Article
A Unifying Frameworkfor Causal Analysisin Set-TheoreticMultimethod Research
Ingo Rohlfing1 and Carsten Q. Schneider2
Abstract
The combination of Qualitative Comparative Analysis (QCA) with processtracing, which we call set-theoretic multimethod research (MMR), is steadilybecoming more popular in empirical research. Despite the fact that bothmethods have an elected affinity based on set theory, it is not obvious how awithin-case method operating in a single case and a cross-case methodoperating on a population of cases are compatible and can be combined inempirical research. There is a need to reflect on whether and how set-theoretic MMR is internally coherent and how QCA and process tracingcan be integrated in causal analysis. We develop a unifying foundation forcausal analysis in set-theoretic MMR that highlights the roles and interplay ofQCA and process tracing. We argue that causal inference via counterfactualson the level of single cases integrates QCA and process tracing and assignsproper and equally valuable roles to both methods.
Keywords
causal mechanisms, causal inference, counterfactuals, mixed methods, pro-cess tracing, Qualitative Comparative Analysis, causal explanation
1 BIGSSS, University of Bremen, Jacobs University Bremen, Bremen, Germany2 Department of Political Science, Central European University, Budapest, Hungary
Corresponding Author:
Ingo Rohlfing, University of Bremen, Jacobs University Bremen, Mary-Somerville-Str. 9, Bremen
28359, Germany.
Email: [email protected]
Sociological Methods & Research2018, Vol. 47(1) 37-63ª The Author(s) 2016
Reprints and permission:sagepub.com/journalsPermissions.nav
DOI: 10.1177/0049124115626170journals.sagepub.com/home/smr
Introduction
Multimethod research (MMR) combining regression analysis with process
tracing, also known as nested analysis (Lieberman 2005), has gained increas-
ing attention in the methodological and empirical literature.1 Based on this
and the spread of Qualitative Comparative Analysis (QCA) as a set-theoretic
method usually operating on the cross-case level (Rihoux and Marx 2013),
set-theoretic MMR has recently been added to the toolbox (Schneider and
Rohlfing 2013). Set-theoretic MMR combines QCA with process tracing to
enhance causal inference of claims of necessity and sufficiency.2 Thus far,
the set-theoretic MMR literature has primarily focused on issues related to
methods, including the role of case study-based insights for QCA (Rihoux
and Lobe 2009), the informal and formalized choice of cases on the basis of
QCA results (Ragin and Schneider 2011; Rohlfing and Schneider 2013;
Schneider and Rohlfing 2013, 2018) and the logic of causal inference in
QCA-based case studies (Rohlfing and Schneider 2013; Schneider and
Rohlfing 2016).
The development of set-theoretic MMR is laudable and the combination
of QCA with process tracing is steadily becoming more popular in empirical
research (e.g., Aleman 2010; Samford 2010; Segura-Ubiergo 2007).3 What is
still lacking is a foundational discussion on how the integration of QCA as a
cross-case method and process tracing as a within-case method is possible.
We believe that tackling this issue on a fundamental level is crucial to
understand how we can generate coherent causal arguments in set-
theoretic MMR.
The fact that both QCA and process tracing have been based on set theory
(Goertz and Mahoney 2012:chapter 1) does not automatically imply that they
allow for the formulation of coherent causal inferences. Set theory is only
about the relation between two or more sets, which is not about causation per
se. Set relations are not causation, to paraphrase a common saying.
With regard to the combination of regression analysis and case studies
(Lieberman 2005), which forms the template for set-theoretic MMR (Schnei-
der and Rohlfing 2013), it has been argued that the two methods are difficult,
if not impossible, to reconcile because the supposed regularity conception of
causation that underlies regression analysis does not fit squarely with the
analysis of one case or a small number of cases (Chatterjee 2013). In light of
this and more general concerns about the way in which multiple methods fit
in MMR designs (Ahmed and Sil 2012; Beach and Pedersen 2013:section
8.4; Blatter and Haverland 2012:section 5.5; Creswell and Plano Clark
2011:chapter 2), we see a need to consider whether set-theoretic MMR
38 Sociological Methods & Research 47(1)
allows for valid inferences based on integrated cross-case and within-case
analyses. The primary goal of our article is to demonstrate that QCA and
process tracing share common ground in set-theoretic MMR and that inte-
grated causal analysis is feasible. The core element of our argument is a
single-case, counterfactual notion of causation.
Our article is based on two premises that are widely shared in the social
science literature on set theory. First, QCA is employed for causal analysis,
as opposed to the search for mere associations (Ragin 2008:9).4 Second,
sound causal analysis should establish a causal effect on the cross-case level
and involve a complementary causal explanation via the analysis of causal
mechanisms (Cartwright 2004; Gerring 2005). In principle, causal inference
is feasible with a valid identification strategy and does not need to shed light
on the underlying mechanisms (Gerring 2010). However, causal explana-
tions that focus on the link between a cause and an effect yield theoretical and
policy-related added value and should be an integral component of causal
analysis (Dessler 1991). The importance of opening the black box between
cause and effect has long been acknowledged in the qualitative literature
(Bennett and Elman 2006) and is now also reflected in the concern of the
quantitative literature about mechanisms, where this issue is discussed under
the label of ‘‘mediation’’ (VanderWeele 2009).
Based on these two premises, we develop a unifying foundation for set-
theoretic MMR that highlights the roles and interplay of QCA and process
tracing for causal analyses. We argue that causal analysis on the cross-case
level and within-case level is compatible in set-theoretic MMR and does not
suffer from the alleged problems of nested analysis and, more generally,
MMR. Causal inference via counterfactuals on the level of single cases
integrates QCA and process tracing as two methods operating on different
levels of analysis and assigns proper roles to each method.
The starting point for the mechanismic foundation for set-theoretic MMR is
Machamer, Darden, and Craver’s (henceforth MDC 2000) conception of
mechanisms as a chain of entities and activities. This concept has received
considerable attention in philosophy of science (McKay Illari and Williamson
2012), but has so far barely resonated in the social sciences (Beach and Ped-
ersen 2013; Rohlfing 2012:chapter 2). In the second section, we introduce
MDC’s conception in more detail and elaborate on how it can be anchored in
set theory. In the third section, we summarize the key elements of set-theoretic
MMR and link it to MDC’s mechanismic account. In the fourth section, we
show that causal inference via counterfactuals in single cases is able to unify
causal inference in set-theoretic MMR. The fifth section concludes.
Rohlfing and Schneider 39
Mechanisms as Chains of Actors and Their Behavior
Machamer, Darden and Craver (MDC) define a mechanism as comprising
‘‘entities and activities organized such that they are productive of regular
changes from start or set-up to finish or termination conditions’’ (2000:3).5 In
the social science literature on set theory, the terms ‘‘start-up condition’’ and
‘‘finish or termination condition’’ are usually referred to as the sufficient
term, X (a condition, conjunction, or equifinal conditions or conjunctions)
and the outcome, Y.6
The mechanism linking X to Y includes a chain of entities with corre-
sponding activities. We speak of actors and their behavior in the following
instead of entities and their activities in order to transfer MDC’s conception
to the social science domain and common social science terminology.7 Cap-
turing mechanisms by specifying actors and their behavior can be clarified
with the example of Owen’s (1994) explanation of the democratic peace
phenomenon and a visualization of the two mechanisms that link the pres-
ence of liberal ideas (X) to peace between two democracies (Y).8 In Figure 1,
the upper panel reproduces the original figure from Owen (1994:102). For
the lower panel, we modify his chart in order to render it in line with MDC’s
conception of mechanisms.
