Developing and Using Typological Theories in Case Study Research _*

Typologies and Typological Theories

The Inductive Development of Typological Theories

The Construction and Use of Deductive Typological Theories

The Relationship of Typological Theorizing to Mill's Methods

Conclusions

The relationships among typologies, typological theories, and case study methods are important but underdeveloped topics. Some researchers have noted that the contingent generalizations to which case study researchers aspire are related to typological theories.(George, 1979:59) Others have suggested that typologies can control for specified variables and help establish similar cases for purposes of comparison. (Diesing, 1971) Still others have downplayed the role of typologies, without distinguishing them clearly from typological theories (Keohane, King, and Verba, 1994:48)

We argue that typologies and especially typological theories can serve broader and more ambitious purposes in case study research and the social sciences than is generally acknowledged. First, typologies can spur research by codifying regularities in social phenomena and identifying them as subjects for explanation. Second, the development and refinement of typological theories, as opposed to typologies, is a key goal of theorizing on social phenomena characterized by complex interactions among numerous variables that may be causal even though they are not necessary or sufficient conditions. Third, case studies can be used inductively to develop and refine typological theories through a "building block" approach that maps out the alternative causal paths to the outcome of interest. This is particularly important in new or emerging research programs, but it can also involve the explanation of "deviant" cases that are not sufficiently explained by existing theories. Fourth, deductively-constructed typological theories are a key tool in the design and selection of case studies for the refinement of well-established research programs. Finally, by combining typological theories with methods of within-case analysis, particularly process-tracing and congruence testing, case study researchers can ameliorate (but not eliminate) the limitations of comparative case study methods derived from John Stuart Mill's methods of agreement and difference. After developing each of these topics, we conclude that typological theories and case study methods are powerful tools for causal inference, even though these methods, like all sources of causal inferences, suffer from important and irreducible limitations.

Typologies and Typological Theories

The formulation of typologies is a familiar activity in social science research. Typologies take many different forms and have a number of different uses, some more ambitious than others. Among the less ambitious uses, a typology may do little more than provide a listing of the types of outcomes of a particular kind of activity, or differentiate the variance in the dependent variable (e.g., types of deterrence failure). Alternatively, it can categorize types of variance in an independent variable, or characterize variants of a given phenomenon in terms of conjunctions of variables (e.g. types of social unrest that may or may not lead to revolutions). A more formal definition is that a typology partitions events into types that share specified combinations of factors. (Stinchcombe, 1968: 43-45)

It is important to note that in a typology, in contrast to a typological theory, these "factors" are not theoretical variables and do not link independent and dependent variables in a causal relationship. Thus, while the less ambitious uses of typologies may serve to facilitate the development of theory, they do not in themselves constitute theory. As Paul Diesing has noted, types and typologies are only implicit theories and "typologies must eventually be controlled by [explicit] theory of some sort to be reliable." (Diesing, 1971:189; see also Achen and Snidal, 1989: 157, Stinchcombe, 1968: 44-45). Even if it exhibits a perfect correlation among a set of factors, a typology alone is not a theory and cannot separate causal from spurious factors, or possible from unlikely or impossible combinations of variables.

However, the finding of a typological regularity can spur the search for underlying theoretical explanations, which can then be tested through within-case analysis. The inter-democratic peace research program illustrates the difference between a typology and a typological theory. When it was first put forth, the interdemocratic peace proposition constituted for the most part an observed typological regularity: democracies fight wars with other democracies infrequently or not at all (Babst, 1972; later researchers added a second typological regularity, that democracies do frequently fight non-democracies (Singer and Small, 1976)). The early proponents of this "interdemocratic peace" did suggest some possible theoretical mechanisms that might explain it, but these were underdeveloped. For example, the theory that voters in democracies would not vote the costs of war upon themselves (Babst, 1972) could not explain why democracies have chosen to make war on many states, but just not upon other democracies. In other words, this proposed theory is monadic and cannot explain a dyadic outcome.

The second wave of research on the inter-democratic peace turned to the question of whether the alleged typological regularity held up when researchers controlled for possible spurious correlations with the number of democracies, the number of contiguous democracies, alliance structures, and so on (Chan 1984, Weeds 1984, Maoz and Abdolali 1989, Maoz and Russett 1993). As in the first generation of research on this subject, this second generation relied primarily on statistical methods.

As a consensus (but not unanimity; Spiro 1994) emerged from the second wave correlational studies that the typological regularity is not spurious, a third generation of research has used case study methods to identify potential causal mechanisms and construct and test more sophisticated dyadic theories that might explain the interdemocratic peace, and that would thus advance from mere typological correlations to typological theories (Hoeft 1993, Owen, 1994, Elman, 1997). Throughout, researchers using both statistical and case study methods have developed more differentiated types of the independent variable of "democracy": states embodying various mixes of democratic values and democratic institutions; long-standing democracies versus new or transitional democracies; and so on (Snyder and Mansfield, 1995, Owen, 1994). Researchers have also more fully differentiated the dependent variable into types and scales of wars, such as preventive wars (Schweller 1992), and other "Militarized Interstate Disputes."

