1 Introduction
Configurational comparative methods (CCMs) subsume techniques for the identification of complex causal dependencies in configurational data using the theoretical framework of Boolean algebra and its various extensions (Rihoux and Ragin 2009). For example, Qualitative Comparative Analysis (Ragin 1987, 2000, 2008)—hitherto the most prominent representative of CCMs—has been applied in areas as diverse as business administration (e.g., Chung 2001), environmental science (Vliet et al. 2013), evaluation (Cragun et al. 2014), political science (Thiem 2011), public health (Longest and Thoits 2012) and sociology (Crowley 2013). Besides three stand-alone programs based on graphical user interfaces, three R packages for QCA are currently available, each with a different scope of functionality: QCA (Thiem and Duşa 2013b,c; Dusa and Thiem 2014), QCA3 (Huang 2014) and SetMethods ((Quaranta 2013); an add-on package to (Schneider and Wagemann 2012)).
A novel technique called Coincidence Analysis (CNA) has recently joined the family of CCMs (Baumgartner 2009a,b; 2013b). Like QCA, CNA searches for rigorously minimized sufficient and necessary conditions of causally modeled outcomes, and it implements the same regularity theory of causation as QCA, that is, the theory most prominently advanced by Mackie (1974). Contrary to QCA, however, CNA can treat any number of factors in a processed data set as endogenous (outcomes), and it does not eliminate redundancies from sufficient and necessary conditions by means of Quine-McCluskey optimization (Quine 1959; McCluskey 1965), but by means of an optimization algorithm that is custom-built for causal modeling. As a direct consequence of these differences, CNA can identify common cause and causal chain structures. Moreover, the algorithm does not need to be told which factors are endogenous and which ones exogenous; it can infer that from the data. What is more, limited data diversity does not force CNA to resort to counterfactual additions to the data. Finally, while the QCA programs fs/QCA (Ragin and Davey 2014), fuzzy (Longest and Vaisey 2008), Tosmana (Cronqvist 2011) and Kirq (Reichert and Rubinson 2014) often fail to find all data-fitting models (Thiem and Duşa 2013a; cf. Baumgartner and Thiem 2014; Thiem 2014b), the R implementation of CNA presented in this paper (cna; (Ambuehl et al. 2015)) not only ensures—as does QCA—that all single-outcome models are identified but additionally recovers the whole space of multiple-outcome models that fit the data.
After an introduction to the theoretical and algorithmic background of CNA, we demonstrate the potential of the cna R package by means of an artificial and a real-life data set. In the final section, we outline planned enhancements of cna that will further increase its applicability.
2 Theoretical and algorithmic background
CCMs search for causal dependencies as defined by so-called regularity theories of causation, whose development dates back to David Hume (1711-1776) and John Stuart Mill (1806-1873). By implementing techniques of Boolean algebra, modern regularity theories spell out the notion of causal relevance in terms of redundancy-free (minimized) sufficiency and necessity relations among the elements of analyzed sets of factors (Mackie 1974; Graßhoff and May 2001; Baumgartner 2008, 2013a).
The crucial component of the regularity theoretic definiens of causal relevance is the notion of a minimal theory. A minimal theory of a factor \(Z\) is a minimally necessary disjunction of minimally sufficient conditions of \(Z\). A conjunction \(\Phi\) of coincidently instantiated factors, i.e. \(Y_{1} {\scriptstyle \ast}Y_2{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_{n}\), is a minimally sufficient condition of \(Z\) if, and only if (iff), \(\Phi\) is sufficient for \(Z\) \(\left(\Phi \rightarrow Z\right)\), and there exists no proper part \(\Phi^{\prime}\) of \(\Phi\) such that \(\Phi^{\prime} \rightarrow Z\). A proper part \(\Phi^{\prime}\) of \(\Phi\) is the result of eliminating at least one conjunct from \(\Phi\). A disjunction \(\Psi\) of minimally sufficient conditions, i.e. \(\Phi_{1} + \Phi_2 + \ldots + \Phi_{n}\), is a minimally necessary condition of \(Z\) iff \(\Psi\) is necessary for \(Z\) \(\left(Z\rightarrow \Psi\right)\), and there exists no proper part \(\Psi^{\prime}\) of \(\Psi\) such that \(Z \rightarrow \Psi^{\prime}\). A proper part \(\Psi^{\prime}\) of \(\Psi\) is the result of eliminating at least one disjunct from \(\Psi\). Overall, a minimal theory of \(Z\) has the following biconditional form: \(\Psi\leftrightarrow Z\) (where \(\Psi\) is an expression in disjunctive normal form and \(Z\) is a single factor).
