How can I assess many explanatory variables with a sample small enough for in-depth qualitative (case-study) analysis? The problem:
My study is on the impact of human factor on regional development.
My “subjects” are regions of the European Union. 276 exist in total under NUTS 2 classification. The idea is to use a sample of 16-20, so that I can also do a small case study on each of them and approach them qualitatively as well.
My independent variables are demographic statistics (e.g. % of people with university degrees, % regional GDP spent on R&D and so on). I have around 20 different potential IVs or predictors, although I suppose I can group them or skip some unnecessary ones and narrow them down to around 8 if necessary.
I have a single dependent variable which will be taken from some widely-accepted scale showing the level of regional development so that doesn’t need tampering with.
Obviously, the goal is to check the impact of the IVs (both separately and as a complete model) on the DV. A regression seems to be the right way to go.
My questions:


*

*Is it incorrect to attempt to check the impact of 20 predictors on such a small sample (16-20 regions)?

*If so, will things improve if I narrow down the predictors to 8 or so?

*What is the best way to narrow down or group the predictors into few important ones? Common sense? Factor analysis? Cronbach’s Alpha? A stepwise regression?

*If that is still not enough, should I also increase the sample size to, say, 50 or 100 regions (out of a total “population” of 276) and skip the qualitative part?

*Is a regression the best choice for the “main” test?

*Somebody mentioned an “artificial neural network” to me!? Isn’t that way too complicated for a relatively straightforward study like this one?

 A: *

*No it's not stupid to look at such analyses. Ecological comparisons are very important, but very limited in their implications. I would interpret findings very cautiously, especially the role of unmeasured confounding variables.

*Absolutely. If you are able to prespecify comparisons of key importance, this increases the power and credibility of any possible conclusions.

*Common sense. By leagues more than other approaches. The latter concern prediction but you are interested in inference.

*The choice of including other regions should be determined by the hypotheses you're trying to address. It depends if you're breaking existing regions down to granular levels or incorporating other regions not previously accounted for. Weighting may also be used to standardize some regions to an appropriate demographic composition like GDP (for economic) or population size.

*Simple regression models are identical approaches to t-tests and ANOVAs for continuous outcomes or contingency table test for logistic regression. In journal articles, I always say, "regression was performed" but in my thesis, I go into great depth about the type of regression, what was controlled for, and how the variables were coded. A thesis should be far more on the verbose side with how the model was fit. But yet, you should use regression... everything is regression!

*Way too complicated.
A: I think the idea of conducting both quantitative / statistical analysies (regression models), and qualitative case-study analyses is a good one.  
That said, you cannot fit a model with 20 predictors and 20 data.  Such a model would be saturated1 and basically worthless.  One strategy would be to fit 20 individual simple regression models (one independent variable each), but that isn't an ideal solution.  The relationships amongst your independent predictors will lead you astray2.  From a statistical point of view, you will get a better model if you use more data.  If 276 regions of data are available, you would definitely do best to use all of them.  
Now, you clearly won't be able to do 276 in-depth case studies, so how do you square that circle?  My advice would be to do the qualitative analyses on a small subsample (say, 16-20, just to make up some numbers) of your full dataset.  These could be chosen to pick out different combinations of independent variables (e.g.: high on var 1 & high on var 2; high on var 1 & low on var 2; low on var 1 & high on var 2; low on var 1 & low on var 2; etc.), and/or you could look at the residuals of your multiple regression model and examine regions that seem well explained vs. those that show different kinds of poor fit (e.g., much higher / lower development than expected).  
(Having said the above, which I take to the the primary / underlying issue, I largely agree with @AdamO on his answers to your specific questions.)  

1. See: Maximum number of independent variables that can be entered into a multiple regression equation,
     and: Rules of thumb for minimum sample size for multiple regression.
2. See: Is there a difference between 'controlling for' and 'ignoring' other variables in multiple regression?

