Why are Ratios "Dangerous" in Statistical Modelling? Why are Ratios "Dangerous" in Statistical Modelling?
A friend was telling me today that it is unwise to use the ratio of two variables as a variable in a regression model, and that is better to use these the same two variables as variables in the same regression model. However, when I asked why this was, I did not get an answer.
I spent some time trying to read about this and found the following points:

*

*In the context of probability, ratios are not always defined. For example, the Cauchy Probability Distribution is the ratio of two Normal Distributions. The mean of the Cauchy Distribution is undefined.


*Ratios have a problem of "spurious correlation". For example, if you generate random points from two independent Normal Distributions - and then take the ratio of each of pair : you will find that the ratio can show statistical correlation, even though the data has come from independent and random Normal Distributions.


*Ratios have the potential of becoming very large numbers and run the risk of division by zero.


*Supposedly, if a plot of the ratio between two variables is not a diagonal line (at 45 degrees) and does not pass through the origin - the ratio is meaningless (I don't understand why).
These are some of the points I identified as to why ratios might be "dangerous" in statistical modelling, and that it might be better to use the numerator and denominator variables from the ratio as variables for the regression model (or any statistical model) - but are there any other main reasons as to why it might be considered dangerous to use ratios in statistical models?
Thanks!
References:

*

*https://en.wikipedia.org/wiki/Cauchy_distribution

*https://en.wikipedia.org/wiki/Spurious_correlation_of_ratios

*https://pubmed.ncbi.nlm.nih.gov/28743689/

*https://journals.physiology.org/doi/pdf/10.1152/advan.00053.2013

*https://www.fs.fed.us/pnw/lwm/aem/docs/steel/2004_leirmann_et_al_ratio_paper_final_11_22_04.pdf

*https://www.law.nyu.edu/sites/default/files/The%20Ratio%20Problem.pdf

*https://media.terry.uga.edu/socrates/publications/2018/05/Ratios_in_Strategic_Management_in_press_1.pdf

Actually, it's kind of simple why. Suppose you calculate CV multiple times from bootstrap. CV is $\frac{SD}{Mean}$. Now suppose that the mean value is not close to zero, but could be, let's say one in a million times. What happens then is we might get a CV that could be -1000 times the median of the other CV values. So the problem with ratios of random variables is that the more data we have, the wilder the mean value may be because of the divide by almost zero problem in the denominator.
EDIT: For a more exact example that I am just crudely summarizing here see: Brody JP, Williams BA, Wold BJ, Quake SR (2002) Significance and statistical errors in the analysis of DNA microarray data. Proc Natl Acad Sci 99(20):12975–12978.