We are interested here in this more ambitious use of case studies for the development of typological theory, and in the use of typological theories for the design of case study research and the selection of cases to study. We define typological theory as a typology that is specified in terms of theoretical independent and dependent variables, which are defined in turn by an underlying theory or set of theories that address not only how these variables act singly but how they behave in specified conjunctive combinations. (For similar notions see Diesing, 1971, Ragin, 1987, Little, 1995). Typological theories thus provide contingent generalizations, or generalizations that apply to specific types in the typology, rather than explaining only particular cases. Such theories, for example, might seek to explain not "the Russian revolution," but all revolutions that: 1) follow an international war, and, 2) replace weak state institutions, and, 3) take place amidst an economic crisis. Typological theories may explain the alternative causal paths that lead to the same or similar outcomes, if the goal is to explain outcomes, or they may explain how the same level of a particular variable can lead to different outcomes depending on the values of other variables, if the goal is to explain the causal powers of a single variable.

Typological theories as we define them may be constructed through either inductive or deductive modes of inquiry. In most research projects and research programs, a combination of induction and deduction is necessary, with the relative roles of each varying according to the research objective, the state of development of the research program in question, and the availability of relevant cases to study. Case studies can contribute to the inductive development of typological theories in the early stages of a research program by identifying an initial list of possible theoretical variables. In the later stages of a research program's development, when theories have already been established and tested, the inductive use of "deviant" cases that do not fit the existing theory may refine the typological theory and perhaps add new variables or a new causal path to it. Theoretical arguments derived through these inductive processes must of course be subject to further testing before they are accorded the status of full-fledged theories. In particular, such inductive insights should explain not only the events or anomalies that spawned them, but they should offer insights into new cases, or into previously unexamined evidence from the cases from which they were derived (Lakatos, 1971). Deductive typological theories may also suggest an initial list of variables, and they can point out the cases whose study is most likely to provide theoretical insights. Often, either a single researcher or a succession of researchers working on the same research program will move back and forth between induction and deduction depending on the needs of the research program as it develops.

The construction of typological theories through these processes is not limited to case study methods, and may also benefit from quantitative methods and formal models. In general, typological theories make use of the fact that social phenomena are frequently characterized by equifinality -- a term from general systems theory that refers to phenomena for which similar outcomes occur through different causal processes -- and they contribute to the development of a rich, differentiated theory on the phenomenon in question, rather than a very general theory. The hallmark of a progressive typological theory is the refinement of contingent generalizations that differentiate both independent and dependent variables in ways that produce increasingly close correlations of events within types, as well as sharper distinctions between types. The development of the interdemocratic peace literature as sketched above is an exemplar of the progressive refinement of a typological theory in these respects; other examples include the literatures on deterrence, coercive diplomacy, the security dilemma, revolutions, military interventions, alliance burden-sharing, and democratic transitions. We now turn in more detail to the ways in which the inductive and deductive development of typological theories, in conjunction with case studies, can contribute to progress in social science research programs.

The Inductive Development of Typological Theories

In the early stages of reflection and research on a complex problem, the investigator will be reluctant to begin comparative study by attempting to build a research design and select cases based on a full, logically complete typology. The investigator's research objective may be more modest in that she does not attempt at the outset to articulate a comprehensive typological theory but hopes to develop such a theory through a series of individual case studies. Research at this stage may be of an exploratory nature, relying on feedback from the initial case studies undertaken to assess, refine, or alter the theoretical framework in which explanation of individual cases will be couched and to identify components of a useful typology.

The aim is to develop a typological theory eventually, but the strategy is to proceed incrementally toward that goal. In opting for this more flexible strategy, the investigator seeks to gradually build a typology and a typological theory via empirical analysis of cases within a theoretical framework. Concern over the risk of a premature commitment to a well-defined, comprehensive typology, one that may prove inadequate after much research on a set of cases selected for that typology, plays a role in encouraging a more gradual approach. It should be emphasized, at the same time, that while this strategy relies on induction it is analytical, theory-driven induction and not raw, atheoretical empiricism. Nor does the use of analytical induction exclude making use of deductive or quasi-deductive theoretical ideas, particularly theories on discrete causal mechanisms that may form the building blocks for more ambitious or integrative theories, to help guide the empirical approach.

To amplify on why an empirical approach to development of typologies and associated typological theories is useful a few examples may be helpful. An a priori, "logical" approach to typologizing outcomes of efforts to achieve deterrence is likely to settle for a simple distinction between "success" and "failure." (In fact, this characterization of deterrence outcomes has been and continues to be used, particularly for the purposes of large N statistical studies). An empirical approach relying on explanations for different cases of failure enables the investigator to discover different types of failures and to pinpoint specific (different) explanations for each type of failure. The different causal patterns of deterrence failure become part of a typological theory of deterrence. Such a differentiated theory of failures is significantly different from, and often more useful than, a theory that attempts to provide a single explanation for all deterrence failures. (This example is drawn from the George-Smoke (1974) study of deterrence, to be discussed in more detail in a forthcoming paper).

Note that the development of typological theory via the case method strategy takes advantage of the possibility that equifinality, as noted in our discussion of Mill's methods in the paper on the congruence method, may characterize many aspects of social and political life. Empirically-derived, theory-oriented case studies are particularly suited for discovering and pin-pointing equifinality and for developing typological theory for the phenomenon in question. Each case may turn out to be useful insofar as it permits the investigator to identify a different causal pattern. Differentiated explanations of the outcomes of the cases which are all instances of the class of events that is being investigated becomes a part of a cumulative typological theory, or a "repertoire of causal mechanisms" of that phenomenon. (Dessler, 1991:343, citing Miller, 1987:139)

As this discussion suggests, the typology -- and the ensuing typological theory -- may be derived empirically, not necessarily by starting with 2x2 (or larger) boxes of two (or more) variables. To this end the investigator should avoid a premature, a priori characterization of variance of the dependent and independent variables. Rather, the variance should emerge via differences discovered in the explanation of the cases. And the investigator should avoid overly general ways of characterizing such variance that limit the variance to but a few alternatives. For example, using the George-Smoke deterrence study again, treatment of variance in outcomes of deterrence attempts should not be limited to "success" and "failure"; rather, the case studies and their cumulation into a theory should be sensitive to the presence of equifinality. Hence, the possibility that each case of "failure" may have a somewhat different explanation can lead to a typology of failures; similarly with cases of deterrence "success" (if it were possible to make valid determination of "successful" deterrence).

