Case-study methodology is an important and often overlooked form of reasoning. Case study is inquiry into a particular case for the purpose of describing and explaining it. The key features here are (1) that it is a form of inquiry; (2) that it focuses on a particular case; (3) and its goal is improved understanding of the case through description and explanation. The most important of these three features is (2), since the peculiarities of case study derive entirely from it.
It is possible to have a multi-case case study; however, such case studies are, in effect, particular case studies united together. A common division of types of case studies is into instrumental and intrinsic:
(a) Instrumental case studies are case studies undertaken as the means to understand a larger issue.
(b) Intrinsic case studies are case studies undertaken for the understanding of the case itself.
Either of these may be individual (one case alone) or collective (several connected individual case studies). The particularity of the object of a case study is both the strength and the weakness of case study method: strength, because as a matter of fact we need a way of dealing reasonably and rationally with particular cases (this becomes especially clear in fields like medicine and ecology, but is in fact quite generally important, although more important in some fields than in others); weakness, because if you only work with one case there are many more ways in which your reasoning can go wrong. We can distinguish out several aspects of case study method (I follow Shrader-Frechette and McCoy, with some differences):
(1) The characteristics of the investigator.
(2) The design of the case study.
(3) The types of evidence accepted.
(4) The analysis of the evidence.
(5) The evaluation of the case study itself.
(1) The elimination of bias is a major issue in case study research. Effectively, the investigator needs to display good taste, which (as readers of this weblog are already aware) means that (a) they need to have a familiarity with a broad field of relevant background knowledge; (b) they need to have practice in the skills relevant to the study (since the particular problems to be considered will vary from case to case, the investigator needs phronesis); and (c) they need to exercise good sense, i.e., reasonable self-critique.
(2) & (3) A case study typically has several elements: questions to be investigated, something to be analyzed, criteria for the interpretation of the evidence, and the actual structure of the reasoning. In addition, case studies don't generally look at everything that might conceivably be put forward as evidence; rather, they pick out forms of evidence that are deemed relevant.
(4) The precise sort of analysis used will vary from case to case, but a typical approach would have the following features:
(a) Case Description: The heart and soul of case analysis, case description consists in the identification of regularities and the classification of elements of the case (and sometimes the case itself). Classification is particularly important. In particular, what one aims for is a natural classification. Unfortunately, natural classification is not discussed much in studies of reasoning; it was of massive importance for nineteenth-century philosophy of science, but discussion of natural classification as such has almost completely lapsed in philosophy. The last thinker of any note who discussed natural classification was Duhem, at the beginning of the twentieth century. In a natural classification, to use the common phrase, one tries to "carve nature at the joints"; the idea is that your classification should not merely be arbitrary, nor should it merely be based on particular natural features, but should itself identify something real. The hope is that the classification will not merely be useful but will shed light on the actual nature of what is being classified. The best rule of thumb for natural classification is probably still Whewell's: that classification is natural to which our various classifications (according to different features) converge.
(b) Hypothesis Formation: Even in the process of description, it is sometimes necessary to make reasonable guesses as to the underlying character or structure of the elements of the case. In other words, hypotheses need to be framed. Further, in instances in which the case study is intended to support some sort of policy decision (for example), there are likely to be hypotheses that presuppose the case description. This is part of the discovery and exploration of the case study.
(c) Informal Testing: Where hypotheses are about, testing is not far behind. It is called 'informal' because case studies rarely are in a position to do rigorous testing of hypotheses (sometimes, however, they may be, depending on the sort of hypothesis in question). Instead, hypotheses are informally tested by seeing how they fit the evidence, seeing whether the evidence makes them plausible or implausible, etc.
(d) Composition of Report: The analysis needs to be organized, mapped out.
(5) Because so many things can go wrong with a case study, evaluation is especially important. We can roughly divide the aspects of the study which need to be divided into interpretation, structure and value.
A. Interpretation: Because description plays such an important role in case study, it is not surprising that the interpretation of the case and its elements is one of the most important things to evaluate. A significant issue in interpretation of a case is what is usually known as construct validity, which is (roughly) the appropriateness of your concepts, measurements, and labels to the case at hand. The concepts and labels should allow you to cover (to the degree required for the purpose of the inquiry) the whole case and every part of the case, without bringing anything irrelevant into the picture. Major issues that are important to construct validity are (a) clear definitions; (b) adequate precision of description, particularly in slippery or hard-to-define areas; (c) adequate explanation of potentially controversial interpretations; (d) well-marshalled evidence for the major aspects of the interpretation; (e) proper distinctions among distinct things; (f) clear recognition of the level at which you are inquiring, and the limitations that imposes on the conclusions you can draw; (g) elimination of interfering bias.
B. Structure: An important aspect of the case study is the reasoning by which one goes from the evidence as interpreted to the conclusions. (As noted above, this can happen even within the interpretation itself, since interpretations of cases generally have many layers of interpretation and reasoning.) The chief points here are adequate background information, internal consistency and well-chosen order. In other words, the case study needs to be rich in relevant information, make no contradictory claims, and provide us with an organization that sets things out in a reasonable way. As the Cartesians pointed out, order is the chief part of any method.
C. Value: We don't want a case study that is nicely organized but has no value for our reasoning at large. Major issues in assessing the value of a case study are utility (for both replication and application), durability in the face of objections and alternative explanations, completeness, heuristic power, and so forth. The precise values held to be important will vary depending on the case, the purpose of the inquiry, and the field in which the inquiry is conducted.
Case study method is more common than generally realized, I think. Not only is it found in various sciences, but historians and literary scholars are often effectively engaged in case study. It's not surprising that this is so; after all, much of what historians and literary scholars do is try to reason upon and draw conclusions from particular cases. It's only natural that, in an informal way, good, well-reasoned history, biography, literary scholarship, and the like would often begin to look like it is following (something along the lines of) the typical sort of case study format described above. Perhaps by recognizing more explicitly the elements of case study method used in our disciplines, we can improve our own reasoning and raise the quality of the research.