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design. If exogenous variables can be ruled out as a source of variation in the outcome (admittedly not a simple matter), then there is some basis for inferring that differences in the outcome can be attributed to the one variable in the typology on which the cases differ. This basis for inference can be strengthened by using process-tracing to establish that the variation in the outcome was indeed due to the single independent variable that differed between the cases. Process-tracing can also test whether factors left out of the typological framework and that differed between the two cases were causally related to the variation in the outcome.

MOST-LIKELY, LEAST-LIKELY, AND CRUCIAL CASES

In a third research design, the preliminary analysis of the property space can point to single cases that may be particularly informative for theory development. Such analysis can facilitate the construction of tough tests by identifying which types might constitute most-likely, least-likely, and crucial cases. In a most-likely case, a single variable is at such an extreme value that its underlying causal mechanism, even when considered alone, should strongly determine a particular outcome. If at the same time the other independent variables, considered singly and together, point toward the same outcome as the extreme variable, then this is a crucial case. If the predicted outcome does not occur, then the hypothesized causal mechanism underlying the extreme variable is strongly impugned. The failure of this mechanism cannot be blamed on the operation of the other variables in the framework. Conversely, if a case is weakly determined or least likely for a single causal mechanism, and alternative hypotheses offer different predictions, but the causal mechanism still correctly predicts the outcome, then this constitutes a crucial case that offers the strongest possible support for the mechanism.500

LEAST SIMILAR CASE COMPARISONS

Finally, cases that are similar in their outcomes but differ on all but one independent variable constitute least similar cases. The typological space helps identify such cases, and a researcher who chooses to study such cases can use process-tracing to test whether these cases’ similar outcomes are due to the one independent variable that they have in common.

Integrating Typological Theorizing and Process-Tracing

Once the research design is set and the cases are selected, the researcher can begin the case studies using methods of within-case analysis. This can lead to more accurate measurements of independent variables, which may lead to reclassification of some cases. The case studies may also lead the researcher to refine the cutoff points between types and to add new variables. Such changes to the preliminary typological theory may re-solve anomalies, but they may also create new ones. They can also lead to changes in the research design and in the cases selected for study.

This iteration between theory and data and between within-case analysis and cross-case comparisons is a key advantage of typological theorizing as compared to comparative methods used alone. The aspects of typological theorizing that rely on cross-case comparisons are in some respects vulnerable to inferential problems like those that beset Mill’s methods. Although typological theorizing does not require that a single variable be necessary or sufficient for outcomes, as Mill’s methods do, such theorizing, like all methods, remains vulnerable to erroneous inferences if relevant variables are omitted. Moreover, the most-similar and least-similar case comparisons facilitated by typological theorizing are based on the same logic as Mill’s methods of difference and agreement, respectively. The key difference between typological theorizing and Mill’s methods is that by relying on within-case methods as well as comparative methods, typological theorizing reduces the risks of mistaken inferences.

Process-tracing provides a check on whether the explanations developed from typological comparisons are spurious. While typological