We elaborate the causal mechanisms in the lower panel of Figure 1, since
it exemplifies the conception we advocate as being the basis for set-theoretic
MMR. A closer look at Owen’s verbal formulation of his theory of demo-
cratic peace reveals that the lower panel provides a more precise visualiza-
tion of his arguments with multiple advantages over his original figure that
we discuss below.
The common starting point of the two mechanisms is ‘‘the elite of a
country holding liberal political ideas’’ (Owen 1994:93-012). According to
the first mechanism, these liberal political ideas lead to the pursuit of a liberal
foreign policy by the elite. An important consequence of a liberal foreign
policy is that members of the elite develop the idea that foreign liberal
democracies are trustworthy and peaceful and should not be attacked. Since
this idea is held by many, although not necessarily all, members of the elite,
the government of a liberal democracy is facing tight constraints with regard
to foreign policies toward other liberal democracies. Even in times of crisis
between liberal democracies, these constraints account for the democratic
peace phenomenon because then members of the elite lobby for peace and
mobilize the public to do the same.
The second mechanism is depicted by the lower path of the bottom panel
in Figure 1. Liberal ideas that are held by the elite lead to the creation of
40 Sociological Methods & Research 47(1)
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41
institutions that give the public political rights such as voting and freedom of
speech. Freedom of speech is used to evaluate different foreign policies in
relation to other liberal democracies. It is assumed that the public, as the
actor incurring the costs of war, is not prepared to carry these costs in a
conflict with a liberal democracy. The opportunity to discuss and evaluate
foreign policies and the right to vote constitute a second constraint on the
government. This constraint limits the available foreign policy options
toward liberal democracies to the pursuit of peaceful relations.
The conception of causal mechanisms as a chain of actors displaying a
certain behavior has three important features. First, the lower panel is more
detailed and conveys much more information that is more precise to the
reader. It highlights the difference between common path diagrams in
the social sciences, such as the upper panel in Figure 1, and diagrams of
mechanisms for the purpose of causal explanation. Second, the decomposi-
tion of a mechanism into actors and their behavior establishes productive
continuity (Machamer et al. 2000:3). The goal of causal explanations is to
elucidate whether and why one event is causal for another. In this concep-
tion of mechanisms, why-questions are answered by pointing to an actor
behaving in a certain way, which prompts another actor to perform a certain
behavior, and so on.9 For every step in the sequence, we are able to explain
how we got there and where we will be led in the next step and those that
follow. In the Owen study, he theorizes that elites share the idea that two
democracies should not fight each other because it is a consequence of the
more fundamental commitment to liberal political ideas. Similarly, Owen
conjectures as to how the process develops further by arguing that a peace-
oriented foreign policy toward other democracies is a cause of government
constraints.
Third and related, in confirmatory process tracing, the view on mechan-
isms as a causal sequence of actors and their behavior has advantages over
the derivation of multiple, but isolated observable implications that are
tested. Being a long-standing practice in the social sciences (Campbell
1975; King, Keohane, and Verba 1994:chapter 1; Mahoney 2010), this is
now referred to as the congruence method (see Blatter and Blume 2010).10 In
theorizing the democratic peace phenomenon, we could, for example,
hypothesize that the government faces tight constraints in launching a war
with another democracy and that the public is generally war-averse in crises
involving another liberal democracy (Owen 1994:102-4). These are consti-
tutive elements of the mechanisms in the bottom panel of Figure 1. Yet, this
form of theorizing leaves open questions as to why the public is war-averse in
the first place and why it enjoys the right to prevent the government from
42 Sociological Methods & Research 47(1)
initiating a war. In contrast, a fully specified mechanism is able to answer
these why-questions by pointing to preceding actors and their behavior
(Beach and Pedersen 2013:chapter 3).
Theorizing multiple observable implications is a valuable endeavor, yet
it only captures parts of the mechanism that is able to fully answer the
question of how we get from the sufficient term to the outcome. In addition,
the theoretical specification of a causal mechanism in terms of actors and
their behavior yields the maximum number of causally related observable
implications. This, in turn, makes it easiest to find disconfirming evidence
and offers the highest level of theoretical leverage in a confirmatory
analysis.
Fourth, the entire mechanism and each constitutive step can be con-
ceptualized in set-theoretic terms (Beach and Pedersen 2013:chapter 3;
George and Bennett 2005:chapter 10; Rohlfing 2012:chapter 2).11 Each
actor and its behavior, including the triggering condition, can be framed
as sufficient for the subsequent actor carrying out its behavior and, com-
bined, are jointly sufficient for the outcome. For one causal chain, this
implies that each actor and its behavior are a necessary element of the
mechanism because their absence causes the discontinuity of the process
(see below on causal inference). Whether an actor is also necessary for the
outcome depends on the number of mechanisms that are expected to be in
place. When there is more than one mechanism operative, as in the Owen
example, each actor and its behavior constitute an insufficient, but neces-
sary part of a mechanism that is unnecessary, but sufficient (INUS)
(Mackie 1974).
In closing this section, one might wonder why we specifically use this
definition of mechanism, given that conceptions of mechanisms are abundant
(Hedstrom and Ylikoski 2010). MDC’s account has achieved huge resonance
in philosophy of science and biology because it is simple but powerful, and
we are of the belief that it can be transferred to the social sciences.12 In
relation to this point, we emphasize that we rely on MDC’s conception as
a template for causal explanation and do not necessarily subscribe to MDC’s
view that it is ontologically accurate.13 The claim of ontological accuracy
implies that we can describe every causal process in terms of actors and their
behavior. In biology, it has already been questioned whether this is possible
(Nicholson 2012), and we leave it as an open question as to whether all social
and political processes can be subsumed under MDC’s model of mechan-
isms. Regardless of this, our point of departure that MDC’s conception does
not need to be ontologically accurate still allows for using it as an explana-
tory template when possible.
Rohlfing and Schneider 43
Set-Theoretic MMR in Short
Set-theoretic MMR combines a cross-case method for the analysis of cases
with process tracing (Ragin and Schneider 2011; Rohlfing and Schneider
2013; Schneider and Rohlfing 2013, 2016).14 It is possible to invoke
typological theory on the cross-case level (see Møller and Skaaning 2015)
but we focus on QCA as the standard set-relational cross-case method. For
our purposes, it suffices to broadly refer to ‘‘QCA’’ because the mechanismic
foundation that we develop in the following applies to all of its variants.15
Here, we conceive of QCA as a technique combining set-theoretic reasoning
with the analysis of a truth table using an algorithm, the truth table analysis.
