Jensen’s Inequality in Medicine

Jensen’s Inequality in Medicine

When a response is nonlinear, the variability of an exposure can matter as well as its average—the difference is the “convexity bias.” If you are antifragile (i.e., convex) to a given substance, then you are better off having it randomly distributed, rather than provided steadily.

I’ve found very few medical papers making use of nonlinearity by applying convexity effects to medical problems, in spite of the ubiquity of nonlinear responses in biology. (I am being generous; I actually found only one explicit use of Jensen’s inequality in one single application— thanks to my friend Eric Briys—and only one that used it properly, so the response “we know that” by medical researchers when the consequence nonlinearity is explained to them is rather lame.)

Remarkably, convexity effects work in an identical way with options, innovations, anything convex. Now let us apply it … to lungs.

Taleb describes research on mechanical ventilation for people with lung diseases, including acute respiratory distress syndrome. The belief was that constant pressure and volume were desirable—steadiness seemed a good idea. But the reaction of the patient is nonlinear to the pressure (convex over an initial range, then concave above it), and he suffers from such regularity. Further, people with very sick lungs cannot take high pressure for a long time—while they need a lot of volume. J. F. Brewster and his associates figured out that dispensing higher pressure on occasion, and low pressure at other times, allowed them to provide a lot more volume to the lungs for a given mean pressure and thus decrease patient mortality. An additional benefit is that an occasional spike in pressure helps to open up collapsed alveoli. Actually, that’s how our lungs function when healthy: with variations and “noise” rather than steady airflow. Variability can help within a convex range of the pressure-volume response, but the range and the clinical consequences must be established empirically. Probabilities must add up to 100 percent by definition; a claim about reduced patient mortality requires clinical evidence.2

Not using models of nonlinear effects such as convexity biases while “doing empirical work” is like having to catalog every apple falling from a tree and call the operation “empiricism” instead of just using Newton’s equation.