Defining causal inference in CCMs
CNA is one method within a class of CCMs used to model complex patterns of conditions hypothesized to contribute to an outcome within a set of data. CCMs search for causal relations as defined by a regularity theory of causality, according to which a cause is a “difference-maker” of its effect within a fixed set of background conditions. More specifically, X is a cause of Y if there exists a fixed configuration of background factors Φ such that, in Φ, a change in the value of X is systematically associated with a change in Y. If X does not make a difference to Y in any Φ, X is redundant to account for Y and, thus, not a cause of Y. The most influential theory defining causation along these lines is Mackie’s INUS-theory , with refinements by Graßhoff and May  and Baumgartner . An INUS condition of an outcome Y is an insufficient but necessary part of a condition that is itself unnecessary but sufficient for Y. To use a common example for illustrating INUS conditions: not every fire is caused by a short circuit—fires can also be started by, for example, arson or lightning. However, a short circuit in combination with other conditions—e.g., presence of flammable material and absence of a suitably placed sprinkler—is sufficient for a fire. In this example, the short circuit is an INUS condition: it is a necessary, but itself insufficient, part of a sufficient, but itself unnecessary, condition for a fire. This particular causal path to a fire includes the combination of three specific conditions: presence of a short circuit, presence of flammable material, and absence of a sprinkler. All three of these conditions are difference-makers, for if one of them is missing, the fire does not occur along this causal path.
Regularity theories account for the Boolean properties of causation, which encompass three dimensions of complexity. The first is conjunctivity: to bring about an outcome, several conditions must be jointly present. For example, in a study of high-performance work practices and frontline health care worker outcomes, Chuang and colleagues  found that no single high-performance work practice was alone sufficient to produce the outcome of high job satisfaction. Instead, a configuration consisting of creative input, supervisor support, and team-based work practices together accounted for 65% of highly satisfied frontline health care workers . Chuang and colleagues identified a second configuration that also led to high job satisfaction: supervisor support, incentive pay, team-based work, and flexible work . Both configurations resulted in high job satisfaction independently of each other. These configurations illustrate equifinality, a second dimension of complexity where different paths can lead to the same outcome. The third dimension of complexity is sequentiality: outcomes tend to produce further outcomes, propagating causal influence along causal chains. For instance, high job satisfaction of health care workers may, in turn, promote patient satisfaction .
Why use CCMs in implementation research? CCMs study different properties of causal structures than RAMs and thus are appropriate for exploring different types of hypotheses. RAMs examine statistical properties characterized by probabilistic or interventionist theories of causation. In the probabilistic framework, X is a cause of Y if, and only if, the probability of Y given X is greater than the probability of Y alone and there does not exist a further factor, Z, that explains (i.e., neutralizes) the probabilistic dependence between X and Y [29, 30]. In the interventionist framework, X causes Y if, and only if, there exists an intervention on X that changes the outcome Y while causes on other paths to Y are fixed. The interventionist theory of causation is counterfactual in that a case cannot simultaneously “receive” and “not receive” an intervention; instead, the intervention model maps possible values of Y onto possible values of X, focuses on how variables X and Y relate to one another, and generates average treatment effects over a population [11, 31].
Conversely, CCMs trace Boolean properties of causal structures as described by regularity theories of causation, according to which X is a cause of Y if, and only if, X is an INUS condition of Y (see INUS definition above) [11, 24]. CCMs study implication hypotheses that link specific values of variables as “X = χi is (non-redundantly) sufficient/necessary for Y = γi” [11, 14]. In this way, CCMs, including CNA, model the effect of conditions (e.g., high degree of X) on outcomes. This is a fundamentally different vantage point than the one adopted by RAMs which examine covariation hypotheses that link variables. Further, CCMs are case-oriented methods, in which observations consist of bounded, complex entities (e.g., organizations) that are considered as a whole . A case-based unit of analysis differs from the approach taken in RAMs, where cases are deconstructed into a series of variables, and estimates represent the net effect of a variable for the average case. As CNA and other CCMs employ a case-based approach and thus can be used to identify which interventions work in an array of contexts, they present opportunities for implementation and health services research questions in particular.
Different types of CCMs
While CCMs have a common regularity theoretic foundation, different CCMs rely on different a priori conceptions of outcome and causal factors and build causal models in different ways. For example, Qualitative Comparative Analysis (QCA), in its standard implementation that uses the Quine-McCluskey (QMC) algorithm [33, 34] requires identification of exactly one factor as outcome. It begins by identifying maximal sufficient and necessary conditions of the outcome, which are subsequently minimized using standard inference rules from Boolean algebra to arrive at a redundancy-free solution composed of INUS conditions of the outcome . However, the QMC algorithm was not originally designed for causal inference. One consequence is that the non-observation of cases instantiating empirically possible configurations of the analyzed factors, also known as limited diversity, forces QMC to draw on counterfactual reasoning that goes beyond available data, sometimes requires assumptions contradicting the very causal structures under investigation , and regularly fails to completely eliminate redundancies in the presence of noise . Moreover, QMC has built-in protocols for ambiguity reduction when multiple models fit the data equally well. Viable models are often eliminated to reduce ambiguity without justification, which is problematic for causal discovery .
Advantages of using CNA
Coincidence Analysis (CNA) is a new addition to the family of CCMs [37, 38]. It uses an algorithm specifically designed for causal inference, thus avoiding the problems mentioned above. In particular, it does not build causal models by means of a top-down approach that first searches for maximal sufficient and necessary conditions and then gradually minimizes them using the QMC algorithm. Rather, CNA employs a bottom-up approach that first tests single factor values for sufficiency and necessity, and then tests combinations of two, three, etc. [13, 14]. All sufficient and necessary conditions revealed by this approach are automatically minimal and redundancy-free.
Additionally, CNA is designed to treat any number of factors as endogenous and is therefore capable of analyzing causal chains, or common-cause structures . For example, Baumgartner and Epple (2014) found in a Swiss policy analysis that certain population, economic, or political characteristics in some areas led to a higher rate of prejudice and, in turn, discriminatory policy . Analyzing causal chains may be advantageous if, for example, an intervention A occurs as a result of other factors but is not the ultimate outcome of interest. Identifying the full causal model, including which factors produce A on the path to the ultimate outcome of interest, is valuable when seeking to understand causal complexity. CNA is the only member of the CCM family that builds and evaluates models representing causal chains.