A minimal theory represents the causally interpretable dependencies of sufficiency and necessity among the factors contained in a data set \(\delta\). That is, causal relevance can be defined in terms of membership in a minimal theory, or more specifically: a factor \(A\) is causally relevant to a factor \(B\) in a data set \(\delta\) iff \(\delta\) entails a minimal theory \(\Psi\leftrightarrow B\) such that \(A\) is contained in \(\Psi\).
CNA aims to infer minimal theories from \(\delta\) by, first, identifying sufficient and necessary conditions in \(\delta\) and by, second, minimizing those conditions. To the latter end, CNA tests the redundancy of factors by eliminating them from sufficient and necessary conditions and checking whether the remaining conditions are still sufficient and necessary, respectively. More specifically, to determine whether a sufficient condition \(Y_1{\scriptstyle \ast}Y_2{\scriptstyle \ast}\ldots{\scriptstyle \ast}Y_h\) of a factor \(Z\) is minimally sufficient, CNA systematically eliminates conjuncts from \(Y_1{\scriptstyle \ast}Y_2{\scriptstyle \ast}\ldots{\scriptstyle \ast}Y_h\). For each conjunction that results from such an elimination, say for \(Y_2{\scriptstyle \ast}Y_3{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_h\), CNA then parses the processed data \(\delta\) to check whether \(\delta\) contains \(Y_2{\scriptstyle \ast}Y_3{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_h\) in combination with the negation of \(Z\), i.e. \(z\). If \(\delta\) does not contain such a configuration, \(Y_2{\scriptstyle \ast}Y_3{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_h\) is itself sufficient for \(Z\), which means that \(Y_1\) is redundant. CNA then proceeds to eliminate the next conjunct from \(Y_2{\scriptstyle \ast}Y_3{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_h\) and tests for further redundancies, until no more redundancies are found. By contrast, if \(\delta\) contains the configuration \(Y_2{\scriptstyle \ast}Y_3{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_h\) in combination with \(z\), \(Y_1\) makes a difference to \(Z\) and is, thus, not redundant. Accordingly, CNA re-adds \(Y_1\) to \(Y_2{\scriptstyle \ast}Y_3{\scriptstyle \ast}\ldots {\scriptstyle \ast}Y_h\) and proceeds to eliminate \(Y_2\), and so forth.
Similarly, to determine whether a complex necessary condition \(\Phi_1 + \Phi_2+\ldots + \Phi_h\) of a factor \(Z\) is minimally necessary, CNA systematically eliminates disjuncts from \(\Phi_1 + \Phi_2+\ldots + \Phi_h\) and checks for every resulting disjunction, say for \(\Phi_2 + \Phi_3+\ldots + \Phi_h\), whether it is still necessary for \(Z\), i.e. whether \(\delta\) contains a configuration featuring \(Z\) without any of the disjuncts in \(\Phi_2 + \Phi_3+\ldots + \Phi_h\). If \(\delta\) does not contain such a configuration, \(\Phi_2 + \Phi_3+\ldots + \Phi_h\) is still necessary for \(Z\), which means that the eliminated disjunct \(\Phi_1\) is redundant. Next, \(\Phi_2 + \Phi_3+\ldots + \Phi_h\) is tested for further redundancies, until no more redundancies are found.