The causal relationship between arms races and war provides another example of the need for more discriminating conditional generalizations. A comprehensive analytical assessment of the voluminous literature on this problem tells us that "there is still no well-developed theory that describes the conditions under which arms races will or will not lead to war. Nor is there a theory that provides a reliable guide for policymakers."(Downs, 1991:75) What available scholarship does tell us is that arms races are neither a necessary condition for the occurrence of war nor a sufficient condition for war. Given additional assessment of relevant historical experience, it should be possible to develop a typological theory of different "arms race --->war" causal relationships and to identify a number of conditions which, if present in an arms race, can be said to "favor" or increase the likelihood of war. Such a finding would identify ways in which policymakers might act to reduce or control the likelihood that an ongoing arms race might result in war.

To this one may object that such a procedure will lead to an infinite number of types if each case turns out to be idiosyncratic enough to warrant creation of a new type to encompass it. This danger need not arise since the number of variables in the theoretical framework will limit the variance in explanation. In addition, the investigator can and should exercise judgement as to the extent to which to construct from the cases more and more refined, narrowly circumscribed types (and sub-types of a type). In the George-Smoke deterrence study, three major types of deterrence failure emerged from the cases studied; the possibility of introducing sub-types of the three types was recognized but not pursued since the objectives of the investigation did not require it.

The typological theory that emerges obviously depends on what cases are selected for examination. Therefore, at the outset of the research, development of a typology and the associated theory must be open-ended. That is, new cases that are studied may lead to identification of new types of "success" or "failure." Of course, new cases may or may not turn out to be similar in type to one or another of those already studied.

Given the open-ended nature of theory development, it is appropriate to regard this research method as achieving a cumulation of findings via a "building-block" approach. That is, each case potentially provides a new component in the construction of a comprehensive typological theory. The number of "cells" that will eventually be identified remains indeterminate (although as stated above, not infinite in number) until more cases are examined.

An important difference between this approach to theory development and reliance on large N statistical methods has to do with "cell reduction." Investigators making use of statistical methods often engage in cell reduction -- i.e., enlarging the scope of a cell in order to get more cases in each of a smaller number of new cells so that statistical analysis becomes possible. However, it should be noted that reducing the number of cells moves the analysis to a higher degree of generality and eliminates the possibility of a more discriminating analysis of the differences among cases. To be sure, cell reduction may be undertaken not only to permit statistical analysis but also for theoretical reasons. Investigators may resort to it in order to identify and explain more general characteristics that a large number of cases may have in common. It is certainly possible that even when cell reduction is undertaken largely to satisfy requirements for statistical analysis it may still generate new concepts for the wider and broader cells of the reduced table and stimulate theoretical imagination. But one of the risks of cell reduction arises when it is not guided by theoretical hypotheses and, instead, takes place in an ad hoc opportunistic search for some findings of a general character, to which new conceptual labels can be attached. This approach to theory development risks production of "findings" that are artifacts of the push for cell reduction in order to make statistical analysis possible. The investigator employing an open-ended approach to developing typological theory is not prevented from engaging in cell reduction at a later stage in the inquiry to formulate more general findings, so there is no need to resort to cell reduction prematurely.

The inductive development of typological theories has important limitations. It is not capable of inferring from case findings how frequently each type of causal pattern appears in the universe of cases of that phenomenon. This limitation follows from the fact that construction of typological theory deliberately works without a representative sample. In fact, investigators engaged in developing typological theory explicitly disavow any effort to project frequency distributions from the cases they study. Sometimes, investigators deliberately select the least frequent or "representative" cases, such as deviant cases, to see if they embody previously unexamined causal paths. It is thus not a legitimate criticism of typological theory that it cannot do that which it does not attempt to do. The value of typological theory does not rest upon an ability to project the expected frequency distribution of types in the total universe of cases of a given phenomenon. However, development of a typological theory does not preclude the possibility of projecting the frequency of the different causal patterns it has identified in the total universe later on. If a representative sample of that universe can be drawn, which is not always possible, a large N study can be undertaken to determine how frequently each type is likely to occur.

In sum, in studies that seek to develop a typological theory the research objective is to identify the variety of different causal patterns that can occur for the phenomenon being investigated. In such studies, it would be useful for the investigator to pursue the additional research objective of identifying some of the conditions under which each distinctive type of causal pattern occurs. These research objectives, and not the question of how often each type of outcome and causal pattern occurs, guide the development of typological theory. Thus, this approach is useful for developing a differentiated theory comprised of a number of conditional generalizations rather than frequency distributions.

