Given the hubbub about Blue Cross and paying for anesthesiologists, there was an interesting paper making the rounds, Comparison of Anesthesia Times and Billing Patterns by Anesthesia Practitioners. (Shared via Crémieux.)

Most medical claims are billed via what are call CPT codes (Current Procedural Terminology). So if you go to the ER, you will get billed for a code 99281 to 99285. The final digit encodes different levels of complexity for the case, with 5’s being more complex. It was news to me that anesthesiologists actually bill for time directly, but the above linked paper showed (pretty plainly) that there is strong evidence they round up to every 5 minutes.

Now the paper just selected the anesthesiologists that have the highest proportion of billed times ending in 5’s. Here I will show a better way to flag specific problematic anesthesiologists (using repeated binomial tests and false discovery rate corrections).

Here I simulate 1,000 doctors, and select 40 of them to be bad, and those 40 round all of their claims up to the nearest 5 minute mark. Whereas the other docs are just billing the time as is. And they have varying total number of claims, between 100 and 500.

import numpy as np
from scipy.stats import gamma,uniform
import matplotlib.pyplot as plt
import pandas as pd
from scipy.stats import binomtest,false_discovery_control

np.random.seed(10)

docs = 1000
# pick a random set 
bad_docs = np.random.choice(np.arange(docs),40,replace=False)
doci = []
claims = []

for i in range(docs):
    totn = int(np.ceil(uniform.rvs(100,500))) # number of claims
    doci += [i]*totn
    g = gamma.rvs(6,scale=12,size=totn)
    if i in bad_docs:
        g = np.ceil(g/5)*5
    else:
        g = g.round()
    claims += g.tolist()

dat = pd.DataFrame(zip(doci,claims),columns=['Doc','Time'])

# Histogram
fig, ax = plt.subplots(figsize=(8,4))
dat['Time'].hist(bins=range(201),alpha=0.8,color='k',ax=ax)
plt.savefig('Histogram.png',dpi=500,bbox_inches='tight')

You can see my gamma distribution is not as heavy tailed as the JAMA paper, but qualitatively has the same spike traits. Based on this, you can see the spike is relative to the other density, and so that shows in the JAMA paper there are more anesthesiologists rounding at 60 minutes than there are at the other 5 minute intervals.

In this particular example, it would be trivial to spot the bad docs, since they round 100% of the time, and you would only expect around 20% (since billing in minute intervals).

dat['Round'] = dat['Time'] % 5 == 0
dat['N'] = 1
gs = dat.groupby('Doc',as_index=False)[['N','Round']].sum()
gs['Perc'] = gs['Round']/gs['N']
gs.sort_values(by='Perc',ascending=False,ignore_index=True,inplace=True)
gs

# Upper Quantiles
np.quantile(gs['Perc'],[0.75,0.8,0.85,0.9,0.95,0.99])

But you can see a problem with using a top 5% quantile cut off here. Since I only have 4% bad doctors, using that hard cut-off will result in a few false positive flags. My suggested approach is to create a statistical test (Chi-Square, binominal, KS, whatever makes sense for individual doctors). Run the test for each doctor, get the p-value, and then run a false discovery rate correction on the p-values.

The above where doctors round 100% of the time is too easy, so here I simulate 40 doctors will round up 10% to 30% of the time. I also have fewer cases (more cases and more rounding make it much easier to spot).

# Redoing sim, but it is a smaller percentage of the stats are bad
for i in range(docs):
    totn = int(np.ceil(uniform.rvs(100,500))) # number of claims
    doci += [i]*totn
    g = gamma.rvs(6,scale=12,size=totn)
    if i in bad_docs:
        randperc = int(np.round(totn*uniform.rvs(0.1,0.2)))
        badn = np.random.choice(np.arange(totn),randperc,replace=False)
        g[badn] = np.ceil(g[badn]/5)*5
        g = g.round()
    else:
        g = g.round()
    claims += g.tolist()

dat = pd.DataFrame(zip(doci,claims),columns=['Doc','Time'])
dat['Round'] = dat['Time'] % 5 == 0
dat['N'] = 1
gs = dat.groupby('Doc',as_index=False)[['N','Round']].sum()
gs['Perc'] = gs['Round']/gs['N']
gs.sort_values(by='Perc',ascending=False,ignore_index=True,inplace=True)

Now we can apply the binomial test to each doctor, then adjust for false discovery rate.

# Calculating binom test
def bt(x):
    k,n = x.iloc[0],x.iloc[1]
    b = binomtest(k,n,p=0.2,alternative='greater')
    return b.pvalue

gs['p'] = gs[['Round','N']].apply(bt,axis=1)
gs['q'] = false_discovery_control(gs['p'],method='by')

# Captures 28 out of 40 bad docs, no false positives
gs['BadDocs'] = gs['Doc'].isin(bad_docs)
gs[gs['q'] < 0.05]

If you check out the doctors via gs.head(50), you can see that a few of the bad-docs where adjusted to have q-values of 1, but they ended up being low N and in the range you would expect.

While anesthesiologists billing is different, this same approach would be fine for CPT codes that have the 1-5 modifier (you might use a leave one out strategy and a Chi-Square test). Anesthesiologists if they know they will be scrutinized with exact 5 minutes, they will likely adjust and round up, but to not regular numbers. If that is the case, the default distribution will be expected to be uniform on the 0-9 digits (sometimes people use a Benford like test for the trailing digits, this is the same idea). That will be harder to fake.

I don’t have an issue with Blue Cross saying they will only bill for pre-allotted times. But even without making an explicit policy like that, they can identify bad actors and pursue investigations into those problematic anesthesiologists even without making widespread policy changes.