CNA does not presuppose that certain factors in \(\delta\) can be identified as endogenous prior to applying CNA. In principle, CNA is designed to recover and rigorously minimize all relationships of sufficiency and necessity among the factors in \(\delta\). In practice, however, it is often known from the outset which factors are exogenous and which endogenous. What is more, often enough theoretical knowledge is available to order the factors in \(\delta\) causally, where a causal ordering is a relation \(Y_i\prec Y_j\) entailing that \(Y_j\) cannot be a cause of \(Y_i\) (e.g., because \(Y_i\) is instantiated temporally before \(Y_j\)). That is, an ordering excludes certain causal dependencies but does not stipulate any. Accordingly, in addition to a data set \(\delta\), CNA may be given a subset \(\mathsf{W}\) of endogenous factors (i.e. possible effects) in \(\delta\) and an ordering \(\prec\) over the factors in \(\delta\) as input. Minimally sufficient and necessary conditions are then calculated for the members of \(\mathsf{W}\) in accordance with \(\prec\) only.
Recovered minimal theories of the elements of \(\mathsf{W}\) are issued as so-called atomic solution formulas. If CNA finds an atomic solution formula \(\Psi_1\leftrightarrow Z_i\) and an atomic solution formula \(\Psi_2\leftrightarrow Z_j\) such that \(Z_i\neq Z_j\) and \(\Psi_1\) and \(\Psi_2\) have at least one factor in common or \(Z_i\) appears in \(\Psi_2\) or \(Z_j\) appears in \(\Psi_1\), then CNA builds the complex solution formula \((\Psi_1\leftrightarrow Z_i) {\scriptstyle \ast}(\Psi_2\leftrightarrow Z_j)\). Configurational data regularly underdetermine their own causal modeling, with the effect that multiple atomic and complex solution formulas fit the data equally well. In cases of such model ambiguities, CNA provides an overview over the whole model space by returning all data-fitting solution formulas.
As causally analyzed data tend to be noisy, that is, confounded by uncontrolled (unmeasured) causes of endogenous factors, it often happens that no configuration of factors is strictly sufficient or necessary for a given \(Z \in\mathsf{W}\). To still extract some (tentative) causal information from such data, Ragin (2006) has introduced so-called consistency and coverage measures (with values between 0 and 1). Consistency reproduces the degree to which the behavior of a given outcome obeys a corresponding sufficiency or necessity relationship (or a whole solution formula), whereas coverage reproduces the degree to which a sufficiency or necessity relationship (or a whole solution formula) accounts for the behavior of the corresponding outcome. If data cannot be causally modeled with maximal consistency and coverage scores, CNA invites its users to gradually lower consistency and coverage thresholds until solution formulas can be built.
The cna package by Ambuehl et al. (2015) implements the methodological protocol of CNA as sketched above. For more details on the background assumptions of CNA, its minimization algorithm, and its relation to other configurational methods such as QCA, we refer interested readers to Baumgartner (2009a).
3 Examples
Hypothetical data
In the following, we illustrate the main steps in using the cna package. First, we employ a hypothetical data set from Baumgartner (2009a) to investigate the causal dependencies among five factors hypothesized to constitute a causal structure behind the overall level of education in western democratic countries. These five factors are “strong unions” (\(U\); 1 = strong, 0 = not strong), “high level of disparity” (\(D\); 1 = high, 0 = not high), “strong left parties” (\(L\); 1 = strong, 0 = not strong), “high gross national product” (GNP; \(G\); 1 = high, 0 = not high) and “high level of education” (\(E\); 1 = high, 0 = not high). The data for eight countries are presented in Table 1.
| Case | \(U\) | \(D\) | \(L\) | \(G\) | \(E\) |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 0 | 1 |
| 3 | 1 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 1 | 0 | 1 |
| 5 | 0 | 1 | 1 | 1 | 1 |
| 6 | 0 | 1 | 1 | 0 | 1 |
| 7 | 0 | 0 | 0 | 1 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
The cna package comes with an integrated bundle of six data sets from
various areas of the social sciences. That bundle also includes the data
in Table 1 as the data frame d.educate.
Accordingly, the first step to causally model
Table 1 by means of CNA is to load the cna
package along with the d.educate data.
> library(cna)
> data(d.educate) The heart of the cna package is constituted by the cna() function.
