# Outliers in Distributions

If you google ‘outlier’, all of the results that come up are in terms of individual observation outliers. So if you have a set of transaction data that is 10, 20, 30, 8000, the singlet observation 8000 is an outlier. But for many situations with transaction data, you don’t want to examine individual outlier incidents, but look for systematic patterns. For example, if I am looking at healthcare insurance claims for my work, a single claim that is \$100,000 is actually not that rare. But if we have a hospital that has mostly \$100,000 claims for a specific treatment, whereas another with similar cases has a range of \$50,000 to \$100,000, that may signal there is some funny business going on.

There is no singular way to examine outliers in distribution. A plain old t-test of mean differences may make sense for some situations. But a generally more useful way IMO to think about the problem is to examine the distribution of the outcome in CDF space, as opposed to looking at particular moments of the distribution. A t-test basically only looks at the differences in means for the distributions, whereas examining the CDF we are looking for weird patterns at any point in the distribution.

Here is an example of comparing the cost of hospital stays (per length of stay), for a hospital compared to all others from the same datasource (details on the data in a sec). The way to read this graph is that at 10^3 (so \$1000 per day claims) for facility 1458, we have around 20% of the claims data are below this value. For the rest of the hospital data, a larger proportion of claims are under a thousand dollars, more like 25%. Since the red line is always below the black line, it also means that the claims at this hospital are pretty much always larger than the claims at all the other hospitals.

For this example analysis, I am using data from New York State health insurance claims data (SPARC). I have posted python code to replicate here (note if you cannot access dropbox links, feel free to email and I will forward).

Here I am specifically analyzing medical, in-patient insurance claims (I dropped surgical claims) for around 300+ hospitals. There are quite a few claims in this data, over 2 million, and the majority of hospitals have plenty of claims to examine (so no hospitals with only 10 claims). I also specifically examine costs per length of stay. Initially I just examined costs, but will get to why I changed to the normalized version towards the end of the post.

# Analysis of CDF Outliers

So first what I did was attempt to do a leave-one-out type stat test using the Kolmogorov-Smirnov test. This is a test that looks at the maximum vertical difference between the CDFs I showed earlier. I should have known better though. Given this large of sample size, even with multiple comparison adjustments for false discovery rate, every hospital was considered an outlier. This is sort of the curse of null hypothesis significance testing, it is either underpowered, so you get null results when things should really be flagged, or with large samples everything is flagged.

So what I did first was make a graph of all the different CDFs for each individual hospital. You can see from this plot we have a mass of the distribution that looks very similar in shape, but is shifted left or right. (Hospitals can bill different values, i.e. casemix, so can have the same types of events but have different bills, so that is normal.) But then we have a few outliers really stick out.

To characterize the central mass in this image, what I did was calculate each empirical CDF for each hospital (over 300 in this sample). Then I estimated the CDF for each hospital at a sample of points logspace distributed between \$100 to \$100,000. Then I took the 90% distribution between the ECDF values. This is easier to show than to say, so in the below pic the grey area is the 90% region for the CDFs. Then you can do stats to see how hospitals may fall outside that band.

So here 1320 is looking good until around 60% of the distribution, and then it is shifted right. There is a kink in the CDF as well, so this suggests really a set of different types of claims, and in that second group it is the outlier. 1320 was the hospital that had the most sample points outside of my grey coverage area, but you could also do outliers in terms of the distance between those two lines (again like a KS test stat), or in the area between those two lines (that is like a version of the Wasserstein distance only considering above/below moves). So here is the hospital that has the largest distance below the band (above the band signals that a hospital has lower claims on average):

Flat lines horizontally signal an absence of data, whereas vertical lines signal a set of claims with the exact same bill. So here we have a set of claims around \$1000 per day that look normal, then an abnormal absence of data from \$1,000 to \$10,000. Then a large spike of claims that end up being around \$45k per day.

So this is looking at the distribution relative to other hospitals, but a few examples I am familiar with look for these flat/vertical spikes in the CDF to identify fraud. Mike Maltz has an example of identifying collusion in bids. In another, Chris Stucchio identifies spikes in transaction data signaling potential fraud. Here I am just doing a test relative to other data to identify weird curves, not just flat lines though.

One limitation of this analysis I have conducted here is that it does not take into account the nature of the claims data. So say you had a hospital that specializes in cancer treatment, it may be totally normal for them to have claims that are higher value overall than a more typical hospital that spreads claims among a wider variety of types of visits/treatments. Initially I analyzed just the cost data, and it identified a few big outliers that ended up being hospice/nursing homes. So they had really high dollar value claims, but also really long stays. So when analyzing the claim per length of say, they were totally normal in that central mass.

So ultimately there could be other characteristics in the types of claims hospitals submit that could explain the weird CDF. One way to take that into account is to do a conditional model for the claims, and then do the ECDF tests on those conditional models. One way may be to look at the residuals for each individual hospital, another would be to draw a matched comparison sample. (Greg Ridgeway did this when examining police behavior in the NYPD.)

That would be like making a single comparison line (like my initial black/red line graph). So controlling the false discovery after that will be tough with larger samples (again the typical KS test, even with a matched sample, will likely always reject). So wondering if there is another machine learning way to identify outliers in CDF space, like a mashup of isolation forests and conditional density forests. Essentially I want to fit a model to draw those grey CDF bands, instead of relying on my sample of hospitals to draw the grey band in those latter plots.