirms the hypothesized X/Y relationship in the case and eliminates rival hypotheses about that case. The weakest evidence comes from straw-in-the-wind tests in which passing the test only confirms the relevance of the hypothesis but does not eliminate the rivals. Hoop tests and smoking-gun tests fall in between these extremes. Passing a hoop test affirms the relevance of the hypothesized X/Y relationship in the case, whereas failing it eliminates the hypothesis. Passing a smoking-gun test confirms the relationship but does not eliminate its rivals. These tests are not mutually exclusive; the art of process tracing is combining different types of tests to build a persuasive account of causal processes in a single case. ^{Footnote 2} By using this language, scholars can clarify the intended contributions of case studies in triangulation-based C-MMR and how these insights differ from (or clarify) the findings of the large-N component of their work.

Table 1

Process Tracing for Causal Inference: Necessary and Sufficient X/Y Relationships

Note: Reproduced from Collier Reference Collier2011.

Analytic transparency requires identifying not only the intended contributions of each method but also the value added of each component to the empirical findings. Although table 1 is framed as “tests” for causal inference, they are not about a causal X/Y relationship across cases. In this regard, it is telling that when scholars explain these tests, they commonly resort to analogies of detectives (often Sherlock Holmes) trying to solve a specific crime with multiple suspects (Collier Reference Collier2011) but without concern for making claims about other crimes. By contrast, in C-MMR, scholars are interested not only in whether X causes Y in a specific case (the “whodunit”) but also in the X/Y relationship across cases (Weller and Barnes Reference Weller and Barnes2014). Whether we can make a general inference about X as a cause of Y based on process-tracing evidence from a single case (or set of cases) is a separate question.

The empirical value added of case studies in triangulation-based C-MMR significantly depends on the nature of the underlying causal claim. If we expect that X is necessary for Y, then a case that demonstrates that Y occurs without X can significantly enhance our findings. If X is hypothesized to be sufficient for Y, finding X without Y in a single case is similarly telling. However, if the posited X/Y relationship is probabilistic, then assessing the empirical contributions of case studies is trickier. With probabilistic relationships, we expect to find individual cases in which X does not cause Y. Therefore, it is consistent with the hypothesized relationship to find cases in which Y emerges in the absence of X and those in which we observe X without Y. As a result, even if we had doubly decisive process-tracing evidence in specific cases, generalization is difficult, and case-study evidence alone reveals little about the X/Y relationships implied by a probabilistic relationship across cases. Humphreys and Jacobs (Reference Humphreys and Jacobs2015) demonstrated this problem in a formal Bayesian approach by showing that our belief in the truth of a given hypothesis could increase, decrease, or remain unchanged depending on the number and type of cases chosen for the small-N analysis.

Case selection, of course, also is important. For example, if we believe that a particular case is “most likely” for finding a causal relationship, and we find doubly decisive evidence that X does not cause Y in that case, then the case study weakens our belief in the hypothesis to some degree. (However, a small number of case studies cannot eliminate the hypothesis that X causes Y in a probabilistic sense.) The more general point seems obvious but bears emphasis: analytic transparency requires attention to how cases are selected and how inferences from a case study affect our understanding of the general causal relationship (see Humphreys and Jacobs Reference Humphreys and Jacobs2015 for one approach).

CASE STUDIES AND INTEGRATION-BASED CAUSAL-ORIENTED MIXED-METHODS RESEARCH

Integration-based C-MMR also uses multiple methods to identify a causal effect, but it leverages large-N and small-N work differently (Dunning Reference Dunning2012; Harding and Seefeldt Reference Harding, Seefeldt and Morgan2013; Seawright Reference Seawright2016). Instead of using each method to provide separate evidence of the X/Y relationship, it envisages a division of labor among methods in which the large-N work estimates causal effects whereas the smaller-N work probes the validity of the large-N work’s underlying assumptions. Here, combining methods improves our confidence in the underlying X/Y relationship not because the findings of the large-N and small-N analyses converge but rather because the small-N work demonstrates that the large-N work satisfies the requisites for causal inference.