Furthermore, the mechanismic framework extends to different varieties of set
relations (necessity, sufficiency, INUS, and sufficient, but unnecessary attri-
bute of a term that is insufficient, but necessary16). For ease of presentation
and without a loss of generality, we only develop our arguments for suffi-
ciency and provide the basis for the following two sections by giving a brief
exposition of set-theoretic MMR.17
QCA and process tracing follow the division of labor that is known from
nested analysis (Lieberman 2005; Schneider and Rohlfing 2013). The role
that each method plays in set-theoretic MMR varies conditionally on the
sequence in which they are implemented (see Beach and Rohlfing 2018).18
In a process tracing-first design, the analysis is confined to typical cases with
the aim of identifying a sufficient term for the outcome and the mechanism
located in-between.19 The purpose of a QCA that is run on the basis of an
initial round of process tracing is to test whether the sufficient terms iden-
tified in one or a small number of cases can be generalized to more cases (see
below on causal inference), and whether the truth table yields sufficient
terms in addition to those discovered in process tracing. We are also inter-
ested in the consistency and coverage of each sufficient term and the entire
QCA solution.20
The alternative is a QCA-first design in which process tracing is done on
the basis of the QCA results. We derive the truth table from existing, possibly
case-based empirical research and theory rather than from process-tracing
evidence.21 Process tracing after the truth table analysis fulfills multiple
roles. With regard to the causal analysis, the most obvious is the study of
causal mechanisms that are triggered by the sufficient term. For this purpose,
we select typical cases that allow us to discern the causal mechanism via
exploratory or confirmatory process tracing.22
When a case is a member of a sufficient term and the membership in the term
exceeds membership in the outcome, we are confronted with a deviant case
44 Sociological Methods & Research 47(1)
consistency. Exploratory process tracing in such cases serves to solve this puzzle
by adding an omitted conjunct to the term.23 The missing conjunct is identified
by finding within-case evidence on why the mechanism is not operative in the
deviant case consistency. A deviant case coverage is a member of the outcome,
but not of any sufficient term derived by the truth table analysis. This constitutes
a dilemma in Y-centered research because its aim consists of comprehensively
explaining the outcome. In an analysis of a deviant case coverage, the goal of
exploratory process tracing is to determine a formerly ignored sufficient term
with which the deviant case coverage can be explained.24
A Proposal for Unified Causal Analysis inSet-Theoretic MMR
MMR is now a widely embraced idea because two methods operating on the
cross-case level and within-case level yield complementary inferences on
causal mechanisms and causal effects (e.g., Collier, Brady, and Seawright
2010). However, linking two methods is only possible if the corresponding
inferences follow a template of causal inference that is compatible with the
cross-case and within-case method. The combination of regression analysis
and case studies has been criticized on precisely this point (e.g., Chatterjee
2013).25 It is said to rely on the regularity conception of causation, that is,
causation is inferred from observing a systematic pattern on the cross-case or
within-case level.26 Case studies can be employed based on a regularity
premise, but the uncertainty of the causal inferences derived from one or a
small number of cases tends to be considerable (Baumgartner 2008; Kuhn
and Rohlfing 2010).27 This has led to the criticism that process tracing is
pointless in nested analysis when the cross-case analysis is grounded in a
regularity conception of causation.
Some might argue that the same criticism extends to set-theoretic MMR.
QCA is run on a population of cases (Ragin 2000:chapters 2 and 7), or at least a
medium to large number of cases with the goal to find patterns. Here, ‘pattern’
means that members of the same term are also members of the outcome and that
non-members of any term are non-members of the outcome. Such cross-case
evidence entails the expectation that members of the same term also display the
same mechanism (Falleti and Lynch 2009; Machamer et al. 2000:3).
Search for patterns in QCA seems to be in conflict with the fact that we do
process tracing in single cases and make inferences about mechanisms with-
out incorporating information from all the other cases in the analysis. Like-
wise, in the truth table analysis we usually make arguments about cross-case
relationships without having detailed knowledge about the mechanisms in all
Rohlfing and Schneider 45
cases under scrutiny. The discrepancy between our reliance on multiple cases
in the truth table analysis and a single case in process tracing gives rise to
multiple questions: Should we make inferences based on the QCA finding
that the sufficient term triggering the mechanism is consistently linked to the
outcome? Or should we make our causal inferences based on the deep
knowledge of the process that we reconstruct via a within-case analysis?
Or is it possible that both methods contribute their share to the causal
analysis?
Many philosophers of science who embrace the conception of mechan-
isms, introduced above, as an ontological or epistemological template,
including Machamer et al. (2000), also endorse difference-making as the
basic template for causal inference.28 The specific variant of difference
making that is widely endorsed is Woodward’s (2003) interventionist
account, which rests on type-level counterfactuals (e.g., Kuorikoski 2012).
In contrast, we propose counterfactuals based on single cases as the basis for
causal inference on terms and mechanisms and the actors and their behavior
constituting it.29 We show that single-case counterfactuals assign proper
roles to process tracing and QCA in set-theoretic MMR and invalidate con-
cerns that the cross-case method and within-case method are fundamentally
incompatible.30
The details of single-case counterfactuals have been extensively dealt
with before in philosophy of science and the social sciences.31 Given our
focus on set-theoretic MMR, we limit ourselves to discussing the elements
that are salient for our purposes. A generally appealing property of single-
case counterfactuals is that causal inferences are located where the actual
mechanism and causal relation rest, namely, in the cases. This nicely empha-
sizes the notion of QCA as a case-based method.32
Single-case counterfactuals assess causal claims by asking what the out-
come in a given case would be if one feature of the actual case was different
(Lebow 2000; Lewis 1973a, 1973b; Paul and Hall 2013). Counterfactuals are
based on the premise that a factor (to use a general term here) only qualifies
as causal when it makes a difference to the outcome. A counterfactual con-
firms the causal relevance of a given factor if we can argue that the presence
and absence of this factor make a difference to the presence and absence of
the outcome. For instance, if we want to assess the cross-case hypothesis that
liberal ideas are a sufficient cause of peace, we can analyze the case of
Franco-American relations from 1796 to 1798 for which Owen argues that
liberal ideas maintained by U.S. elites were a cause of peace between the two
countries. In counterfactual-based causal inference, we would make this
inference on the basis of within-case evidence and the conclusion that France
46 Sociological Methods & Research 47(1)
and the United States would not have maintained peaceful relations if liberal
ideas had not been entertained by the U.S. elite.33
The generation of inferences on an entire mechanism requires that we
make an inference on every actor and its associated behavior that constitute
the mechanism. We concur with MDC that an explanation is only meaningful
if we consider an actor together with its behavior (2000:3); an activity must
be carried out by an actor in order to contribute to productive continuity, and
an actor must engage in some behavior. For purposes of causal inference,
however, we must detach actors from their behavior and ask two counter-
factuals. Would the process break down and the outcome be absent if a given
actor did not behave as she did in the actual case? And, would the process
break down and the outcome be absent if a given behavior were not per-
formed by the actor in the actual case, but by another actor?34 If both ques-
tions are answered in the affirmative, we can conclude that the outcome
depends on this component of the mechanism and is causally relevant.
We believe that tracing the actual process is in itself not sufficient for
establishing explanatory relevance (Hitchcock 1995), although it is of course
integral for counterfactual reasoning (Lebow 2000).35 Tracing a process is
not sufficient for causal inference because a traceable process can be non-
causal in fact (Woodward 2011).36 Neither is a process necessary for attri-
buting causal influence because of cases of negative causation in which the
absence of an event is a cause (Paul and Hall 2013). This is worth emphasiz-
ing because one might think that explanatory relevance is achieved by ‘‘peer-
ing deeply into the box of causation’’ (Gerring 2008:160). In practice, one
might make a plausible argument by tracing an actual process. However,
natural scientists who work with mechanisms on a much lower level of
analysis than social scientists do not accept this as a standard for explanatory
relevance (Woodward 2004), so it is not clear to us why social scientists
should settle for this in principle.
Regarding the democratic peace phenomenon, take for example the claim
that ‘‘elite holding liberal ideas’’ is sufficient for the elite’s adherence to a
peaceful foreign policy toward other democracies. We can assign explana-
tory relevance to this element of the process if we are able to substantiate two
claims. First, the elite would not have subscribed to a peaceful foreign policy
if it had not entertained liberal political ideas, that is, we counterfactually
consider the consequences of the absence of the observed behavior. Second,
the elite would not have subscribed to a peaceful foreign policy if another
actor or group of actors had subscribed to liberal political ideas, that is, we
counterfactually evaluate the consequence of the absence of the actor. If both
counterfactuals lead us to conclude that a nonliberal elite would not have
Rohlfing and Schneider 47
adhered to a peaceful foreign policy, we have established the explanatory
relevance of this part of the mechanism.
From a set-relational point of view, a twofold counterfactual on an actor
and its behavior is necessary for the assessment of causal relevance. In
empirical research, however, we consider it legitimate to focus on either the
actor or the behavior, depending on the theoretical interest of the study at
hand. In the example just given, we would deem it more interesting to look at
the causal relevance of the elites’ behavior because, at least implicitly,
Owen’s research interest seems to be that liberal ideas, as opposed to other
ideas, trigger the mechanism that results in democratic peace.
