How can I choose the 10 variables that explain the most variation in a wealth index?

I have household survey data with 32 questions about assets the household has or doesn't. I assume that taken together the answers to these asset questions (e.g. how many televisions does the household own) are an indication of wealth, and could be used to make a good index of wealth, e.g. using the first component in a principal components analysis.

What I want to do, however, is to choose 10 of these variables that jointly explain the largest possible proportion of the variation in wealth and use those as the questions in a shorter questionnaire that I am developing. What is the best way of doing this?

One possibility that has occurred to me is to calculate the wealth index using PCA then regress this on every possible combination (60 million or so I think) of 10 variables from the 32, and see which gets the highest R-squared. I'm hoping there's an easier way.

Ideally I'm looking to implement this in Stata.

• You might also consider looking into the penalized regression literature (e.g. Lasso) – bdeonovic Dec 11 '13 at 23:05
• by the way, what is wrong with using the PCA approach? That seems to be the standard approach in such a scenario. – bdeonovic Dec 11 '13 at 23:16
• @Benjamin can you explain what you mean by the PCA approach? Do you mean the PCA with 64 million regressions as mentioned in my question? – Stuart Dec 11 '13 at 23:18
• well ideally, if you just did PCA and looked at the first couple PCs, the variables with nonzero loadings would represent the subset of your 32 questions that explain the most variance. This might not necesarrily be 10...of course your PC loadings might not be so sparse either. You should look into Sparse Principal component analysis which combines PCA with penalized regression (paper: stanford.edu/~hastie/Papers/spc_jcgs.pdf). – bdeonovic Dec 12 '13 at 0:30
• @Benjamin Thanks - that paper has a ref to McCabe 1982/1984 stat.purdue.edu/research/technical_reports/pdfs/1982/… which uses the term 'principal variables' for exactly what I'm trying to do. That term didn't seem to catch on though... – Stuart Dec 12 '13 at 1:12

This is very commonly done in Sabermetrics (think Moneyball) to see, for example, what are the $n$ factors that most influcence a baseball team's winning percentage. See this book chapter for a good introduction to linear regression that uses a similar example: http://www.stat.wisc.edu/~wardrop/courses/371chapter14.pdf
• Even when the scales are commensurate, this often does not work anyway. A counterexample is shown at stats.stackexchange.com/a/14528: there, the variable $y_8$ has the second highest absolute coefficient (out of $10$ variables) but is not part of a good selection of $5$ of those variables (and is unlikely to be part of the $5$ best). – whuber Dec 11 '13 at 20:37