Ratio between two numbers allowing positive and negative values

Question

I'm working on a project analyzing how many research projects every EU region has been involved in in 2020. My lists show (1) the number of projects each region joined as coordinator, and (2) how many projects the region joined as participant. I want to generate a ratio between these two values to check to what extent they are more geared to coordinating or to participating.

I'm trying to get a value above zero if they are more coordinating than participating, and below zero if the opposite happens. Zero means they coordinate as much as they participate. To do that, I'm using the formula (#PARTICIPANT - #COORDINATOR)/#COORDINATOR. For instance, if they were participants in 30 projects and coordinators in 9, I get (30-9)/9 = 2.333. So far so good.

However, in cases where #COORDINATOR is bigger than #PARTICIPANT (i.e. the end result is negative), the results vary only between $0$ and $-1$. Using the values above as illustration, but inverted, I get $(9-30)/30 = 0.7$. Therefore, I get results which are impossible to mutually compare although they should basically communicate the same - in both cases, a region is "equally" more playing one role than the other.

So, in short, is there a way to make these results consistent both for negative and positive values? A way to immediately see that no matter what the direction of the imbalance, the ratio between both roles is the same. Thanks in advance.

Answer 1

I assume you mean you want not only to separate positive and negative in this way, but to do so in such a way that swapping the coordinating and participating numbers gives the same magnitude, but opposite signs.

One way to do this is to start by looking instead at the ratio of participant to total, so $\frac{p}{p+c}$. If $p>c$ this is bigger than $1/2$ and if $c>p$ it is smaller. Now you can double it and subtract $1$ to get $$\frac{p-c}{p+c}.$$ This is always between $-1$ and $+1$, and has the properties you want.

NB: I am not sure whether you want this or the reverse $\frac{c-p}{c+p}$ - you say at the start that you want positives to mean "more coordinating than participating", but then all your examples do the opposite.

Answer 2

As an alternative to Especially Lime's proposal, $\log(p/c)$ (or, as EL points out, $\log(c/p)$, if you want positive values to correspond to more coordinating than participating) would also do the trick. This would, as in the OP, stretch things out to $\pm\infty$ when $pc=0$ (i.e., for regions that only participate or only coordinate).

In general, any monotonic function satisfying $f(1/r)=-f(r)$ will work, by letting $r=p/c$ (or $c/p$, as appropriate, to get the desired agreement of signs).

Answer 3

Especially Lime's answer may be better than mine. It depends on what you want. Letting $p,c$ denote the participants and coordinators, respectively, an alternative approach is

$$\frac{p - c}{\min\left( ~|p|, |c| ~\right)}.$$

However, you are then forced to consider what happens if $p=0$ or $c = 0$. One artificial remedy is that when the denominator would otherwise equal $0$, change the denominator to $(+1)$.