If causal inferences are generated in process tracing, one might wonder
what role is left for the simplification of conjunctions in QCA. In an ideal
process-tracing scenario, we identify all sufficient terms and corresponding
mechanisms via process tracing which is, however, highly improbable to
ever happen in research practice. Most, if not all, social phenomena can
come about by multiple sufficient terms and causal mechanisms, and we
consider it unlikely that we would be able to select a case for process tracing
in which all the terms and mechanisms are present. Even if we leave this
issue aside, the availability and quality of sources create epistemic uncer-
tainty and constrain our causal inferences in process tracing (George and
Bennett 2005:chapters 5 and 6).
The first reason why it is beneficial to run a QCA stems from these
challenges of causal inference in single-case studies. A relation of counter-
factual dependence is taken as a sufficient criterion for separating causal
from noncausal relationships, but not a necessary one (Hall 2009). Two
prominent problems that render counterfactuals nonnecessary are overde-
termination and preemption which come in various types that are not
relevant here (see Paul and Hall 2013). In a set-relational setting, over-
determination describes a constellation in which two terms seem to be
causally related to one outcome. Preemption means that one term brought
about the outcome, but that another term would have brought about the
outcome just as well if it had not been for the first term. A classic example
from philosophy of science is two bullets hitting a vase simultaneously
(overdetermination) and one bullet hitting the vase earlier than a second
bullet (preemption; Northcott 2010).37 Both constellations are a problem
for counterfactual inference because it is evident that in both scenarios
both bullets are causes of the vase shattering. In a naive counterfactual
analysis, however, we would infer that neither bullet is a cause because if
one bullet had not been shot, the vase still would have been shattered by
the other bullet.
48 Sociological Methods & Research 47(1)
Philosophers of science committed to single-case counterfactuals submit
various proposals for enhancing causal inference in the face of overdetermi-
nation or preemption (Paul and Hall 2013). In single-case settings, the fun-
damental problem is one of a low-information environment because we only
have one case at hand. In set-theoretic MMR, in contrast, the truth table
analysis has the potential to ameliorate causal inference because it introduces
additional cases that might increase the amount of information. With regard
to the two-bullets example, more cases would solve the problem if they
included a case in which only one of the two bullets was shot and we observe
that the vase shattered, and cases in which no bullet at all was fired and the
vase remained intact. It still holds that the causal inferences are made in
process tracing, but, under overdetermination and preemption, we are left
uncertain about what the proper inference is and QCA, by looking at more
cases, can help us sort out which of the possible inferences are accurate.
The second added value of QCA relates to its exploratory nature and the
fact that it can point our attention toward sufficient terms that, based on
process tracing, we did not infer to be present (see Beach and Rohlfing
2018). This might be because we overlooked these terms in process tracing
or they simply were not operative in the selected cases.
The third important task QCA performs is that it allows us to assess the
degree to which a causal inference on a sufficient term travels from the exam-
ined case to similar cases under analysis. Although this step is not about causal
inference, it is equally valuable because MMR is concerned with populations or
samples of cases and rests on the assumption of causal homogeneity. We address
this assumption by calculating the consistency value of a term. We cannot
directly evaluate the degree to which the mechanism is operative in other cases
because a QCA only involves data on the term and the outcome. However, when
the QCA results indicate that the sufficient relationship between X and Y we
derived from process tracing generalizes to more members of X and Y, we can
invoke the causal homogeneity assumption and be more confident than in stand-
alone process tracing that the same mechanism is present in all cases sharing the
same X and Y. Just because in set-theoretic MMR causal inferences are made in
process tracing, QCA should not be considered to be of second-order impor-
tance. It performs important complementary functions in causal analysis.
Advantages of the Unified Framework forSet-Theoretic MMR
There are multiple templates for causal inference (Brady 2008), which gives
rise to the legitimate question of why we advocate single-case
Rohlfing and Schneider 49
counterfactuals. We do not claim that case-level counterfactuals are fool-
proof. We are aware that they involve challenges, such as establishing the
truth value of the counterfactual and preemption and overdetermination (see
above). However, all templates for causal inference have their share of
problems and the search for criteria that are necessary and sufficient for valid
causal inference has so far proven to be fruitless (Paul and Hall 2013:chapter
2). Against this background, we argue that causal inference via single-case
counterfactuals has four advantages in the context of set-theoretic MMR.
First, counterfactual inferences based on single cases are fully in line with
the case-centered nature of QCA (Ragin 1987:chapter 1). They reinforce
QCA’s case-based and qualitative component by assigning the task of causal
inference to process tracing. Case-based counterfactuals dispel the notion
that QCA hampers qualitative research by distracting qualitative scholars
from their focus on cases (Collier 2014a, 2014b). Second, single-case coun-
terfactuals address long-standing criticisms of, and questions about, the basis
for causal inference in QCA (e.g., Lieberson 2001). The truth table analysis
can invoke a regularity-centered criterion based on observing actual patterns
even when a term only has one case as a member (Baumgartner 2013).
‘‘However, the type-level based inferences of regularity theories can hardly
be reconciled with process tracing in single cases where token-level features
such as time and space matter.’’ We therefore argue that the truth table
analysis is important for causal analysis in set-theoretic MMR, but it does
not directly contribute to causal inference, our issue of interest here. Causal
inference via single-case counterfactuals can more easily accommodate suf-
ficient terms having only one case as a member.
Third and related, when we use the Quine–McCluskey algorithm, we
simplify conjunctions by determining redundant conjuncts the presence and
absence of which do not make a difference to the outcome (Ragin 1987:chap-
ter 6).38 Single-case counterfactuals invoke the difference-making criterion
with the primary goal of confirmation. Causal inference via case-based
counterfactuals establishes difference-making as a unifying principle for
set-theoretic MMR and additionally takes into account that the intermediate
and parsimonious QCA solution also rely on counterfactuals (Schneider and
Rohlfing 2014, 2016). This would be different if causal inferences on
mechanisms were based on any criterion that does not entail the idea of
difference making, such as, for instance, the process tracing of spatiotempo-
rally ordered actual events (Waskan 2011).39 Besides the problem of estab-
lishing causal relevance without difference-making (Hitchcock 1995), this
would entail that set-theoretic MMR required a twofold standard of causal
inference, that is, separate criteria would apply to QCA and to process
50 Sociological Methods & Research 47(1)
tracing. Twofold criteria are not unknown to theories of causation (Russo and
Williamson 2011). However, one sacrifices parsimony by relying on two
criteria, which, so far, have been only discussed for probabilistic theories
of causation that face many challenges. Parsimony is not an end in itself, but
if one favors a dual criterion, one has to carefully consider whether the loss in
parsimony is worth the inferential gain.
Fourth, our proposal aligns the idea of difference-making and counter-
factual reasoning, on the one hand, with set relations, on the other. This is not
an end in itself, but worth emphasizing because it hints at counterfactuals as a
common basis for set-theoretic and quantitative research. The development
of the potential outcomes framework has the individual treatment effect
(ITE) as its basis but then moves on to the average treatment effect (ATE)
and its variants because the ITE cannot be estimated (Morgan and Winship
2014). Based on our proposal, the differences between set-relational and
quantitative analyses boil down to the focus on different types of ‘‘treatment
effect’’. This we consider an important difference because it sheds light on
why it is difficult to conceptualize set relations in terms of the ATE.