It is the function that identifies and minimizes dependencies of
sufficiency and necessity in the data, which can be given to cna()
either in terms of a Boolean data frame or of a truth table as produced
by the truthTab() function. Essentially, truthTab() simply merges
multiple rows of a data frame featuring the same configuration into one
row, such that each row of the resulting truth table corresponds to one
determinate configuration. The number of occurrences (cases) and an
enumeration of the cases are saved as attributes n and cases,
respectively. As Table 1 does not contain
multiple rows with identical configurations, the application of
truthTab() is uncalled for and we can directly pass d.educate on to
cna(). Moreover, let us assume that we have no prior causal knowledge
about the underlying causal structure, such that we cannot additionally
supply a causal ordering. The following is the default output returned
by cna().
> cna(d.educate)
--- Coincidence Analysis (CNA) ---
Factors: U, D, L, G, E
Minimally sufficient conditions:
--------------------------------
Outcome D:
condition consistency coverage
L*u -> D 1.000 0.500
E*g*u -> D 1.000 0.250
Outcome E:
condition consistency coverage
L -> E 1.000 0.857
D -> E 1.000 0.571
G -> E 1.000 0.571
U -> E 1.000 0.571
Outcome G:
condition consistency coverage
d*E*u -> G 1.000 0.250
E*l -> G 1.000 0.250
Outcome L:
condition consistency coverage
D -> L 1.000 0.667
U -> L 1.000 0.667
E*g -> L 1.000 0.500
Outcome U:
condition consistency coverage
d*L -> U 1.000 0.500
d*E*g -> U 1.000 0.250
Atomic solution formulas:
-------------------------
Outcome E:
condition consistency coverage
D + G + U <-> E 1.000 1.000
G + L <-> E 1.000 1.000
Outcome L:
condition consistency coverage
D + U <-> L 1.000 1.000
Complex solution formulas:
--------------------------
condition consistency coverage
(D + G + U <-> E) * (D + U <-> L) 1.000 1.000
(G + L <-> E) * (D + U <-> L) 1.000 1.000First, cna() lists all minimally sufficient conditions of all factors
in d.educate, second, it reports the atomic solution formulas for the
factors that can be modeled as endogenous factors, and third, it
specifies the resulting complex solutions. All solution types come with
corresponding consistency and coverage scores. In case of
Table 1, these scores reach maximal values for
both atomic and complex solution formulas. Thus, the d.educate data
are as good as configurational data can possibly get.
The above results show that the causal structure generating
Table 1 features two endogenous factors,
viz. “strong left parties” (\(L\)) and “high level of education” (\(E\)).
Moreover, there is one atomic solution for \(L\) and there are two for
\(E\). Overall, cna() infers that the d.educate data can be modeled in
terms of the two complex structures depicted in
Figure 1.
Graph 1(left) represents a common cause
structure, in which “high level of disparity” (\(D\)) and “strong unions”
(\(U\)) appear as direct common causes of \(L\) and \(E\), whereas
Graph 1(right) depicts a causal chain such that
\(D\) and \(U\) are direct causes of \(L\), which in turn is a direct cause of
\(E\). As the data in Table 1 are optimal by all
standards of configurational modeling, there is no way to determine
which of these two structures is the true or correct one.
|
|
| Common cause structure | Causal chain |
d.educate data.
Real-life data
This subsection illustrates further functionalities of the cna package
on the basis of a real-life data set. To this end, we choose the study
by Lam and Ostrom (2010), who analyze the effects of an irrigation experiment in the
course of development interventions on the Indrawati River watershed in
the central hills of Nepal. Among other things, the authors investigate
the causal relevance of five exogenous factors on “persistent
improvement in water adequacy at the tail end in winter” (\(W\)), which
takes the value 1 when farmers at the tail end of the watershed
persistently receive the water they need in winter, and the value 0
otherwise. The five exogenous factors are: “continual assistance on
infrastructure improvement” (\(A\)), “existence of a set of formal rules
for irrigation operation and maintenance” (\(R\)), “existence of
provisions of fines” (\(F\)), “existence of consistent leadership” (\(L\)),
and “existence of collective action among farmers for system
maintenance” (\(C\)), for all of which the values 1 and 0 represent “yes”
and “no”, respectively. The relevant data set, which comprises 15 cases,
is included in the cna package as the data frame d.irrigate.
