# Ratios in Regression, aka Questions on Kronmal

Recently, randomly browsing questions triggered a memory of on off-hand comment from one of my professors a few years back warning about the usage of ratios in regression models. So I started reading up on this, leading eventually to Kronmal 1993.

I want to make sure that I'm correctly interpreting his suggestions on how to model these.

1. For a model with a ratio with the same denominator on both the dependent and independent side:
$$Z^{-1}Y = Z^{-1}1_n\beta_0 + Z^{-1}X\beta_X + \beta_Z + Z^{-1}\epsilon$$

• Regress dependent ratio on the (inverse) denominator variable in addition to the other ratios
• Weight by the (inverse) denominator variable
2. For a model with dependent variable as a ratio:
$$Y = \beta_0 + \beta_XX + Z1_n\alpha_0 + ZX\alpha_X + Z^{-1}\epsilon$$

• Regress numerator by original variables, denominator, and denominator times original variables [what about categorical variables?]
• Weight by (inverse) denominator
3. For model with only independent variable ratios: $$Y = \beta_0 + X\beta_X + Z^{-1}1_n\beta_{Z^{-1}} + W\beta_W + Z^{-1}W\beta_{Z^{-1}W} + \epsilon$$

• Include numerator and (inverse) denominator as main effects, ratio as interaction term.

Are my interpretations here correct?

data(stork, package="TeachingDemos")

I will leave the fun for the readers, but one interesting plot is this coplot: