# Analysing ratios of variables

Ratios (X/Y, e.g. body mass index) are variables with odd distributions. They have no means or moments (e.g. variances, skewness or kurtosis). Thus, my questions are:

1. How to compare between (or among) 2 or more groups with ratio variables?
2. How to use ratio variables as the independent variable or dependent variable in a regression analyses?
3. If:
• X/Y is an independent variable: is it better to use X and Y (or 1/Y) as independent variables separately?
• X/Y is a dependent variable: is it better to use X as a dependent variable and Y (or 1/Y) as an independent variable?
4. Are most studies on ratios statistically wrong?
5. Is the method described by a recent article Error propagation in calculated ratios. Clinical biochemistry, Vol. 40, No. 9-10. (June 2007), pp. 728-734) the way to a solution?
• if $X$ is bounded away from zero the ratio has moments. The only problem is with $X$ which have non-zero probability of being close to zero. For example with body mass index, height is never zero (or close to it), so there are no moment problems. – mpiktas Jul 26 '11 at 9:56
• To continue @mpiktas' thoughts: my study concerns a normally distributed variable $Z$. My measuring apparatus, however, is able to measure only $Y = Z\cdot X$ where $X$ is a multiplicative, normally distributed error. If I can independently measure $X$, then analyzing the ratio $Y/X$ = $Z$ (which is normally distributed) is exactly the right thing to do! This hints at the answers to questions 1, 2, 3, and 4. – whuber Jul 26 '11 at 13:42
• – David LeBauer Jul 27 '11 at 5:13
• Ratios of random variables occur almost everywhere (no pun intended) in statistics. Just take the $t$-statistic as an example. Except for tiny sample sizes, it definitely has a well-defined mean and variance. :) – cardinal Jul 27 '11 at 20:00
• – kjetil b halvorsen Aug 16 '18 at 11:40