Juan-Jose Gibaja-Martins writes,

In our recent research we have been working with data on per capita GDP, Human Development Index, etc in european countries and we are interested in performing a Principal Components Analysis (PCA). So far we have been weighting our data -based on the idea that we should allow a bigger weight to a bigger economy- My question is: does it make sense to weight our data -for instance with the population or the GDP of each country- or should we keep all the weights equal to 1? Is it advisable to use weighted PCA? If so, when?

Oddly enough, a related question came up in our quantitative political science research group a couple weeks ago: Rebecca and Matt were talking about a cross-national study that had something like 1000 respondents from each of several European countries, and the question was how to weight the results to account for the fact that some countries are larger than others. A simple data analysis would implicitly count all countries equally, which doesn’t sound right.

My solution (as usual) is multilevel modeling plus poststratification: in short, you only need to worry about these weights to the extent that the estimand of interest varies from country to country. Rather than weighting the data, my recommendation is to use a multilevel model to get estimates for all the countries and then to poststratify to get continent-wide estimates, if these are desired.