Choosing cases for small n analysis is hard. The goal of analysis is to make generalizations beyond the sample you study, but to do that, you have to have a representative sample. In large n methods, we achieve this by random selection or sometimes by studying the entire population. In small n methods, we use a purposively-drawn sample – we deliberately choose a small number of cases to study – and this strategy doesn’t allow us to use either the random or full-population justifications for generalizations.
For small n research, the short form is, you can only make inferences to the population represented by your sample. So you want to maximize the representativeness of your sample as much as you possibly can. That said, what ‘representative’ means can vary based on the context. You have two main strategies to choose from: an ‘interior’ strategy, and an ‘exterior’ strategy.
In the interior strategy, you pick typical cases – ones that are on the interior of the distribution of the independent variable of interest – that nonetheless show variation on that independent variable. You aim for cases whose values are toward the center of the distribution, where presumably most of the population lies, so that you can generalize to a greater number of cases. Unfortunately, this strategy does very poorly in explaining outlier cases and cases at the extremes of the distribution.
In the exterior strategy, you pick cases that are further out on the distribution of the IV of interest. By choosing to maximize variation on the IV, you hope to be able to generalize to a greater range of IV values. The exterior strategy is in some ways a more encompassing strategy, but a significant risk exists that features that make the case take on such extreme values (i.e., omitted variables) are driving the results; this is the outlier problem.
So the major point to make is that a tradeoff exists between explaining a greater number of more similar cases (interior strategy) or a perhaps smaller (or perhaps larger) number of more dissimilar cases (ones that exhibit a greater range of IV values) (exterior strategy). The interior strategy may explain a greater number of cases, but those cases will be largely similar or at least exhibit only a limited range of variation. You will learn less about the effect of the IV with the interior strategy, but with more confidence. The exterior strategy will give you a better sense of the IV’s effect over a range of values, but with somewhat less confidence because of the outlier problem described above.
Which strategy do you choose? The answer to this question comes entirely from your theory, and in particular from its scope conditions. If your theory’s scope conditions state that it applies primarily to extreme events, then you’re looking at an exterior strategy. If your theory’s scope conditions imply that the theory holds only for some narrow critical range of values, then you have a justification for an internal strategy bounded by the values your theory specifies. If your theory’s scope conditions say it should be universally applicable, then you’re looking for something at the broad end of the interior strategy (maximizing reasonable variation and ignoring outliers).