Appendix/Stats
The lifetime incidence for appendicitis is about 7%. We want to take out around 15-20% of appendices that are normal, which has been decided upon as a sufficiently high proportion of false positives to make sure we don't miss many.
I'll skip the background stats lecture, but will offer a few basic definitions. Sensitivity measures how likely test x is to catch the disease we're looking for. Specificity measures how likely test x is to catch
only that disease.
Those are useful numbers by themselves, but we can use the Bayesian Theorem to calculate Likelihood Ratios, which allow us to gauge how much
more likely an individual is to have the disease based on a positive test (or contra). We multiply what we think the individual's probability of having the disease before the test was by this likelihood ratio and thus get a modified probability of disease.
So let's say that someone walking into your office with vague sort-of-appendixy symptoms has a 10% chance of having appendicitis. This happens a lot. What do you do? Most docs do the standard tests and then make an iffy sort of "well, let's hope this is probably right" judgment based on that. The family doc, or whoever, makes the first clinical iffy judgment, then the surgeon makes a somewhat more precise, but still iffy judgment. By the time the person gets to the OR or is turned way, the iffiness factor has been compounded by the decisions of the person choosing to go into the doctor, the doctor referring, and the doctor making the final call on the need for surgery.
Likelihood ratios allow you not to eliminate, but to
quantify the iffiness. Let's take our 10% patient again. I found an article on sensitivities and specificities for a few appendicitis-related tests and calculated LR's for them since the article didn't (LR's aren't extremely popular or well-known yet).
So: Patient's white blood cells are above 10,000. This has a sensitivity of .77 (it will catch 77% of true appendicitis cases) and specificity of .63 (37% of the time it will indicate something else entirely). Vaguely useful, but let's convert it into a likelihood ratio. NOW we have a really useful number: A positive LR of about 2. In other words, our patient with a 1:9 chance of having appendicitis based on our initial impression now has a chance more like 2:9, or about 20%.
Let's do an ultrasound--we find fluid in the appendiceal lumen, which has sens of 0.5 (only catches half of appendicitis cases) and a great spec of .92 (not many other diseases where you'll see this). Our positive LR for this is over 6. In other words, our patient is now six times more likely to have appendicitis as before...2:9 odds before * 6= 12:9 = 4:3. In other words, our patient more likely than not has appendicitis now.
At this point you might use some clincial judgment: Sure, we're not at the 80% certainty we'd like to get to if we want only 20% of our appendices removed to be normal, but it's awfully rare that the ultrasound results are going to be positive when the person
doesn't have appendicitis (rarer than 1/5, for sure) and we might at this point consider the possibility that our pretest probability was too low...since that's generally an intuitive/anecdotal call rather than something based on careful evidence, it's the weak link in the chain. The next time we see a patient with the same symptoms, we might bump the pretest probability up to 20%, which given the same series of tests would put us much closer to our 80% cutoff.
Lots of wiggle room, and the stats, like logic and theology and such, are only as good as our initial assumptions. So we'll still be wrong a lot, but at least we'll have quantified the wrongness, allowing us to maybe be less likely wrong, more precise, the next time around.
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