you analyse each case separately, and then explore patterns of similarity or difference. Another method is the most similar and most difference approaches; and also QCA, all of which compare the cases systematically from the beginning, focusing on the factors. I have used these methods in my study. It has a chapter of Methodology and an appendix explaining QCA. (See my book: Ethno-=territorial Conflict and coexistence....)
The question is, What is the logic of case selection in comparative cases study research? Because of the theoretical issues to be explored and tested, the small-n comparative case study is the appropriate approach to research (Lijphart, 1971; 1975). Keeping in mind the benefits, in terms of internal validity, that experimentation offers and the confidence in causal inferences that it provides, the proposed research strategy optimizes control and effectively isolates the relationships of interest, given the constraints created by our need to observe the phenomenon contextually. One should try to articulate such a method by relying on a logic of case selection that, within the limits inherent in the well-designed small-n comparative case-study (Verba, 1967; Eckstein, 1975; Yin, 1984), allows the researcher to maximize the internal and external validity possible given his/her contextual interests, thus increasing the confidence and generalizability of our causal explanations. Careful attention to the issues of case selection, in case study research, is a critical component of defensibility. The challenge is to determine the research strategy and case selection method most appropriate to the investigators’ theoretical concerns and to their desire to make confident inferences from the findings. How should we do it?
The answer is purposeful sampling of cases for comparison from the universe of cases. Purposeful sampling enables researchers to move away from the indeterminacy that makes generalizations from case studies problematic and in the direction of valid causal explanations of social and political phenomena; for example, the role that research-based knowledge has played in preparing for a probable disaster. Where these concerns pertain, selection of cases in small-n comparative case study research should be guided by some theoretically-driven decision rule. Such a selection process is referred to as theoretical sampling (Mitchell, 1983; Scott, 1987: 158). Cases selected in this manner vary on identified characteristics of theoretical interest. As Fernandez 2005) notes, if researchers hope to explain variation in a dependent variable, the choice of cases must allow for variation in the dependent variable. King and his co-authors (1994, 129) put it bluntly when they ask, “How can we explain variations on a dependent variable if it does not vary?” (p. 129).
I (Goggin, 1986) have identified one common problem that plagues many who design and carry out comparative policy research, the “too few cases/too many variables problem”. One remedy for the too few cases/too many variables problem is to select cases with an eye toward maximizing similarities among cases except for the phenomenon to be explained or maximizing differences among cases except for the phenomenon to be explained (Goggin, 1986:333-34). Przeworski and Tuene (1970:39) label these "most similar" and "most different" systems designs, respectively. The authors describe the differences in these two logics of case selection as follows: “The most similar systems design is based on a belief that a number of theoretically significant differences will be found among similar systems and that these differences can be used in explanation. The alternative design, which seeks maximal heterogeneity in the sample of systems, is based on a belief that in spite of intersystem differentiation, the population will differ with regard to only a limited number of variables or relationships.” The logic of the two designs is identical, with co-variation or the lack thereof, as the instrument for distinguishing relevant from irrelevant variables. It is only the incidence of variability in the dependent variable – some variability in the most similar systems approach and no variability in the most dissimilar systems design – that differentiates these two approaches to case selection in comparative research.
I would strongly recommend the Fifth edition of Yin's Case Study Research. His book explains exactly how you should analyse a multiple case study. Hope this helps!
Yin (2003) is the leading book. Robert Stake and Merrium other three leading authors. Eisenhardt, also had given useful guidance. However, y Case study method has no standard protocol to claim its legitimacy as a a standard method according to Yin.
You should be aware of your paradigm. Yin takes a positivistic point of view. Stake an Merriam take a more constructivistic view. This affects the way you approach the case study, the type of (f.i.) interviewing technique, and also methodological quality criteria.