Gerring (2012:chapter 12) proposes to unify social science methodology
by transferring the potential outcomes framework, which has a solid counter-
factual foundation (Morgan and Winship 2014:chapter 2), to the realm of set
relations. He argues that patterns of necessity and sufficiency can be inferred
by calculating the ATE or 2 � 2 tables, as they are known from crisp-set
QCA. As has been shown before, this transfer of the potential outcomes
framework is problematic because the ATE can produce significant results
when a visual inspection of the 2 � 2 table shows that no set relation
whatsoever is involved, and, vice versa, a set relation can be in place when
the ATE is not significant (Schneider and Rohlfing 2012).
Table 1, reproduced from Schneider and Rohlfing (2012), exemplifies for
a sufficiency relation the perils of relying on the ATE. The left panel in Table
1 represents an ATE of 0 because the presence and absence of the condition
(elite holding liberal ideas) do not make a difference to the outcome on the
cross-case level. In the right panel, in contrast, we get an ATE of 0.5 that is
significant at .05. Yet, the presence of elites holding liberal ideas is definitely
not sufficient for democratic peace, as a look at the right-hand column with a
consistency score of 0.5 shows.
Instead of comparing the right and left column of a 2 � 2 table, as is
required for calculating the ATE, in process tracing at least one case in the
upper-right cell is subject to counterfactual causal inference. The criterion of
single-case counterfactuals implies that a symmetric relation between X and
Y on the within-case level, which follows from the requirement that a term
Rohlfing and Schneider 51
must make a difference to the outcome, can be reconciled with an asym-
metric X-Y relation on the cross-case level.
When the two tables represent results of a truth table analysis that was
run before process tracing, we of course would only do process tracing for
one or few of the 100 cases in the upper-right cell of the left table because
the condition is consistently linked to the outcome. A follow-up cross-case
analysis producing the result in the right-hand table would show us that the
causal inference derived from process tracing does not generalize because
50 cases that are similar to the examined case do not display the outcome.
This does not automatically invalidate our process-tracing inference
because it is about a single case, but we would learn that it is not general-
izable to all purportedly similar cases, that is, all cases in the right column
of the table.
Conclusion
MMR is becoming increasingly popular among empirical researchers. At the
same time, some methodologists remain skeptical as to whether MMR can
work on a principled level. In this article, we build on Machamer et al.’s
conception of mechanisms as entities and activities for the formulation of a
unifying framework for causal analysis in set-theoretic MMR that takes
single-case counterfactuals as the template for causal inference. Causal infer-
ence via single-case counterfactuals is certainly not without problems, just as
no theory of causation is infallible (Paul and Hall 2013:chapter 2). We have
shown, though, that inferences based on single-case counterfactuals have
attractive characteristics that render them highly compatible with QCA using
the Quine-McCluskey algorithm and the way in which it aims to substantiate
causal claims.
Table 1. Average Treatment Effects and Sufficiency.
DemocraticPeace
Yes 100 100 DemocraticPeace
Yes 0 50
No 0 0 No 100 50
No Yes No Yes
Elite holdingliberal ideas
Elite holdingliberal ideas
Note: Reproduced from Schneider and Rohlfing (2012).
52 Sociological Methods & Research 47(1)
We want to stress that our framework for set-theoretic MMR is also
relevant for single-method research invoking either process tracing or QCA.
Even if, for whatever reason, we are not combining two methods in one
design, it might still be possible that the process-tracing insights form the
basis for a QCA study at some point in the future and that the QCA results are
the starting point for future process tracing. Under the assumption that the
combination of both methods is generally appropriate for the research inter-
est at hand (Ahmed and Sil 2012), our framework clarifies what and how
exactly process tracing and the truth table analysis contribute to set-theoretic
research even when carried out by multiple researchers in a sequential order.
In the division of labor we propose, single-case counterfactuals might
seem to put process tracing in the driver’s seat because it is the method that
is used for the generation of inferences. This impression is misleading
because set-theoretic MMR involves a population or sample of cases. It is
important to determine how far the single-case inferences travel to other
cases that are members of the same sufficient term, and how many cases are
members of the outcome but do not fall under the causal explanation found
via process tracing. Taken together, we are not only convinced that process
tracing and QCA can be combined without friction, but also that their com-
bination represents a fruitful combination for causal analysis in MMR.
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) disclosed receipt of the following financial support for the research,
authorship, and/or publication of this article: Ingo Rohlfing received financial support
by Deutsche Forschungsgemeinschaft as part of Excellence Initiative.
Notes
1. See Goerres and Prinzen (2012), Rohlfing (2008), Rohlfing and Starke (2013),
and Seawright and Gerring (2008) for method-centered discussions.
2. See Møller and Skaaning for an extension of set-theoretic MMR to typological
theory.
3. We agree with Ahmed and Sil (2012) who argue that MMR is not a panacea and
that some research questions are better answered with single-method designs.
4. We understand ‘‘causal inference’’ to be about the inference of a causal relation-
ship based on an empirical association. ‘‘Causal analysis’’ has a broader meaning
Rohlfing and Schneider 53
and subsumes causal inference as well as generalization of inferences to a larger
population because the goal is to make claims about a population.
5. We do not follow a definition of mechanism such that it necessarily involves the
physical transfer of power or energy (Beach and Pedersen 2013:25; George and
Bennett 2005:chapter 7). The reason is that it is not necessarily involved in causal
relationships. An example from philosophy of science is that not watering a plant
is sufficient for its death. The absence of watering is an instance of causation by
omission and lacks a transfer of energy or power (see McGrath 2005). Similar
examples can be constructed for the social sciences. It is true that watering a plant
is necessary for its survival and watering involves physical transfer. However, the
goal is to formulate an explanation for the death of the plant which cannot refer to
the physical transfer of anything in this case.
6. Sets are put in italics throughout this article.
7. Beach and Rohlfing (2018) speak of entities and activities instead of actors and
their behavior. In discussions of mechanisms independently of MMR, this is
defensible because reference to actors and their behavior entails philosophical
and theoretical commitments to micro-level explanations that are not mandated
in research on mechanisms. In MMR, however, where process tracing takes
place on a level of analysis lower than that of Qualitative Comparative Anal-
ysis (QCA), we find it appropriate to rely on the actor-behavior terminology.
8. See Beach and Rohlfing (2018) for a similar visualization of another
mechanism.
9. A frequently raised criticism of causal mechanisms is that of infinite regress
(Gerring 2010:1506), meaning that every link between two entities can be broken
down into a more fine-grained sequence involving more entities and activities.
Machamer, Darden, and Craver (MDC) have two answers to this criticism (2000:
13-18). First, multilevel mechanisms allow to systematically model mechanisms that
are nested in each other. Second, one does not always need to rely on multilevel
mechanisms because the research interest and theory determine the level at which
the causal explanation bottoms out. In the social sciences, explanations generally
bottom out at the level of individual or collective actors (thus leaving aside research
on genopolitics and neuropolitics, e.g., Hatemi and McDermott 2012).
10. Owen (1994) theorizes the two mechanisms, but then also derives multiple,
observable implications that do not constitute a causal chain.
11. MDC does not explicitly model mechanisms in a set-theoretic way (see the fifth
section). Fagan’s (2012) account of joint explanation uses different terminology
but goes into this direction.
12. The explicit conception of mechanisms as a chain of entities and activities is
unique to MDC’s framework. However, a review of multiple definitions of
54 Sociological Methods & Research 47(1)
mechanisms shows that it shares elements with other conceptions (McKay Illari
and Williamson 2012).
13. On this dimension, our use of this conception of mechanisms differs from that of
Beach and Petersen (2013) who make the stronger ontological claim.
14. See Ragin (2000:chapters 2 and 7) for populations in QCA. Rihoux and Lobe
(2009) give a broader discussion of the role of cases and cases knowledge in
QCA. See also Beach and Rohlfing (2018).
15. The commonly used variants are crisp-set (Ragin 1987), multivalue (Cronqvist
and Berg-Schlosser 2008).
