Variable selection in multiple regression

In the R notebook on principal component regression I noted that interpreting principal components can be a challenge. When I wrote that, I hadn’t seen a nice paper by Chong et al.^{1} The method they describe is presented specifically in the context of interpreting selection gradients after a principal components regression, but the idea is general. Once you’ve done the regression on principal components, transform the regression coefficients back to the original scale. Doing this does require, however, fitting * all* of the principal components, not just the first few.

Here’s the citation and abstract:

Chong, V. K., H. F. Fung, and J. R. Stinchcombe. 2018. A note on measuring natural selection on principal component scores. *Evolution Letters*. 2-4: 272–280 dos: 10.1002/evl3.63

Measuring natural selection through the use of multiple regression has transformed our understanding of selection, although the methods used remain sensitive to the effects of multicollinearity due to highly correlated traits. While measuring selection on principal component (PC) scores is an apparent solution to this challenge, this approach has been heavily criticized due to difficulties in interpretation and relating PC axes back to the original traits. We describe and illustrate how to transform selection gradients for PC scores back into selection gradients for the original traits, addressing issues of multicollinearity and biological interpretation. In addition to reducing multicollinearity, we suggest that this method may have promise for measuring selection on high‐dimensional data such as volatiles or gene expression traits. We demonstrate this approach with empirical data and examples from the literature, highlighting how selection estimates for PC scores can be interpreted while reducing the consequences of multicollinearity.

- John Stinchcombe pointed me to the paper. Thanks John. ↩