Of course, not all outcomes are equal. Most patients would agree that the best outcome is for a patient who has strep to get antibiotics. In a decision analysis, we talk about “utility”. The utility is a single, global estimate of the preference for the outcome. For example, patients with a true positive test result who are treated with antibiotics have the best possible outcome. Those with a true negative test result who do not receive antibiotics share this good outcome. Those with a false positive result receive antibiotics unnecessarily, and risk adverse effects like diarrhea or allergic reaction with no benefit. Similarly, patients with a false negative do not receive antibiotics when they should. These two groups have lower utilities than the true positive and negative groups. Utility is measured on a scale from 0 to 1.0. In the next tree, I have changed the final nodes in the tree to “terminal nodes” and assigned utilities of 1.0 for appropriate treatment of strep, 0.6 for not giving antibiotics to a patient with strep, and 0.8 for giving antibiotics to a patient who doesn’t have strep:
So, which is the best management option? “Rolling back” a decision tree calculates the overall utility of a particular management option. To fold back the tree, you multiply the probability that a patient arrives at a particular outcome by that outcomes utility, and then add all of the products for a particular management option. In this case, let’s calculate the utility for the second management option, empiric antibiotics for all patients:
Utility = (0.2 * 1.0) + (0.8 * 0.8)
Calculating the utility for the second option is more complicated, because there are more possible outcomes:
Utility = (0.2 * 0.9 * 1.0) + (0.2 * 0.1 * 0.6) + (0.8 * 0.1 * 0.8) + (0.8 * 0.9 * 1.0)
Of course, we don’t have to do all of that multiplication – that’s why computers were invented! Here, the software has “rolled back” the tree:
The total “expected utility” for ordering a rapid antigen test is 0.98, and for empiric treatment is only 0.84. Therefore, we should use the rapid antigen test.
Note that in this example, an error of omission (not treating a patient with strep due to a false negative test) was assigned a lower utility than an error of commission (giving a patient without strep an unnecessary antibiotic): 0.6 vs 0.8. This is typical of the way we tend to think of errors in American society. In a society less oriented toward intervention, the utilities may be assigned differently. Let’s see what happens if we reverse the utilities, and assign a lower utility to overtreatment than to undertreatment:
Empiric treatment now looks even worse, which is what you would expect.
Of course, the above example is overly simplistic. I’m sure you’ve already started to poke holes in it:
To answer these questions, we have to learn about sensitivity analysis.