Making small samples of large populations useful

I was recently contacted by someone who is working for a consulting firm that has a contract to evaluate the implementation of a large-scale health program covering a huge number of countries.  Their client had questioned their choice of 6 countries as case studies.  They were encouraging the consultancy firm to expand the number of country case studies, apparently because they thought this would make this sample of country cases more representative of the population of countries as a whole.  However, the consulting firm wasn't planning to aggregate results of the six country case studies and then make a claim about generalisability of findings across the whole population of countries.  Quite the opposite, the intention was that each country case study would provide a detailed perspective on one or more particular issues that was well exemplified by that case.

In our discussions, I ended up suggesting a strategy that might satisfy both parties in that it addressed to some extent the question of generalisable knowledge at the same time was designed to exploit the particularities of individual country cases.  My suggestion was relatively simple, although implementing might take a bit of work making use of whatever data is available on the full population of countries.  The suggestion was that for each individual case study the first step in the process would be to identify and explain the interesting particularities of that case, within the context of the evaluation's objectives.    Then the evaluation team would look through whatever data is available on the whole population of countries, with the aim of identifying a sub-set of other countries that had similar characteristics (perhaps both generic {political, socio-economic indicators} and issue specific) with the case study country. These would then be assumed to be the countries where the case study findings and recommendations could be most relevant. 

A shown in the diagram below, it is possible that the sub-set of countries relevant to each case study county might overlap to some extent. Even when one case study country is examined it is possible that it might have more than one particularity of interest, each of whose analysis might be usefully generalised to a limited number of other countries. And those different sub-sets of countries may themselves overlap to some extent (not shown below).  

Green nodes = case study countries
Red nodes = remainder of the whole population 
Red nodes connected to green nodes = countries that might find green node country case study findings relevant
Unconnected red nodes = Parts of whole population where case study findings not expected to have any relevance

Another possibility, perhaps seen as unadvisable in normal circumstances, would be to identify the relevant countries to any case study analysis after the fact, not necessarily or only before.  After the case study had actually been carried out there would be much more information available on the relevant particularities of the case study country that might make it easier to identify which other countries these finding were most relevant to. However the client of the evaluation might need to be given some reassurance in advance. For example, by ensuring that at least some of these (red node) countries were identified at the beginning, before the case studies were underway.

PS: It is possible to quantify the nature of this kind of sampling. For example, in the above diagram

Total number of cases =  37  (red and green). 

Case study cases = 5 (14%)  of all cases

Relevant-to-case-study cases = 17 (46%) of all cases

Relevant-to->1-case-study cases = 3 (8%) of all cases

Not-relevant-to-case-study* cases = 15 (40%) of all cases 

*Bear in mind also that in many evaluations case studies will not be the only means of inquiry. For example, there are probably internal and external data sets that can be gathered and analysed re the whole set of 37 countries.

Conclusion: We should not be thinking in terms of binary options. It is not true that either  a case is part of a representative sample of a whole population, or it is representative and of interest only to itself. It can be relevant to a sub-set of the population.