> data(d.irrigate)
> d.irrigate
A R F L C W
1 0 1 0 1 1 1
2 0 1 0 1 1 0
3 0 1 1 1 1 1
.. . . . . . .
<rest omitted> Lam and Ostrom (2010) assume that \(W\) is the ultimate outcome of the data-generating
causal structure. This background assumption can be given to cna() by
means of the argument ordering, which takes a list of character
vectors referring to the factors in the data frame as input. In case of
d.irrigate, the intended ordering is this:
ordering = list(c("A", "R", "F", "L", " C"), "W"). It determines that
\(W\) is causally located after \(A\), \(R\), \(F\), \(L\), \(C\), meaning that
the former cannot be a cause of the latter. Moreover, as this data frame
does not comprise all relevant factors for \(W\), it is no longer possible
to reach perfect coverage scores. In the following analysis, we set the
coverage threshold (cov) to \(0.9\) and extract only the complex
solution formulas from the resulting solution object via the function
csf().
> sol1 <- cna(d.irrigate, ordering = list(c("A", "R", "F", "L", "C"), "W"), cov = 0.9)
> csf(sol1)
condition consistency coverage
1 (a + f*R + L <-> C) * (A*C + a*f*r + F*R + l*R <-> W) 1.000 0.917
2 (a + f*R + L <-> C) * (A*C + a*l + F*R <-> W) 1.000 0.917
3 (a + f*R + L <-> C) * (A*C + C*f*r + F*R + l*R <-> W) 1.000 0.917
4 (a + f*R + L <-> C) * (A*C + C*l + F*R <-> W) 1.000 0.917
5 (a + f*R + L <-> C) * (a*f*r + A*L + F*R + l*R <-> W) 1.000 0.917
6 (a + f*R + L <-> C) * (a*f*r + A*R + F*R + l*R <-> W) 1.000 0.917
7 (a + f*R + L <-> C) * (a*l + A*L + F*R + l*R <-> W) 1.000 0.917
8 (a + f*R + L <-> C) * (a*l + A*R + F*R <-> W) 1.000 0.917
9 (a + f*R + L <-> C) * (A*L + C*f*r + F*R + l*R <-> W) 1.000 0.917
10 (a + f*R + L <-> C) * (A*L + C*l + F*R <-> W) 1.000 0.917
11 (a + f*R + L <-> C) * (A*R + C*f*r + F*R + l*R <-> W) 1.000 0.917
12 (a + f*R + L <-> C) * (A*R + C*l + F*R <-> W) 1.000 0.917This output of cna() shows that not only \(W\) can be modeled as an
endogenous factor, but also \(C\)—a fact which is overlooked by Lam and Ostrom (2010)
due to their reliance on QCA with its focus on single-outcome
structures. cna() returns one atomic solution formula for \(C\) and 12
for \(W\), yielding a total of 12 complex solution formulas that fare
equally well with respect to all parameters of model fit. According to
some of these models, the behavior of \(C\) and \(W\) is regulated by a
common cause structure; according to others, the underlying structure is
a causal chain.
To generate models for negative outcomes, cna() provides the argument
notcols, which takes a character vector of factors to be negated as
input. In the following analysis, we set cov to \(0.66\) and negate the
factors \(C\) and \(W\) (which then must also appear negatively in the
ordering). Moreover, we pass the solution object on to the print()
function, which provides arguments determining the number of solutions
to print (nsolutions) and what elements of the solution to print
(what). The what argument takes a character vector as input, where
"t" prints the truth table, "m" the minimally sufficient conditions,
"a" the atomic solution formulas, "c" the complex solution formulas,
and "all" returns all solution elements.