16. See Mahoney, Kimball, and Koivu (2009) for sufficient but unnecessary attri-
butes of a condition that is insufficient, but necessary for the outcome (SUIN).
17. See Beach (2015) and Casal Bertoa (2017) for applications.
18. See below and Beach and Rohlfing (2018).
19. See Rohlfing (2012:chapter 3) and Beach and Rohlfing for the choice of typical
cases in stand-alone process tracing.
20. See Ragin (2006) and Schneider and Wagemann (2012:chapters 5 and 8) for
discussions of these parameters.
21. A QCA-first design might also draw on case-based knowledge when constructing
the truth table, but this case knowledge tends to be more informal.
22. Whether the choice of cases is based on informal or formal techniques is not
relevant here.
23. We focus on model-related reasons for deviance, that is, the inclusion of too
many conditions in the QCA or the exclusion of causally relevant terms. Other
reasons for deviance are a misdelineated population, concept misformation,
measurement error, and miscalibrated sets.
24. Rohlfing and Schneider (2013) and Schneider and Rohlfing (2016) discuss
case selection in more detail. See Mikkelsen (2017) for negative case
selection.
25. See also Sale, Lohfeld, and Kevin (2002).
26. We leave aside here whether this criticism of nested analysis is correct. A second,
related criticism is that causation is reductionist, which is not relevant for our
purposes.
27. We say ‘‘tends to be’’ because the uncertainty depends on the size of the popu-
lation (Rohlfing 2012:chapter 9).
28. Craver and Darden (2012) favor a difference-making account and experiments
which are the natural method for assessing type-level counterfactuals (Wood-
ward 2003). Likewise, Machamer (2004) endorses a difference-making, inter-
ventionist account on an epistemological level, which is the dimension in which
we are interested. Bogen (2005) criticizes type-level counterfactuals and the
notion that mechanisms have to be regular, but he does not explicitly address
Rohlfing and Schneider 55
token-level counterfactuals (we also believe Bogen is not concerned with epis-
temology). In general, we agree that mechanisms do not have to be regular, but it
is difficult to avoid the causal homogeneity assumption for mechanisms in MMR,
where this assumption is made on the cross-case level.
29. See Schneider and Rohlfing (2016) for an exposition of counterfactual inference
which does not elaborate on how QCA and process tracing are linked to this
template. Our account of causal inference differs from Mahoney’s Unified The-
ory of Causality (2008). Mahoney links set-relational inferences on the case level
to regression-based inferences on the cross-case level. Instead, we establish the
more straightforward connection between set-relational analysis on the cross-
case and within-case levels.
30. In his criticism of the ontological and epistemological foundations of MMR,
Chatterjee (2013) considers and discards the idea of causal inference via
single-case counterfactuals, which he refers to as the singularist view of causa-
tion (see Russo and Williamson 2011). His refusal of single-case counterfactuals
suffers from two problems. First, he finds it incompatible with nested analysis as
developed by Lieberman (2005). In addition to Lieberman not explicitly formu-
lating his standard of causal inference, the point is that one can develop a dif-
ferent procedure for nested analysis that is in line with single-case
counterfactuals as the underlying principle (or type-level counterfactuals). Chat-
terjee ignores the potential outcomes framework as a possible basis for nested
analysis which has its anchorage in case-level counterfactuals (Morgan and Win-
ship 2014).
31. Lewis (1973a, 1973b) and Stalnaker (2011) have done seminal work on counter-
factuals and possible worlds in philosophy of science. For the social sciences,
see, for example, Lebow (2010), Levy (2008), Emmenegger (2011), and Gryna-
viski (2013) for more hands-on discussions of counterfactual inference in the
social sciences. Woodward (2003) proposes an interventionist account of coun-
terfactuals. The interventionist theory is applicable to single cases (2013:section
2.7) but is based on structural equations involving variables. Woodward’s theory
might be transferable to a set-relational setting, but this has not happened yet. The
idea of interventions solves some problems of Lewis’ account (see also Paul and
Hall 2013), but our understanding is that Lewis’ focus on closest-possible worlds
is closer to the case orientation of set-theoretic MMR (see Lebow 2010).
32. We discuss a criterion for causal inference which does not, in our view, auto-
matically imply that now we all must engage in single-case counterfactuals. If
there is an empirical case at hand that resembles the counterfactual and the actual
case of interest, we would prefer a comparison rather than a counterfactual.
33. We assume that we follow the unique membership principle when choosing cases
(Schneider and Rohlfing 2013).
56 Sociological Methods & Research 47(1)
34. In constructing the counterfactual, we should be careful in constructing the
proper contrast class (Lebow 2010:chapter 2; Steglich-Petersen 2012), that is,
negated set in terms of set theory. We need to consider what other behavior
would be in place if not the actual one, and what other actor might show the
behavior if not the actual one.
35. Waskan (2011) makes a plea for pursuing such an account without having an
actual proposal for a working approach.
36. For example, German chancellor Gerhard Schroder decided to call early federal
elections in 2005. He declared this repeatedly after his party lost the state-level
elections in North-Rhine Westphalia and confronted a huge opposition majority
in the Bundesrat (the second chamber). We could trace a process between the loss
of the elections and his call for new elections, but it turned out the process is not
causal. Later on, it became known that Schroder’s primary reason for his decision
was an increasing opposition among the members of parliament of his own party.
The actual reason for calling snap elections became public eventually, but we
cannot always count on this in empirical research.
37. There are additional constellations that potentially bedevil causal inference such
as prevention and double prevention which we do not address in more detail.
38. Coincidence Analysis (Baumgartner 2009) is an alternative based on a regularity
understanding of causation and does not require counterfactuals.
39. Actualism could also go along with a difference-making criterion (Baumgartner
2013), but Waskan (2011) adopts a nondifference-making perspective (as do
Beach and Pedersen 2013).
References
Ahmed, Amel and Rudra Sil. 2012. ‘‘When Multi-method Research Subverts Meth-
odological Pluralism—or, Why We Still Need Single-method Research.’’ Per-
spectives on Politics 10:935-53.
Aleman, Jose A. 2010. Labor Relations in New Democracies: East Asia, Latin Amer-
ica, and Europe. Basingstoke, UK: Palgrave Macmillan.
Baumgartner, Michael (2008). Regularity Theories Reassessed. Philosophia 36(3):
327-354.
Baumgartner, Michael. 2009. ‘‘Inferring Causal Complexity.’’ Sociological Methods
& Research 38:71-101.
Baumgartner, Michael. 2013. ‘‘A Regularity Theoretic Approach to Actual Causa-
tion.’’ Erkenntnis 78:85-109.
Beach, Derek. 2015. ‘‘Achieving Methodological Alignment When Combining QCA
and Process-tracing in Practice.’’ Typescript, R&R Under Review.
Beach, Derek and Rasmus Brun Pedersen. 2013. Process-tracing Methods. Ann
Arbor: University of Michigan Press.
Rohlfing and Schneider 57
Beach, Derek and Ingo Rohlfing. 2018. ‘‘Integrating Cross-case Analyses and Process
Tracing in Set-theoretic Research: Strategies and Parameters of Debate.’’ Socio-
logical Methods & Research 47:3-36.
Bennett, Andrew and Colin Elman. 2006. ‘‘Qualitative Research: Recent Develop-
ments in Case Study Methods.’’ Annual Review of Political Science 9:455-76.
Blatter, Joachim and Till Blume. 2010. ‘‘In Search of Co-variance, Causal Mechan-
isms or Congruence? Towards a Plural Understanding of Case Studies.’’ Swiss
Political Science Review 14:315-56.
Blatter, Joachim and Markus Haverland. 2012. Designing Case Studies: Explanatory
Approaches in Small-N Research. Basingstoke, UK: Palgrave Macmillan.
