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Third, even when a plausible argument can be made that a factor is necessary to the outcome in a particular case, this does not automatically translate into a general claim for its causal role in other cases. If equifinality is present, the factor’s necessity and causal weight may vary considerably across cases or types of cases.58

THE ‘DEGREES OF FREEDOM PROBLEM’ AND CASE STUDIES: MISAPPLICATION OF A STATISTICAL VERSION OF UNDERDETERMINATION

Analysts have occasionally criticized case studies for having a “degrees of freedom problem.” This is the statistical term for the broader issue of underdetermination, or the potential inability to discriminate between competing explanations on the basis of the evidence. In our view, the statistical concept and nomenclature of “degrees of freedom” has often led to a misunderstanding of how the more generic problem of underdetermination can pose a challenge to case study methods.

In statistical methods—we focus for purposes of illustration on the example of multiple regression analysis—the term “degrees of freedom” refers to the number of observations minus the number of estimated parameters or characteristics of the population being studied (such as mean or variance). In a multiple regression analysis, the number of observations is taken as the number of cases (or the sample size) and the number of parameters is the number of independent variables and one additional parameter for the value of the intercept (the point at which the estimated regression line intercepts the axis on a graph). Thus, a study with 100 cases and 6 variables would have 100 - (6+1) or 93 degrees of freedom.

In a statistical study, degrees of freedom are crucial because they determine the power of a particular research design or the probability of detecting whether a specified level of explained variance is statistically significant at a specified significance level. In other words, as the sample size increases or the number of variables decreases—either of which would increase the degrees of freedom—lower and lower levels of explained variance are necessary to conclude with some confidence that the relationship being studied is unlikely to have been brought about by chance.

It is easy to see why this important consideration in the design of statistical research might seem directly applicable to case study research, which also uses the terms “case” and “variables.” In a strictly literal sense, any study of a single case using one or more variables might seem to have zero or even negative degrees of freedom and be hopelessly indeterminate apart from simple tests of necessity or sufficiency. This is a fundamentally mistaken interpretation.

We have criticized above the definition of a case as a phenomenon in which we report only one measure on any pertinent variable. It is this definition that leads to the conclusion that case studies suffer from an inherent degrees of freedom problem. In fact, each qualitative variable has many different attributes that might be measured. Statistical researchers tend to aggregate variables together into single indices to get fewer independent variables and more degrees of freedom, but case study researchers do the reverse: they treat variables qualitatively, in many of their relevant dimensions. Statistical databases, for example, have created indices for “democracy,” while qualitative researchers have been more active in measuring different attributes of or types of democracy, or what has been called “democracy with adjectives.”59

In addition, within a single case there are many possible process-tracing observations along the hypothesized causal paths between independent and dependent variables. A causal path may include many necessary steps, and they may have to occur in a particular order (other causal paths, when equifinality is present, might involve different steps in a different order.) Some analysts emphasize that defining and observing the steps along the hypothesized causal path can lead to “a plethora of new observable