Dimension reduction techniques for very small sample sizes I have 21 socio-economic and attitudinal macro-level variables (such as percentage of mothers aged 24-54 not employed, percentage of children aged 3-5 in nursery schools and so on). I also have data on the proportions of grandparents which provided intensive childcare. Most of the socio-economic variables which I selected are highly correlated with childcare provision (for instance, there is a negative correlation between proportion of mothers employed part-time and provision of grandparental childcare). 
Ideally, I would like to create a typology of different kinds of countries. My hope would be to use some kind of dimension reduction technique whose components or factors would make some intuitive sense (e.g. attitudes towards family and gender, labour market structure, family policies). Or, alternatively, assess which of the 21 macro-level indicators best explain the variability in childcare provision across countries.
My main problem is that I only have 12 European countries. I reckon that PCA and factor analyses are not appropriate techniques with so few cases. Am I correct? I was told to try use qualitative comparative analysis or multiple correspondence analysis, although to my understanding the latter techniques are more appropriate for binary (or categorical) macro-level indicators (whereas mine are percentages or continuous variables).
 A: I would go for co-inertia analysis, which is an unspoken variant of canonical analysis. This would give you a linear combination of the 21 variables that has the highest co-inertia with a linear combination of childcare data (or with child care if it is a single quantitative variable). The trick of working with co-inertia instead of correlation is that you can still perform the computations when there are more variables than observations.
Unfortunately, CIA is not very wide-spread. It was developed for ecology, where there is usually more variables than observation sites. You can find some technical information in Dray, Chessel and Thioulouse, Ecology 84(11), 3078-89, 2003.
That said, the other comments/answers are right that 12 is a relatively small number and you will have to live with that...
A: As Peter Ellis' comment/answer suggests you are talking about dimensionality reduction and not data reduction.  You have changed the number of data points just the size of the space of covariates. Now Peter Flom is right that the PCA and FA methods can be tried with small sample sizes but it is not only the correlations that are likely to be poorly estimated but also that you could be fooled into dropping into too low dimensions because features may appear more highly correlated than they would have turn out to be with a larger sample.  I would not recommend it.
A: Regularized exploratory factor analysis was designed with this problem in mind. The authors have Matlab code available.