Bogen, Jim. 2005. ‘‘Regularities and Causality; Generalizations and Causal Explana-
tions.’’ Studies in History and Philosophy of Science Part C: Studies in History
and Philosophy of Biological and Biomedical Sciences 36:397-420.
Brady, Henry A. 2008. ‘‘Causation and Explanation in Social Science.’’ Pp. 217-70 in
The Oxford Handbook of Political Methodology, edited by J. M. Box-Steffens-
meier, H. Brady, and D. Collier. Oxford, UK: Oxford University Press.
Campbell, Donald T. 1975. ‘‘Degrees of Freedom and the Case Study.’’ Comparative
Political Studies 8:178-93.
Cartwright, Nancy. 2004. ‘‘From Causation to Explanation and Back.’’ Pp. 230-45 in
The Future for Philosophy, edited by B. Leiter. Oxford, UK: Oxford University Press.
Casal Bertoa, Fernando. 2017. ‘‘It’s Mostly about Money! A Set-theoretic Multi-
method Approach to the Sources of Institutionalization in Post-communist Party
Systems.’’ Sociological Methods & Research 46:683-714.
Chatterjee, Abhishek. 2013. ‘‘Ontology, Epistemology, and Multimethod Research in
Political Science.’’ Philosophy of the Social Sciences 43:73-99.
Collier, David. 2014a. ‘‘Comment: QCA Should Set Aside the Algorithms.’’ Socio-
logical Methodology 44:122-26.
Collier, David. 2014b. ‘‘Problematic Tools: Introduction to Symposium on Set The-
ory in Social Science.’’ Newsletter of the American Political Science Association
Organized Section for Qualitative and Multi-Method Research 12:2-9.
Collier, David, Henry E. Brady, and Jason Seawright. 2010. ‘‘Outdated Views of
Qualitative Methods: Time to Move on.’’ Political Analysis 18:506-13.
Craver, Carl F. and Lindley Darden. 2012. In Search of Mechanisms: Discoveries
across the Life Sciences. Chicago: University of Chicago Press.
Creswell, John W. and Vicki L. Plano Clark. 2011. Designing and Conducting Mixed
Methods Research. Los Angeles, CA: Sage.
Cronqvist, Lasse and Dirk Berg-Schlosser. 2008. ‘‘Multi-value QCA (mvQCA).’’ Pp.
69-86 in Configurational Comparative Methods, edited by B. Rihoux and C.
Ragin. Thousand Oaks, CA: Sage.
58 Sociological Methods & Research 47(1)
Dessler, David. 1991. ‘‘Beyond Correlations: Toward a Causal Theory of War.’’
International Studies Quarterly 35:337-55.
Emmenegger, Patrick. 2011. ‘‘How Good Are Your Counterfactuals? Assessing
Quantitative Macro-comparative Welfare State Research with Qualitative Cri-
teria.’’ Journal of European Social Policy 21:365-80.
Fagan, Melinda Bonnie. 2012. ‘‘The Joint Account of Mechanistic Explanation.’’
Philosophy of Science 79:448-72.
Falleti, Tulia G. and Julia F. Lynch. 2009. ‘‘Context and Causal Mechanisms in
Political Analysis.’’ Comparative Political Studies 42:1143-66.
George, Alexander L. and Andrew Bennett. 2005. Case Studies and Theory Devel-
opment in the Social Sciences. Cambridge, MA: MIT Press.
Gerring, John. 2005. ‘‘Causation: A Unified Framework for the Social Sciences.’’
Journal of Theoretical Politics 17:163-98.
Gerring, John. 2008. ‘‘The Mechanismic Worldview: Thinking Inside the Box.’’
British Journal of Political Science 38:161-79.
Gerring, John. 2010. ‘‘Causal mechanisms: Yes, but . . . ’’ Comparative Political
Studies 43:1499-526.
Gerring, John. 2012. Social Science Methodology: A Unified Framework. Cambridge,
UK: Cambridge University Press.
Goerres, Achim and Katrin Prinzen. 2012. ‘‘Using Mixed Methods for the Analysis of
Individuals: A Review of Necessary and Sufficient Conditions and an Application
to Welfare State Attitudes.’’ Quality & Quantity 46:415-50.
Goertz, Gary and James Mahoney. 2012. A Tale of Two Cultures: Contrasting Qua-
litative and Quantitative Paradigms. Princeton, NJ: Princeton University Press.
Grynaviski, Eric. 2013. ‘‘Contrasts, Counterfactuals, and Causes.’’ European Journal
of International Relations 19:823-46.
Hall, Ned. 2009. ‘‘Counterfactual Theories.’’ Pp. 158-84 in The Oxford Handbook of
Causation, edited by H. Beebee, C. Hitchcock, and P. Menzies. Oxford, UK:
Oxford University Press.
Hatemi, Peter K and Rose McDermott. 2012. ‘‘The Political Psychology of Biology,
Genetics, and Behavior.’’ Political Psychology 33:307-12.
Hedstrom, Peter and Petri Ylikoski. 2010. ‘‘Causal Mechanisms in the Social
Sciences.’’ Annual Review of Sociology 36:49-67.
Hitchcock, Christopher Read. 1995. ‘‘Salmon on Explanatory Relevance.’’ Philoso-
phy of Science 62:304-20.
King, Gary, Robert O. Keohane, and Sidney Verba. 1994. Designing Social Inquiry:
Scientific Inference in Qualitative Research. Princeton, NJ: Princeton University Press.
Kuhn, David and Ingo Rohlfing. 2010. ‘‘Causal Explanation and Multi-method
Research in the Social Sciences.’’ IPSA Committee on Concepts and Methods,
Working Paper Series Political Methodology, no. 26.
Rohlfing and Schneider 59
Kuorikoski, Jaakko. 2012. ‘‘Mechanisms, Modularity and Constitutive Explanation.’’
Erkenntnis 77:361-80.
Lebow, Richard Ned. 2000. ‘‘What’s So Different about a Counterfactual?’’ World
Politics 52:350-85.
Lebow, Richard Ned. 2010. Forbidden Fruit: Counterfactuals and International
Relations. Princeton, NJ: Princeton University Press.
Levy, Jack. 2008. ‘‘Counterfactuals and Case Studies.’’ Pp. 627-44 in Oxford Hand-
book of Political Methodology, edited by J. Box-Steffensmeier, H. Brady, and D.
Collier. New York: Oxford University Press.
Lewis, David. 1973a. ‘‘Causation.’’ The Journal of Philosophy 70:556-67.
Lewis, David. 1973b. Counterfactuals. Cambridge, MA: Harvard University Press.
Lieberman, Evan S. 2005. ‘‘Nested Analysis as a Mixed-method Strategy for Com-
parative Research.’’ American Political Science Review 99:435-52.
Lieberson, Stanley. 2001. ‘‘Book Review: Fuzzy-set Social Science.’’ Contemporary
Sociology 30:331-34.
Machamer, Peter. 2004. ‘‘Activities and Causation: The Metaphysics and Epistemol-
ogy of Mechanisms.’’ International Studies in the Philosophy of Science 18:27-39.
Machamer, Peter, Lindley Darden, and Carl F. Craver. 2000. ‘‘Thinking about
Mechanisms.’’ Philosophy of Science 67:1-25.
Mackie, John L. 1974. The Cement of the Universe: A Study of Causation. Oxford,
UK: Clarendon Press.
Mahoney, James. 2008. ‘‘Toward a Unified Theory of Causality.’’ Comparative
Political Studies 41:412-36.
Mahoney, James. 2010. ‘‘After KKV: The New Methodology of Qualitative
Research.’’ World Politics 62:120-47.
