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I'm trying to build a regression model that predicts Trump's vote share in a county in the 2016 election, given demographic data about that county. One of the demographic variables I would like to use is the distribution of income in that county; i.e. the % of people in the county earning <$10k, $10k-20k,etc. If I use the % of people in each bin as an independent variable, then intuitively the coefficient estimates for "nearby" bins would have high collinearity and be prone to overfitting. How do I use the prior knowledge that "%<10k" and "%10k-20k" variables should have a similar (but unknown) effect on Trump vote share in my regression?

Edit: To be clear, I would use n-1 variables to represent n income bins.

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    $\begingroup$ Actually, the collinearity would be perfect since all bins in each county should sum to 100%. Also, welcome to CV! $\endgroup$ – JTH Aug 2 '20 at 23:08
  • $\begingroup$ Oh I was planning to use n-1 variables for n bins. Should I edit the question to make that clearer? $\endgroup$ – KD89042 Aug 2 '20 at 23:21
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You want to use a variable that can be considered as compositional (https://en.wikipedia.org/wiki/Compositional_data). As stated in previous comments there is the issue of collinearity if you try to include all the percentages in the model. There exist possible solutions to this for example using the Additive logratio transform. Interpretation of estimates of parameter regression would slightly difficult however.

I would ask also why do you use those income “bins” as a way to summarize the distribution. If a “complete” distribution is available (for example as census data of income for each subject in the county) you could try different statistics to synthetize the distribution, for example the mean, the median or any other quantile, standard deviation, etc. Each of these summary statistics could be used in your model and check which one performs better.

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  • $\begingroup$ Using compositional data techniques doesn't solve the key issue, which is building in the prior knowledge that the coefficients for neighboring bins are similar. Compositional data techniques would work for eg race, where there's no ordering, but they wouldn't work for income as far as I can tell. Re. "why those bins": as far as I know the census doesn't release the full distribution. You're right that I could use median income as well, but I already know how to do that :) What I'm wondering about is how to use the income bins provided by the census in my regression. $\endgroup$ – KD89042 Aug 3 '20 at 4:35
  • $\begingroup$ I guess that similarity between bins arises as consequence of assuming “smooth” distributions (for example unimodal) for income, therefore nearby income bins have similar values. This would lead to the assumption of “nearby bins with similar effects”. Also, you are correct pointing out that the compositional data framework ignores the natural ordering of the compositions. I “feel” that this is not very important since the bins are covariates and not the response variable in your model. $\endgroup$ – Nicolas Molano Aug 3 '20 at 11:38
  • $\begingroup$ I have made a quick search of compositional data where compositions are ordered, but have not found anything interesting. Perhaps it could be a new field of research in compositional data analysis. Let’s wait if there are other more satisfactory answers to your question. However, have you tried this approach (alr transform) in your data? $\endgroup$ – Nicolas Molano Aug 3 '20 at 11:39
  • $\begingroup$ I've used an ilr transform for other similar projects, but not for this data set. If I don't get any other answers, I'll probably do that, but I feel like there must be a better way to deal with ordered bins. $\endgroup$ – KD89042 Aug 6 '20 at 6:40

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