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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?
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    $\begingroup$ 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. $\endgroup$ – mpiktas Jul 26 '11 at 9:56
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    $\begingroup$ 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. $\endgroup$ – whuber Jul 26 '11 at 13:42
  • $\begingroup$ related: stats.stackexchange.com/questions/3640/… $\endgroup$ – David LeBauer Jul 27 '11 at 5:13
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    $\begingroup$ 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. :) $\endgroup$ – cardinal Jul 27 '11 at 20:00
  • $\begingroup$ Related: stats.stackexchange.com/questions/299722/… $\endgroup$ – kjetil b halvorsen Aug 16 '18 at 11:40
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For question 1 you can use standard methods if the denominator is bounded away from 0 (as mentioned in the comments). If your sample size is not large relative to the potential skewness in the ratios then you probably do not want to use normal based methods (t-tests, anova), but resampling methods (bootstrap, permutation tests) would be worth investigating.

For questions 2, 3, & 4 read "Spurious Correlation and the Fallacy of the Ratio Standard Revisited" by Richard Kronmal (1993) Journal of the Royal Statistical Society. Series A (vol 156, no 3, pp. 379-392).

Someone else will need to comment on 5.

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