Mahoney, James, Erin Kimball, and Kendra L. Koivu. 2009. ‘‘The Logic of Historical
Explanation in the Social Sciences.’’ Comparative Political Studies 42:114-46.
McGrath, Sarah. 2005. ‘‘Causation by Omission: A Dilemma.’’ Philosophical Studies
123:125-48.
McKay Illari, Phyllis and Jon Williamson. 2012. ‘‘What Is a mechanism? Thinking
about Mechanisms across the Sciences.’’ European Journal for Philosophy of
Science 2:119-35.
Mikkelsen, Kim Sass. 2017. ‘‘Negative Case Selection: Justifications and Conse-
quences for QCA-Based MMR.’’ Sociological Methods & Research 46:739-771.
Møller, Jørgen and Skaaning Svend-Erik. 2015. Explanatory Typologies as a Nested
Strategy of Inquiry Combining Cross-case and Within-case Analyses. Sociologi-
cal Methods & Research. doi:10.1177/0049124115613778.
Morgan, Stephen L. and Christopher Winship. 2014. Counterfactuals and Causal
Inference: Methods and Principles for Social Research. 2nd ed. New York: Cam-
bridge University Press.
60 Sociological Methods & Research 47(1)
Nicholson, Daniel J. 2012. ‘‘The Concept of Mechanism in Biology.’’ Studies in
History and Philosophy of Science Part C: Studies in History and Philosophy of
Biological and Biomedical Sciences 43:152-63.
Northcott, Robert. 2010. ‘‘Natural-born Determinists: A New Defense of Causation
as Probability-raising.’’ Philosophical Studies 150:1-20.
Owen, John M. 1994. ‘‘How Liberalism Produces Democratic Peace.’’ International
Security 19:87-125.
Paul, L. A. and Ned Hall. 2013. Causation: A User’s Guide. Oxford, UK: Oxford
University Press.
Ragin, Charles C. 1987. The Comparative Method: Moving Beyond Quantitative and
Qualitative Strategies. Berkeley: University of Berkeley Press.
Ragin, Charles C. 2000. Fuzzy-set Social Science. Chicago: University of Chicago
Press.
Ragin, Charles C. 2006. ‘‘Set Relations in Social Research: Evaluating Their Con-
sistency and Coverage.’’ Political Analysis 14: 291-31.
Ragin, Charles C. 2008. Redesigning Social Inquiry. Chicago: Chicago University
Press.
Ragin, Charles C and Garrett Andrew Schneider. 2011. ‘‘Case-oriented Theory Building
and Theory Testing.’’ Pp. 150-66 in The SAGE Handbook of Innovations in Social
Research Methods, edited by M. Williams and W. P. Vogt. London, UK: Sage.
Rihoux, Benoıt and Bojana Lobe. 2009. ‘‘The Case of Qualitative Comparative
Analysis (QCA): Adding Leverage for Cross-Case Comparison.’’ Pp. 222-42 in
The Sage Handbook of Case-based methods, edited by D. Byrne and C. C. Ragin.
Thousand Oaks, CA: Sage.
Rihoux, Benoıt and Axel Marx. 2013. ‘‘QCA, 25 Years after ‘‘The Comparative
Method’’: Mapping, Challenges, and Innovations—Mini-symposium.’’ Political
Research Quarterly 66:167-235.
Rohlfing, Ingo. 2008. ‘‘What You See and What You Get: Pitfalls and Principles of
Nested Analysis in Comparative Research.’’ Comparative Political Studies 41:
1492-514.
Rohlfing, Ingo. 2012. Case Studies and Causal Inference: An Integrative Framework.
Basingstoke, UK: Palgrave Macmillan.
Rohlfing, Ingo and Carsten Q. Schneider. 2013. ‘‘Improving Research on Necessary
Conditions: Formalized Case Selection for Process Tracing after QCA.’’ Political
Research Quarterly 66:220-35.
Rohlfing, Ingo and Peter Starke. 2013. ‘‘Building on Solid Ground: Robust Case
Selection in Multi-method Research.’’ Swiss Political Science Review 19:
492-512.
Russo, Federica and Jon Williamson. 2011. ‘‘Generic versus Single-case Causality:
The Case of Autopsy.’’ European Journal for Philosophy of Science 1:47-69.
Rohlfing and Schneider 61
Sale, Joanna E. M., Lynne H. Lohfeld, and Brazil Kevin. 2002. ‘‘Revisiting the
Quantitative-qualitative Debate: Implications for Mixed-methods Research.’’
Quality & Quantity 36:43-53.
Samford, Steven. 2010. ‘‘Averting ‘‘Disruption and Reversal’’: Reassessing the
Logic of Rapid Trade Reform in Latin America.’’ Politics & Society 38:
373-407.
Schneider, Carsten Q and Ingo Rohlfing. 2012. ‘‘Does Set-relational Causation Fit
into a Potential Outcomes Framework? An Assessment of Gerring’s Proposal.’’
Newsletter of the APSA Section on Qualitative Methods and Multi-Methods
Research 10:8-14.
Schneider, Carsten Q and Claudius Wagemann. 2012. Set-theoretic Methods for the
Social Sciences. A Guide to Qualitative Comparative Analysis.Cambridge: Cam-
bridge University Press.
Schneider, Carsten Q and Ingo Rohlfing. 2013. ‘‘Combining QCA and Process Tra-
cing in Set-theoretic Multi-method Research.’’ Sociological Methods & Research
42:559-97.
Schneider, Carsten Q and Ingo Rohlfing. 2016. ‘‘Case Studies Nested in Fuzzy-set
QCA on Sufficiency: Formalizing Case Selection and Causal Inference.’’ Socio-
logical Methods & Research 45:526-568.
Seawright, Jason and John Gerring. 2008. ‘‘Case Selection Techniques in Case Study
Research: A Menu of Qualitative and Quantitative Options.’’ Political Research
Quarterly 61:294-308.
Segura-Ubiergo, Alex. 2007. The Political Economy of the Welfare State in Latin
America: Globalization, Democracy, and Development. Cambridge, UK: Cam-
bridge University Press.
Stalnaker, Robert. 2011. Mere Possibilities: Metaphysical Foundations of Modal
Semantics. Princeton, NJ: Princeton University Press.
Steglich-Petersen, Asbjørn. 2012. ‘‘Against the Contrastive Account of Singular
Causation.’’ The British Journal for the Philosophy of Science 63:115-43.
VanderWeele, Tyler J. 2009. ‘‘Mediation and Mechanism.’’ European Journal of
Epidemiology 24:217-24.
Waskan, Jonathan. 2011. ‘‘Mechanistic Explanation at the Limit.’’ Synthese 183:
389-408.
Woodward, James. 2003. Making Things Happen: A Theory of Causal Explanation.
New York: Oxford University Press.
Woodward, James. 2004. ‘‘Counterfactuals and Causal Explanation.’’ International
Studies in the Philosophy of Science 18:41-72.
Woodward, James. 2011. ‘‘Mechanisms Revisited.’’ Synthese 183:409-27.
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Author Biographies
Ingo Rohlfing is a professor for Qualitative Methods, Political Science at the Bremen
International Graduate School of Social Sciences (BIGSSS; hosted by University of
Bremen and Jacobs University Bremen). He is working on qualitative methods, multi-
method research, and political parties and has published the monograph Case Studies
and Causal Inference with Palgrave Macmillan.
Carsten Q. Schneider is a professor and head of the Department of Political Science
at Central European University (CEU). His research focuses on regime transitions,
inequalities, and social science methodology, especially set-theoretic methods. He
has published articles in, among others, Sociological Methods and Research, Eur-
opean Journal of Political Research, Political Research Quarterly, Socio-economic
Review, and the book Set-theoretic Methods for the Social Sciences with Cambridge
University Press.
Rohlfing and Schneider 63