# 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).

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Because you want a typology, this sounds like a cluster analysis problem rather than dimension reduction. With your limited data, you could use that and a few basic plots to tell the story - but you're almost into qualitative research methods rather than quant here. –  Peter Ellis Jul 26 '12 at 11:03
Thanks. I thought about cluster analysis as well although the problem of having so many variables and so few cases remains. I guess i'll stick to basic plots then and convince my boss that there's nothing more exciting to be done (as i've always secretly suspected). –  Giorgio Jul 26 '12 at 11:16
I think @PeterEllis is right about what sort of thing you want to do. However, you can do PCA and FA on small data sets. Both these methods depend on correlations and a correlation is valid, even with 12 observations. However, the correlations may not be estimated very well. –  Peter Flom Jul 26 '12 at 11:16

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.

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Thanks. Sorry, I meant indeed dimension reduction! Also, I agree that PCA and FA are best to be avoided with only 12 cases. –  Giorgio Jul 26 '12 at 14:34
+1 for pointing out that, with very small sample sizes, sample correlations are usually quite high. As an extreme example, if $n=3$, you have a very good chance of getting nearly perfect correlation. Typing cor( rnorm(3), rnorm(3) ) repeatedly in R will reveal that. Also, I noticed you reviewed an edit today - thanks for pitching in! –  Macro Jul 26 '12 at 18:26
@Macro and with n=2 correlation of +1 or -1 is guaranteed. –  Michael Chernick Jul 26 '12 at 18:33

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...

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