StATS: Parametric tests for a ratio (October 27, 2006)

Dear Professor Mean, I computed a variable, Y3, which is the ratio of two other variables, Y1 and Y2. Can I use a parametric test on this ratio?

One of the amendments to the U.S. Bill of Rights talks about this. I think it is the Seventh Amendment. It says, the government shall not infringe on the right of its citizens to analyze their data anyway that they darn well please.

Well, no the Seventh Amendment doesn't say that. But my point is that there is no mandate written somewhere that dictates what types of analyses are legitimate. So when you ask, "can I use a parametric test?" what you really mean to ask is "can I use a parametric test and still get my research published?"

If I am the one fortunate enough to be the referee on your paper, you will get it published (or if it doesn't get published it will be for a reason other than the fact that you used a parametric test on a ratio).

If someone else reviews your paper, I cannot say what will happen. Some reviewers are very picky about this sort of thing and others are more flexible.

The concern that a picky reviewer will raise is that a ratio of two variables is not likely to be normally distributed, and will probably be highly skewed. There's no empirical evidence that I know of that ratios are likely to follow any distributional pattern.

What we do know is that if your numerator is normal and your denominator is normal, it is impossible for the ratio to be normal. In many situations it can be close to normal, but in other situations, it can be disastrously non-normal.

One thing to look for is the behavior of the denominator. Exceedingly small values in the denominator cause the ratio to explode, leading to problems with outliers. If your denominator is bounded well away from zero, this is not a problem. One thing to look for is if the coefficient of variation (or relative standard deviation) is small (less than 0.3, for example).

At the other extreme, if your denominator includes some negative values, I would worry about what the ratio really means. And I presume that you do not have any zero denominators, because if you try to divide by zero, your computer usually complains.

But all of this discussion makes a very big assumption--that your numerator and denominator were normally distributed to begin with. Do you know this for a fact? I guess you could check this by drawing a normal probability plot, but why not bypass the middleman. You have all the individual data values for the ratio, so why not draw a normal probability plot for the ratio values.

This is what you do for any variable prior to running a parametric test, right? So why should a ratio be any different?

Keep in mind that a parametric test is helped out by the central limit theorem, so normality is increasingly irrelevant as your sample size increases.

So if a referee tries to get picky, just show them the normal probability plot and remind them of the value of the central limit theorem.

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