> sol2 <- cna(d.irrigate, ordering = list(c("A", "R", "F", "L", "c"), "w"),
notcols = c("C", "W"), cov = 0.66)
> print(sol2, nsolutions = 3, what = "a,c")
--- Coincidence Analysis (CNA) ---
Causal ordering:
A, R, F, L, c < w
Atomic solution formulas:
-------------------------
Outcome R:
condition consistency coverage
A*C + f*L <-> R 1.000 0.667
A*F + f*L <-> R 1.000 0.667
A*L + f*L + F*l <-> R 1.000 0.667
Outcome w:
condition consistency coverage
A*r + F*r <-> w 1.000 0.667
A*r + L*r <-> w 1.000 0.667
c*f + F*r <-> w 1.000 0.667
... (total no. of formulas: 6)
Complex solution formulas:
--------------------------
condition consistency coverage
(A*C + f*L <-> R) * (A*r + F*r <-> w) 1.000 0.667
(A*F + f*L <-> R) * (A*r + F*r <-> w) 1.000 0.667
(A*L + f*L + F*l <-> R) * (A*r + F*r <-> w) 1.000 0.667
... (total no. of formulas: 18)Finally, the condition() function provides assistance to inspect the
properties of sufficient and necessary conditions in a data frame, most
notably, of minimally sufficient and necessary conditions that appear in
solution formulas returned by cna(). It takes a vector of strings
specifying Boolean functions as input, reveals which configurations and
cases instantiate a given condition or solution, and lists consistency,
coverage, as well as unique coverage scores (cf. Ragin 2008 63–68).
Below, we investigate the properties of the first atomic solution for
outcome \(w\) from the previous analysis.
> condition("A*r + F*r <-> w", d.irrigate)
A*r+F*r -> w :
A*r+F*r w n cases
0 0 1 1
0 1 1 2
0 0 2 3,4
0 0 2 5,6
0 0 2 7,8
0 0 1 9
0 0 1 10
1 1 1 11
1 1 1 12
0 0 1 13
0 0 1 14
0 0 1 15
Consistency: 1.000 (2/2)
Coverage: 0.667 (2/3)
Total no. of cases: 15
Unique Coverages: A*r : 0.333 (1/3)
F*r : 0.333 (1/3)The first two columns of the table returned by condition() indicate
the configurations instantiating (1) and not instantiating (0) the
disjunction \(A{\scriptstyle \ast}r + F{\scriptstyle \ast}r\) and the
outcome \(w\), respectively. The third column specifies how many cases in
the associated data feature a corresponding configuration, and the forth
column lists these cases. According to the above output, hence,
\(A{\scriptstyle \ast}r + F{\scriptstyle \ast}r\) covers the instances of
\(w\) in cases 11 and 12 and leaves the occurrence of \(w\) in case 2
uncovered. Consequently, the overall solution coverage is \(\frac{2}{3}\),
with each disjunct uniquely covering one of the instances of \(w\).
4 Summary and outlook
We have presented cna, an R package implementing Coincidence Analysis (CNA), which is a method for the identification of multi-outcome structures in configurational data. CNA not only differs from QCA—the standard method of configurational causal modeling—by relaxing the single-outcome restriction but also by not drawing on Quine-McCluskey optimization for the elimination of redundancies from sufficient and necessary conditions. Instead, CNA employs its own minimization algorithm that is custom-built for causal modeling purposes.
At this stage of development, cna still requires bivalent variables. Planned future enhancements include the capability to process multivalent factors that generate crisp sets (Thiem 2013) and bivalent factors with fuzzy sets (Smithson and Verkuilen 2006). Possibilities to merge these constructs in multivalent factors with fuzzy sets, as has recently been suggested in the context of QCA (Thiem 2014c), will be explored as well. In this connection, aspects of alternative procedures proposed in the context of minimization with fuzzy sets may be incorporated where appropriate (Eliason and Stryker 2009). Finally, functionality for sensitivity diagnostics that facilitates robustness tests is envisaged (Thiem 2014a).
Complex causal structures are communicated most effectively to readers of scientific articles in the form of graphs rather than formulas. This is all the more true for multivalent factors. In this regard, functionality that translates cna solutions into corresponding graphs enjoys high priority on the list of future enhancements.
Acknowledgment
This work was generously supported by the Swiss National Science Foundation, grant number PP00P1_144736.
CRAN packages used
QCA, SetMethods, cna
CRAN Task Views implied by cited packages
Note
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