The Construction and Use of Deductive Typological Theories

For research programs that are more advanced and theories that have already survived some empirical testing, the deductive use of typological theories can help explain cases and guide research design and case selection. The deductive use of typological theory proceeds first from the designation of the research objective -- what the phenomenon to be studied is and whether the focus is on the causal powers of particular factors or the explanation of a particular outcome or class of outcomes. Once the phenomenon to be studied is specified, the researcher can specify the relevant theories, causal mechanisms, and variables that purport to explain the causal properties of the phenomenon. The researcher can draw here upon formal deductive theories and/or the explanatory "theories in use" of participants and experts, rendered into theoretical form. (Little, 1995: 51, Lepgold, 1997, Van Evera, 1996) As the researcher assembles the relevant theories and variables, it is important to focus on the predicted effects of interactions among combinations of variables.

As with the inductive development of typological theory, a key set of issues concerns how many independent variables to include, whether to partition these variables into two or more types, and how finely to differentiate the dependent variable. As new variables are added, typological categories multiply. For example, a typology with n dichotomous variables has 2 raised to the power of n possible types. Thus, a theory with four independent variables and one dependent variable, all dichotomous, would have 2 raised to the power of five or 32 possible types. Clearly, it quickly becomes difficult for the researcher to remember, use, and articulate more than a few of the most important types of a typology of five or more variables. The researcher might respond by focusing on one or a few well-specified sub-types, or focusing on a few variables that are hypothesized to have the greatest causal weight. Alternatively, as detailed below, it is often possible to reduce substantially the number of relevant types of complex typological theories.

As noted in the discussion above on cell reduction, the tradeoffs involved in adding variables to a typology are fundamentally different from those involved in adding an independent variable to a statistical research design. Statistical methods require positive degrees of freedom, narrowly defined, for a meaningful result. In such methods, each additional independent variable requires a corresponding increase in the number of cases to be included in order to establish coefficient estimates. This creates considerable pressure to keep the number of independent variables low unless data is extremely abundant, particularly if interactions effects (which also require a larger sample size to estimate) are to be taken into account.

This reasoning is appropriate to statistical methods, but it can be misleading on the issue of whether additional variables or cells should be included in a typological theory that is to be explored through case studies. Case study research seeks to differentiate independent variables that rely on different causal mechanisms and thus have different effects. The case study researcher may be able to address the case study equivalent of the "degrees of freedom" problem by generating detailed process tracing evidence on each new variable, if such process tracing data is accessible. Adding variables increases the complexity of the research design, but even though each new variable does require additional process tracing evidence if it is to be tested, new variables do not automatically raise a degrees of freedom problem and the number of cases to be studied (as opposed to the number of observations in each case) does not set an upper limit on the number of independent variables that can be considered.

The more general tradeoff for the case study researcher is whether the problem at hand requires added complexity, whether process tracing evidence is available to deal with this complexity, and whether the problem is important enough to merit a complex theory -- we will tolerate more sub-types of war than of the common cold. Parsimony and simplicity are always preferable, but they should not be sacrificed when complexity is necessary for adequate explanatory theory. (Keohane, King, and Verba, 1994)

As in the inductive development of typological theories, it is important for researchers to give as much thought to differentiation of the dependent variable in deductive theories as they do to that of the independent variables. Far too many research designs provide nuanced attention to the independent variables while lumping the dependent variable into a few vaguely-defined categories. Often, careful characterization of the dependent variable proves to be one of the most important and lasting contributions to research.

Once the specification of the variables is complete, it defines the "typological space," or the complete set of all possible combinations of the specified independent and dependent variables (this is analogous to a "truth table" in mathematics). This constitutes the relevant universe of all possible cases. The typological space thus answers the important question, "what is this a case of?" However, some of the possible cases in the typological space may not exist in nature, and a good theory may be able to specify hypothetical cases, or combinations of variables, that should not exist or should at least be highly unlikely or infrequent.

The specification of the relevant theories and variables, and analysis of their possible interactions effects, allows the researcher to narrow the typological space and select cases for study in three ways.

First, the typological theory may suggest that some types are not socially possible. (Little, 1995) In other words, a particular outcome may be impossible when the independent variables over-determine a different outcome. For example, we do not expect deterrence to fail when the deterrer has overwhelming and usable instruments of force, is far more committed to success than the opponent, communicates its intentions clearly, and faces a rational, unified, and attentive opponent. If we do find a failure in such circumstances, it constitutes a "deviant case" and may suggest new variables that need to be added to our typological theory.

Defining the typological space and developing a typological theory enables researchers to check whether they have given sufficient attention to the cells of the typology that should not, according to the theory, exist in the social world. In other words, rather than merely assuming that the cells which the theory predicts to be empty are in fact empty, the researcher should carefully consider whether there might be historical cases that fit these cells. Researchers sometimes make the basic mistake of looking only at cases where the hypothesized cause and the hypothesized effect are both present. While there are valid research designs that use only one case study, or that focus on all the possible paths to a given effect or all the possible effects to a given cause, ideally in these designs the researcher should examine or at least invite others to propose and study cases where the hypothesized cause and effect are absent. More generally, in a given typological space, causal inferences will be strongest if the researcher attempts to study or at least considers cases of every type.

Consider, for example, a simple version of the interdemocratic peace hypothesis, in which states are either democracies or non-democracies, and in which dyads have either engaged in war or maintained a peace. With these three dichotomous variables (democracy or non democracy for the first state, the same for the second, and either war or peace for the outcome) there are eight possible types. Most of the research on the interdemocratic peace has focused on one type of case: democratic dyads that go to war (or the close cousins of these cases, near democracies that go to war, and near wars between democracies). The focus on these cases is appropriate, but it is also important to make comparisons to other kinds of cases. For example, how do conflict resolution mechanisms between democracies differ from those between other kinds of dyads, including democracy/non-democracy dyads and non-democracy/non-democracy dyads?

