Techniques for analyzing ratios

Question

I am looking for advice and comments that deal with the analysis of ratios and rates. In the field in which I work analysis of ratios in particular is widespread but I have read a few papers that suggest this can be problematic, I am thinking of:

Kronmal, Richard A. 1993. Spurious correlation and the fallacy of the ratio standard revisited. Journal of the Royal Statistical Society Series A 156(3): 379-392

and related papers. From what I have read so far it seems ratios can generate spurious correlations, force regression lines through the origin (which is not always appropriate) and modelling them may violate the principle of marginality if not done correctly (Using Ratios in Regression, by Richard Goldstein). However, there must be occasions when the use of ratios is justified and I wanted some opinions from statisticians on this topic.

Answer 1

I wouldn't call observed correlations spurious, but rather false causal inferences drawn from those correlations. Problems with ratios are of a kind with other types of confounding.

If you define random variables $U=\frac{X}{Q}$ & $V=\frac{Y}{Q}$, where $X$, $Y$, & $Q$ are independent, then $U$ & $V$ are correlated. This could mislead you into thinking there's a causal relationship from $X$ to $Y$ or vice versa, or from something other than $Q$ to both of them. It's no use however simply deciding to eschew the use of ratios. Observations aren't essentially ratios, & if $U$, $V$, & $Q$ are independent you will introduce "spurious" correlations by using the ratios $X=\frac{U}{1/Q}$ & $Y=\frac{V}{1/Q}$. Including $Q$ in your analysis—& it's important to note that 'scaling' by $Q$ is not the same thing^†—protects you whichever you use; but not from other confounders $R,S,T,\ldots$

Aldrich (1995), ""Correlations Genuine and Spurious in Pearson and Yule", Statistical Science, 10, 4 provides an intesting historical perspective.

† See Including the interaction but not the main effects in a model.