Monday, February 12, 2007

Going Solo?: Become familiar with Bayes Theorem

If you remember having taken statistics 101 class, you may remember Bayes Theorem. If you don’t remember it, I’ll tell you about it. Bayes Theorem deals with probabilities and allows you to calculate, mathematically, the likelihood of possible outcomes. Bayes Theorem has been around for many years, and has many useful applications, from gambling to medicine to business; even to the military.

Mathematically, Bayes Theorem has a formula that for an outcome involving 2 variables, A and B, is as follows:

Outcome = Probability A x Probability B or P(A) x P(B)

For 3 variables, the formula would read:

Outcome = P(A) x P(B) x P(C)

For example, with a coin toss, there are 2 possible outcomes, head or tails, and each outcome has a probability of 0.5, or 50%. Therefore, the probability of flipping 2 heads in a row is:

Outcome = 0.5(H) x 0.5(H) = 0.25 or a 25% probability that you will flip heads twice in a row.

Flipping it 3 times in a row can be calculated to have a probability of:
Outcome = 0.5(H) x 0.5(H) x 0.5(H) = 0.125 or a 12.5% chance that you will successfully calls “head” thrice in row.

So what? How can this be applied in a real world setting. Well here are some examples.

In the 1960’s, a military transport jet exploded over the mid-Atlantic ocean. The plain happened to be carrying 3 nuclear weapons. Not good! The military had to find them. So using a complex Bayesian formula, the scientists imputed multiple probabilities to output an outcome, in this case the most likely location of the nuclear devices. The inputs were variables such as likely wind speed and direction, flight speed and altitude, weapon trajectory, debris scatter, etc. And you know what. They found the 3 devices. True story.

In medicine, Bayes theory is quite helpful as well. We use it to help guide patients toward or away from treatment/evaluation plans as part if an informed consent discussion. We may not actually do the math, but the logic is Bayesian. For example, let’s say a patient has a 3mm ureter stone. We know that this stone will have a 75% chance of passing spontaneously, or on its own, and that some forms of surgical intervention can have a 5% complication likelihood. We therefore use our mental Bayes Theorem calculation to guide the patient towards an observational approach.

In the business side of medicine, we use this as well, or at least you should if you want to be successful. Lets say you want to buy a CT scanner (I don’t, but other urologist have been). The CT costs $250K. You look back through your records and determine that you order 1000 CT scans per year for a variety of diagnoses. Therefore, in theory, by charging $500 per CT scan, you can get $500K in revenue in year 1. Hey!!! Not bad, I may do dat! But here is where Bayes Theorem ought to be invoked, in my view.

You know that Medicare will pay this year for the CT scans, but you figure that, let’s say 20% chance, they may not pay next year. That’s just how it goes. And you have 200 potential Medicare patients. In addition, you don’t know whether the commercial payers will pay at all, but you figure that 20% of patients will have insurance that pays off the bat, and after an extensive application process, insurance for another 50% will ultimately pay within the first year, but not before you spend 50K on staff salaries to get credentialed. For 30% of patients, you will not get paid, period. You then figure that the carriers will pay on average $250 per scan, including Medicare. Therefore, here is the calculation:

Outcome = 200x250(Medicare) + 800x250x0.20 + 800x250x 0.50 -50,000 = $140K in year one.

Outcome year 2 = 200 x 250 x 0.8 – 200 x 250 x 0.2 + 800 x 250 x 0.75 = $150K in year 2.

See, you can tell for a medical practice that orders 1000 scans per year, it may be worth it to buy a CT scanner for $250K since you can make back your investment in 2 years. You can play with the numbers for best, and even more importantly, worse case scenarios. For example, let’s change the Medicare number to 100 scans per year and the commercial number to 200 scans per year, with the dollar amount per scan held constant.

Outcome year 1 = 100 x 250 + 200 x 250 x 0.2 + 200 x 250 x 0.5 -50K = negative $2500.
Obviously, here it is not worth it. In fact, even if you decided not do scans on the 50% of patients whose insurers would ultimately pay, given the expensive and timely application process, it is still not a worthwhile endeavor, since you’d only recoup $22,500 in year one. This number might not even cover a yearly payment if you financed the CT scanner with a bank loan. And, you would not even make back your investment for 10 years. Since every year, reimbursements go down or are cut entirely, you would likely never make back the investment.

Of course, most of us don’t actually do the math, the actual math, in Bayes Theorem, but we think this way. We perform what are called heuristic algorithms. It is how we determine whether or not to invest in a certain piece of office equipment or software etc. If you find yourself on the fence and unsure about whether or not to invest in an expensive piece of equipment, you may want to use Bayes Theorem to run the numbers and help you decide.

Hope you liked the post.