This example suggests a second criterion for reducing the typological space and choosing cases to study. When an outcome is overdetermined by existing theories and it turns out as expected, it is less likely to be theoretically informative (although even here process tracing might point out that causal mechanisms did not operate exactly as expected). Such "most likely" cases are usually useful only when a theory unexpectedly fails to explain them, and a typological theory can point to other cases that offer tougher tests and are more likely to uncover new theoretical implications. In the interdemocratic peace example, wars between two non-democracies may be worth considering for purposes of comparison, but their existence is not too surprising and they may not merit full case studies.

Third, a typological theory can reduce the typological space and help in case selection by identifying which types and cases are suited to the research objective. This is true whether this objective is the testing of existing theories, comparison of typologically similar cases, identification and study of deviant cases, or use of a plausibility probe (or a preliminary and relatively easy test of a new theory on a case to which it is most likely to apply). The research objective and the case study research design should be devised with a view toward the stage of development of the research program in question. New and relatively untested research programs are more likely to be advanced by plausibility probes and inductive studies of deviant cases, for example, while more advanced research programs are amenable to theory-testing case studies, and studies of typologically similar cases with slightly different outcomes that might yield new sub-types or more finely differentiated variables.

Typological theories contribute to specific case study research designs and the selection of cases in the following ways. If two cases are to be regarded as typologically similar, or are common in the values of all the independent variables and thus occupy the same cell in the typology, then they should have similar outcomes. This offers the most basic test of the validity of the specification of the type. If the outcomes of typologically similar cases differ, then the researcher should check for measurement error and for the presence of probabilistic causal processes. To the extent that these can be ruled out, a new variable (or several) in which the cases differ may need to be added to the typology and linked to a generalizable causal theory.

A second research design involves two cases that occupy adjacent cells in the typology and thus differ only in the outcome and in one independent variable. If exogenous variables can be ruled out as a source of variation in the outcome (admittedly not a simple matter), then differences in the outcome can be attributed to the one variable in the typology on which the cases differ. This method uses a typological theory to establish a controlled comparison of similar cases, but it is important to remember that this research design, which has heretofore received a disproportionate amount of attention -- and criticism, most of which focuses on the impossibility of finding cases that are truly comparable in all but one respect -- is only one of many possible case study research designs..

Typological theories can also point to single cases that may be theoretically informative. Such theories facilitate the construction of "tough tests" by identifying which cells might constitute most-likely, least-likely, and "crucial" cases. Most likely cases are those at which a theory's key variables assume values that should strongly determine a particular outcome. If alternative hypotheses also predict this outcome but it does not occur, then this is a crucial case. The theory cannot blame its failure on exogenous conditions and it is strongly impugned. Conversely, if a case is weakly determined or least-likely for a theory, alternative hypotheses offer different predictions, but the theory still correctly predicts the outcome, then this constitutes a crucial case that offers the strongest possible support for the theory.

These three criteria for reducing the typological space and selecting cases -- identifying cases that are socially possible, theoretically informing, and suited to the research objective -- make it possible to reduce significantly the number of typological categories and cases to be studied. The use of a well-developed typological theory for case selection is in fact one of its most important functions.

An example, involving two related studies of alliance burden-sharing in the 1990-1991 Persian Gulf conflict by Bennett, Lepgold, and Unger, illustrates these points (Bennett, Lepgold, Unger, 1994; Bennett, Lepgold, and Unger, 1997). The first study used existing theories to identify five variables that should affect alliance contributions: 1) ability to contribute (collective action theory), 2) the specific threats Iraq presented to the potential contributor (balance of threat theory), 3) the potential contributor's security dependence on the U.S. (alliance security dilemma theory), 4) the issue-specific strength of the state vis-a-vis that of the society (strong state/weak state theory) and, 5) the power and interests of top government officials (bureaucratic politics theory). The later study added a sixth variable, "lessons that leaders drew from previous alliance experiences" (learning theory). The dependent variable in both studies was differentiated into three kinds of alliance contributions: military, political, and economic.

This yields a potentially complex typology, with 32 cells of different combinations of independent variables, and three outcome variables, or 256 possible cells if all of the variables are treated dichotomously (and 512 possible cells in the second study due to the added independent variable). Yet it was possible to reduce substantially the number of cells of interest. Cases that were overdetermined by a mix of variables or by a few variables at extreme values, such as the Kuwaiti and Saudi contributions to the Desert Storm coalition, were deemed unlikely to be theoretically informative (though if such states had failed to contribute, they would have constituted potentially useful deviant cases). The same was true of overdetermined non-contributors -- distant states not threatened by Iraq, dependent on oil or the world economy, or reliant on the U.S. for their security. The first study thus selected case studies of the leading contributors: the U.S., Britain, France, Germany, Japan, and Egypt. These varied substantially in the kind of contributions they made, and they included most-likely cases for all of the theories (except that on the balance of threats, and its overdetermined most-likely cases of Saudi Arabia and Kuwait). However, as the authors noted, this relaxed the ideal criteria for case selection, as it did not include studies of states that might have contributed but did not do so. The second and much lengthier study, which included additional case studies by regional experts, improved on case selection by studying a non-contributing "free-rider," Iran. This second study also included an abbreviated examination, or "mini-case study," of China, a state whose failure to make a substantial or costly contribution appeared to be (and upon closer study indeed was) overdetermined. This illustrates how abbreviated cases can be used to fill in typological spaces that are unlikely to be surprising but may turn out to be so.

