Posts Tagged stats

Exact p-value versus Information Value

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):

6 29
10 15
p = 0.0751, IV = 0.332  
5 84
7 37
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.


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Using median rather than mean

Analysts in consumer finance are used to dealing with large sets of incoming customer data, and are often asked to provide summary statistics to show whether application quality is changing over time: Are credit scores going down? What’s the average monthly income? What’s the average age of the customers applying this quarter, compared to last?

Early on in the process, the data is likely to be messy, unsanitised, and potentially chock-full of errors. (Once the data has been processed to the point of considering credit-worthiness, then it’s all clean and perfect, yeah..?) Therefore, you have to be careful when you report on this data, as it’s easy to get nonsense figures that will send your marketing and risk people off in the wrong direction.

Two really common errors seen in numerical application data:

1. Putting gross yearly salary into the net monthly income field, and it’s not always easy to spot which one it ought to be: if it says ‘£50000’, then it’s very likely to be an annual figure; but what about ‘£7000’? If monthly, that’s a really good wage; but I can assure you, people earning that much are still applying for credit.

2. Dates of birth: if unknown, it’s common to see ‘1st January 1900’ and similar. So when you convert it to age, the customer is over 100.

Also, if you happen to get credit scores with your data, you have to watch out for magic numbers like 9999 – which to Callcredit means “no [credit] score could be generated”, not that the customer has the credit-worthiness of Bill Gates or George Soros.

Hence, it’s fairly obvious, that if you’re including these figures in mean averages, you’re going to be giving a misleading impression, and people can infer the wrong thing. For example, say you’ve 99 applications with an average monthly income of £2000, but there’s a also an incorrect application with a figure of £50,000. If you report the mean, you’ll get an answer of £2480, instead of the correct £2010 (assuming that £50k salary translates to ~£3k take-home per month). However, if you report the median, you’ll get an answer of £2000, whether the incorrect data is in there or not.

In statistical parlance, the median is “a robust measure of central tendency”, whereas the mean is not. The median isn’t affected by a few outliers (at either end).

End note: Credit scores can be (roughly) normally distributed; for a normal distribution, the median and mean (and mode) are the same. But data doesn’t have to be normally distributed: e.g. call-waiting times follow the exponential distribution, where the median and mean are not the same.

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