The potential-outcomes model provides language for fleshing out this approach’s underlying intuitions and highlighting key contributions of case studies (Imbens and Rubin Reference Imbens and Rubin2015). The crux of the potential-outcomes model is that causal inference requires solving a missing-data problem. The causal effect is the difference between the outcome in the treatment condition (Y_treatment) and the outcome in the control condition (Y_control). For any case or unit of analysis, it is only possible to observe a case in either the treatment or the control condition. To identify the causal effect, we must estimate what the outcome would have been for cases that received the treatment if they had not received the treatment and the outcome for nontreatment cases had they received the treatment. However, we cannot observe both states of the world. Solving the missing-data problem typically requires identifying and comparing a treatment case and a control (or nontreatment) case. The comparison rests on two assumptions: Strongly Ignorable Treatment Assignment (SITA) and Stable Unit Treatment Value Assumption (SUTVA). ^{Footnote 3}

SITA allows us to “ignore” how units were assigned to treatment because the assignment mechanism is unrelated to any factors that would be correlated with both the treatment and the potential outcome (but not the observed outcome because the essence of a causal effect is that the treatment will affect the observed outcome). ^{Footnote 4} There are three basic ways to satisfy SITA: (1) using randomization via experimental manipulation; (2) taking advantage of a natural or artificial intervention (i.e., a natural experiment or a regression discontinuity); and (3) modeling the assignment mechanism (i.e., creating “as if” randomization or conditional independence). Regardless of which method is used, the credibility of causal inference depends on the assumption’s plausibility.

The second major assumption of the potential-outcomes approach—SUTVA—encompasses two assumptions needed to make a causal inference based on a relatively simple comparison between the treatment and control groups. The first aspect of SUTVA is noninterference between units, which requires that the treatment received by the treatment group does not affect units in the control group. The second aspect is that the treatment is consistent within groups, which means that subjects and/or units grouped together in the analysis received equivalent treatments. Under this framework, the role of cases studies in integration-based C-MMR is providing evidence related to SITA and SUTVA in a particular project.

As a practical matter, the large-N or experimental work in integration-based C-MMR must already provide a reasonable basis for drawing causal inferences because if it is accepted that the large-N work fails SITA or SUTVA, then there is little reason to conduct any follow-up work. If the assumptions of causal inference seem plausible on their face, then case studies can provide further evidence regarding whether the assumptions are satisfied. This also implies that, in the language of table 1, researchers will be most interested in hoop tests, which reject the hypothesis that a particular assumption is met in a given case (or set of cases).

Particularly promising avenues of inquiry for small-N analysis in integration-based C-MMR center on treatment assignment, treatment spillovers, and treatment consistency. Regarding treatment assignment, case studies can probe the presence or absence of confounders associated with both treatment assignment and potential outcomes. For this purpose, researchers will want to select cases intentionally knowing the values of both treatment and outcome and to look for unmeasured variables correlated with both treatment assignment and the observed outcome (Dunning Reference Dunning2012).

Case studies also can help researchers understand whether SUTVA has been met by focusing on two crucial issues: noninterference between units and treatment consistency. Noninterference can be violated even if the treatment is assigned randomly, and case studies can be useful in addressing this concern (Miguel and Kremer Reference Miguel and Kremer2004; Nickerson Reference Nickerson2008). Regarding treatment spillover, researchers must identify cases in which the treatment applied to one (or more) of the cases also would have affected the cases that did not receive the treatment. Treatment consistency relates to variation in the treatment among those assigned to the same experimental group. For example, in an experiment studying the effect of aspirin on headaches, SUTVA is violated if some subjects take different quantities of aspirin even though they were all assigned to take the same amount. For studying treatment consistency, researchers should select multiple cases that are expected to have received similar treatments and then investigate whether the units, in fact, received the same treatment.

In summary, case studies can play a range of roles in integration-based C-MMR. Each role implies different case-selection strategies and is quite distinct from how case studies are used in triangulation-based C-MMR. For researchers addressing analytic transparency, it is crucial to locate the logic of the case studies and case-selection strategies within the broader logic of the underlying type of C-MMR being used.