The case selection in these burden-sharing studies allowed for a test of the key assertion that cases in the same box of the typology should have similar outcomes. Germany and Japan were most similar cases, as they were both dependent on the U.S. for security, relatively distant from the Middle East, dependent on foreign oil, and domestically constrained on the use of force. The great similarities in these states' contributions -- both provided over $8 billion but sent no combat troops -- were consistent with the predictions of the typological theory. Other states, such as Syria and Iran, had similar values on many of their independent variables but very different outcomes, allowing an examination of which independent variables might account for the difference in outcomes. This illustrates how case study researchers, after constructing a typological theory, should be alert to "targets of opportunity," identifying potential case studies that might fit various research designs, including most similar cases, least similar cases, deviant cases, most-likely cases, least-likely cases, and so on. It is also often possible to carry out more than one of these kinds of case study or case comparison within a single study -- cases that are most similar with respect to one another, for example, may be least similar to a third case, or a case may be most-likely for one hypothesis and least-likely for another.

Finally, these burden-sharing studies demonstrate how a complex typological theory can be developed inductively as well as deductively, and how it can be reduced into a simplified causal diagram. While the first study used deductive theories to define the variables of interest, it used case studies inductively to build these theories into an integrated typological theory on the contingent outcomes of combinations of variables. The second study then tested this theory against additional cases. The causal diagram in Figure 1 groups together similar outcomes while still allowing for alternative paths to these outcomes (i.e., equifinality). It depicts four possible outcomes depending on whether a state's contribution, or lack thereof, matched the value it placed on the public good of reversing the Iraqi invasion. A state "rides free" if it values the good but does not contribute, "keeps its distance" if it does not value the good and does not contribute, "reveals its preferences and pays up" if it values the good and contributes, or is "entrapped" if it does not value the good but contributes anyway. As both studies make clear, most of these outcomes can arise through more than one causal path. In addition, the second study included at least one case that fit each of the four outcome types fairly well, improving upon the case selection in the first study (which had no studies of states that "rode free" or "kept their distance").

Figure 1

While this example illustrates the potential power of typological theories and their utility in selecting cases, Daniel Little has used a hypothetical example to indicate the limitations of such methods. (Little, 1995) As Little's argument is tightly reasoned and extensive, the details are relegated to an appendix and only a brief version follows. For purposes of illustration, Little uses a hypothetical deterministic causal model similar to Theda Skocpol's theory of social revolutions. Little then addresses the following question: assuming this model is true and complete, what would a researcher who is not certain about the underlying causal relations conclude by studying social revolutions based on a limited number of cases? Little concludes that the researcher would be able to make preliminary assessments of whether some variables are either necessary or sufficient in causing a revolution, but that they might well miss some possible causal relationships. The main reason for this is that extant historical cases may represent only a few of the combinations of variables that are possible in the social world. Little also concludes that left-out variables and probabilistic causal relations, which he assumes away in his simple example, can further weaken causal inferences from case studies. However, Little also notes that one partial remedy to these problems is the use of empirically supported social theories, and the causal mechanisms that they posit, to establish hypothesized causal linkages in the cases under study that can then be tested empirically through process tracing.

We agree in principle with these conclusions, but we argue that in practice the severity of these limitations may be reduced through rigorous case study methods. First, as noted above, not all cases are equally theoretically informing, and a single crucial or nearly crucial case can strongly support or undermine a theory. Second, good case study researchers, cognizant of the limits of their methods, have been careful to avoid over-generalizing their conclusions or claiming to have uncovered all possible causal paths. Finding cases that represent previously undocumented causal paths has always been a priority for case study researchers. Third, as our example of alliance burden sharing indicates, much of the typological space in a given study can be set aside as unlikely or uninformative -- a point that Little notes but does not apply in his example -- allowing relatively strong inferences from even a small number of cases. Fourth, like Little, we argue that the use of prior theories, process tracing and other methods of within-case analysis can strengthen the inferences that would otherwise have to be made on the basis of comparative methods alone. Fifth, it is important to distinguish between instances where the range of extant historical cases is insufficient for strong causal inferences, and instances in which the researcher does not have the resources to study all of the potentially informative cases. In the former instance, case study methods will be weak but may be the only methods available. In the latter, researchers may focus their efforts on a sub-set of the typological space, where even a few cases may exhaust the causal paths of most interest, and/or they may add "mini-case studies" of otherwise unexamined types to test and strengthen their inferences. Finally, while we fully agree that low probability causal relations pose the toughest challenge for case study methods, we would add that such relations are the toughest challenge for any methods. When probabilistic relations are hypothesized but the extant number of cases is insufficient for statistical inquiry, case study methods are of limited utility but they are the only alternative.

The Relationship of Typological Theorizing to Mill's Methods

Several of the possible case study designs consistent with our view of typological theory bear a resemblance to John Stuart Mill's "method of agreement" or his "method of difference." However, this superficial similarity has caused many commentators wrongly to attribute to case studies and typological theories all of the well-known limits that Mill and others have identified regarding these methods. In fact, typological theorizing, together with methods of within-case analysis such as congruence testing and process tracing, can compensate for many of the limits of Mill's methods, even though it cannot eliminate them entirely.

