I've read Kronmal's 1993 paper regarding using ratios in regression, specifically using the ratio as an IV. The case he makes is that the ratio is an interaction term and thus should include it's components, and further suggest including the components, and the ratio itself as an interaction.I want to make sure I understand correctly.
As an illustrative example, suppose I think bmi is predictive of head circumference in children.
library(mice) # for the data data(boys) boys$hgt <- boys$hgt/100 # convert to cm mo <- lm(hc~bmi,data=boys) # (1) mo <- lm(hc~ I(1/(hgt^2)) + wgt + bmi, data=boys) # (2) bmi without considering it's an interaction mo <- lm(hc~ I(1/(hgt^2)) + wgt + wgt*I(1/(hgt^2)), data=boys) # (3) bmi set as an interaction term, very similar to above mo <- lm(hc~ hgt + wgt + bmi, data=boys) # (4) simpler and performs marginally less
From the data it seems all models are superior to (1) (better $R^2$, lower residuals etc), and while bmi as ratio is significant, it actually provides very little as a measure over using height (which is the main influence factor) and weight.
To sum: 1) which is the best modelling form? should I always make these examination and infer based on data, or is form (3) always preferred? 2) Am I correct in the last paragraph?