A couple years ago, I used a question by Benjamin Kay as an excuse to write that it’s usually a bad idea to study a ratio whose denominator has uncertain sign. As I wrote then:

Similar problems arise with marginal cost-benefit ratios, LD50 in logistic regression (see chapter 3 of Bayesian Data Analysis for an example), instrumental variables, and the Fieller-Creasy problem in theoretical statistics. . . . In general, the story is that the ratio completely changes in interpretation when the denominator changes sign.

More recently, Kay sent in a related question:

I [Kay] wondered if you have any advice on handling ratios when the signs change as a result of a parameter.

I have three functions, one C * x^a, another D * x^a, and a third f(x,a) in my paper such that:

C * x^a, < f(x,a) < D * x^a C,D and a all have the same signs. We can divide through by C * x^a but the results depend on the sign of C either 1< f(x,a) / C * x^a < D * x^a / C * x^a, or 1 / f(x,a) / C * x^a > D * x^a / C * x^a,

That is, when the sign on a changes, the inequalities flip.

I want to say something about the ratio C/D being close to one so that I can say something about how tight the bounds are on f(x,a) / C * x^a.

So being close (say within 5%) has a confusing presentation.

a>0 : 1< f(x,a) / C * x^a < 1.05 a<0 : 1> f(x,a) / C * x^a >.95

Which no one who has read my paper likes. They mostly find it confusing. I cannot be the first person to have to deal with this, so I wondered if you had any suggestions?

My reply:

I have to admit I can’t understand much of your notation but I think I get the general picture. In other settings what I’ve done is to try to reformulate the problem. For example, instead of looking at C/D, look at C – D, or perhaps delta*(C-D), where delta is a positive quantity set to a reasonable value (of the order of magnitude of |C+D|).

My feeling is that if you carefully express these things as decision problems, ultimately it’s differences rather than ratios that really matter. We use ratios because they are conveniently scale-free, but really it shouldn’t be hard to scale a difference in a reasonable way. The small amount of effort placed into scaling can pay off big-time in clean and direct interpretation.

Thanks for the speedy reply. Sorry if the notation is confusing.

C, D, and a are constants determined elsewhere.

C * x^a is the constant C times x raised to the power a. If anyone thinks they have any more advice but can't understand what I'm doing, please let me know.

The reason why I want to study C/D is because C and D are complicated constants determined by model parameters whereas C/D is a quite simple quantity that just happens to have this sign problem.

I'll have to try a reformulation, I just haven't seen a way to do it with some sort of distance measure like (C-D)^2 or |C-D|. Back to the drawing board I guess.

take logs? ratio becomes a difference and no more sign flips…

Thanks Joshua Vogelstein but that doesn't work because for a

Sorry, something happened to my 11:23 comment. I think it mistook my less than sign for a HTML code and ended my post. Here it is again without the symbols replaced with words.

Thanks Joshua Vogelstein but that doesn't work because for a less than 0, the three functions are negative. To use the log you'd have to multiply by negative one, which would flip all the inequalities. Therefore you have the same problem as before, where you have a flip of the inequalities at a equal to 0.