. In general, this is because typological theorizing, even in the instance of the inductive development of typological theory, is theory-driven rather than blindly empirical. Without prior theories, or at least proto-theories, it is impossible to select which of the thousands of facts about cases should be analyzed for purposes of comparison or within-case analysis. Case comparisons, congruence testing, and process tracing are all set up by prior theories, and their results should be weighted or discounted by our existing level of confidence in these theories. This, together with methodological standards for progressive theorizing, such as Lakatos's insistence that theories must explain not only existing anomalies but "new facts," provides safeguards against the potential pitfalls of Mill's methods.

The key limitation of Mill's methods, which Mill himself identified, is that they cannot work well in the presence of equifinality. Put another way, Mill's methods can work well at identifying underlying causal relations only under three very strong conditions. First, the causal relations being investigated must be deterministic regularities involving conditions that are either necessary or sufficient for a specified outcome. Second, all causally-relevant variables must be identified prior to the analysis. Third, there must be available for study cases that represent the full range of all logically and socially possible causal paths. (Little, 1996; Lieberson 1994, George and McKeown, 1985)

Clearly, these strong assumptions seldom hold true. However, typological theorizing and case study methods do not require such stringent conditions. Let us consider each condition in turn. First, typological theories address the problem of equifinality directly, acknowledging and even taking advantage of the fact that there may be different causal paths to similar outcomes. The inductive development of typological theory attempts to map out the different causal paths, while the deductive development of typological theory attempts to provide theoretical reasons why particular conjunctions of variables lead to particular outcomes. Case study methods do not require causal relations of necessity and sufficiency, although case study methods -- like all methods -- offer stronger inferences on the existence of such relations than on that of equifinality or probabilistic causality. (Dion, 1997)

In addition, as long as all relevant variables are included in a typology, that typology inherently reflects interactions effects, even when those effects are not fully identified or understood by the researcher. Some critics of case study methods have suggested otherwise, arguing that these methods cannot incorporate interactions effects. (Lieberson, 1992:109-113) In fact, the logic of case study methods and the notions of causality associated with them have made case study researchers very attentive to interactions effects. (Ragin, 1987). If there are no measurement errors and there are deterministic or very high probability processes involved, admittedly two big assumptions, then two typologically similar cases, or cases with highly similar values on their independent variables, will have the same outcome, even if the interactions among the variables are caused that outcome are not fully understood or specified. Thus, we can have accurate predictions without accurate explanations, or the problem of spuriousness. For some research objectives, such as policy-makers' use of typologies, this may be acceptable, while for others, such as explanation by reference to causal mechanisms, it is not. Typological theorizing, as opposed to the simple use of typologies, pushes theorists to try to anticipate and explain interactions effects, although there is no guarantee that they will do so adequately. Process tracing and cross-case comparisons, though still fallible, may help identify which interactions are causal and which are spurious.

The second limitation of Mill's methods, the problem of left-out variables, is a potential threat to all methods of causal inference. Some critics have argued that the omission of causal variables is more likely to result in spurious inferences in case study methods than in research using other methods. (Lieberson, 1992:113). The problem with this critique is that it conflates Mill's methods with case study methods, and it does not acknowledge that process-tracing can test whether seemingly causal variables are spurious and to uncover supposedly unrelated variables that may in fact be causal. Moreover, the likelihood that relevant variables will be left out is lower for case studies than for statistical methods. Case study methods allow for the inductive identification of variables as well as their deductive specification, and they do not face a narrowly-defined degrees of freedom problem on how many independent variables to include. Indeed, one of the most visible and important contributions of case study methods has been to identify causal variables that have been left out or insufficiently examined by studies relying on purely deductive theories or correlational methods. This is evident in the literatures noted above on deterrence, where case studies have added variables on psychological dynamics and domestic politics, and alliance burden-sharing, where case studies have added domestic political variables and given greater weight to states' security dependence on an alliance leader.

The third limitation on Mill's methods, the requirement of having available for study cases representing all logically and socially possible causal paths, is a more binding constraint on case study methods. Causal inferences are indeed stronger when extant cases cover more of the typological space. Even so, having all possible types of cases available for study, while desirable, is not necessary. Not all research designs or research objectives require a fully inhabited typological space. Single cases, if they are most-likely, least-likely, or especially crucial cases, can be quite revealing about the strength of a theory. Comparisons of a few cases, if they are most similar or least similar, can also be revealing. Some cases provide more information than others on the theoretical issues of interest to a particular researcher. Moreover, for some research objectives, there may be cases for study representing most or even all of the possible types. The extant cases may also provide diverse causal paths even if the cases for any one causal path are not numerous enough for statistical methods.

Perhaps the most important difference between Mill's methods and case study methods in all three of the areas discussed immediately above is that case study methods can use within-case analyses, particularly process tracing, to ameliorate the limits of Mill's methods. Process tracing can identify different causal paths to an outcome, point out variables that otherwise might be left out, check for spuriousness, and allow causal inferences on the basis of a few cases or even a single case. These potential contributions of process tracing make case studies worthwhile even when sufficient cases exist for the use of statistical methods. Sophisticated critiques of case study methods acknowledge the value of process tracing. For example, Daniel Little, while more pessimistic than we are on the possibilities for typological theorizing, notes that such theorizing can be strengthened by the use of empirically supported social theories to establish hypothesized causal linkages that can then be tested through process tracing. As an example, Little notes that Theda Skocpol's work on social revolutions, in addition to traditional comparative analysis based on Mill's methods, uses established social theories in this manner. (Little, 1995:54)

Conclusions

When effectively accomplished, typological theory makes a distinctive contribution to the enrichment of theory. In contrast to a general explanatory theory of a given phenomenon, a richer and more differentiated depiction of that phenomenon is provided by a typological theory comprised of a number of contingent generalizations. This gives typological theory a capability for providing more discriminating explanations than does general theory. Contrast, for example, a general explanatory theory such as "war is the result of miscalculation" with a typological theory that distinguishes the conditions under which different types of miscalculations may lead to war. The latter type of differentiated theory not only provides the possibility of making more discriminating explanations, it also has greater practical value for policymakers because it enables them to make more discriminating diagnoses of emerging situations in which some kind of miscalculation might contribute to the outbreak of war.

