Working with ratios: standardizing problem

Question

I have a dataset, let's say of counts of apples and oranges. I would like to have a metric that equally reflects the ratio of apples to oranges but if I simply use the ratio:

apple count + 1 : orange counts +1

I get very large numbers when I have more apples than oranges but very small numbers when I have more oranges than apples. I would like a ratio that is more normally distributed so that when apples = oranges my ratio = 1 but my max value is 2 and min value is 0.

Any suggestions on how to achieve this?

Answer 1

Let's forget apples and oranges and generalise painlessly to counts $n_1 + n_2 =: n$ and work with the proportions $p_1 = n_1/n, p_2 = n_2/n$.

As your own answer implies, the problem with $\ln (n_1/n_2) = \ln n_1 - \ln n_2 = \ln p_1 - \ln p_2$ is that either count $n_1, n_2$ could be zero. Hence over many years there have been proposals of various fudges of the form $\ln [(n_1 + c)/(n_2 + k)] = \ln (n_1 + c) - \ln (n_2 + k)$ for positive $c, k$ where $c$ and $k$ need not be different.

However, a fudge-free alternative is to use some folded power, say $p_1^\lambda - p_2^\lambda$, so that for example $\lambda = 1/2$ gives folded square roots and $\lambda = 1/3$ gives folded cube roots. Positive features of such folded powers include

1. symmetry of definition, so that folded powers for $\lambda > 0$ range from $-1$ to $1$ (and so could be translated to the interval from $0$ to $2$ simply by adding $1$).

2. being defined simply for $n_1 = 0$ or $n_2 = 0$ or both.

There is no guarantee that the results will be normally distributed, nor is there for any transform of $n_1$ and $n_2$. However, the results are likely to be closer to a normal distribution, and indeed to symmetry, than the results of $n_1/n_2$ would be.

More on folded powers at What is the most appropriate way to transform proportions when they are an independent variable? which in turn gives further references.

Answer 2

Consider the transformation $$ \theta = \text{log}\,\frac{\pi_\text{oranges}}{\pi_\text{apples}} $$ where $\pi_{\text{oranges}}$ is the probability of seeing an orange, or if you prefer, the true proportion of oranges in a population of oranges and apples.

This is just the population version of the log of your ratio without the extra 1s. We'll get to those in a minute.

$\theta$ is a logit and thus zero when there are the same proportion of apples as oranges, and it increases (decreases) with the proportional increase (decrease) in oranges. Naturally it's not defined when there are no oranges and/or apples.

Assume a prior over $\pi_\text{oranges}$ $$ p(\pi_\text{oranges}) = \text{Beta}(a,b) $$ Setting $a=b=1$ would imply that all values of $\pi_\text{oranges}$, from 0 to 1 are equally likely.

With this prior, the posterior distribution $p(\pi_\text{oranges} \mid \text{apples}, \text{oranges})$ is exactly $$ \text{Beta}(\text{oranges}+a,\text{apples}+b) $$ and when there are more than a handful each of apples and oranges, the corresponding posterior for $\theta$ is, to good approximation (Lindley, 1965), Normal with mean $$ \text{log}\,\frac{\text{oranges}+a}{\text{apples}+b} $$ which really is your ratio, and variance $$ \frac{1}{\text{oranges}+a} + \frac{1}{\text{apples}+b} $$ So $\theta$ is a transformation to Normality that also makes sense.

If you don't like the whole prior-posterior business, you can think about the log of your original measure simply as a regularized estimator of $\theta$. It's shrunk towards 0, but decreasingly so as more fruit appears.

Answer 3

I am trying: $$ \text{log}\,\Big[\frac{apple count + 1}{orange count +1}\Big] $$ but is this transformation justifiable?