As I think I’ve mentioned before, one of the ‘go-to’ stats in my scorecard-building toolkit is the p-value that results from performing Fisher’s Exact Test on contingency tables. It’s straightforward to generate (in most cases), and directly interpretable: it’s just a sum of the probabilities of ‘extreme’ tables. When I started building credit risk scorecards, and using the Information Value (IV) statistic, I had to satisfy myself that there was a sensible relationship between the two values. Now, my combinatoric skills are far too lacking to attempt a rigorous mathematical analysis, so naturally I turn to R and the far easier task of simulating lots of data!
I generated 10,000 2-by-2 tables at random, with cell counts between 5 and 100. Here’s a plot of the (base e) log of the resulting exact p-value, against the log of the IV:
(I’ve taken logs as the relationship is clearer.) As you can see, I’ve drawn in some lines for the typical levels of p-value that people care about (5%, 1% and 0.1%), and the same for the IV (0.02, 0.1, 0.3 and 0.5). In the main, it looks like you’d expect, no glaring outliers.
For fun, I’ll look at those that fall into the area (p_exact > 0.05) and (0.3 < IV < 0.5):
|p = 0.0751, IV = 0.332||
|p = 0.0613, IV = 0.321|
In both cases, the exact p-value says there’s not much evidence that the row/column categories are related to each other — yet the IV tells us there’s “strong evidence”! Of course, the answer is that there’s no one single measure of independence that covers all situations; see, for instance, the famous Anscombe’s Quartet for a visual representation.
Practically, for the situations in which I’m using these measures, it doesn’t matter: if I have at least one indication of significance, I may as well add another candidate variable to the logistic regression that’ll form the basis of my scorecard. If the model selection process doesn’t end up using it, that’s fine.
Anyway, I end with a minor mystery. In my previous post, I came up with an upper bound for the IV, which means I can scale my IV to be between zero and one. I presumed that this new scaled version would be more correlated with the exact p-value; after all, how can a relationship with an IV of 0.25, but an upper bound of 5, be less significant than one with an IV of 0.375, but an upper bound of 15 (say)? Proportionally, the former is twice as strong as the latter, no?
What I found was that the scaled version was consistently less correlated! Why would this be? Surely, the scaling is providing more information? I have some suspicions, but nothing concrete at present — hopefully, I can clear this up in a future post.