The use of case studies for the development of typological theories, and the use of these theories in turn to design case study research and select cases, are iterative processes that involve both inductive study and deductive theorizing. An inductive, building-block approach to developing typological theories can identify causal paths and variables relevant to a given outcome. Such an approach is particularly useful in new or emerging research programs and in the study of deviant cases. Ultimately, this process aims to outline a comprehensive map of all of the causal paths to an outcome. A deductive approach to typological theorizing can help test established theories and propose integrative theories that incorporate interactions effects and address the problem of equifinality. Combining these modes of inductive and deductive development of typological theories with methods of within-case analysis, particularly process tracing, can substantially reduce the limitations of Mill's methods.

A greater awareness of the strengths and uses of typological theories and case studies, however, also provides a sharper understanding of their limits. Typological theories, case studies, process tracing, and congruence tests reduce but do not eliminate the limits similar to those that afflict Mill's methods of agreement and difference. Left-out variables and measurement errors can undermine causal inferences no matter what methods are used. Case study researchers should be sensitive to interactions effects but there is no guarantee that they will incorporate and explain such effects adequately. Finally, when low probability causal relations hold and there are only a few cases, no methods of causal inference can work well. Still, in this event, case study methods are the only available alternative.

APPENDIX

A Demonstration of The Limits of Typological Theories and Case Studies (Little, 1995)

Daniel Little has provided a useful demonstration of the potential limitations of case study methods and typological theorizing. Little posits a hypothetical model regarding the causes of social revolutions, similar to but not the same as that advanced by Theda Skocpol, and assumes for purposes of illustration that the argument is ontologically true. This causal model is as follows:

Figure 2 (Little, 1995:32)

This model indicates that variables A or E in conjunction with B, working together with variables C or F in conjunction with D, will cause both social unrest and a state crisis and hence a revolution. Little then addresses the question: assuming this model is true, and that it includes all causally relevant variables and involves exceptionless regularities, what would a researcher who is not certain about the underlying causal relations conclude by studying social revolutions based on a limited number of cases? To explore this, Little uses the model to generate values for the cases included in a hypothetical study, as follows:

Figure 3 (Little, 1995: 46)

Little points out that the researcher, studying these cases but unaware of the causal relations that generated their outcomes, would be able to reject some variables are either necessary or sufficient in causing a revolution. For example, he notes, a food crisis or war are not necessary for revolution, as some revolutions occurred without one or the other. Also, local organization, weak state institutions, exploitation, and economic crisis cannot be excluded as being jointly sufficient for a revolution as they led to a revolution by themselves in Cuba and were present in all the other revolutions. In all, Little offers four such inferences that are consistent with the underlying causal model used to generate the cases; many other equally accurate inferences are also possible in his example. As Little points out, however, the problem is that other combinations are also sufficient to cause revolutions but are not indicated by the cases studied, including BDAF, BDAC, and BDEC. Moreover, a limited number of cases does not allow the researcher to conclude that any variable is a necessary condition or to eliminate some variables or combinations of variables as unnecessary or insufficient.

The main reason for these limited conclusions is that the cases studies represent only a few of the combinations of variables that are possible in the social world. Little illustrates this by presenting a truth table (or in our terms, a full typology that exhausts the typological space). With six independent variables, there are 2 raised to the power of 5, or 64, possible combinations:

Figure 4 (Little, 1995: 48-49)

As in our other examples, some of these combinations can be excluded as unlikely to exist in the social world. With regard to the cases that are socially possible, if an actual case could be found that instantiates every possible type, then the researcher could indeed uncover all of the causal relations expressed in the model. Of course the problem, as Little notes, is that history almost never provides cases that fit every possible type in the truth table or typological theory. Little's conclusions from this exercise, and our view of these conclusions, are presented in the text above at pages 26-28.

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*: Not for quotation or additional circulation.

This paper is a preliminary discussion of material that will appear in a book on the use of case studies for theory development, co-authored by Andrew Bennett, Georgetown University, and Alexander George, Stanford University. The book will also include material from two other papers being presented in Toronto: A. L. George, "The Role of the Congruence Method for Case Study Research," and A. Bennett, "Lost in the Translation: Big N Misinterpretations of Case Study Research." Additional material is being prepared for the book as well -- e.g. on structured focused comparison, research design, criteria for selection of cases, lessons learned from past uses of case studies for theory development, and other topics. Back.

**: Andrew Bennett, Georgetown University, Alexander L. George, Stanford University. A. L. George authored the section of this paper on "The Inductive Development of Typological Theories," and Andrew Bennett authored the sections on "The Construction and Use of Deductive Typological Theories" and "The Relationship of Typological Theorizing to Mill's Methods," and the appendix. The remaining sections were jointly authored. Back.